CORNELL UNIVERSITY LIBRARY BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND GIVEN IN 1891 BY HENRY WILLIAMS SAGE Cornell University Library QA 931.L89 1920 A treatise on the mathematical theory of 3 1924 001 072 259 Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924001072259 A TREATISE ON THE MATHEMATICAL THEORY OF ELASTICITY CAMBRIDGE UNIVERSITY PRESS C. F. CLAY, Manager LONDON : FETTER LANE, E.G. 4 LONDON : H. K. LEWIS & CO., Ltd., 136, Gower Street, W.C. i LONDON : WILLIAM WESLEY & SON, 28, Essex Street, Strand, W.C. 2 NEW YORK : G. P. PUTNAM'S SONS BOMBAY 1 CALCUTTA V MACMILLAN AND CO., Ltd. MADRAS J TORONTO : J. M. DENT AND SONS, Ltd. TOKYO : MARUZEN-KABUSHIKI-KAISHA ALL RIGHTS RESERVED A TREATISE ON THE MATHEMATICAL THEORY OF ELASTICITY BY A. E. H. LOVE, M.A., D.Sc, F.R.S. FORMERLY FELLOW OF ST JOHn's COLLEGE, CAMBRIDGE HONORARY FELLOW OF QUEEn's COLLEGE, OXFORD SEDLEIAN PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF OXFORD THIRD EDITION CAMBRIDGE AT THE UNIVERSITY PRESS 1920 PEEFACE THIS book is the third edition of one, with the same title, which was originally published in two volumes in 1892, 1893. The second edition, published in 1906, was practically a new book, but on the present occasion no such extensive revision appeared to be neces- sary. Some new researches have been incorporated, some of the old matter ha« been transferred from notes to text, and some Articles have been wholly or partially re- written. It seemed to be desirable to retain the numbering of Articles in the second edition, and, with this object in view, new Articles have been numbered thus: "226a, 226b." One of those ^o numbered, Article 332a, is an extract from an Article now omitted. An Appendix to Chapters viii and IX deals with Volterra's theory of dislocations. A new chapter, dealing with the equilibrium of thin shells, has been added at the end of the book, and the Articles in it are numbered consecutively with those in the preceding chapter. It is unnecessary to dwell on the importance of the science of Elas- ticity, either arising from the physical notions involved in its develop- ment, or from its applications to the mechaiiics of solid bodies as they really are, or from the illustrations it affords of analytical methods that are widely used in other branches of theoretical physics. The object of the book is to present a reasonably complete account of the science from the points of view of the physicist, the engineer and the student of mathe- matics. To avoid undue expansion many matters of interest to one or other of these three Classes of readers had to be omitted, or treated very briefly, but it is hoped that the numerous references to autho- rities will assist them in finding at least the beginnings of the relevant literature. In a science, to which many additions are made annually, it is impossible for a treatise of moderate size to be other than very incomplete. A student approaching the subject for the first time may find the long preliminary discussions somewhat forbidding. He is therefore recommended to turn as early as possible to Chapter v, where he will find a recapitulation of some of the most important results obtained in previous chapters, solutions of some of the most interesting problems. VI PREFACE and a number of results, of which the verification, or direct investiga- tion, will be useful to him as exercises. The practice, followed in this and other chapters, of leaving details of pieces of work for the student to supply, may compensate for the lack of a collection of examples, such as is frequently given in text-books on other departments of mathe- matical physics. Several friends very kindly helped me by reading the proofs of pre- vious editions, but there seemed to be no need to inflict such a labour upon them for the present edition. All the more are my thanks due to the Authorities of the Press for the readiness with which they have met all my wishes, and the unfailing vigilance which they have bestowed upon the proofs. A. E. H. LOVE. Oxford, November, 1919. CONTENTS Historical Introduction PAGE 1 Scope of History. Galileo's enquiry. Enunciation of Hooke's Law. Mariotte's investigations. The problem of the elastica. Euler's theory of the stability of struts. Researches of Coulomb and Young. Euler's theory of the vibrations of bars. Attempted theory of the vibrations of bells and plates. Value of the researches made before 1820. Navier's investigation of the general equations. Impulse given to the theory by Presnel. Cauohy's first memoir. Cauchy and Poisson's investigations of the general equations by means of the "molecular" hypothesis. Green's introduction of the strain- energy-function. Kelvin's application of the laws of Thermodynamics. Stokes's criticism of Poisson's theory. The controversy concerning the number of the "elastic constants.'' Methods of solution of the general problem of equilibrium. Vibrations of solid bodies. Propagation of waves. Technical problems. Saint- Venant's theories of torsion and flexure. Equi- pollent loads. Simplifications and extensions of Saint- Venant's theories. Jouravski's treatment of shearing stress in beams. Continuous beams. KirchhoflPs theory of springs. Criticisms and applications of KirchhoiPs theory. Vibrations of bars. Impact. Dynamical resistance. The problem of plates. The Kirchhofi'-Gehring theory. Clebsch's modification of this theory. Later researches in the theory of plates. The problem of shells. Elastic stability. Conclusion. Chapter I. Analysis of strain ART. 1. Extension 32 2. Pure Shear • . . ... 33 3. Simple Shear .... 33 4. Displacement . .... 35 5. Displacement in simple extension and simple shear . . . .35 6. Homogeneous strain . . 36 7. Rblative displacement .... 37 8. Analysis of the relative displacement 38 9. Strain corresponding with small displacement ...... 39 10. Components of strain 40 11. The strain quadric 41 12. Transformation of the components of strain 42 13. Additional methods and results 43 14. Types of strain, {a) Uniform extension, (6) Simple extension, (c) Shearing strain, {d) Plane strain . 44 15. Relations connecting the dilatation, the rotation and the displacement . 46 16. Resolution of any strain into dilatation and shearing strains . . . 47 17. Identical relations between components of strain . . . .49 18. Displacement corresponding with given strain .... .50 a 5 VUl CONTENTS ART. 19. Curvilinear orthogonal coordinates . . . • • ■ 20. Components of strain referred to curvilinear orthogonal coordinates 21. Dilatation and rotation referred to curvilinear orthogonal coordinate.s 22. Cylindrical and polar coordinates Appendix to Chapter I. The general theory of strain PAGE . 51 53 . 54 . 56 23. Introductory 24. Strain corresponding with any displacement 25. Cubical dilatation 26. Reciprocal strain ellipsoid . 27. Angle between two curves altered by strain 28. Strain ellipsoid ... 29. Alteration of direction by the strain . 30. Application to cartography . 31. Conditions satisfied by the displacement 32. Finite homogeneous strain . 33. Homogeneous pure strain . 34. Analysis of any homogeneous strain into a i 35. Eotation 36. Simple extension .... 37. Simple shear 38. Additional results relating to shear 39. Composition of strains .... 40. Additional results relating to the composition of strains pure strain and a rotation 57 57 59 60 60 61 62 63 63 64 65 67 67 68 68 69 69 70 Chapter II. Analysis ob' stress 41. Introductory . . 72 42. Traction across a plane at a point 72 43. Surface tractions and body forces . . .... 73 44. Equations of motion ... 74 45. Equilibrium 75 46. Law of equilibrium of surface tractions on small volumes . . .75 47. Specification of stress at a point 75 48. Measure of stress . ... 77 49. Transformation of stress-components 78 50. The stress quadric . ' 79 51. Types of stress, (a) Purely normal stress, (6) Simple tension or pressure, (c) Shearing stress, (d) Plane stress 79 52. Resolution of any stress-system into uniform tension and shearing stress . 81 53. Additional results . . . .81 54. The stress-equations of motion and of equilibrium .... 83 55. Uniform stress and uniformly varying stress . . . 84 56. Observations concerning the stress-equations . . 85 57. Graphic representation of stress .... .... 86 58. Stress-equations referred to cui-vilinear orthogonal coordinates . . .87 59. Special cases of stress-equations referred to curvilinear orthogonal co- ordinates . . . . . ... 89 CONTENTS IX Chapter III. The elasticity of solid bodies ART. 60. Introductory 61. Work and energy .... 62. Existence of the strain-energy-function 63. Indirectness of experimental results .. 64. Hooke's Law 65. Form of the strain-energy-function . 66. Elastic constants .... 67. Methods of determining the stress in a body 68. B'orm of the strain-energy -function for isotropic solids 69. Elastic constants and moduluses of isotropic solids .... 70. Observations concerning the stress-strain relations in isotropic solids 71. Magnitude of elastic constants and moduluses of some isotropic solids 72. Elastic constants in general . . 73. Moduluses of elasticity . , . .... 74. Thermo-elastic equations . . 75. Initial stress . . . . . . ^ . . . PAGE 90 90 92 94 95 96 97 98 99 100 101 103 103 104 106 107 Chapter IV. The relation between the mathematical THEORY OF ELASTICITY AND TECHNICAL MECHANICS 76. Limitations of the mathematical theory 77. Stress-strain diagrams 78. Elastic limits 79. Time-effects. Plasticity 79a. Momentary stress 80. Viscosity of solids 81. .^olotropy induced by permanent set 82. Repeated loading 82a. Elastic hysteresis 83. Hypotheses concerning the conditions of rupture 84. Scope of the mathematical theory of elasticity . 110 111 113 114 115 115 116 117 117 119 121 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. Chapter V. The equilibrium of isotropic elastic solids Kecapitulation of the general theory 123 Uniformly varying stress, {a) Bar stretched by its own weight, (6) Cylinder immersed in fluid, (c) Body of any form immersed in fluid of same density, {d) Round bar twisted by couples . . 124 Bar bent by couples 127 Discussion of the solution for the bending of a bar by terminal couple . 128 Saint-Venant's principle . . ., - 129 Rectangular plate bent by couples . 130 Equations of equilibrium in terms of displacements ... . 131 Equilibrium under surface tractions only 132 Various methods and results 133 Plane strain and plane stress 135 Bending of narrow rectangular beam by terminal load .... 136 Equations referred to orthogonal curvilinear coordinates . . .138 \ CONTENTS ART. 97. 99. 100. 101. 102. Polar coordinates • ■ Radial displacement. Spherical shell under internal and external pressure. Compression of a sphere by its own gravitation Displacement symmetrical about an axis . Tube under pressure Application to gun construction Rotating cylinder. Rotating shaft. Rotating disk PAGE 139 139 141 141 143 144 Chapter VI. Equilibrium of jEOLotropic elastic solid bodies 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. Symmetry of structure Geometrical symmetry Elastic symmetry Isotropic solid . Symmetry of crystals Classification of crystals Elasticity of crystals . Various types of symmetry Material with three rectangular planes of symmetry. Extension and bending of a bar Elastic constants of crystals. Results of experiments Curvilinear eeolotropy Moduluses Chapter VII. General theorems 147 148 149 153 153 155 157 158 159 160 161 162 The variational equation of motion . . 164 Applications of the variational equation . ..... 165 The general problem of equilibrium 167 Uniqueness of solution .■* . . . .168 Theorem of minimum energy . . . . . . . . .169 Theorem concerning the potential energy of deformation . . .171 The reciprocal theorem 171 Determination of average strains 172 Average strains in an isotropic solid body 173 The general problem of vibrations. Uniqueness of solution . . . 174 Flux of energy in vibratory motion . . . . . . . .175 Free vibrations of elastic solid bodies 176 General theorems relating to free vibration ... . . 178 Load suddenly applied or suddenly reversed 179 Chapter VIII. The transmission of force Introductory Force operative at a point First type of simple solutions Typical nuclei of strain Local perturbations . Second type of simple solutions Pressure at a point on a plane boundary . Distributed pressure Pressure between two bodies in contact. Geometrical preliminaries 181 181 183 184 187 188 189 190 191 CONTENTS XI ART. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. PAGE Solution of the problem of the pressure between two bodies in contact . 193 Hertz's theory of impact jgg Impact of spheres jgg Effects of nuclei of strain referred to polar coordinates .... 199 Problems relating to the equilibrium of cones 201 Chapter IX. Two-dimensional elastic systems Introductory . ... ..... 202 Displacement corresponding with plane strain 202 Displacement corresponding with plane stress ... . 204 Generalized plane stress 206 Introduction of nuclei of strain 206 Force operative at a point 207 Force operative at a point of a boundary 208 Case of a straight boundary 209 Additional results : (i) the stress function, (ii) normal tension on a segment of a straight edge, (iii) force at an angle, (iv) pressure on faces of wedge 209 Typical nuclei of strain in two dimensions 211 Transformation of plane strain 213 Inversion 213 Equilibrium of a circular disk under forces in its plane, (i) Two opposed forces at points on the rim. (ii) Any forces applied to the rim. (iii) Heavy disk resting on horizontal plane . . 215 Examples of transformation 217 Appendix to Chapters VIII and IX. Volterea's Theory of Dislocations 156a. Introductory, (a) Displacement answering to given strain. (6) Discon- tinuity at a barrier, (c) Hollow cylinder deformed by removal of a slice of uniform thickness, {d) Hollow cylinder with radial fissure 219 Chapter X. Theory of the integration op the equations OF equilibrium of an isotropic elastic solid body 157. Nature of the problem 158. R&ume of the theory of Potential . 159. Description of Betti's method of integration . 160. Formula for the dilatation .... 161. Calculation of the dilatation from surface data 162. Formulse for the components of rotation . 163. Calculation of the rotation from surface data . 164. Body bounded by plane — Formulae for the dilatation 165. Body bounded by plane — Given surface displacements 166. Body bounded by plane — Given surface tractions 167. Historical Note 168. Body bounded by plane — Additional results . 169. Formulse for the displacement and strain 170. Outlines of various methods of integration 227 228 230 231 233 234 235 235 237 239 241 242 243 245 Xll CONTENTS Chapter XL The equilibrium of an elastic sphere and RELATED PROBLEMS ART. PAGE 171. Introductory . . . ■ 247 172. Solution in spherical harmonics of positive degrees .... 247 173. The sphere with given surface displacements . . . . • 249 174. Greneralization of the foregoing solution, (i) Integration by means of polynomials, (ii) Body force required to maintain a state of strain with zero surface displacement, (iii) General method for integrating the equations by meatis of series ...... 250 175. The sphere with given surface tractions . .... 251 176. Conditions restricting the prescribed siu-face tractions . . 254 177. Surface tractions directed normally to the boundary .... 255 178. Solution in spherical harmonics of negative degrees . . . 256 179. Sphere subjected to forces acting through its volume. Particular solution 257 180. Sphere deformed by body force only 258 180a. Sphere and spherical shell under body force. Alternative method . . 259 181. Gravitating incompressible sphere 263 182. Deformation of gravitating incompressible sphere by external forces 265 183. Gravitating nearly spherical body 268 184. Rotating sphere 268 185. Tidal deformation. Tidal eflfective rigidity of the Earth . . . 270 185a. Applications of the theory of Elasticity to Geophysical questions . 272 186. Plane strain in a circular cylinder 273 187. Applications of curvilinear coordinates .... . . 274 188. Symmetrical strain in a solid of revolution .... . 276 189. Symmetrical strain in a cylinder 278 Chapter XII. Vibrations of spheres and cylinders 190. Introductory 191.' Solution by means of spherical harmonics 192. Formation of the boundary-conditions for a vibrating sphere 193. Incompressible material 194. Frequency equations for vibrating sphere 195. Vibrations of the first class 196. Vibrations of the second class .... 197. Further investigations of the vibrations of spheres 198. Radial vibrations of a hollow sphere 199. Vibrations of a circular cylinder 200. Torsional vibrations . 201. Longitudinal vibrations .... 202. Transverse vibrations ... 281 282 284 287 287 288 289 290 290 291 292 292 294 Chapter XIII. The propagation of waves in elastic solid media 203. Introductory .... 297 204. Waves of dilatation and waves of distortion . . ; . . 297 205. Motion of a surface of discontinuity. Kinematical conditions 299 206. Motion of a surface of discontinuity. Dynamical conditions . 300 CONTENTS xiii AET. PAGE 207. Velocity of waves in isotropic medium 301 208. Velocity of waves in reolotropic solid medium . . . 302 209. Wave-surfaces 3qq 210. Motion determined by the characteristic equation .... 305 211. Arbitrary initial conditions 307 212. Motion due to body forces 3O8 213. Additional results relating to motion due to body forces . 310 214. Waves propagated over the surface of an isotropic elastic solid body . 311 ChAPTEK XIV. TOBSION 215. stress and Strain in a twisted prism 216. The torsion problem .... 217. Method of solution of the torsion problem 218. Analogies with Hydrodynamics 219. Distribution of the shearing stress ... 220. Strength to resist torsion 221. Solution of the torsion problem for certain boundaries . 222. Additional results 223. Graphic expression of the results 224. Analogy to the form of a stretched membrane loaded uniformly 225. Twisting couple 226. Torsion of seolotropic prism 226a. Bar of varying circular section . 315 . 316 318 319 . 320 . 321 . 322 324 325 . 327 . 327 . 329 330 226b. Distribution of traction over terminal section . . . 332 Chaptee XV. The bending of a beam by terminal TRANSVERSE LOAD 227. Stress in a bent beam ..... . . 334 228. Statement of the problem .... ... 335 229. Necessary type of shearing stress .... . . 336 230. FormulsB for the displacement 338 231. Solution of the problem of flexure for certain boundaries : (a) The circle, (6) Concentric circles, (c) The eUipse, (d) Confocal ellipses, (e) The rectangle, (/) Additional results 340 232. Analysis of the displacement : (a) Curvature* of the strained central-line, (6) Neutral plane, (a) Obliquity of the strained cross-sections, (rf) De- flexion, (e) Twist, (/) Anticlastic curvature, (g) Distortion of the cross- sections into curved surfaces . 343 233. Distribution ,of shearing stress ... .... 34V 234. Generalizations of the foregoing theory : {a) Asymmetric loading, (6) Com- bined strain, (c) .^olotropic material . 348 235. Criticisms of certain methods : (a) A method of determining the shearing stress in the case of rectangular sections, (6) Extension of this method to curved boundaries, (c) Form of boundary for which the method gives the correct result, (d) Defectiveness of the method in the case of an elliptic section, (e) Additional deflexion described as "due to shearing," (/) Defective method of calculating this additional deflexion . 351 XIV CONTENTS Chapi'er XVI. The bending of a beam loaded uniformly ALONG ITS LENGTH 236. Introductory 237. Stress uniform along the beam 238. Stress varying uniformly along the beam ■ •'•'" 239. Uniformly loaded beam. Reduction of the problem to one of plane strain 359 240. The constants of the solution .362 241. Strain and stress in the elements of the beam 363 242. Eolation between the curvature and the bending moment . . 365 243. Extension of the central-line 3" ' 244. Illustrations of the theory— (a) Form of solution of the related problem of plane strain, (b) Solution of the problem of plane strain for a beam of circular section bent by its own weight, (c) Correction of the curvature in this case, (d) Case of narrow rectangular beam loaded along the top treated as a problem of "generalized plane stress." (e) Narrow rectangular beam supported at the ends and loaded along the top 367 Chapter XVII. The theory of continuous beams 245. Extension of the theory of the bending of beams 371 245a. Further investigations . 374 246. The problem of continuous beams 375 247. Single span, (a) Terminal forces and couples. (6) Uniform load. Sup- ported ends, (c) Uniform load. Built-in ends, (rf) Concentrated load. Supported ends, (e) Concentrated load. Built-in ends . . . 376 248. The theorem of three moments, (a) Uniform load. (6) Equal spans. (c) Uniform load on each span, (d) Concentrated load on one span . 380 249. Graphic method of solution of the problem of continuous beams . . 382 250. Development of the graphic method 384 Chapter XVIII. General theory of the bending and TWISTING of thin RODS 251. Introductory 387 252. Kinematics of thin rods 387 253. Kinematical formulte . 388 254. Equa,tions of equilibrium 392 255. 1 The ordinary approximate theory 394 256. Nature of the strain in a bent and twisted rod 395 257. Approximate formulae for the strain 398 258. Discussion of the ordinary approximate theory . . . . 399 258a. Small displacement ... 401 259. Rods naturally curved 402 CONTENTS XV Chapter XIX. Problems concerning the equilibrium of THIN rods ART. PAGE 260. Kirchhoffs kinetic analogue .... ... 405 261. Extension of the theorem of the kinetic analogue to rods naturally curved 406 262. The problem of the elastica * 407 263. Classification of the forms of the elastica. (a) Inflexional elastica, (6) Non- inflexional elastica 408 264. Buckling of long thin strut under thrust 411 265. Computation of the strain-energy of the strut .... . 41.3 266. Eesistance to buckling 414 267. Elastic stability 415 267a. Southwell's method . • 416 268. Stability of inflexional elastica 417 269. Rod bent and twisted by terminal forces and couples ... 419 270. Rod bent to helical form ... 420 271. Theory of spiral springs 421 272. Additional results, (a) Rod subjected to terminal couple, (6) Straight rod with initial twist, (c) Rod bent into circular hoop and twisted uniformly, (rf) Stability of rod subjected to twisting couple and thrust, (e) Stability of flat blade bent in its plane 423 273. Rod bent by forces applied along its length 273a. Influence of stiffness on the form of a suspended wire 274. Rod bent in one plane by uniform normal pressure 275. Stability of circular ring under normal pressure 276. Height consistent with stability .... 427 428 429 430 431 Chapter XX. Vibrations of rods. Problems of dynamical RESISTANCE 277. Introductory ^^^ 278. Extensional vibrations 434 279. Torsional vibrations ^^^ 280. Elexural vibrations ^^^ 281. Rod fixed at one end and struck longitudinally at the other . . .437 282. Rod free at one end and struck longitudinaUy at the other . . .441 283. Rod loaded suddenly ^^^ 284. Longitudinal impact of rods ^'^^ 284a. Impact and vibrations ^^^ 285. Problems of dynamical resistance involving transverse vibration . . 447 286. The whirling of shafts '^^ Chapter XXI. Small deformation of naturally curved rods 287. Introductory •■''''''''' a kq 288. Specification of the displacement 289. Orientation of the principal torsion-flexure axes 451 290. Curvature and twist XVI CONTENTS PAGE 453 ART. 291. Simplified formulse . . 292. Problems of equilibrium, (a) Incomplete circular ring bent in its plane. (6) Incomplete circular ring bent out of its plane .... 454 293. ^'ibrations of a circular ring, (a) Flexural vibrations in the plane of the ring. (6) Flexural vibrations at right angles to the plane of the ring. (c) Torsional and e.\tensional vibrations 457 Chapter XXII. The stretching and bending of plates 294. Specification of stress in a plate ■ 461 295. Transformation of stress-resultants and stress-couples .... 462 296. Equations of equilibrium • 463 297. Boundary-conditions 464 298. Relation between the flexural couples and the curvature . . . 469 299. Method of determining the stress in a plate . . ". . . 471 300. Plane stress ... 473 301. Plate stretched by forces in its plane 473 302. Plate bent to a state of plane stress .... 476 303. Generalized plane stress ... 477 304. Plate bent to a state of generalized plane stress .... 479 305. Circular plate loaded at its centre 481 306. Plate in astate of stress which isuniformor varies uniformly over its plane 481 307. Plate bent by pressure imiform over a face 483 308. Plate bent by pressure varying uniformly over a face .... 485 309. Circular plate bent by uniform pressure and supported at the edge . 487 310. Plate bent by uniform pressure and clamped at the edge . . . 488 311. Plate bent by uniformly varying pressure and clamped at the edge . 490 312. Plate bent by its own weight ... 491 313. Approximate theory of the bending of a plate by transverse forces . 492 314. Illustrations of the approximate theory, (a) Circular plate loaded sym- metrically. (6) Application of the method of inversion, (c) Rectangular plate, (d) Transverse vibrations of plates, (e) Extensional vibrations of plates 493 Chapter XXIII. Inextensional deformation of curved plates or shells 315. Introductory 500 316. Changes of curvature in inextensional deformation 501 317. Typical flexural strain 503 318. Method of calculating the changes of curvature ..... 505 319. Inextensional deformation of a cylindrical shell, (a) Formulae for the dis- placement. (6) Changes of curvature 50g 320. Inextensional deformation of a spherical shell, (a) Formulse for the dis- placement. (6) Changes of curvature . . .... 508 321. Inextensional vibrations, (i) Cylindrical shell, (ii) Spherical shell . 514 CONTENTS XVll Chapter XXIV. General theory of thin plates and shells ART. PAGE 322. Formulae relating to the curvature of surfaces 517 323. Simplified formulse relating to the curvature of surfaces .... 519 324. Extension and curvature of the middle surface of a plate or shell . . 520 325. Method of calculating the extension and the changes of curvature . . 521 326. Formulse relating to small displacements 523 327. Nature of the strain in a bent plate or shell . . . . 527 328. Specification of stress in a bent plate or shell . .... 530 329. Approximate formulse for the strain, the stress-resultants, and the stress- couples 531 330. Second approximation in the case of a curved plate or shell . . . 535 331. Equations of equilibrium 536 332. Bomidary-conditions ... 539 332 a. Buckling of a rectangular plate under edge thrust 540 333. Theory of the vibrations of thin shells . 541 334. Vibrations of a thin cylindrical shell, (a) General equations. (6) Ex- tensional vibrations, (c) Inextensional vibrations, (d) Inexactness of the inextensional displacement, (e) Nature of the correction to be applied to the inextensional displacement ...... 546 335. Vibrations of a thin spherical shell .... ... 552 Chapter XXIV a. The equilibrium of thin shells 336. Small displacement .... 556 337. The middle surface a surface of revolution . .... 557 338. Torsion .... 559 CYLINDRICAL SHELL 339. Symmetrical conditions, (a) Extensional solution. (6) Edge-efiect . . 560 340. Tube under pressure . . 562 341. Stability of a tube under external pressure 563 342. Lateral forces, (a) Extensional solution, (b) Edge-efiect . . . 567 343. General unsymmetrical conditions. Introductory, (a) Extensional solution. (6) Approximately inextensional solution, (c) Edge-effect . 571 SPHERICAL SHELL 344. Extensional solution 575 345. Edge-effect. Symmetrical conditions . . ... 578 CONICAL SHELL 346. Extensional solution. Symnietrical conditions 582 347. Edge-effect. Symmetrical conditions . 583 348. Extensional solution. Lateral forces .... . . 586 349. Edge-effect. Lateral forces. Introductory, (a) Integrals of the equations of equilibrium. (6) Introduction of the displacement, (c) Formation of two linear differential equations, (d) Method of solution of the equations ... 587 XVlll CONTENTS ART. PAGE 350. Extenaional solution. Unsymmetrical conditions 593 351. Approximately inextensional solution 595 352. Edge-effect. Unsymmetrical conditions— Introductory, (a) Formation of the equations. (6) Preparation for solution, (c) Solution of the equations . 597 NOTES A. Terminology and Notation . . .... . 606 B. The notion of stress 608 C. Applications of the method of moving axes 611 INDEX Authors cited 616 Matters treated ... 619 HISTORICAL INTRODUCTION The Mathematical Theory of Elasticity is occupied with an attempt to reduce to calculatipn the state of strain, or relative displacement, within a solid body which is subject to the action of an equilibrating system of forces, or is in a state of slight internal relative motion, and with endeavours to obtain results which shall be practically important in applications to archi- tecture, engineering, and all other useful arts in which the material of con- struction is solid. Its history should embrace that of the progress of our experimental knowledge of the behaviour of strained bodies, so far as it has been embodied in the mathematical theory, of the development of our con- ceptions in regard to the physical principles necessary to form a foundation for theory, of the growth of that branch of mathematical analysis in which the process of the calculations consists, and of the gradual acquisition of practical rules by the interpretation of analytical results. In a theory ideally worked out, the progress which we should be able to trace would be, in other par- ticulars, one from less to more, but we may say that, in regard to the assumed physical principles, progress consists in passing from more to less. Alike in the experimental knowledge obtained, and in the analytical methods and results, nothing that has once been discovered ever loses its value or has to be discarded; but the physical principles come to be reduced to fewer and more general ones, so that the theory is brought more into accord with that of other branches of physics, the same general dynamical principles being ultimately requisite and sufficient to serve as a basis for them all.. And although, in the case of Elasticity, we find frequent retrogressions on the part of the experimentalist, and errors on the part of the mathematician, chiefly in adopting hypotheses not clearly established or already discredited, in push- ing to extremes methods merely approximate, in hasty generalizations, and in misunderstandings of physical principles, yet we observe a continuous progress in all the respects mentioned when we survey the history of the science from the initial enquiries of Galileo to the conclusive investigations of Saint-Venant and Lord Kelvin. The first mathematician to consider the nature of the resistance of solids to rupture was Galileo ^ Although he treated solids as inelastic, not being in ' Galileo Galilei, Discorsi e Dimostrazioni viatematiche, Leiden, 1638. L. B. 1 2 HISTORICAL INTRODUCTION possession of any law connecting the displacements produced with the forces producing them, or of any physical hypothesis capable of yielding such a law, yet his enquiries gave the direction which was subsequently followed by many investigators. He endeavoured to determine the resistance of a beam, one end of which is built into a wall, when the tendency to break it arises from its own or an applied weight; and he concluded that the beam tends to turn about an axis perpendicular to its length, and in the plane of the wall. This problem, and, in particular, the determination of this axis, is known as Galileo's problem. In the histor}' of the theory started by the question of Galileo, undoubtedly the two great landmarks are the discovery of Hooke's Law in 1660, and the formulation of the general equations by Navier in 1821. Hooke's Law pro- vided the necessary experimental foundation for the theory. When the general equations had been obtained, all questions of the small strain of elastic bodies were reduced to a matter of mathematical calculation. In England and in France, in the latter half of the l7th century, Hooke and Mariotte occupied themselves with the experimental discovery of what we now term stress-strain relations. Hooke ^ gave in 1678 the famous law of proportionality of stress and strain which bears his name, in the words " Ut tensio sic vis; that is, the Power of any spring is in the same proportion with the Tension thereof." By "spring'' Hooke means, as he proceeds to explain, any "springy body," and by "tension" what we should now call "extension," or, more generally, "strain." This law he discovered in 1660, but did not publish until 1676, and then onlyunder the form of an a.na,gTa.m,ceiiinosssttuii. This law forms the basis of the mathematical theory of Elasticity, and we shall hereafter consider its generalization, and its range of validity in the light of modem experimental research. Hooke does not appear to have made any application of it to the consideration of Galileo's problem. This application was made by Mariotte', who in 1680 enunciated the same law independently. He remarked that the resistance of a beam to flexure arises from the exten- sion and contraction of its parts, some of its longitudinal filaments being extended, and others contracted. He assumed that half are extended, and half contracted. His theory led him to assign the position of the axis, required in the solution of Galileo's problem, at one-half the height of the section above the base. In the interval between the discovery of Hooke's law and that of the general differential equations of Elasticity by Navier, the attention of those mathematicians who occupied themselves with our science was chiefly directed to the solution and extension of Galileo's problem, and the related theories of the vibrations of bars and plates, and the stability of columns. The first investigation of any importance is that of the elastic line or elastica by James \ Robert Hooke, De Potentia restitutiva, London, 1678. ^ E. Mariotte, Traite du mouvement des eaux, Paris, 1686. HISTORICAL INTRODUCTION 3 Bernoulli*' in 1705, in which the resistance of a bent rod is assumed to arise from the extension and contraction of its longitudinal filaments, and the equation of the curve assumed by the axis is formed. This equation practi- cally involves the result that the resistance to bending is a couple proportional to the curvature of the rod when bent, a result which was assumed by Euler in his later treatment of the problems of the elastica, and of the vibrations of thin rods. As soon as the notion of a flexural couple proportional to the curvature was established it could be noted that the work done in bending a rod is proportional to the square of the curvature. Daniel Bernoulli' sug- gested to Euler that the differential equation of the elastica could be found by making the integral of the square of the curvature taken along the rod a minimum; and Euler", acting on this suggestion, was able to obtain the differential equation of the curve and to classify the various forms of it. One form is a curve of sines of small amplitude, and Euler pointed out' that in this case the line of thrust coincides with the unstrained axis of the rod, so that the rod, if of sufficient length and vertical when unstrained, may be bent by a weight attached to its upper end. Further investigations* led him to assign the least length of a column in order that it may bend under its own or an applied weight. Lagrange^ followed and used his theory to determine the strongest form of column. These two writers found a certain length which a column must attain to be bent by its own or an applied weight, and they concluded that for shorter lengths it will be simply compressed, while for greater lengths it will be bent. These researches are the earliest in the region of elastic stability. In Euler's work on the elastica the rod is thought of as a line of particles which resists bending. The theory of the flexure of beams of finite section was considered by Coulomb". This author took account of the equation of equilibrium obtained by resolving horizontally the forces which act upon the part of the beam cut off by one of its normal sections, as well as of the equation of moments. He was thus enabled to obtain the true position of the "neutral line," or axis of equilibrium, and he also made a correct calcu- lation of the moment of the elastic forces. His theory of beams is the most exact of those which proceed on the assumption that the stress in a bent '' Bernoulli's memoir is entitled, ' Veritable hypoth^se de la resistance des solides, avee la demonstration de la courbure des corps qui font ressort,' and will be found in his collected works, t. 2, Geneva, 1744. ^ See the 26th letter of Daniel Bernoulli to Euler (October, 1742) in Fuss, Correspondanee matMnatique et physique, t. 2, St Petersburg, 1843. ' See the Additamentum 'De curvis elastiois' in the Methodus inveniendi Uneas curvas maximi minimive proprietate gaudentes, Lausanne, 1744. ' Berlin, Histoire de VAcaMmie, t. 13 (1757). 8 Acta Acad. Petropolitance of 1778, Par's prior, pp. 121—193. 8 Miscellanea Taurinensia, t. 5 (1773). " 'Essai sur une application des regies de Maximis et Minimis i quelques Probl&mes de Statique, relatifs 4 I'Architecture,' Mim....par divers savans, 1776. 1—2 4 HISTORICAL INTRODUCTION beam arises wholly from the extension and contraction of its longitudinal filaments, and is deduced mathematically from this assumption and Hooke's Law. Coulomb was also the first to consider the resistance of thin fibres to torsion", and it is his account of the matter to which Saint- Venant refers under the name I'ancienne theorie, but his formula for this resistance was not deduced from any elastic theory. The formula makes the torsional rigidity of a fibre proportional to the moment of inertia of the normal section about the axis of the fibre. Another matter to which Coulomb was the first to pay attention was the kind of strain we now call shear, though he considered it in connexion with rupture only. His opinion appears to have been that rupture''' takes place when the shear of the material is greater than a certain limit. The shear considered is a permanent set, not an elastic strain. Except Coulomb's, the most important work of the period for the general mathematical theory is the physical discussion of elasticity by Thomas Young. This naturalist (to adopt Lord Kelvin's name for students of natural science) besides defining his modulus of elasticity, was the first to consider shear as an elastic strain" He called it "detrusion," and noticed that the elastic resistance of a body to shear, and its resistance to extension or contraction, are in general different; but he did not introduce a distinct modulus of rigidity to express resistance to shear. He defined " the modulus of elasticity of a substance"" as "a column of the same substance capable of producing a pressure on its base which is to the weight causing a certain degree of com- pression, as the length of the substance is to the diminution of its length." What we now call "Young's modulus" is the weight of this column per unit of area of its base. This introduction of a definite physical concept, associated with the coefficient of elasticity which descends, as it were from a clear sky, on the reader of mathematical memoirs, marks an epoch in the history of the science. Side by side with the statical developments of Galileo's enquiry there were discussions of the vibrations of solid bodies. Euler^ and Daniel Bernoulli'^ obtained the differential equation of the lateral vibrations of bars by variation of the function by which they had previously expressed the work done in " Histoire de VAcacUmie for 1784, pp. 229—269, Paris, 1787. 12 See the introduction to the memoir first quoted, Mgn....par divers savam, 1776. 'S A Course of Lectures on Natural Philosophy aiul the Mechanical Arts, London, 1807, Lecture xiii. It is in Kelland's later edition (1845) on pp. 105 et seq. '* hoc. cit. (footnote 13). The definition was given in Section ix of Vol. 2 of the first edition, and omitted in Kelland's edition, but it is reproduced in the Miscellaneous Works of Dr Younp. 15 'De vibrationibu8...1aminarum elasticarura,.,,' and ' De sonis multifariis quos laminae elasticae...edunt...' published in Gommentarii Academice Scientiarum Imperialis Petropolitana t. 13 (1751). The reader must be cautioned that in writings of the 18th century a "lamina" means a straight rod or curved bar, supposed to be cut out from a thin plate or cylindrical shell by two normal sections near together. This usage lingers in many books. HISTORICAL INTRODUCTION 5 bending '^ They determined the forms of the functions which we should now call the "normal functions," and the equation which we should now call the "period equation," in the six cases of terminal conditions which arise accord- ing as the ends are free, clamped or simply supported. Chladni" investig'ated these modes of vibration experimentally, and also the longitudinal and torsional vibrations of bars. The success of theories of thin rods, founded on special hypotheses, appears to have given rise to hopes that a theory might be developed in the same way for plates and shells, so that the modes of vibration of a bell might be deduced from its form and the manner in which it is supported. The first to attack this problem was Euler. He had already proposed a theory of the resistance of a curved bar to bending, in which the change of curvature played the same part as the curvature does in the theory of a naturally straight bar'* In a note "De Sono Campanarum^'" he proposed to regard a bell as divided into thin annuli, each of which behaves as a curved bar. This method leaves out of account the change of curvature in sections through the axis of the bell. James Bernoulli^" (the younger) followed. He assumed the shell to consist of a kind of double sheet of curved bars, the bars in one sheet being at right angles to those in the other. Reducing the shell fco a plane plate he found an equation of vibration which we now know to be incorrect. James Bernoulli's attempt appears to have been made with the view of discovering a theoretical basis for the experimental results of Chladni con- cerning the nodal figures of vibrating plates ''^ These results were still unex- plained when in 1809 the French Institut proposed as a subject for a prize the investigation of the tones of a vibrating plate. After several attempts the prize was adjudged in 1815 to Mdlle Sophie Germain, and her work was published in 1821=''- She assumed that the sum of the principal curvatures of the plate when bent would play the same part in the theory of plates as the curvature of the elastic central-line in the theory of rods, and she pro- posed to regard the work done in bending as proportional to the integral of the square of the sum of the principal curvatures taken over the surface. From this assumption and the principle of virtual work she deduced the equation of flexural vibration in the form now generally admitted. Later investigations IS The form of the energy-function and the notion of obtaining the differential equation by varying it are due to D. Bernoulli. The process was carried out by Euler, and the normal functions and the period equations were determined by him. " B. F. P. Chladni, Die Akustik, Leipzig, 1802. The author gives an account of the history of his own experimental researches with the dates of first publication. 18 In the MethQdus inveniendi . . . p. 274. See also his later writing 'Genuina principia... de statu sequilibrii et motu corporum...,' Nov. Gamm. Acad. Petropolitanee, t. 15 (1771). 19 Nov. Gamm. Acad. Petropolitanee, t. 10 (1766). •-0 'Essai tWorique sur les vibrations des plaques Clastiques...,' Nov. Acta Petrnpolitarue, t. 5 (1789). 21 First published at Leipzig in 1787. See Die Akustik, p. vii. 32 Becherches sur la tlieorie des surfaces elastiques. Paris, 1821. 6 HISTORICAL INTRODUCTION have shown that the formula assumed for the work done in bending was incorrect. During the first period in the history of our science (1638—1820), while these various investigations of special problems were being made, there was a cause at work which was to lead to wide generalizations. This cause was physical speculation concerning the constitution of bodies. In the eighteenth century the Newtonian conception of material bodies, as made up of small parts which act upon each other by means of central forces, displaced the Cartesian conception of a plmum pervaded by " vortices." Newton regarded his "molecules" as possessed of finite sizes and definite shapes^, but his successors gradually simplified them into material points. The most definite speculation of this kind is that of Boscovich'*, for whom the material points were nothing but persistent centres of force. To this order of ideas belong Laplace's theory of capillarity^ and Poisson's first investigation of the equi- librium of an "elastic surface^"," but for a long time no attempt seems to have been made to obtain general equations of motion and equilibrium of elastic solid bodies. At the end of the year 1820 the fruit of all the ingenuity ex- pended on elastic problems might be summed up as — an inadequate theory of flexure, an erroneous theory of torsion, an unproved theory of the vibrations of bars and plates, and the definition of Young's modulus. But such an estimate would give a very wrong impression of the value of the older researches. The recognition of the distinction between shear and extension was a preliminary to a general theory of strain; the recognition of forces across the elements of a section of a beam, producing a resultant, was a step towards a theory of stress ; the use of differential equations for the deflexion of a bent beam and the vibrations of bars and plates, was a foreshadowing of the employment of difierential' equations of displacement ; the Newtonian conception of the con- stitution of bodies, combined with Hooke's Law, offered means for the formation of such equations; and the generalization of the principle of virtual work in the Mecanique Analytique threw open a broad path to discovery in this as in every other branch of mathematical physics. Physical Science had emerged from its incipient stages with definite methods of hypothesis and induction and of observation and deduction, with the clear aim to discover the laws by which phenomena are connected with each other, and with a fund of analytical pro- cesses of investigation. This was the hour for the production of general theories, and the men were not wanting. Navier" was the first to investigate the general equations of equilibrium and vibration of elastic solids. He set out from the Newtonian conception of ^ See, in particular, Newton, Optiks, 2n(l Edition, London, 1717, the 31st Query. " E. J. Boscovich, Theoria Philosophia Naturalis redacta ad unicam lepem virium in natura e.rixtentium, Venice, 1743. 2S Mecanique Ce'lcate, SuppUment au 10« Litre, Paris, 1806. ^ Paris, MSm. de I'Institut, 1814. ' " Paris, Mem. Anid. Sciences, t. 7 (1827). The memoir was read in May, 1821. HISTORICAL INTRODUCTION 7 the constitution of bodies, and assumed that the elastic reactions arise from variations in the intermolecular forces which result from changes in the mole- cular configuration. He regarded the molecules as material points, and assumed that the force between two molecules, whose distance is slightly increased, is proportional to the product of the increment of the distance and some function of the initial distance. His method consists in forming an expression for the component in any direction of all the forces that act upon a displaced molecule, and thence the equations of motion of the molecule. The equations are thus obtained in terms of the displacements of the molecule. Themaierial is assumed to be isotropic, and the equations of equilibrium and vibration contain a single constant of the same nature as Young's modulus. Navier next formed an expression for the work done in a small relative displacement by all the forces which act upon a molecule; this he described as the sum of the moments (in the sense of the Micanique Analytique) of the forces exerted by all the other molecules on a particular molecule. He deduced, by an application of the Calculus of Variations, not only the differential equations previously obtained,! but also the boundary conditions that hold at the surface of the body. This memoir is very important as the first general investigation of its kind, but its arguments have not met with general acceptance. Objection has been raised against Navier 's expression for the force between two "molecules," and to his method of simplifying the expressions for the forces acting on a single " mole- cule." These expressions involve triple summations, which Navier replaced by integrations, and the validity of this procedure has been disputed^'. In the same year, 1 821, in which Navier 's memoir was read to the Academy the study of elasticity received a powerful impulse from an unexpected quarter. Fresnel announced his conclusion that the observed facts in regard to the interference of polarised light could be explained only by the hypothesis of transverse vibrations^". He showed how a medium consisting of "molecules" connected by central forces might be expected to execute such vibrations and to transmit waves of the required type. Before the time of Young and Fresnel such examples of transverse waves as were known — waves on water, transverse -8 For criticisms of Navier's memoir and an account of the discussions to which it gave rise, see Todhunter and Pearson, History of the Theory of Elasticity , vol. 1, Cambridge, 1886, pp. 139, 221, 277: and cf. the account given by H. Burkhardt in his Eeport on ' Entwickelungen nach oscillirenden Functionen ' published in the Jahresbericht der Deutschen, Mathematiker- Vereinigung, Bd. 10, Heft 2, Lieferung 3 (1903). It may not be superfluous to remark that the conception of molecules as material points at rest in a state of stable equilibrium under their mutual forces of attraction and repulsion, and held in slightly displaced positions by external forces, is quite different from the conception of molecules with which modern Thermodynamics has made us familiar. The " molecular " theories of Navier, Poisson and Cauchy have no very intimate relation to modern notions about molecules. 2" See E. Verdet, (Euvres computes d'Augustin Fresnel, t. 1, Paris, 1866, p. Ixxxvi, also pp. 629 et seq. Verdet points out that Fresnel arrived at his hypothesis of transverse vibrations in 1816 {lac. cit. pp. lv, 385, 394). Thomas Young in his Article ' Chromatics ' (Encycl. Brit. Supplement, 1817) regarded the luminous vibrations as having relatively feeble transverse com- ponents. 8 HISTORICAL INTRODUCTION vibrations of strings, bars, membranes and plates— were in no case examples of waves transmitted through a medium; and neither the supporters nor the opponents of the undulatory theory of light appear to have conceived of light waves otherwise than as "longitudinal" waves of condensation and rarefaction, of the type rendered familiar by the transmission of sound. The theory of elasticity, and, in particular, the problem of the transmission of waves through an elastic medium now attracted the attention of two mathematicians of the highest order: Cauchy» and Poisson'^— the former a discriminating supporter, the latter a sceptical critic of Fresnel's ideas. In the future the developments of the theory of elasticity were to be closely associated with the question of the propagation of light, and these developments arose in great part from the labours of these two savants. By the Autumn of 1822 Cauchy'-' had discovered most of the elements of the pure theory of elasticity. He had introduced the notion of stress at a point determined by the tractions per unit of area across all plane elements through the point. For this purpose he had generalized the notion of hydro- static pressure, and he had shown that the stress is expressible by means of .six component stresses, and also by means of three purely normal tractions across a certain triad of planes which cut each other at right angles — the " principal planes of stress." He had shown also how the differential coefficients of the three components of displacement can be used to estimate the extension of every linear element of the material, and had expressed the state of strain near a point in terms of six components of strain, and also in terms of the extensions of a certain triad of lines which are at right angles to each other — the " principal axes of strain." He had determined the equations of motion (or equilibrium) by which the stress-components are connected with the forces that are distributed through the volume and with the kinetic reactions. By means of relations between stress-components and strain-components, he had eliminated the stress-components from the equations of motion and equilibrium, and had arrived at equations in terms of the displacements. In the later published version of this investigation Cauchy obtained his stress-strain rela- tii)ns for isotropic materials by means of two assumptions, viz.: (1) that the relations in question are linear, (2) that the principal planes of stress are '" Cauchy's studies in Elasticity were first prompted by his being a member of the Commission appointed to report upon a memoir by Navier on elastic plates which was presented to the Paris Academy in August, 1820. ■" We have noted that Poisson had already written on elastic plates in 1814. ■- Cauchy's memoir was communicated to the Paris Academy in September 1822, but it was not published. An abstract was inserted in the Bulletin des Sciences a la Societe y/iiJo- tiuithique, 1823, and the contents of the memoir were given in later publications, viz. in two Articles in the volume for 1827 of Cauchy's Exercices de matMmatique and an Article in the volume for 1828. The titles of these Articles are (i) ' De la pression ou tension dans un corps solide,' (ii) ' Sur la condensation et la dilatation des corps solides,' (iii) ' Sur les Equations qui expriment les conditions d'i5quilibre ou les lois de mouvement int^rieur d'un corps solide.' The la.'^t of these contains the correct equations of Elasticity. HISTORICAL INTRODUCTION 9 normal to the principal axes of strain. The experimental basis on which these assumptions can be made to rest is the same as that on which Hooke's Law rests, but Cauehy did not refer to it. The equations obtained are those which are now admitted for isotropic solid bodies. The methods used in these investigations are quite different from those of Navier's memoir. In particular, no use is made of the hypothesis of material points and central forces. The resulting equations differ from Navier's in one important respect, viz. : Navier's equations contain a single constant to express the elastic behaviour of a body, while Cauchy's contain two such constants. At a later date Cauehy extended his theory to the case of crystalline bodies, and he then made use of the hypothesis of material points between which there are forces of attraction or repulsion. The force between a pair of points was taken to act in the line joining the points, and to be a function of the distance between them ; and the assemblage of points was taken to be homo- geneous in the sense that, ii A, B, G are any three of the points, there is a point D of the assemblage which is situated so that the line CD is equal and parallel to AB, and the sense from C to D is the same as the sense from A to B. It was assumed further that when the system is displaced the relative displacement of two of the material points, which are within each other's ranges of activity, is small compared with the distance between them. In the first memoir^ in which Cauehy made use of this hypothesis he formed an expression for the forces that act upon a single material point in the system, and deduced differential equations of motion and equilibrium. In the case of isotropy, the equations contained two constants. In the second memoir^* expressions were formed for the tractions across any plane drawn in the body. If the initial state is one of zero stress, and the material is isotropic, the stress is expressed in terms of the strain by means of a single constant, and one of the constants of the preceding memoir must vanish. The equations are then identical with those of Navier. In like manner, in the general case of aeolotropy, Cauehy found 21 independent constants. Of these 15 are true "elastic constants," and the remaining 6 express the initial stress and vanish identically if the initial state is one of zero stress. These matters were not fully explained by Cauehy. Clausius'°, however, has shown that this is the meaning of his work. Clansius criticized the restrictive conditions which Cauehy imposed upon the arrangement of his material points, but he argued that these conditions are not necessary for the deduction of Cauchy's equations. '^'■' Exercices de matMmatique, 1828, ' Sur I'equilibre et le mouvement d'un systSme de points mat^riels sollicites par des forces d'attraction ou de repulsion mutuelle.' This memoir follows immediately after that last quoted and immediately precedes that next quoted. "•* Exercices de mathivuitique, 1828, ' De la pression ou tension dans un syst^me de points materiels. ' 35 ' Ueber die Veranderungen, welche in den bisher gebrauchlichen Formeln fiir das Gleichge- wicht und die Bewegung elastischer fester Korper durch neuere Beobachtungen nothwendig geworden sind,' Ami. Phys. Chem. {Poggendorff), Bd. 76 (1849). 10 HISTORICAL INTRODUCTION The first memoir by Poisson^" relating to the same subject was read before the Paris Academy in April, 1828. The memoir is very remarkable for its numerous applications of the general theory to special problems. In his in- vestigation of the general equations Poisson, like Cauchy, first obtains the -equations of equilibrium in terms of stress-components, and then estimates the tractioii across any plane resulting from the "intermolecular" forces. The expressions for the stresses in terms of the strains involve summations with respect to all the "molecules," situated within the region of "molecular" ac- tivity of a given one. Poisson decides against replacing all the summations by integrations, but he assumes that this can be done for the summations with respect to angular space about the given "molecule," but not for the summations with respect to distance from this "molecule." The equations of equilibrium and motion of isotropic elastic solids which were thus obtained are identical with Navier's. The principle, on which summations may be replaced by integrations, has been explained as follows by Cauchy'*^:— The number of molecules in any volume, which contains a very large number of molecules, and whose dimensions are at the same time small compared with the radius of the sphere of sensible molecular activity, may be taken to be proportional to the volume. If, then, we make abstraction of the molecules in the immediate neighbourhood of the one considered, the actions of all the others, contained in any one of the small volumes referred to, will be equiva- lent to a force, acting in a line through the centroid of this volume, which will be proportional to the volume and to a function of the distance of the particular molecule from the centroid of the volume. The action of the remoter molecules is said to be "regular," and the action of the nearer ones, "irregular"; and thus Poisson assumed that the irregular action of the nearer molecules may be neglected, in comparison with the action of the remoter ones, which is regular. This assumption is the text upon which Stokes''? afterwards founded his criticism of Poisson. As we have seen, Cauchy arrived at Poisson's results by the aid of a different assumption^*. Clausius^' held that both Poisson's and Cauchy's methods could be presented in unexception- able forms. The theory of elasticity established by Poisson and Cauchy on the then accepted basis of material points and central forces was applied by them and * ' M^moire sur I'equilibre et le mouvement des corps ^lastiques,' Mini. Paris Acad., t. & (1829). " 'On the Theories of the... Equilibrium and Motion of Elastic Solids,' Cambridge Phil. Soe. Tram. vol. 8 (1845). Reprinted in Stokes's Math, and Phys. Papers, vol. 1, Cambridge, 1880, p. 75. ** In a later memoir presented to the Academy in 1829 and published in J. de I'Bcolepoly- technique, t. 13 (1831), Poisson adopted a method quite similar to that of Cauchy (footnote 34). Poisson extended his theory to seolotropie bodies in his • M^moire sur I'equilibre et le mouve- ment des corps cristallisees,' read to the Paris Academy in 1839 and published after his death in Mem. de I'Acad., t. 18 (1842). HISTORICAL INTRODUCTION 11 also by Lam6 and Clapeyron"" to numerous problems of vibrations and of statical elasticity, and thus means were provided for testing its consequences experimentally, but it was a long time before adequate experiments were made to test it. Poisson used it to investigate the propagation of waves through an isotropic elastic solid medium. He found two types of waves which, at great distances from the sources of disturbance, are practically "longitudinal" and "transverse," and it was a consequence of his theory that the ratio of the velocities of waves of the two types is v/3 : 1* Cauchy" applied his equations to the question of the propagation of light in crystalline as well as in isotropic media. The theory was challenged first in its application to optics by Green ^^ and afterwards on its statical side by Stokes ^^ Green was dissatisfied with the hypothesis on which the theory was based, and he sought a new foundation ; Stokes's criticisms were directed rather against the process of deduction and some of the particular results. The revolution which Green effected in the elements of the theory is comparable in importance with that produced by Navier's discovery of the general equations. Starting from what is now called the Principle of the Conservation of Energy he propounded a new method of obtaining these equations. He himself stated his principle and method in the following words : — "In whatever way the elements of any material system may act upon each "other, if all the internal forces exerted be multiplied by the elements of their " respective directions, the total sum for any assigned portion of the mass will "always be the exact differential of some function. But this function being "known, we can immediately apply the general method given in the Mecanique " Analytique, and which appears to be more especially applicable to problems "that relate to the motions of systems composed of an immense number of "particles mutually acting upon each other. One of the advantages of this "method, of great importance, is that we are necessarily led by the mere " process of the calculation, and with little care on our part, to all the equations "and conditions which are requisite and sufficient for the complete solution of "any problem to which it may be applied." The function here spoken of, with its sign changed, is the potential energy of the strained elastic body per unit of volume, expressed in terms of the components of strain; and the differential coefficients of the function, with 39 'M^moire sur I'^quilibre int^rieur d^s corps solides homogtoes,' Paris, Mim....par divers savants, t. 4 (1833). The memoir was published also in J.f. Math. (Crelle), Bd. 7 (1831); it had been presented to the Paris Academy, and the report on it by Poinsot and Navier is dated 1828. In regard to the general theory the method adopted was that of Navier. *> See the addition, of date November 1828, to the memoir quoted in footnote 36. Cauchy recorded the same result in the Exercices de mathematique, 1830. ^1 Exercices de mathematique, 1880. <" 'On the laws of reflexion and refraction of light at the common surface of two non- erystallized media,' Cambridge Phil. Soc. Trans, vol. 7 (1839). The date of the memoir is 1837. It is reprinted in Mathematical Paper.-! of the late George Green, London, 1871, p. 245. 12 HISTORICAL INTRODUCTION respect to the components of strain, are the components of stress. Green supposed the function to be capable of being expanded in powers and products of the components of strain. He therefore arranged it as a sum of homogeneous functions of these quantities of the first, second and higher degrees. Of these terms, the first must be absent, as the potential energy must be a true minimum when the body is unstrained; and, as the strains are all small, the second term alone will be of importance. From this principle Green deduced the equations of Elasticity, containing in the general case 21 constants. In the case of isotropy there are two constants, and the equations are the same as those of Cauchy's first memoir^''. Lord Kelvin*' has based the argument for the existence of Green's strain- energy-function on the First and Second Laws of Thermodynamics. From these laws he deduced the result that, when a solid body is strained without alteration of temperature, the components of stress are the differential co- eflficients of a function of the components of strain with respect to these components severally. The same result can be proved to hold when the strain is effected so quickly that no heat is gained or lost by any part of the body. Poisson's theory leads to the conclusions that the resistance of a body to compression by pressure uniform all round it is two-thirds of the Young's modulus of the material, and that the resistance to shearing is two-fifths of the Young's modulus. He noted a result equivalent to the first of these^, and the second is virtually contained in his thBory of the torsional vibrations of a bar^° The observation that resistance to compression and resistance to shearing are the two fundamental kinds of elastic resistance in isotropic bodies was made by Stokes* and he introduced definitely the two principal moduluses of elasticity by'wlTich these resistances are expressed — the "modulus of com- pression " and the " rigidity," as they are now called. From Hooke's Law and from considerations of symmetry he concluded that pressure equal in all directions round a point is attended by a proportional compression without shear, and that shearing stress is attended by a corresponding proportional shearing strain. As an experimental basis for Hooke's Law he cited the fact that bodies admit of being thrown into states of isochronous vibration. By a method analogous to that of Cauchy's first memoir'^ but resting on the above-stated experimental basis, he deduced the equations with two constants which had been given by Cauchy and Green. Having regard to the varying degrees in which different classes of bodies— liquids, soft solids, hard solids — « Sir W. Thomson, Quart. J. of Math. vol. 5 (1855), reprinted in Phil. Mag. (Ser. 5), vol. 5 (1878), and also in Mathematical and Physical Paperx by Sir William Thomson, vol. 1, Cambridge, 1882, p. 291. " Armales de Chimie et de Physique, t. 36 (1827). « This theory is given in the memoir cited in footnote 36. ** See footnote 37. The distinction between the tvfo kinds of elasticity had been noted by Poncelet, Introduction a la Mecanique imhiHtrielle, physique et exp&rimentaU, Metz, 1839. HISTORICAL INTRODUCTION , 13 resist . compression and distortion, he refused to accept the conclusion from Poisson's theory that the modulus of compression has to the rigidity the ratio 5 : 3. He pointed out that, if the ratio of these moduluses could he regarded as infinite, the ratio of the velocities of "longitudinal" and "transverse" waves would also be infinite, and then, as Green had already shown, the application of the theory to optics would be facilitated. The methods of Navier, of Poisson, and of Cauchy's later memoirs lead to equations of motion containing fewer constants than occur in the equations obtained by the methods of Green, of Stokes, and of Cauchy's first memoir. The importance of the discrepancy was first emphasized by Stokes. The questions in dispute are these — Is elastic seolotropy to be characterized by 21 constants or by 15, and is elastic isotropy to be characterized by two constants or one ? The two theories are styled by Pearsonf the " multi-con- stant" theory and the "rari-constant" theory respectively, and the controversy concerning them has lasted almost down to the present time. It is to be understood that the rari-constant equations can be included in the multi- constant ones by equating certain pairs of the coefficients, but that the rari- constant equations rest upon a particular hypothesis concerning the constitu- tion of matter, while the adoption of multi-constancy l^as been held to imply denial of this hypothesis. Discrepancies between the results of the two theories can be submitted to the test of experiment, and it might be thought that the verdict would be final, but the difficulty of being certain that the tested material is isotropic has diminished the credit of many experimental investigations, and the tendency of the multi-constant elasticians to rely on experiments on such bodies as cork, jelly and india-rubber has weakened their arguments, ' ^uch of t he discussion has turned upon the value of the ratio ^ of lateral contraction to longitudinal extension of a bar under terminal tractive load. This ratio is often called "Poisson's ratio." Poisson =" deduced from his theory the result that this ratio must be j. The experiments of Wertheim on glass and brass did not support this result, and Wertheim^' proposed to take the ratio to be ^ — a value which has no theoretical foundation. The experimental evidence led Lam6 in his treatise^' to adopt the multi-constant equations, and after the publication of this book they were generally employed. Saint- Venant, though a firm believer in rari-constancy, expressed the results of his researches on torsion and flexure and on the distribution of elasticities round a point™ in terras of the multi-constant theory. Kirchhoffi adopted the same theory in his investigations of thin rods and plates, and supported « Toihniitera.nAVearsoD,HistoryoftheTheoryof Elasticity, Yo\. 1, Cambridge, 1886, p. 496. ^8 Annates de Chimie, t. 23 (1848). « Legons sur la thiorie mathematique de Velasticite des corps solides, Paris, 1852. ™ The memoir on torsion is in M^m. des Savants etrangers, t. 14 (1855), that on flexure is in J. deMath. [Liouville), (S^r. 2), t. 1 (1856), and that on the distribution of elasticities is in J. de Math. (LUmville), (S^r. 2), t. 8 (1863). 01 J.f. Math. (Crelle), Bd. 40 (1850), and Bd. 56 (1859). I 4 HISTORICAL INTRODUCTION it by experiments on the torsion and flexure of steel bars"''; and Clebsch in his treatise^ used the language of bi-constant isotropy. Kelvin and Tait" dismissed the controversy in a few words and adopted the views of Stokes. The best modern experiments support the conclusion that Poisson's ratio can differ sensibly from the value J in materials which may without cavil be treated as isotropic and homogeneous. But perhaps the most striking experimental evidence is that which Voigt"^ has derived' from his study of the elasticity of crystals. The absence of guarantees for the isotropy of the tested materials ceased to be a difficulty when he had the courage to undertake experiments on materials which have known kinds of seolotropy'^. The point to be settled is, however, more remote. According to Green there exist, for a material of the most generally asolotropic character, 21 independent elastic constants. The molecular hypothesis, as worked out by Cauchy and supported by Saint- Venant, leads to 15 constants, so that, if the rari-constant theory is correct, there must be 6 independent relations among Green's 21 coefficients. These relations I call Cauchy's relations"'. Now Voigt's experiments were made on the torsion and flexure of prisms of various crystals, for most of which Saint- Venant's formulae for seolotropic rods hold good, for the others he supplied the required formulae. In the cases of beryl and rocksalt only were Cauchy's relations even approximately verified; in the seven other kinds of crystals examined there were very considerable differences between the coefficients which these relations would require to be equal. Independently of the experimental evidence the rari-constant theory has lost ground through the widening of our views concerning the constitution of matter. The hypothesis of material points and central forces does not now hold the field. This change in the tendency of physical speculation is due to many causes, among which the disagreement of the rari-constant theory of elasticity with the results of experiment holds a rather Subordinate position. Of much greater importance have been the development of the atomic theory in Chemistry and of statistical molecular theories in Physics^ the growth of the doctrine of energy, the discovery of electric radiation. It is now recognized that a theory of atoms must be part of a theory of the aether, and that the confidence which was once felt in the hypothesis of central forces between material points was premature. To determine the laws of the elasticity of solid bodies without knowing the nature of the aethereal medium or the nature 52 Ann. Phys. Chem. (Poggendorff), Bd. 108 (1859). •" Theorie der Elasticitdt fester K'nrper, Leipzig, 1862. " Thomson and Tait, Natural Philosophy, Ist edition Oxford 1867, 2nd edition Cambridge 1879—1883. =" W. Voigt, Ann. Phys. Chem. (Wiedemann), Bde. 31 (1887), 34 and 35 (1888), 38 (1889). =« A certain assumption, first made by P. E. Neumann, is involved in the statement that the fflolotropy of a crystal as regards elasticity is known from the crystallographio form. " They appear to have been first stated explicitly by Saint-Venant in the memoir on torsion of 1855. (See footnote 50.) HISTORICAL INTRODUCTION 15 of the atoms, we can only invoke the known laws of energy as was done by Green and Lord Kelvin ; and we may place the theory on a firm basis if we appeal to experiment to support the statement that, within a certain range of strain, the strain-energy-function is a quadratic function of the components of strain, instead of relying, as Green did, upon an expansion of the function in series. The problem of determining the state of stress and strain within a solid i body which is subjected to given forces acting through its volume and to given tractions across its surface, or is held by surface tractions so that its surface is deformed into a prescribed figure, is reducible to the analytical problem of finding functions to represent the components of displacement. These functions must satisfy the differential equations of equilibrium at all points within the surface of the body and must also satisfy certain special conditions at this surface. The methods which have been devised for in- tegrating the equations fall into two classes. In one class of methods a special solution is sought and the boundary conditions are satisfied by a solution in the form of a series, which may be infinite, of special solutions. The special solutions are generally expressible in terms of harmonic functions. This class of solutions may be regarded as constituting an extension of the methods of ^ expansion in spherical harmonics and in trigonometrical series. In the other class of methods the quantities to be determined are expressed by definite integrals, the elements of the integrals representing the effects of singularities distributed over the surface or through the volume. This class of solutions constitutes an extension of the methods introduced by Green in the Theory of the Potential. At the time of the discovery of the general equations of Elasticity the method of series had already been applied to astronomical problems, to acoustical problems and to problems of the conduction of heat=*; the method of singularities had not been invented^'. The application of the method of series to problems of equilibrium of elastic solid bodies was initiated by Lam6 and Clapeyron^. They considered the case of a body bounded by an unlimited plane to which pressure is applied according to an arbitrary law. Lam^"" later considered the problem of a body bounded by a spherical surface and deformed by given surface tractions. The problem of the plane is essentially that of the transmission into a solid body of force applied locally to a small part of its surface. The problem of the sphere has been developed by Lord Kelvin'S who sought to utilize it for the purpose of investigating ^^ See Burkhardt , ' Bnt wickelungen nach oscillirenden Functionen , ' Jahresbericht der Deutschen Mathematiker-Vereinigung, Bd. 10, Heft 2. ™ It was invented by Green, An Essay on the Application of Mathematical Analysis to the Tlieories of Electricity and Magnetism, Nottingham, 1828. Reprinted in Mathematical Papers of the late George Green, London, 1871. en J. de Math. (Liouville), t. 19 (1854). 61 Phil. Trans. Boy. Soc, vol. 153 (1863). See also Math. andPhys. Papers, vol. 3 (Cambridge, 1890), p. 351, and Kelvin and Tait, Nat. Phil., Part 2. 16 HISTORICAL INTRODUCTION the rigidity of the Earth "^ and by G. H. Darwin in connexion with other problems of cosmical physics'®. The serial solutions employed are expressed in terms of spherical harmonics. Solutions of the equations in cylindrical coordinates can be expressed in terms of Bessel's functions", but, except for spheres and cylinders, the method of series has not been employed very successfully. The method of singularities was first applied to the theory of Elasticity by E. Betti«", who set out from a certain reciprocal theorem of the type that is now familiar in many branches of mathematical physics. From this theorem he deduced incidentally a formula for determining the average strain of any type that is produced in a body by given forces. The method of singularities has been developed chiefly by the elasticians of the Italian school. It has proved more effective than the method of series in the solution of the problem of transmission of force. The fundamental particular solution which expresses the displacement due to force at a point in an indefinitely extended solid was given by Lord Kelvin'*. It was found at a later date by J. Boussinesq"^ along with other particular solutions, which can, as a matter of fact, be derived by synthesis from it. Boussinesq's results led him to a solution of the problem of the plane, and to a theory of " local perturbations," according to which the effect of force applied in the neighbourhood of any point of a body falls off very rapidly as the distance from the point increases, and the application of an equilibrating system of forces to a small part of a body produces an effect which is negligible at a considerable distance from the part. To estimate the effect produced at a distance by forces applied near a point, it is not necessary to take into account the mode of application ( )f the forces but only the statical resultant and moment. The direct method of integration founded upon Betti's reciprocal theorem was applied to the problem of the plane by V. Cerruti"". Some of the results were found in- dependently by Hertz, and led in his hands to a theory of impact and a theory of hardness"'. A different method for determining the state of stress in a body has been developed from a result noted by Gt. B. Airy™. He observed that, in the case 82 Brit. Assoc. Rep. 1876, Math, and Phys. Papers, vol. 3, p. 312. «3 Phil. Trans. Boy. Soc, vol. 170 (1879), and vol. 173 (1882). Eeprinted in G. H. Darwin's Siientijio Papers, vol. 2, Cambridge 1908, pp. 1, 459. '■■' L. Pochhammer, J.f. Math. (Grelle), Bd. 81 (1876), p, 38. •^5 II Suoro Cimentn (Ser. 2), tt. 6—10 (1872 et seq.). « Sir W. Thomson, Gavihridjie and Dublin Math. J., 1848, reprinted in Math, and Phys. Papers, vol. 1, p. 97. "' For Boussinesq's earlier researches in regard to simple solutions, see Paris, C. R., tt. 86—88 (1878—1879) and tt. 93—96 (1881—1883). A more complete account is given in his book, Appli- ratiuns des potentiels a VHude de Vequilibre et du mouvement des soUdes elastiques, Paris, 1885. '" Rome, Ace. Livcei, Mem. fis. mat., 1882. '» -T. f. Math. (Crelle), Bd. 92 (1882), and Verhandlungen des Vereins znr Bef&rderung des Geioe.rhefleisses, Berlin, 1882. The memoirs are reprinted in Ges. Werke von Heinrich Hertz, Bd. 1, Leipzig, 1895, pp. 155 and 174. ■" Brit. Assoc. Rep. 1862, and Phil. Trans. Roy. Soc, vol. 153 (1863), p. 49. HISTORICAL INTRODUCTION 1 7 of two dimensions, the equations of equilibrium of a body deformed by surface tractions show that the stress-components can be expressed as partial differential coefiScients of the second order of a single function. Maxwell" extended the result to three dimensions, in which case three such "stress-functions" are required. It appeared later that these functions are connected by a rather complicated system of, differential equations'^. The stress-components must in fact be connected with the strain-components by the stress-strain relations, and the strain-components are not independent; but the second differential coefficients of the strain-components with respect to the coordinates are con- nected by a system of linear equations, which are the conditions necessary to secure that the strain-components shall correspond with a displacement, in accordance with the ordinary formulas connecting strain and displacement". It is possible by taking account of these relations to obtain a complete system of equations which must be satisfied by stress-components, and thus the way is open for a direct determination of stress without the intermediate steps of forming and solving differential equations to determine the components of dis- placement". In the case of two dimensions the resulting equations are of a simple character, and many interesting solutions can be obtained. The theory of the free vibrations of solid bodies requires the integration^ of the equations of vibratory motion in accordance with prescribed boundary conditions of stress or displacement. Poisson^* gave the solution of the problem of free radial vibrations of a solid sphere, and Clebsch'^' founded the general theory on the model of Poisson's solution. This theory included the extension of the notion of "principal coordinates" to systems with an infinite number of degrees of freedom, the introduction of the corresponding "normal functions," and the proof of those properties of these functions upon which the expansions of arbitrary functions depend. The discussions which had taken place before and during the time of Poisson concerning the vibrations of strings, bars, membranes and plates had prepared the w,ay for Clebsch's gene- ralizations. Before the publication of Clebsch's treatise a different theory had been propounded by Lame*"- Acquainted with Poisson's discovery of two types of waves, he concluded that the vibrations of any solid-body must fall into two corresponding classes, and he investigated the vibrations of various b^odieson this assumption. The fact that his solutions do not satisfy the conditions which hold at the boundaries of bodies free from surface traction is a sufficient disproof of his theory; but it was finally disposed of when all the modes of fi-ee vibration of a homogeneous isotropic sphere were determined, " Edinburgh Roy. Soc. Trans., vol. 26 (1870). Beprinted in Maxwell's Scientific Papers, vol. 2, p. 161. '2 W. J. Ibbetson, An Elementary Treatise on the Mathematical Theory of perfectly Elastic Solids, London, 1887. '3 Saint-Venant gave the identical relations between strain-components in his edition of Navier's Resume des Legons sur I'application de la Mecanique, Paris, 1864, ' Appendiee 3.' '* J. H. Michell, London Math. Soc. Proc, vol. 31 (1900), p. 100. L. E. - 2 18 HISTORICAL INTRODUCTION and it was proved that the classes into which they fall do not verify Lamp's supposition. The analysis of the general problem of the vibrations of a sphere was first completely given by P. Jaerisch", who showed that the solution could be expressed by means of spherical harmonics and certain functions of the distance from the centre of the sphere, which are practically Bessel's functions of order integer + ^. This result was obtained indepen- dently by H. Lamb'", who gave an account of the simpler modes of vibration and of the nature of the nodal division of the sphere which occurs when any normal vibration is executed. He also calculated the more important roots of the frequency equation. L. Pochhammer'" has applied the method of normal functions to the vibrations of cylinders, and has found modes of vibration analogous to the known types of vibration of bars. The problem ol tracing, by means of the equations of vibratory motion, the propagation of waves through an elastic solid medium requires investi- gations of a different character from those concerned with normal modes of vibration. In the case of an isotropic medium Poisson'* and Ostrogradsky'' adopted methods which involve a synthesis of solutions of simple harmonic type, and obtained a solution expressing the displacement at any time in terms of the initial distribution of displacement and velocity. The investi- gation was afterwards conducted in a different fashion by Stokes*", who showed that Poisson's two waves are waves of irrotational dilatation and waves of equivoluminal distortion, the latter involving rotation of the elements of the medium. Cauchy" and Green*' discussed the propagation of plane waves -through a crystalline medium, and obtained equations for the velocity of propa- gation in terms of the direction of the normal to the wave-front. In general the wave-surface has three sheets ; when the medium is isotropic all the sheets are spheres, and two of them are coincident. Blanchet*^ extended Poisson's results to the case of a crystalline medium. Christoffel** discussed the advance through the medium of a surface of discontinuity. At any instant, the sur- face separates two portions of the medium in which the displacements are expressed by different formulae; and Christoffel showed that the surface moves normally to itself with a velocity which is determined, at any point, by the direction of the normal to the surface, according to the same law as holds « J.f. Math. (Crelle), Bd. 88 (1880). ™ London Math. Soc. Proc, vol. 13 (1882). " J. f. Math. (Crelle), Bd. 81 (1876), p. 324. ™ Paris, Mem. de I' Acad., t. 10 (1831). ™ St Petersburg, Mem. de I' Acad., t. 1 (1831). w ' On the Dynamical Theory of Diffraction,' Cambridge Phil. Soc. Trans., vol. 9 (1849). Eeprinted in Stokes's Math, and Phys. Papers, vol. 2 (Cambridge, 1883). " Cambridge Phil. Soc. Trans., vol. 7 (1839). Eeprinted in Green's Mathematical Papers, p. 293. «i J. de Math. (Liouville), t. 5 (1840), t. 7 (1842). 8» Ann. di Mat. (Ser. 2), t. 8 (1877). Eeprinted in E. B. Christoffel Ges. math. Abhandlmgen, Bd. -2, p. 81, Leipzig 1910. HISTORICAL INTRODUCTION 19 for plane waves propagated in that direction^ Besides the waves of dilata- tion and distortion which can be propagated througl* an isotropic solid body Lord Eayleigh^^ has investigated a third type which can be propagated over the surface. The velocity of waves of this type is less than that of either of the other two. Before the discovery of the general equations there existed the'ories of the torsion and flexure of beams starting from Galileo's enquiry and a sug- gestion of Coulomb's. The problems thus proposed are among the most important for practical applications, as most problems that have to be dealt with by engineers can, at any rate for the purpose of a rough approximation, be reduced to questions of the resistance of beams. Cauchy was the first to attempt to apply the geiieral equations to this class of problems, and his investigation of the torsion of a rectangular prism*', though not correct, is historically important, as he recognized "that the normal sections do not remain plane. His result had little influence on practice. The practical treatises of the earlier half of the last century contain a theory of torsion with a result that we have already attributed to Coulomb, viz., that the resistance to torsion is the product of an elastic constant, the amount of the twist, and the moment of inertia of the cross-section. Again, in regard to flexure, the practical treatises of the time followed the BemouUi-Eulerian (really Coulomb's) theory, attributing the resistance to flexure entirely to extension and contraction of longitudinal filaments. , To Saint-Venant belongs the credit of bringing the problems of the torsion and flexure of beams under the general theory. . Seeing the difiiculty of obtaining general solutions, the .pressing need for practical purposes of some theory that eould be applied to the strength of structures, and the improbability of the precise mode of application of the load to the parts of any apparatus being known, he was led to reflect on the methods used for the solution of special problems before ^e formulation of the general equations. These reflexions l^ed him to the invention of the semi-inverse method of solution which bears his name. Some of the habitual assumptions, or some of the results commonly deduced from them, may be true, at least in a large majority of cases; and it may be possible by retaining some of these assumptions or results to simplify the equations, and thus to obtain solutions — ^not indeed such as satisfy arbitrary surface con- ditions, but such as satisfy practically important types of surface conditions. The first problem to which Saint-Venant applied his method was that of the torsion of prisms, the theory of which he gave in the famous memoir on torsion of 1855™. For this application he assumed the state of strain to consist of a simple iwist about the axis of the prism, such as is implied in Coulomb's theory, combined with the kind of strain that is implied by a longitudinal displacement variable over the cross-section of the prism. The 8* Londm Math. Soc. Froc, vol^ 17 (1887), or Scientific Papers, vol. 2, Cambridge, 1900, p. 441. ■ 85 Exercices de mathSmatique, 4me Ann^e, 1829. 2—2 20 HISTORICAL INTRODUCTION effect of the latter displacement is manifested in a distortion of the sections" into curved surfaces. He showed that a state of strain having this character can he maintained in the prism by forces applied at its ends only, and that the forces which must be applied to the ends are statically equivalent to a couple about the axis of the prism. The magnitude of the couple can be expressed as the product of the twist, the rigidity of the material, the sqiiare of the area of the cross-section and a numerical factor which depends upon the shape of the cross- section. For a large class of sections this numerical factor is very nearly proportional to the ratio of the area of the section to the square of its radius of gyration about the axis of the prism. Subsequent investigations have shown that the analysis of the problem is identical with that of two distinct problems in hydrodynamics, viz., the flow of viscous liquid in a narrow pipe of the same, form as the prism**, and the motion produced in frictionless liquid filling a vessel of the same form as the prism when the vessel is rotated about its axis*'. These hydrodynamical analogies have resulted in a considerable simplification of the analysis of the problem. The old theories of flexure involved two contradictory assumptions: (1) that the strain consists of extensions and contractions of longitudinal filaments, (2) that the stress consists of tension in the extended filaments (on the side remote from the centre of curvature) and pressure along the contracted fila- ments (on the side nearer the centre of curvature). If the stress is correctly given by the second assumption there must be lateral contractions accom- panying the longitudinal extensions and also lateral extensions accompanying the longitudinal contractions. Again, the resultant of the tractions across any normal section of the bent beam, as given by the old theories, vanishes, and these tractions are statically equivalent to a couple about an axis at right angles to the plane of bending. Hence the theories are inapplicable to am case of bending by a transverse load. Saint- Venant^ adopted from the oluer theories two assumptions. He assumed that the extensions and con- tractions of the longitudinal filaments are proportional to their distances from the plane which is drawn through the line of centroids of the normal sections (the "central-line") and at right angles to the plane of bending. He assumed also that there is no normal traction across any plane, drawn parallel to the central-line. The states of stress and strain which satisfy these conditions in a prismatic body can be maintained by forces and couples applied at the ends only, and include two cases. One case is that of uniform bending of a bar by couples applied at its ends. In this case the stress is correctly given by the older theories and the curvature of the central-line is proportional to the bending couple, as in those theories; but the lateral contractions and extensions have the effect of distorting those longitudinal sections which are ** J. BouBsinesq, .7. de Math. (Liouville), (S^r. 2), t. 16 (1871). ^ Kelvin and Tait, Nat. Phil., Part 2, p. 242. ^ See the memoirs of 1855 and 1856 cited in footnote 50. HISTORICAL INTRODUCTION 21 at right angles to the plane of bending into anticlastic surfaces. The second case of bending which is included in_Saint-Venant's^1Jieory is that of a canti- lever, or beam 'fixed in a horizontal position at one end, and bent by a vertical load applied at the other end. In this case the stress given by the older theories requires to be corrected by the addition of shearing stresses. The normal tractions across any normal section are statically equivalent to a couple, which is proportional to the curvature of the central-line at the section, as in the theory of simple bending. The tangential tractions across any normal section are statically equivalent to the terminal load, but the magni- tude and direction of the tangential traction at any point are entirely deter- minate and follow rather complex laws. The strain given by the older iheories requires to be corrected by the addition of JateraLcoiilractions and extensions, as"in the'^thedi-y of simple bending, and also by shearing strains corresponding with the shearing stresses. In Saint-Venant's theories of torsion and flexure the couples and forces applied to produce twisting and bending are the resultants of tractions exerted across the terminal sections, and these tractions are distributed in perfectly definite ways. The forces and couples that are applied to actual structures are seldom distributed in these ways. The application of the theories to practical problems rests upon a principle introduced by Saint- Venant which has been called the "principle of the elastic equivalence of statically fequi- ■pollent systems of load." According to this principle the effects produced by deviations from the g,ssigned laws of loading are unimportant except near the ends of the bent beam or twisted, bar, and near the ends they produce merely "local perturbations." The condition for the validity of the results in practice is that the length of the beam should be a considerable multiple of the greatest diameter of its cross-section. Later researches by A. Clebsch'*' and W. Voigt^' have resulted in con- siderable simplifications of Saint-Venant's analysis. Clebsch showed that the single assumption that there is no normal traction across any plane parallel to the central- line leads to four cases of equilibrium of a prismatic body, viz., (1) simple extension under terminal tractive load, (2) simple bending by couples, (3) torsion, (4) bending of a cantilever by terminal transverse load. Voigt showed that the single .assumption that the stress at any point is inde- pendent of the coordinate measured along the bar led to the first three cases, and that the assumption that the stress is a linear function of that coordinate leads to the fourth case. When a quadratic function is taken instead of a linear one, the case of a beam supported at the ends and bent by a load which is distributed uniformly along its length can be included""- The case where the load is not uniform but is applied by means of surface tractions which, so far as 89 ' Theoretische Studien fiber die Elastioitatsverhiiltnisse der Krystalle,' Gnttingen Abhahd- lungen, Bd. 34 (1887). M J. H. Michell, Quart. J. of Math., vol. 32 (1901). ■22 HISTORICAL INTRODUCTION they depend on the coordinate measured along the beam, are rational integral functions, can be reduced to the case where the load is uniform'^ It appears from these theories that, when lateral forces are applied to the beam, the relation of proportionality between the curvature of the central-line and the bending moment, verified in Saint-Venant's theory, is no longer exact*''- Unless the conditions of loading are rather unusual, the modification that ought to be made in this relation is, however, of little practical importance. Saint-Venant's theories of torsion and of simple bending have found their way into technical treatises, but in most current books on applied Mechanics the theory of bending by transverse load is treated by a method invented by Jouravski'^ and Rankine'^ and subsequently developed by Grashof^ The components of stress determined by this method do not satisfy the conditions which are necessary to secure that they shall correspond with any possible displacement"l The distribution of stress that is found by this method is, however, approximately correct in the case of a beam of which the breadth is but a small fraction of the depth"'. The most important practical application of the theory of flexure is that which was made by Navier" to the bending of a beam resting on supports. The load may consist of the weight of the beam and of weights attached to the beam. Young's modulus is usually determined by observing the deflexion of a bar supported at its ends and loaded at the middle. All such applications of the theory depend upon the proportionality of the curvature to the bending moment. The problem of a continuous beam resting on several supports was at first very difficult, as a solution had to be obtained for each span by Navier's method, and the solutions compared in order to determine the constants of integration. The analytical complexity was very much diminished when Clapeyron" noticed that the bending moments at three consecutive supports are connected by an invariable relation, but in many particular cases the analysis is still formidable. A method of graphical solution has, however, been invented by Mohr*®, and it has, to a great extent, superseded the calculations " E. Almansi, Rmia, Ace. Lined Rend. (Ser. 6), t. 10 (1901), pp. 333, 400. In the second of these papers a solution of the problem of bending by uniform load is obtained by a method which differs from that used by Michell in the paper just cited. ^ This result was first noted by K. Pearson, Quart. J. of Math., vol. 24 (1889), in con- nexion with a particular law for the distribution of the load over the cross-section, 9' Ann. deg ponti et chaussees, 1856. " Applied Mechanics, 1st edition, London, 1858. The method has been retained in later editions. »' FJasticit/it mid Festiijkeit, 2nd edition, Berlin, 1878. Grashof gives Saint-Venant's theory as well. * Saint- Venaut noted this result in his edition of Navier's LeQom, p. 394. '■" In the second edition of his Lcfoiu (1833). "8 Paris, C. R., t. 45 (1857). The history of Clapeyron's theorem is given by J. M. Heppel, Proc. Roy. Soc., London, vol. 19 (1871). «• 'Beitrag zur Theorie des Fachwerks,' Zeitxchrift des Arehitekten- und Ingenimr-Vereins 211 Hannover, 1874. This is the reference given by Miiller-Breslau. L^vy gives an account of the HISTORICAL INTRODUCTION 23 that were formerly conducted by means of Clapeyron's "Theorem of Three Moments." Many other applications of the theory of flexure to problems of frameworks will be found in such books as Miiller-Breslau's Die Neueren Methoden der Festigkeitslehre (Leipzig, 1886), Weyrauch's Theorie Elastisoher Korper (Leipzig, 1884), Ritter's Anwendungen der graphischen, Statik {Zarich, 1888). A considerable literature has sprung up in this subject, but the use made of the Theory of Elasticity is small. The theory of the bending and twisting of thin rods and wires — including the theory of spiral springs — was for a long time developed, independently of the general equations of Elasticity, by methods akin to those employed by Euler. At first it was supposed that the flexural couple must be in the osculating plane of the curve formed by the central-line ; and, when the equation of moments about the tangent was introduced by Binet"", Poisson"' concluded from it that the moment of torsion was constant. It was only by slow degrees that the notion of two flexural couples in the two principal planes sprang up, and that the measure of twist came to be understood. When these elements of 'the theory were made out it could be seen that a know- ledge of the expressions for the flexural and torsional couples in terms of the curvature and twist"* would be sufficient, when combined with the ordinary conditions of equilibrium, to determine the form of the curve assumed by the central-line, the twist of the wire around that line, and the tension and shearing forces across any section. The flexural and torsional couples, as well as the resultant forces across a section, must arise from tractions exerted across the elements of the section, and the correct expressions for them must be sought by means of the general theory. But here a difficulty arises from the fact, that the general equations are applicable to small displacements only, while the displacements in such a body as a spiral spring are by no means small. Kirchhoff'"' was the first to face this difficulty. He pointed out that the general equations are strictly applicable to any small portion of a thin rod if all the linear dimensions of the portion are of the same order of magnitude as the diameters of the cross-sections. He held that the equa,tions of equilibrium or motion of such a portion could be simplified, for a first approximation, by the omission of kinetic reactions and forces dis- tributed through the volume. The process by which Kirchhoff developed his theory was, to a great extent, kinematical. When a thin rod is bent and, method in his Statique Graphique, t. 2, and attributes it to Mohr. A slightly different account isgiven by Canevazzi in Merrwrie delV Accademia di Bologna (Ser. 4), t. 1 (1880). The method has been extended by Culman, Die graphische Statik, Bd. 1. Zurich, 1875. See also Bitter, Die elastische Linie imd ihre Jnwendung auf den continuirlichm Balken, Ziirich, 1883. loo ,/. de V :^cole polytechnique, t. 10 (1815). iw Gorrespondance sur I'^^colepolytechniqw, t. 3 (1816). 102 They are due to Saint- Venant, Paris, G. R.. tt. 17, 19 (1843, 1844). 103 ' tjber das Gleiohgewicht und die Bewegung eines unendlioh diinnen elastischen Stabes,' J. f. Math. (Grelle), Bd. 56 (1859). The theory is also given in Kirchhoff 's Vorlesungen Uher maLh. Physik, Mechanik (3rd edition, Leipzig, 1883). 24 HISTORICAL INTRODUCTION twisted, every element of it undergoes a strain analogous to that in one of Saint- Venant's prisms, but neighbouring elements must continue to fit. To express this kind of continuity certain conditions are necessary, and these conditions take the form of differential equations connecting the relative displacements of points within a small portion of the rod with the relative coordinates of the points, and with the quantities that define the position of the portion relative to the rod as a whole. From these differential equa- tions Kirchhoff deduced an approximate account of the strain in an elemejit of the rod, and thence an expression for the potential energy per unit of length, in terms of the extension, the components of curvature and the twist. He obtained the equations of equilibrium and vibration by varying the energy-function. In the case of a thin rod subjected to terminal forces only he showed that the equations by which the form of the central-line is deter- mined are identical with the equations of motion of a heavy rigid body about a fixed point. This theorem is known as " Kirchhoff's kinetic analogue." Kirchhoff's theory has given rise to much discussion. Clebsch"'' proposed to replace that part of it by which the flexural and tojsional couples can be evaluated by an appeal to the results of Saint- Venant's theories of flexure and torsion. Kelvin and Tait" proposed to establish Kirchhoff's formula for the potential energy by general reasoning. J. Boussinesq"* proposed to obtain by the same kind of reasoning Kirchhoff's approximate expression for the extension of a longitudinal filament. Clebsch^' gave the modified formulae for the flexural and torsional couples when the central-line of the rod in the unstressed state is curved, and his results have been confirmed by later independent investigations. The discussions which have taken place have cleared up many difficulties, a;nd the results of the theory, as distinguished firom the methods by which they were obtained, have been confirmed by the later writers"* The applications of Kirchhoff's theory of ,thin rods include the theory of the elastica which has been investigated in detail by means of the theorem of the kinetic analogue™, the theory of spiral springs worked out in detail by Kelvin and Tait=^ and various problems of elastic stability. Among, the latter we may mention the problem of the buckling of an elastic ring sub- jected to pressure directed radially inwards and the same at all points of the circumference "'. The theory of the vibrations of thin rods was brought under the general equations of vibratory motion of elastic solid bodies by Poisson^. He regarded the rod as a circular cylinder of small section, and expanded all •»' J. de Math. {Limville), (S^r. 2), t. 16 (1871). ">■> See, for example, A. B. Basset, London Math. Soc. Pioc, vol. 23 (1892), and Aiiwr. J. of Math., vol. 17 (1895), and J. H. Michell, London Math. Soc. Proc, vol. 31 (1900), p. 130. >** W. Hess, .l/a«;i. .-Inii., Bde. 23 (1884) and 25 (1885). '"' This problem appears to have been discussed first by Bresse, Cour» de mecaniqiie appliquge, J'leiiiU're partie, Paris, 1859. HISTORICAL INTRODUCTION 25 the quantities that occur in powers of the distance of a particle from the axis of the cylinder. When terms above a certain order (the fourth power of the radiils) are neglected, the equations for flexural vibrations are identical with Euler's equations of lateral vibration. The equation found for the longitudinal vibrations had been obtained by Navier"^. The equation for the torsional vibrations was obtained first by Poisson"" The chief point of novelty in Poisson's results in regard to the vibrations of rods is that the coeflficients on which the frequencies depend are expressed in terms of the constants that occur in the genei^al equations; but the deduction of the generally admitted special differential equations, by which these modes of vibration are governed, from the general equations of Elasticity constituted an advance in method. Reference has already been made to L. Pochhammer's more complete investigation ''. Poisson's. theory Js- verified as an approxi- mate theory by an application of Kirchhoff's_results. This application has "lieen' extended toHthe vibrations of curved bars, the first problem to be solved being that of the flexural vibrations of a circular ring which vibrates in its own plane"*. An important problem arising in connexion with the theory of longitudinal vibrations is the problem of impact. When two bodies collide each. is thrown into a state of internal vibration, and it appears to have been hoped that a solution of the problem of the vibrations set up in two bars which impinge longitudinally would throw light on the laws of impact. Jnisson"" was the first to attempt a solution of the problem from this point of view. His method of , integration in trigonometric series vastly increases the difficulty of deducing general results, and, by an unfortunate error in the analysis, he arrived at the paradoxical conclusion that, when the bars are of the same material and section, they never separate unless they are equal in length. Saint- Venant"* treated the problem by means of the solution of the equation of vibration in terms of arbitrary functions, and arrived at certain results, of which the most important relate to the duration of impact, and to the existence of an apparent " coefficient of restitution " for perfectly elastic bodies"^ This theory is not confirmed by experiment. A correction sug- gested by Voigt"', when worked out, led to little better agreement, and it thus appears that the attempt to trace the phenomena of impact to vibra- tions must be abandoned. Much more successful was the theory of Hertz"'', i"8 Bulletin des Sciences a la Societe philnmathiqtie, 1824. ™"E. Hoppe, J.f. Math. {Crelle), Bd. 73 (1871). i"> In his Traite de M4canique, 1833. 1" ' Sur le choc longitudinal de deux barres elastiques...,' J. de Jlfai7^. {F.iouville), {S&. 2), 1. 12 (1867). '"^ Cf. Hopkinson, Messmiger of Mathematics, vol. 4, 1874. i>» Ann. Phys. Chem. (Wiedemann), Bd. 19 (1882). See also Hausmaninger in the same Annalen, Bd. 25 (1885). i" ' Uebei- die Beriihrung fester elastischer Korper,' J. f. Math. (Crelle), Bd. 92 (1882). 26 HISTORICAL INTRODUCTION obtained from a solution of the problem which we have named the problem of the transmission of force. Hertz made an independent investigation of a particular case of this problem — that of two bodies pressed together. He proposed to regard the strain produced in each by impact as a local statical effect, produced gradually and subsiding gradually ; and he found means to determine the duration of impact and the size and shape of the parts that come into contact. The theory yielded a satisfactory comparison with experiment. The theory of vibrations can be applied to problems concerning various kinds of shocks and the effects of moving loads. The inertia as well as the elastic reactions of bodies come into play in the resistances to strain under rapidly changing conditions, and the resistances called into action are sometimes described as "dynamical resistances." The special problem of the longitudinal impact of a massive body upon one end of a rod was discussed by Sebert and Hugoniot"' and by Boussinesq"'' The conclusions which they arrived at are tabulated and illustrated graphically by Saint- Venant"' But problems of dynamical resistance under impulses that tend to produce flexure are perhaps practically of more importance. When a body strikes a rod perpendicularly the rod will be thrown / into vibration, and, if the body moves with the rod, the ordinary solution in terms of the normal functions for the vibrations of the rod becomes inapplicable. Solutions of several problems of this kind, expressed in terms of the normal functions for the compound system consisting of the rod and the striking body, were given by Saint-Venant"'. Among problems of dynamical resistance we must note especially Willis's problem of the travelling load. When a train crosses a bridge, the strain is not identical with the statical strain which is produced when the same train is standing on the bridge. To illustrate the problem thus presented Willis"" proposed to consider the bridge as a straight wire and the train as a heavy particle deflecting it. Neglecting the inertia of the wire he obtained a certain differential equation, which was subsequently solved by Stokes™- Later writers have shown that the effects of the neglected inertia are very important. A more complete solution has been obtained by E. Phillips'^' "° Paris, G. B., t. 95 (1882). "" Applicatiom des Potentiels . . . , Paris, 1885. The results were given in a note in Paris, C. R., t. 97 (1883). ™ In papers in Paris, C. R., t. 97 (1883), reprinted as an appendix to his Translation of Clebsch's Treatise (Paris, 1883). "8 In the ' Annotated Clebsch ' just cited. Note du § 61. Cf. Lord Eayleigh, Theory of Sornid, Chapter VIII. "^ Appendix to the Report of the Commissioners .. .to enquire into the Application of Item to Railway Structures (1849). 120 Cambridge Phil. Soc. Tranx., vol. 8 (1849), or Stokes, Math, and PJiys. Papers, vol. 2 (Cambridge, 1883), p. 178. 121 Paris, Ann. des Mines, t. 7 (1855). HISTORICAL INTRODUCTION 27 and Saint- Venant'^, and an admirable precis of their results anay be read in the second volume of Todhunter and Pearson's History (Articles 373 et seq.). We have seen already how problems of the equilibrium and vibrations of plane plates and curved shells were attempted before the discovery of the general equations of Elasticity, and how these problems were among those which led to the ' investigation of such equations. After the equations had been formulated little advance seems to have been made in the treatment of the problem of shells for many years, but the more special problem of plates attracted much attention. Poisson^^^ and Cauchy^^ both treated this problem, proceeding from the general equations of Elasticity, and supposing that all the quantities which occur can be expanded in powers of the distance from the middle-surface. The equations of equilibrium and free vibration which hold when the displacement is directed at right angles to the plane of the plate were deduced. Much controversy has arisen concerning Poisson's boundary conditions. These expressed that the resultant forces and couples applied at the edge must be equal to the forces and couples arising from the strain. In a famous mSmoir Kirchhoff'^ showed that these conditions are too numerous and cannot in general be satisfied. His method rests on two assumptions : (1) that linear filaments of the plate initially normal to the rniddle-surface remain straight and normal to the middle- surface after strain, and (2) that all the elements of the middle-surface remain unstretched. These assumptions enabled him to express the potential energy of the bent plate in terms of the curvatures produced in its middle- surface. The equations of motion and boundary conditions were then deduced by the principle of virtual work, and they were applied to the problem of the flexural vibrations of a circular plate. The problem of plates can be attacked by means of considerations of the same kind as those which were used by Kirchhoff in his theory of thin rods. An investigation of the problem by this method was made by Gehring^''* and was afterwards adopted in an improved form by Kirchhoff"". The work is very similar in detail to that in Kirchhoff's theory of thin rods, and it leads to an expression for the potential energy per unit of area of the middle-surface of the plate. This expression consists of two parts : one a quadratic function of the quantities defining the extension of the middle-surface with a coefficient proportional to the thickness of the plate, and the other a quadratic function 122 In the ' Annotated Clebseh,' Note du § 61. 1''* In the memoir of 1828. A large part of the investigation is reproduced in Todhunter and Pearson's History. '^^ In an Article ' Sur I'^quilibre et le mouvement d'une plaque solide ' in the Exercices de math4matique, vol. 3 (1828). Most of this Article also is reproduced by Todhunter and Pearson. i2« J.f. Math. (Crelle), Bd. 40 (1850). 126 I j)e ^quationibus differentialibus quibus iequilibrium et motus laminse crystallinse definiuntur' (Diss.), Berlin, 1860. The analysis may be read in Kirchhoff's Vorlesungen iiber math. Phys., Mechanik, and parts of it also in Clebsch's Treatise. 127 Vorlesungen iiber math. Phys., Mechanik. •28 HISTORICAL INTRODUCTION of the quantities defining the flexure of the middle-surface with a coefficient proportional to the cube of the thickness. The equations of small motion are deduced by an application of the principle of vii-tual work. When the displacement of a point on the middle-surface is very small the flexure depends only on displacements directed at right angles to the plane of the plate, and the extension only on displacements directed parallel to the plane of the plate, and the equations fall into two sets. The equation of normal vibration and the boundary conditions are those previously found and dis- cussed by Kirchhofi^^'. As in the theory of rods, so also in that of plates, attention is directed rather to tensions, shearing forces and flexural couples, reckoned across the whole thickness, than to the tractions across elements of area which give rise to such forces and couples. To fix ideas we may think of the plate as hori- zontal, and consider the actions exerted across an imagined vertical dividing plane, and on this plane we may mark out a small area by two vertical lines near together. The distance between these lines maybe called the "breadth" of the area. The tractions across the elements of this area are statically equivalent to a force at the centroid of the area and a couple. When the "breadth" is very small, the magnitudes of the force and couple are pro- portional to the breadth, and we estimate them as so much per unit of length of the line in which our vertical dividing plane cuts the middle plane of the plate. The components of the force and couple thus estimated we call the "stress-resultants" and the "stress-couples." The stress-resultants consist of a tension at right angles to the plane of the area, a horizontal shearing force and a vertical shearing force. The stress-couples have a component about the normal to the dividing plane which we shall call the "torsional couple,". and a component in the vertical plane containing this normal which we shall call the "flexural couple." The stress-resultants and stress-couples depend upon the direction of the dividing plane, but they are known for all such directions when they are known for two of them. Clebsch °' adopted from the K-irchhoff- Gehring theory the approximate account of the strain and stress in a small portion of the plate bounded by .vertical dividing planes, and he formed equations of equilibrium of the plate in terms of stress-resultants and stress- couples. His equations fall into two sets, one set involving the tensions and horizontal shearing forces, and the other s6t involving the stress-couples and the vertical shearing forces. The latter set of equations are those which relate to the bending of the plate, and they have such forms that, when the expres- sions for the stress-couples are known in terms of the deformation of the middle plane, the vertical shearing forces can be determined, and an equation can be formed for the deflexion of the plate. The expressions for the couples can be obtained from Kirchhoff's theory. Clebsch solved his equation for the deflexion of a circular plate clamped at the edge and loaded in an arbi- trary manner. HISTORICAL INTRODUCTION '29 All the theory of the equations of equilibrium in terms of stress-resultants and stress-couples was placed beyond the reach of criticism by Kelvin and Tait"- These authors noticed also, that, in the case of uniform bending, the expressions for the stress-couples could be deduced from Saint- Venant's theory of the anticlastic flexure of a bar; and they explained the union of two of Poisson's boundary, conditions in one of Kirchhoff's as an example of the principle of the elastic equivalence of statically equipollent systems of load. More recent researches have assisted in removing the difficulties which had been felt in respect of Kirchhoff's theory i'''. One obstacle to progress has been the lack of exact solutions of problems of the bending of plates analogous to those found by Saint- Venant for beams. The few solutions of this kind which have been obtained'^' tend to confirm the main result of the theory which has* not been proved rigorously, viz. the approximate expression of the stress-couples in terms of the curvature of the middle-surface. The problem of curved plates or shells was first attacked from the point of view of the general equations of Elasticity by H. Aron"°. He expressed the geometry of the middle-surface by means of two parameters after the manner of Gauss, and he adapted to the problem the method which Clebsch ° had used for plates. He arrived at an expression for the potential energy of the strained shell which is of the same form as that obtained by Kirchhoff for plates, but the quantities that define the curvature of the middle-surface were replaced by the differences of their values in the strained and unstrained states. E. Mathieu"' adapted to the problem the method which Poisson had used for plates. He observed that the modes of vibration possible to a shell do not fall into classes characterized respectively by normal and tangential displacements, and he adopted equations of motion that could be deduced from Aron's formula for the potential energy by retaining the terms that depend on the stretching of the middle-surface only. Lord Eayleigh'^ pro- posed a different theory. He concluded from physical reasoning that the middle-surface of a vibrating shell remains unstretched, and determined the character of the displacement of a point of the middle-surface in accordance with this condition. The direct application of the Kirchhoff-Gehring method"^ led to a formula for the potential energy of the same form as Aron's and to equations of motion and boundary conditions which were difficult to recon- cile with Lord Rayleigh's theory. Later investigations have shown that the 128 gee, for example, J. Boussinesq, /. &R Math. (Liouville), (S6r. 2), t. 16 (1871) and (S^r. 3), t. 5 (1879) ; H. Lamb, London Math. Soc. Proc, vol. 21 (1890); J. H. Michell, London Math. Soc. Proc, vol. 31 (1900), p. 121 ; J. Hadamard, Trans. Amer. Math. Soc, vol. 3 (1902). i2» Some solutions were given by Saint- Venant in the 'Annotated Clebseh,' pp. 337 et seq. Others will be found in Chapter XXII of this book. 130 J.f. Math. (Crelle), Bd. 78 (1874). '31 J. de I'tcole polytechnique, t. 51 (1883). 132 London Math. Soc. Proc, vol. 13 (1882). 133 A. E. H. Love, Phil. Trans. Boy. Soc. (Ser. A), vol. 179 (1888). 30 HISTORICAL INTEODUCTION extensional strain which was thus proved to be a necessary concomitant of the vibrations may be practically confined to a narrow region near the edge of the shell, but that, in this region, it may be so adjusted as to secure the satisfaction of the boundary conditions while the greater part of the shell vibrates according to Lord Rayleigh's type. Whenever very thin rods or plates are employed in constructions it becomes necessary to consider the possibility of buckling, and thus there arises the general problem of elastic stability. We have already seen that the first investigations of problems of this kind were made by Euler and Lagrange. A number of isolated problems have been solved. In all of them two modes • of equilibrium with the same type of external forces are possible, and the ordinary proof"* of the determinacy of the solution of the equations of Elas- ticity is defective. A general theory of elastic stability has beeii proposed by G. H. Bryan^'^ He arrived at the result that the theorem of determinacy cannot fail except in cases where large relative displacements can be accom- panied by very small strains, as in thin rods and plates, and in cases where displacements differing but slightly from such as are possible in a rigid body can take place, as when a sphere is compressed within a circular ring of slightly smaller diameter. In all cases where two modes of equilibrium are possible the criterion for determining the mode that will be adopted is given by the condition that the energy must be a minimum. The history of the mathematical theory of Elasticity shows clearly that the development of the theory has not been guided exclusively by considerations of its utility for technical Mechanics. Most of the men by whose researches it has been founded and shaped have been more interested in Natural Phi- losophy than in material progress, in trying to understand the world than in trying to make it more comfortable. From this attitude of mind it may possibly have resulted that the theory has contributed less to the material advance of mankind than it might otherwise have done. Be this as it may, the intellectual gain which has accrued from the work of these men must be estimated very highly. The discussions that have taken place concerning the number and meaning of the elastic constants have thrown light on most recondite questions concerning the nature of molecules and the mode of their interaction. The efforts that have been made to explain optical phenomena by means of the hypothesis of a medium having the same physical character as an elastic solid body led, in the first instance, to the understanding of a concrete example of a medium which can transmit transverse vibrations, and, at a later stage, to the definite conclusion that the luminiferous medium has not the physical character assumed in the hypothesis. They have thus issued in an essential widening of our ideas concerning the nature of the aether and the nature of luminous vibrations. The methods that have been devised for '•'* Kirchhoff, Vurlesungm iiber math. Phys., Mechanik. "5 Cambridge Phil. Soc. Proc, vol. 6 (1889), p. 199. HISTORICAL INTRODUCTION 31 solving the equations of equilibrium of an isotropic solid body form part of an analytical theory which is of great importance in pure mathematics. The application' of these methods to the problem of the internal constitution of the Earth has led to results which must influence profoundly the course of speculative thought both in Geology and in cosmical Physics. Even in the more technical problems, such as the transpoission of force and the resist- ance of bars and plates, attention has been directed, for the most part, rather to theoretical than to practical aspects of the questions. To get insight into what goes on in impact, to bring the theory of the behaviour of thin bars and plates into accord with the general equations — these and sueh-like aims have been more attractive to most of the men to whom we owe the theory than endeavours to devise means for effecting economies in engineering construc- tions or to ascertain the conditions in which structures become unsafe. The fact that much.material progress is the indirect outcome of work done in this spirit is not without significance. The equally significant fact that most great advances in Natural Philosophy have been made by. men who had a first-hand acquaintance with practical needs and experimental methods has often been emphasized; and, although the names of Green, Poisson, Cauchy show that the rule is not without important exceptions, yet it is exemplified well in the history of our science. CHAPTER I ANALYSIS OF STRAIN 1. Extension. Whenever, owing to any cause, changes take place in the relative positions of the parts of a body the body is said to be "strained." ^A very simple example of a strained body is a stretched bar. Consider a bar of square section suspended vertically and loaded with a weight at its lower end. Let a line be traced on the bar in the direction of its length, let two points of the line be marked, and let the distance between these points be measured. When the weight is attached the distance in question is a little greater than it was before the weight was attached. Let l„ be the length before stretching, and I the length when stretched. Then (I - Qjlf, is a number (generally a very small fraction) which is called the extension of the line in question. If this number is the same for all lines parallel to the length of the bar, it may be called "the extension of the bar." A steel bar of sectional area 1 square inch (=6-4515 cm.'') loaded with 1 ton (= 1016-05 kilogrammes) will undergo an extension of about 7x1 0~'. It is clear that for the measurement of such small quantities as this rather elaborate apparatus and refined methods of observation are required*. Without attending to methods of measurement we may consider a little more in detail the state of strain in the stretched bar. Let e denote the extension of the bar, so that its length is increased in the ratio 1 +e : 1, and consider the volume of the portion of the bar con- tained between any two marked sections. This volume is increased by stretch- ing the bar, but not in the ratio 1 -I- e : 1. When the bar is stretched longi- tudinally .it contracts laterally. If the linear lateral contraction is e', the sectional area is diminished in the ratio (1 - e'f : 1, and the volume in question is increased in the ratio (1 + «) (1 -e'y : 1. In the case of a bar under tension e is a certain multiple of e, say ae, and a is about ^ or ^ for very many materials. If e is very small and e^ is neglected, the areal contraction is ^ae, and the cubical dilatation is (1 - 2o-)e. For the analytical description of the state of strain in the bar we should take an origin of coordinates x, y, z on the axis, and measure the coordinate z * See, for example, Ewing, Strength of MateriaU, Cambridge, 1899, pp. 73 et seq., or G. P. C. Searle, Experimental Elanticity, Cambridge, 1908. i-3] ; EXTENSION AND SHEAR 35 along the length of the bar. Any particle of the bar which has the eoordi- nates x, y, z when the weight- is not attached will move after the attachment of the weight into a new position. Let the particle which was at the origin move through a distance z^,, then the particle which was at (*■, y, z) moves to ■the point of which the coordinates are «(l-o-e), y(l-o-e), ^„ + (^_2„)(i+e). ( The state of strain is not very simple. If lateral forces could be applied to the bar to prevent the lateral contraction the state of strain w6uld be very much simplified. It would then be described as a "simple extension." 2. Pure- shear. .As a second example of strain let us suppose that lateral forces are applied to the bar so as to produce extension of amount Cj of lines parallel to, the axis of X and extension of amount eg of lines parallel to the axis oiy, and that longitudinal forces are applied, if any are required, to prevent any extefision or Contraction parallel to the axis of z. The particle which was at (x, y,,,z)> will move to {x + ei*, y + e^y, z) and the area of the section will be increasedl in the ratio (1 + d) (1 + e^ : 1. If e^ and eg are related so that this ratio is- equal to unity th^re will be no change in the area of the section or in the volume of any portion of the bar, but the shape of the section will be distorted. Either e^ or 62 is then negative, or there is contraction of the corresponding set of lines. The strain set up in the bar is called "pure shear." Fig. 1 below shows a square ABCB distorted by pure shear into a rhombus A'B'C'D' of the same area. 3. Simple shear. As a third example of strain let us suppose that the bar after being dis- torted by pure~~shear is turned bodily about its axis. We suppose that the axis of X is the direction in which contraction takes place, and we put ea -€1 = 2 tan «. ^ -^ i. '^ " '1 Then we can show that, if the rotation is of amount a in the sense from y to x, the position reached by any particle is one that could have been reached by the sliding of all the particles in the direction of a certain line through .distances proportional to the distances of the particles from a certain plane containing this line. Since (l + fi)(l+e2) = l, and £3,— ei = 2tana, we have l+ei=seca-tana, l+e2=seoa+tana. By the pure shear, the particle which was at {x, y) is moved to (xi, y-^, where i»i = 30 (sec a — tan a), 5/1 = y (sec a + tan a) ; and by the rotation it is moved again to {302, y^, where a?2=a;jCosa+yisin(i, ^2= -a;iSina+yiCosa; so that we have ^2=a;+tana{-a;cosa+y (l+sina)}, y^=y+\iS,u a{-x(\- sin a) +y cos a}. L. E. ■ 3 34 SPECIFICATION OF STRAIN [CH. I Now, writing /3 for Jir - a, we have a;j=a;+ 2 tan a cos ^j3 ( - ^sin il3+y cos ^^), i/2=y + 2 tan a sin J/3 ( - « sin ^/3+y cos J|8) ; and we can observe that - ^2 sin 4 3 +^2 cos ii3 = - ^ sin i^ +y cos J/3, and that x^ cos J/3+y2 sin J0 =.r cos J^ +y sin J^ + 2 tan a ( - « sin Jj3+y cos J0). Hence, taking axes of X and F which are obtained from those of x and y by a rotation through ^tt — Ja in the sense from a; towards y, we see that the particle which was at (X, Y) is moved by the pure shear followed by the rotation to the point (Xa, Y^), where Za = Z + 2 tan « . F, Y^=Y. Thus every plane of the material which is parallel to the plane of (Z, z) slides along itself in the direction of the axis of Z through a distance proportional to the distance of the plane from the plane of (Z, z). The kind of strain just described is called a "simple shear," the angle a is the "angle of the shear," and 2 tan a is the "amount of the shear." Fig. 1. 3-5] BY MEANS OF DISPLACEMENT 35 Fig. 1 shows a square ABGD distorted by pure shear into a rhombus A'B'Ciy oitYie same area, which is then rotated into the position A"B"0"D". The angle of the shear is A'OA", and the angle AOX is half the complement of this angle. The lines AA", BE', CO", DD" are parallel to OX and propor- tional to their distances from it. We shall find that all kinds of strain can be described in terms of simple extension and simple shear, but for the discussion of complex states of strain and for the expression of them by means of simpler strains we require a general kinematical theory *. 4. Displacement. We have, in every case, to distinguish two states of a body — a first state and a second state. , The particles of the body pass from their positions in the first state to their positions in the second state by a displacement. The displacement may be such that the line joining any two particles of the body has the same length in the second state as it has in the first; the displace- ment is then one which would be possible in a rigid body. If the displacement alters the length of any line, the second state of the body is described as a "strained state," and then the first state is described as the "unstrained state." In what, follows we shall denote the coordinates of the point occupied by a particle, in the unstrained state of the body, by x, y, z, and the coordinates of the point occupied by the same particle in the strained state by x + u, y-\-v, z + w. Then u, v, w are the projections on the axes of a vector quantity — the dis- placement. We must take u, v,w to be continuous functions of x, y, z, and we shall in general assume that they are analjrtic functions. It is clear that, if the displacement (u, v, w) is given, the strained state is entirely determined; in particular, the length of the line joining any two particles .can be determined. 5. Displacement in simple extension and simple shear. The displacement in a simple extension parallel to the axis of x is given by the equations u=ex, v=0, w=0, where e is the amount of the extension. If e is negative there is contraction. The displacement in a simple shear of amount «(=2tano), by which lines parallel to the axis of x slide along themselves, and particles in any plane parallel to the plane of (X, y) remain in that plane, is given by the equations u=sy, ■»=0, w=0. * The greater part of the theory is due to Cauohy (see Introduction). Somg^mprovements were made by Clebsch in his treatise of 1862, and others were made by Kelvin and Tait, Nat. Phil. Part I. 3—2 36 SPECIFICATION OF STRAIN [CH. In Fig. 2, ^S is a segment of a line parallel to the axis of x, which subtends an angle 2a at and is bisected by Oy. By the simple shear particles lying on the line OA are displaced so as to he on OB. The particle at any point P on ABM displaced to § pn AB so that Pq=AB, and the particles on OP are displaced to points on 0§. A parallelogram such as OPNM becomes a parallelogram such as OQKM. Fig. 2. If the angle xOP=6 we may prove that „_- 2tanatan2 5 ia,nPOQ= — iaxixOQ= tan 6 1 + 2 tan a tan 6 sec^ fl + 2 tan a tan 6 ' In particular, if 6 = ^17, cot xOQ=s, so that, if s is small, it is the complement of the angle in the strained state between two lines of particles which, in the unstrained state, were at right angles to each other. 6. Homogeneous strain. In the cases of simple extension and simple shear, the component dis- placements are expressed as linear functions of the coordinates. In general, if a body is strained so that the component displacement can be expressed in this way, the strain is said to be homogeneous. Let the displacement corresponding with a homogeneous strain be given by the equations u = a^x + ai22/ + (ht^^K ^ = ^^^ + ^■is.y + <^wZ> w' = C'^"" + o-^y + a^z. Since x, y, z are changed into x+u,y + v, z + w, that is, are transformed by a linear substitution, any plane is transformed into a plane, and any ellipsoid is transformed, in general, into an ellipsoid. We infer at once the following, characteristics of homogeneous strain: (i) — Straight lines remain straight, (ii) Parallel straight lines remain parallel, (iii) All straight lines in the same direction are extended, or contracted, in the same ratio, (iv) A sphere is transformed into an ellipsoid, and any three orthogonal diameters of the sphere are transformed into three conjugate diameters of the ellipsoid, (v) Any ellipsoid of a certain shape and orientation is transformed, into a sphere, and any set of conjugate diameters of the ellipsoid is transformed 5-7] BY MEANS OF DISPLACEMENT 37 into a set of orthogonal diameters of the sphere, (vi) There is one set of three orthogonal lines in the unstrained state which remain orthogonal after the strain; the directions of these lines are in general altered by the strain. In the unstrained state they are the principal axes of the ellipsoid referred to in (v); in the strained state, they are the principal axes of the ellipsoid referred to in (iv). The ellipsoid referred to in (iv) is called the strain ellipsoid; it has the property that the ratio of the length of a line, which has a given direction in the strained state, to the length of the corresponding line in the unstrained state, is proportional to the central radius vector of the surface drawn in the given direction. The ellipsoid referred 'to in (v) may be called the reciprocal strain ellipsoid; it has the property that the length of a line, which has a given direction in the unstrained state, is increased by the strain in a ratio inversely proportional to the central radius vector of the surface drawn in the given direction. The principal axes of the reciprocal strain ellipsoid are called the principal axes of the strain. The extensions of lines drawn in these directions, in the unstrained state, are stationary for small variations of direction, One of them is the greatest extension, and another the smallest. 7. Relative displacement. Proceeding now to the general case, in which the strain is not necessarily homogeneous, we take (oc + x, y + y, z + z) to he a point near to (x, y, z), and (m + u, w Tf V, w + Tv) to be the corresponding displacement. There will be expressions for the components u, v, w of the relative displacement as series in powers of x, y, z, viz. we have du du du , dx dy oz dv dv dv ~ dx dy dz dw dw dw W = X;r- + y^ + Z^+.. dx dy oz ■(1) where the terms that are not written contain powers of x, y, z above the first. When X, y, z are sufficiently small, the latter terms may be neglected. The quantities u, v, w are the displacements of a particle which, in the unstrained state, is at {x + -K,y + y, z+z), relative to the particle which, in the same ■state, is at («, y, z). We may accordingly say that, in a sufficiently small neighbourhood of any point, the relative displacements are linear functions of the relative coordinates. In other words, the strain about any point is sensibly homogeneous. All that we have Said about the effects of homogeneous strain upon straight lines will remain true for linear elements going out from a point. In particular, there will be one set of three orthogonal linear elements, in the unstrained state, which remain orthogonal afte~r the strain, but the ■(2) 38 IRROTATIONAL DISPLACEMENT [CH. I directions of these lines are in general altered by the strain. The directions, in the unstrained state, of these linear elements at any point are the "principal axes of the strain" at the point. 8. Analysis of the relative displacement*. In the discussion of the formulae (1) we shall confine our attention to the displacement near a point, and shall neglect terms in x, y, z above the first. It is convenient to introduce the following notations: — du _ 9y _dw ^ dw dv _du dw _'^ du ^^'^d^'^dz' ^''''dz^d^' ^^~dx^dy' dw dv „ du dw ^ dv du \ ^^^=3^~8^' ^"^'^^di'dx' ^^'-dx~dy-' The formulae (1) may then be written u= e^x^+^e^Y+^ezxZ-Tirz7 + vryZ,\ v = ^ea:yX+ ej,j,y + JeyzZ-OT^z+tCTjX, I (3) w = |e2^x + ^63,^7+ e^zZ-'STyX + vT^y.] The relative displacement is thus represented as the resultant of two dis- placements, expressed respectively by such forms as exx^ + ^exyj + ^e^x^ and — ■BTjy + Wj,z; and there is a fundamental kinematical distinction between the cases in which the latter displacement vanishes and the cases in which it does not vanish. When it vanishes, that is when ■stx, ■a^y, ar^ vanish, the component displacements are the partial differential coefficients, with respect to the coordinates, of a single function (f), so that 9d) dd> d ^ = 3^^ ' = d^' ""^Tz' and the line-integral of the tangential component of the displacement taken round any closed curve vanishes, provided that the curve can be contracted to a point without passing out of the space occupied by the body. Such a function as would be called a "displacement-potential." Through each point (x, y, z) there passes one quadric surface of the family exx'3i^ + Byy T + ezz2.^ + By^jz + Bzx^x + BxyX^ = const (4) and the displacement that is derived, as above, from a displacement-potential, is, at each point, directed along the normal to that surface of the family (4) which passes through the point. The linear elements that lie along the principal axes of these quadrics m the unstrained state continue to do so in the strained state, or the three orthogonal linear elements which remain orthogonal retain their primitive directions. The strain involved in such ^ * Stokes, Cambridge Phil. Soc. Tram. vol. 8 (1845), Math, and Phys. Papers, vol. 1, p. 75. 7-9] EXTENSION OF A LINEAR ELEMENT 39 displacements is described as a "pure strain." We learn that the relative displacement is always compounded of a displacement involving a pure strain and a displacement represented by such expressions as — vs^y + ^y*- The line-integral of the latter displacement taken round a closed qurve does not vanish (cf Article 15, infra). If the quantities istx, ^y, ■n^z are small, the terms such as — ra-jy + •sTyZ represent a displacement that would be possible in a rigid body, viz. a small rotation of amount '^{■Br/ + ts^ + sr/) about an axis in direction (waj : issy : Wg). For this reason the displacement corresponding with a pure strain is often described as "irrotational." 9. Strain corresponding with small displacement*. It is clear that the changes of size and shape of all parts of a' body will be determined when the length, in the strained state, of every line is known. Let I, m, n be the direction cosines of a line going out from the point (a;, y, z). Take a very short length r along this line, so that the coordinates of a neighbouring point on the line are x-^lr, y + mr, z + nr. After strain the particle that was at («, y, z) comes to {x + u, y + v, z + w), and the particle that was at the neighbouring point comes to the point of which the coordinates are y + mr+v + r[l^^ +m- +n ^^j , j (5) provided r is so small that we may neglect its square. Let r^ be the length after strain which rt^rresponds with r before strain. Then we have l- + m.^ + n{l + ^)'^^ (6) When the relative displacements are very small, and squares and products of such quantities as ^ , ... can be neglected, this formula passes over into ri = r [1 + exxl' + eyym' + e^in" + ey^mn + ea>nl + e^lm] (7) where the notation is the same as that in equations (2). * In the applications of the theory to strains in elastic solid bodies, the displacements that have to be considered are in general so small that squares and products of first differential coefficients of u, v, w with respect to x, y, z can be neglected in comparison with their first powers. The more general theory in which this simplification is not made will be discussed in the Appendix, to this Chapter. 40 SPECIFICATION OF STRAIN [CH. I 10. Components of strain*. By the formula (7) we know the length r^ of a line which, in the unstrained state, has an assigned short length r and an assigned direction (l, m, n), as soon as we know the values of the six quantities exx, Syy, ^zz. Syz, ^zx, ^xy- These six quantities are called the " components of strain." In the case of hopaogeneous strain they are constants ; in the more , general case they are variable from point to point of a body. The extension e of the short line in direction {I, m, n) is given at once by (7) in the form e = Bxxl^ + eyytn'' + e^n^ + ey^mn + 62x^1 + eg^lm, (8) so that the three quantities exx, ^yy, ^zz are extensions of linear elements which, in the unstrained state, are parallel to axes of coordinates. Again let {li, m^, n-^ be the direction in the strained state of a linear element which, in the unstrained state, has the direction (I, m, n), and let e be the corresponding extension, and let the same letters with accents refer to a second linear element and its extension. From the formulae (5) it appears that with similar expressions for mi, n,. The cosine of the angle between the two elements in the strained state is easily found in the form IJi + miWii' + Kirii' = (W + mm' + nn')(l —e — e')+2 {BxxU' + Cyymm' + Czgnn') + Byz {mn' + m'n) + e^x {nV + n'J) + exy {Im' + I'm) (9) If the two lines in the unstrained state are the axes of x and y the cosine of the angle between the corresponding lines in the strained state is e^. In like manner By^ and e^x are the cosines of the angles, in the strained state, between pairs of lines which, in the unstrained state, are parallel to pairs of axes of coordinates. Another interpretation of the strain-components of type e^y is afforded immediately by such equations as _ dv du from which it appears that exy is made up of two simple shears. In one of these simple shears planes of the material which are at right angles to the * When the relative displacement is not small the strain is not specified completely by the quantities e^, ... ey^, .... This matter is considered in the Appendix to this Chapter. Lord Kelvin has called attention to the unsymmetrical character of the strain-components here specified. Three of them, in fact, are extensions and the remaining three are shearing strains. He has worked out a symmetrical system of strain-components which would be the extensions of lines parallel to the edges of a tetrahedron. See Edinburgh, Proc. Roy. Soc, vol. 24 (1902), and Phil. May. (Ser. 6), vol. 3 (1902), pp. 95 and 444. 10> 11] BY MEANS OF EXTENSION 41 axis of X slide in the direction of the axis of y, while in the other these axes are interchanged. The strain denoted by e^y will be called the "shearing strain corresponding with the directions of the axes of x and y." The change of volume of any small portion of the body can be expressed in terms of the components of strain. The ratio of corresponding very small volumes in the strained and unstrained states is expressed by the functional determinant ^ 9m du du dx ' dy' dz dv T i^'" 9" dx' dy' dz dw dw dw dx ' dy ' dz and, when squares and products of du/dx, , . . are neglected, this becomes 1+0-+3-+3-. or say 1 + A. The quantity A which is defined by the equation A . du dv dw ,,„, , ^ = «- + %. + «- = 9-^ + 9^+9^ (10) is the increment of volume per unit of volume, or the " cubical dilatation," often called the "dilatation." With the introduction of the components of strain, the interpretation of these components and the expression of the . cubical dilatation in terms of them, we have achieved a general kinematical theory of the strains that accompany small displacements. The rest of this Chapter will be devoted to theorems and methods, relating to small strains, which will be useful in the development of the theory of Elasticity. 11. The Strain Quadric. Through any point in the neighbourhood of (x, y, z) there , passes one, and only one, quadric surface of the family e^xT^ + eyyY'' + e^T? + ej,syz + ggajZx + e^/gics = const. ..... .(4 his) Any one of these quadrics is called a strain quadric; such a surface has the property that the reciprocal of the square of its central radius vector in any direction is proportional to the extension of a line in that direction. If the quadric is an ellipsoid, all lines issuing from the point {x, y, z) are extended, or else all are contracted ; if the quadric is an hyperboloid, some lines are extended and others contracted; and these sets of lines are separated by the common asymptotic cone of the surfaces. Lines which undergo no extension or contraction are generators of this cone. The directions of lines, in the unstrained state, for which the extension is a maximum or a minimum, or is stationary without being, a true maximum 42 GEOMETRICAL CHARACTER [CH. I or minimum, are the principal axes of the quadrics (4). These axes are therefore the principal axes of the strain (Article 7), and the extensions ia the directions of these axes are the " principal extensions." When the quadrics are referred to their principal axes, the left-hand member of (4) takes the form wherein the coefficients ei, Cj, 63 are the values of the principal extensions. We now see that, in order to specify completely a state of strain, we require to know the directions of the principal axes of the strain, and the magnitudes of the principal extensions at each point of the body. With the point we may associate a certain quadric surface which enables us to express the strain at the point. The directions of the principal axes of the strain are determined as follows : — let I, m, n be the direction cosines of one of these axes, then we have I ~ m n ' and, if e is written for either of these three quantities, the three possible values of e are the roots of the equation Sxx~ ^ 2^xy 2^zx these roots are real, and they are the values of the principal extensions ej, 621 «3. 12. Transformation of the components of strain. The same state of strain may be specified by means of its components referred to any system of rectangular axes ; and the components referred to any one system must therefore be determinate when the components referred to some other system, and the relative situation of the two systems, are known. The determination can be made at once by using the property of the strain quadric, viz. that the reciprocal of the square of the radius vector in any direction is proportional to the extension of a line in that direction. We shall take the coordinates of a point referred to the first system of axes to be, as before, x, y, z, and those of the same point referred to the second system of axes to be a! , 3/', /, and we shall suppose the second system to be connected with the first by the orthogonal scheme X y z v h OTl «i y h mi «2 2- h ma »3 Further we shall suppose that the determinant of the transformation is 1 (not — 1), so that the second system can be derived from the first by an 11-13] OF COMPONENTS OF STRAIN 43 operation of rotation*. We shall write e^^., e^'^, e^^, e^^, e^^, e^y, for the components of strain referred to the second system. The relative coordinates of points in the neighbourhood of a given point may be denoted by x, y, z in the first system and x', y', z' in the second system. These quantities are transformed by the same substitutions as X, y, z and x', y', z'. When the form e^x^? + eyyY^ + e^jZ^ + e^^yz + e^^^-x. + e^^^xy is transformed by the above substitution, it becomes e«'a^x'2+ ej/yy'' + e^-^'Z'" + ey'^j'z' + e^f^z'-x! + e^^yx'y'. It follows that ^iCi' = e^xli + eyyTTii" + e^^n^^ + e^jmiWi + es^fhl-^ + exylima,\ 65^2-= 2eaa;4?s + 2ej,!/««2W3 + 2622% W3 + ej,j(m2Ws + mgWa) r (H) + e^ (wa ^s + W3 ^2) + Bxy {hrih + km^, 1 These are the formulae of transformation of strain-components. 13. Additional methods and resnlts. (a) The formulae (11) might have been inferred from the interpretation of e^^' ^^ the extension of a linear element parallel to the axis of a;', and of ej,v ^^ the cosine of the angle between the positions after strain of the linear elements which before strain are parallel to the axes of 3/' and /. (6) The formulae (11) might also have been obtained by introducing the displacement (m'j ■»', icf) referred to the axes of {of, y, /), and forming du'jdaf, .... The displacement being a vector, u, v, w are cogredient with x, y, z, and we have for example This method may be applied to the transformation of cr^, ra-j, vi^. We should find for example C7a;' = Z,orj;+TOiW„+rai53r2, (12) and we might hence infer the vectorial character of {■ss„ ^y, 'm^. The same inference might be drawn from the interpretation of ■ss^j 'i^v ^» ^^ components of rotation. (c) According to a well-known theorem t concerning the transformation of quadratic expressions, the following quantities are invariant in respect of transformations from one set of rectangular axes to another : *^yy^Ba'r^B3^xx'^xx^yy~tK^yz'''^sx "r^xy )y i ^ The first of these invariants is the expression for the cubical dilatation. * This restriction makes no difference to the relations between the components of strain referred to the two systems. It affects the components of rotation ■Bt^, I3y, w,- t Salmon, Geometry of three dimensions, 4th ed. , Dublin, 1882, p. 66. 44 INVARIANTS OF A STRAIN [CH. I (d) It may be shown directly that the following quantities are invariants : (i) nr^2 + c7,2+nT/, (ii) e^m^^ + Byyis;^ + e«ii7,2 + ey^Wym, + e^w^^m^ + e^m^iSy ; and the direct verification may serve as an exercise for the student. These invariants could be inferred from the fact that w^, ■aiy, or, are cogredient with x, y, z. (e) It may be shown also that the following quantities are invariants* :— . fdwdr dwdv\ /du dw _dudw\ fdvdu _dv_du\ ("'^ [d^di~di.d^)'^\dz dx dxdz)^ \dx dy dy dxj ' (iv) eJ + eyy' + eJ+i{e,J' + eJ+e^^) + 2{'ujJ' + ^y' + ^,^): (/) It may be shown t also that, in the notation of Article 7, the invariant (iv) is equal to f [ [ (u2 + v2 + w2) dxdydz 1 1 1 {x^+Y^+z'^)dxdydz where the integrations are taken through a very small sphere with its centre at the point (■^■, y, z). (g) The following result is of some importance} :— If the strain can be expressed by shears e„, e,^ only, the remaining components being zero, then the strain is a shearing strain e„. ; and the magnitude of this fehear, and the direction of the axis of in the plane of X, y, are to be found from e^ and e^^ by treating these quantities as the projections of a vector on the axes of x and y. 14. Types of strain. (a) Uniform, dilatation. When the strain quadric is a sphere, the principal axes of the strain are indeterminate, and the extension (or contraction) of all linear elements issuing from a point is the same ; or we have where A is the cubical dilatation, and the axes of x, y, z are any three orthogonal lines. In this case the linear extension in any direction is one-third of the cubical dilatation — a result which does not hold in general, (6) Simple extension. We may exemplify the use of the methods and formulae of Article 12 by finding the components, referred to the axes of x, y, z, of a strain which is a simple extension, of amount e, parallel to the direction (I, m, n). If this direction were that of the axis of x' the form (4) would be ezf^ ; and we have therefore 6j^ ^ fit , Byy ^ etn , 622 ^ en , ey^ = 2emn, e^='2,enl, e^ = 2elm. A simple extension is accordingly equivalent to a strain specified by these six com- ponents. It has been proposed § to call any kind of quantity, related to directions, which is equivalent to components in the same way as a simple extension, a tensor. Any strain is, * The invariant (iii) will be useful in a subsequent investigation (Chapter VII.). t E. Betti, II Nuovo Cimento (Ser. 2), t. 7 (1872). i Cf. Chapter XIV. infra. § W. Voigt, Gottingen S'achr. (1900), p. 117. Cf. M. Abraham in Ency. d. math. Wise,, Bd. 4, Art. 14. 13, 14] TYPES OP STRAIN 45 as we have already seen, equivalent to three simple extensions parallel to the principal axes of the strain. It has been proposed to call any kind of quantity, related to directions, which is equivalent to components in the same way as a strain, a tensor-triad. The discussion in Articles 12 and 13 (6) brings out clearly the distinction between tensors and vectors. (c) Shearing strain. The strain denoted by e^y is called "the shearing strain corresponding with the direc- tions of the axes of x and y." We have already observed that it is equal to the cosine of the angle, in the strained state, between two linear elements which, in the unstrained state, are parallel to these axes, and that it is equivalent to two simple shears, consisting of the relative sliding, parallel to each of these directions, of planes at right angles to the other. The "shearing strain" is measured by the sum of the two simple shears and is independent of their ratio. The change in the length of any hue and the change in the angle between any two lines depend upon the sum of the two simple shears and not on the ratio of their amounts. The components, of a strain, which is a shearing strain corresponding with the direc- tions of the axes of V and y', are given by the equations exc=«hh, eyy=sm-^i, e^=sn^n^, eyg=s{min2+m2ni), e^x = «{'"-ik+'>>-2li), exy=s(ljm2+l2ifh\ where s is the amount of the shearing strain. The strain involves no cubical dilatation. If we take the axes of a/ and y' to be in the plane of x, y, and suppose that the axes of x, y, z are parallel to the principal axes of the strain, we find that e^^ vanishes, or there is no extension at right angles to the plane of the two directions concerned. In this case we have the form sx'y' equivalent to the form e^x^'^ + ^yy^^- ' It follows that ex^= -eyy= +5S, and that the principal axes of the strain bisect the angles between the two directions concerned. In other words equal exterision and contraction of two linear elements at right angles to each other are equivalent to shearing strain, which is numerically equal to twice the extension or •contraction, and corresponds with directions bisecting the angles between the elements. We may enquire how to choose two directions so that the shearing strain corresponding with them may be as great as possible. It may be shown that the gi'eatest shearing strain is equal to the difference between the algebraically greatest and least principal extensions, and that the corresponding directions bisect the angles between those principal axes of the strain for which the extensions are the maximum and minimum extensions*. (d) Plane strain. A more general type, which includes simple extension and shearing strain as particular cases, is obtained by assuming that one of the principal extensions is zero. If the corre- sponding principal axis is the axis of i, the strain quadrio becomes a cylinder, standing on a conic in the plane of x, y, which may be called the strain conic ; and its equation can be written «MX2+e„j,y2 + ej.„xy = oonst. ; so 'that the shearing strains e^, and e^ vanish, as well as the extension e^^. In the particular case of simple extension, the conic consists of two parallel lines ; in the case of shearing strain, it is a rectangular hyperbola. If it is a circle, there is extension or con- traction, of the same amount, of all linear elements issuing from the point (x, y, z) in directions at right angles to the axis of z. * The theorem here stated is due to W. Hopkins, Cambridge Phil. Soc. Trans., vol. 8 (1849). 46 THEOREMS CONCERNING [CH. I The relative displacement corresponding with plane strain is parallel to the plane of the strain ; or we have w=const., while u and ■;; are functions of .r and y only. The axis of the resultant rotation is normal to the plane of the strain. The cubical dilatation, A, and the rotation, ■m, are connected with the displacement by the equations du dv' g _^ ^^ dx dy' dx dy' We can have states of plane strain for which both A and w vanish ; the strain is pure shear, i.e. shearing strain combined with such a rotation that the principal axes of the strain retain their primitive directions. In any such state the displacement components V, u are conjugate functions of x and y, or v + m is a function of the complex variable x+iy. 15. Relations connecting the dilatation, the rotation and the dis- placement. The cubical dilatation A is connected with the displacement (m, v, w) by the equation 3m 3v dw dx dy dz ' A scalar quantity derived from a vector by means of this formula is described as the divergence of the vector. We write A = div(M, V, w) (14) This relation is independent of coordinates, and may be expressed as follows: — Let any closed surface S be drawn in the field of the vector, and let N denote the projection of the vector on the normal drawn outwards at any point on S, also let dr denote any element of volume within S, then jJMS=jjJAdr, (15) the integration on the right-hand side being taken through the volume within S, and that on the left being taken over the surface S*. The rotation (■sr^, my, ct,) is connected with the displacement (u, v, w) by the equations Jdw "bv du dw dv du A vector quantity derived from another vector by the process here indicated is described as the curl of the other vector. We write ^{■^x, ■^v, ■^i) = cnr\{u,v,w) (16) This relation is independent of c6ordinatest, and may be expressed as follows:— Let any closed curve s be drawn in the field of the vector, and let any surface S be described 80 as to have the curve s for an edge ; let T be the resolved part of the vector {u, v, w) along the tangent at any point of s, and let Scr, be the projection of the vector 2 {w^, w^, or,) on the normal at any point of S, then \ Tds= lJ2m^dS, (17) * The result is a particular case of the theorem known as ' Green's theorem ' See Encv- d math. Wise. n. A 2, Nos. 45 — 47. ' t It is assumed that the axes of x, y, z form a right-handed system. If a transformation to a left-handed system is admitted a convention must be made as to the sign of the curl of a vector. 14-16] DILATATION AND ROTATION 47 the integration on the right being taken over the surface S, and that on the left being taken along the curve s*. 16. Resolution of any strain into dilatation and shearing strains. When the strain involves no cubical dilatation the invariant exx + eyy + e^ vanishes, and it is possible to choose rectangular axes of «', i/, z so that the form e-ca;*^ + eyy'f + e^z^^ + eyzyz + e^^zx + e^xy is transformed into the form ey V y'z' + e/a;- /a;' + e^y x'y', in which there are no terms in «'", y'^, z!^. The strain is then equivalent to shearing strains corresponding with the pairs of directions {y\z'), (z',x'), (x'.y'). When the strain involves cubical dilatation the displacement can be analysed into two constituent displacements, in such a way that the cubical dilatation corresponding with one of them is zero; the strains derived from this constituent are shearing strains only, when the axes of reference are chosen suitably. The displacement which gives rise to the cubical dilatation is the gradient f of a scalar potential (), and the remaining part of the displacement is the curl of a vector potential {F, G, H), of which the divergence vanishes. To prove this statement we have to show that any vector {u, v, w) can be expressed in the form (m, V, w) = gradient of ^ + curl {F, 0, H) (18) involving the three equations of the type dx dy dz'' ^ in which F, G, H satisfy the equation ' f -l^f =« <-> In the case of displacement in a body this resolution must be valid at all points within the surface bounding the body. There are many different ways of effecting this resolution of (m, v, w)|. * The result is generally attributed to Stokes. Cf. Ency. d. math. Wiss. n. A 2, No. 46. It implies that there is a certain relation between the sense in which the integration along ds is taken and that in which the normal v is drawn. This relation is the same as the relation of rotation to translation in a right-handed screw. ^ /d

\ t The gradient of (p is the vector l^, g- , g^ j . J See, e.g., E. Betti, II Nuovo Cimmto (Ser. 2), t. 7 (1872), or P. Duhem, J. de Math. (Liouville), (S^r. 5), t. 6 (1900). The resolution was first effected by Stokes m his memoir on Diffraction. (See Introduction, footnote 80.) 48 DILATATION AND ROTATION [CH. I We observe that if it is effected the dilatation and rotation will be expressed in the forms A=V^(^, 2«T»=-V2jf, ■I'BTy^-V'G, 2isr,= -V''H, (21) the last three holding good because dF/dx + dOjdy + dHjdz = 0. Now solu- tions of (21) can be written in the forms '^^'Llllv ^*w^^'' ^= i jjlv ^'"'^y'^''' (22) where r is the distance between the point {x, y, z') and the point {x, y, z) at which , F, G, H in every case, that is to say one mode of resolution is always given by the equations 16, 17] CONDITIONS OF COMPATIBILITY , 1 [[ff , 9r-i ,9r-i ,'br-^\ ,,,,,, 1 fff/ 9*""^ dr~^\ 49 .(23) 47rJ where the integration extends throughout the body; for it is clear that these make div (F, G, H) = and also make 9^ dH_dG dx dy dz W^'///^'^''''^^''^^'""' 17. Identical relations between components of strain*. The values of the components of strain Bxx, ■■■ Syg, ... as functions of x, y, z cannot be given arbitrarily; they must be subject to such relations as will secure that there shall be functions u, v, w, which are connected with them by the six equations du dw dv .(24) du , = i^xy + ^z The relations in question may be obtained by taking account of the three equations dw dv for all the differential coefficients of u, v, w can be expressed in terms of Sxx, ••■ Byz> ■■• '^x, ■■■ ■ We have in fact three pairs of equations such as dv dx and the conditions that these may be compatible with the three equations such as 9M/9a; = e^x, are nine equations of the type 9^_ 19e^_9CTz dy 2 dx dx ' and these equations express the first differential coefficients of ■sr^, nry, ■as^ in terms of those of e^x,--- V' •••• If we write down for example the three * These relations were given by Saint- Venant in his edition of Navier's Lemons, Appendix m. ' The proof there indicated was developed by Kirohhoff, Mechanik, Vorlesvmg 27. The proof in the text is due to Beltrami, Paris, C. R., t. 108 (1889), of. Koenigs, Lesons de Cin4matiqve, Paris 1897, p. 411. L.E. 4 50 DISPLACEMENT [p^- ^ equations that contain iir^ we can see at once how to obtain the conditions that they may be compatible. These three equations are gro-a; ^ dezx dexy dx dy dz dy dy dz 2?^ = 2— - — dz dy dz ' and from the set of nine equations of this type we can eliminate nr^, vty, zj^ and obtain the six identical relations between the components of strain. They are d^eyy a^z^aV ^d^ ^ ^ fdey, _^de^ ^de^\ dz'~ "^ dy' dydz ' dydz dx \ dx dy dz J ' d^ d^^d^ 2^=—/' ^^^+^ da? dz'' dzdx' dzdx dy\ dx dy dz d%^ d%iy^d^ 2^=-f 9^ , 3!??_?!?yA dy' da? dxdy ' dxdy dz V dx dy dz ) .(25) 18. Displacement corresponding with given strain*. When the components of strain are given functions, which satisfy the identical relations of the last Article, the components of displacement are to be deduced by solving the equations (24) as differential equations for u, v, w. These equations are linear, and the complete solutions of them are compounded of (1) any set of particular solutions, (2) complementary solutions containing arbitrary constants. The complementary solutions satisfy the equations du _dv dw _ dw dv _du dw _dv 9^ _ r, /n^\ dx dy dz dy dz dz dx dx dy If we differentiate the left-hand members of these equations with respect to X, y, z we shall obtain eighteen linear equations connecting the eighteen second differential coefficients of u, v, w, from which it follows that all these second differential coefficients vanish. Hence the complementary u, v, w are linear functions of x, y, z, and, in virtue of equations (26), they must be expressed by equations of the forms u ^v^ — ry -\-qz, v = VQ — pz + rx, 'w = w„ — qx + py (27) which are the formulae for the displacement of a rigid body by a translation (m„, v„, Wo) and a small rotation {p, q, r). In the complementary solutions thus obtained, the constants p, q, r must be small quantities of the same order of magliitude as the given functions Cxx, •■■> as otherwise the equations (6) of Art. 9 show that these functions * Cf . Kirchhoff, Mechanik, Vorlesung 27. 17-19] DETERMINED BY STRAIN 51 would not express the strain in the body correctly, and the terms of (27) that contain p, q, r would not represent a displacement possible in a rigid body. Bearing this restriction in mind, we conclude that, if the six components of strain are given, the corresponding displacement is arbitrary to the extent of an additional displacement of the type expressed by (27) ; but, if we impose SIX independent conditions, such as that, at the origin, the displacement (u, V, w) and the rotation (ct^, -ssy, ot^) vanish, or again that, at the same point ..0, ..0,„.0, |.0,|=0,,^.0 (28) the expression for the displacement with given strains will be unique. The particular set of equations (28) indicate that one point of the body (the origin), one linear element of the body (that along the axis of z issuing from the origin) and one plane-element of the body (that in the plane of z, x containing the origin) retain their positions after the strain. It is mani- festly possible, after straining a body in any way, to bring it back by trans- lation and rotation so that a given point, a given linear element through the point and a given plane-element through the line shall recover their primitive positions. 19. Curvilinear orthogonal coordinates*. For many problems it is convenient to use systems of curvilinear co- ordinates instead of the ordinary Cartesian coordinates. These may be introduced as follows : — Let f{oc, y, z) = a, some constant, be the equation of a surface. If a is allowed to vary we obtain a family of surfaces. In general one surface of the family will pass through a chosen point, and a neighbouring point will in general lie on a neighbouring surface of the family, so that a is a function of a;, y, z, viz., the function denoted hyf. If a + da is the parameter of that surface of the family which passes through (x + dx, y 4- dy, z ■\- dz), we have da = J-dx + i^dy+^dz = :r-dx + —-du + — dz. Ox oy oz ox dy Oz If we have three independent families of surfaces given by the equations fi{x,y,z) = a, f^{x,y,z) = ^, fz(x,y,z) = y, so that in general one surface of each family passes through a chosen point, then a point may be determined by the values of a, /8, y which belong to the surfaces that pass through itf, and a neighbouring point will be determined * The theory is due to Lam^. See his Legons sur les coordonnges curvilignes, Paris, 1859. t The determination of the point may not be free from ambiguity, e.g., in elliptic coordinates, an ellipsoid and two confooal hyperboloids pass through any point, and they meet in seven other points. The ambiguity is removed if the region of space considered is suitably limited, e g., in the case of elliptic coordinates, if it is an octant bounded by principal planes. 4—2 52 STRAIN REFERRED TO [CH. I by the neighbouring values a + da, /9 + d0, y + dy. Such quantities as a, /8, y are called " curvilinear coordinates " of the point. The most convenient systems of curvilinear coordinates for applications to the theory of Elasticity are determined by families of surfaces which cut each other everywhere at right angles. In such a case we have a triply-orthogonal family of surfaces. It is well known that there exists an infinite number of sets of such surfaces, and, according to a celebrated theorem due to Dupin, the line of intersection of two surfaces belonging to different families of such a set is a line of curvature on each*. In what follows we shall take a, 0, y to be the parameters of such a set of surfaces, so that the following relations hold: 9^87 9^97 9^87 _ dx dx dy dy dz dz ' dydoL dydoi dyda. _^ dx dx dy dy dz dz ' d_a.d^ SadlS da.d^_ dx dx dy dy dz dz The length of the normal, dn^, to a surface of the family a intercepted between the surfaces a and a + da is determined by the observation that the direction-cosines of the normal to a at the point (x, y, z) are 1 3a 1 3a 1 9a hidx' hidy' hidz ' ' ^ ^ where fh is expressed by the first of equations (31) below. For, by projecting the line joining two neighbouring points on the normal to a, we obtain the equation ^».4(£^-l*4>^)4: w In like manner the elements dn^, dn, of the normals to /3 and 7 are d^/h^ and dy/fhi, where The distance between two neighbouring points being {d'rii' + dn^" + dns'')K we have the expression for the " line-element," ds, i.e. the distance between the points (a, ^, 7) and (a + dci,/3 + d/3, y + dy), in the form {dsf = (da/h,y + id0/h,y + (dy/h,y (32) In general h^, h.,,^^ are regarded as functions of a, /3, 7. Salmon, Geometry of three dimensions, 4th ed., p. 269. 19, 20] CURVILINEAE COORDINATES 53 20. Components of strain referred to curvilinear orthogonal co- ordinates*. The length in the unstrained state of the line joining the points («, P, y) and (a + da, /S + d/8, 7 + ^7) is given by (32); we seek the length in the strained state of the line joining the same pair of particles. Let «», %, Uy be the projections of the displacement of any particle on the normals to the surfaces «, /8, 7 that pass through its position in the unstrained state. When the displacement is small the coordinates of the point occupied by a particle are changed from a, y8, 7 to a + /iiii„, /S + h^u^ , 7 + h^Uy. The a-coordinate (a + da.) of a neighbouring point is changed into AiM„ + da^ (JiiUc) +d^^ (hjU„,) +T>i .. ^/3 dd U^-r-+dl 3 / dei\ Fig. 3. On adding these contributions, we obtain ^(1)}- This must be the same as 'i'ssydad^ jk^h^, and we have thus an expression for xsy which is given in the third of equations (38) ; the other equations of this set can be obtained in the same way. The formulse* are --'■MK?)-!©)') 2cr^=A3Ai \dy\hj da\hs)r 2ts'v -k,h, Jg^ 3/3 (a.7/ ■ , .(38) * The formulsB (38), as also (36) and (37), are due to Lam^. The method here used to obtain (38), and used also in a slightly more analytical form by CesSro, Introduzione alia teoria matematica della Elasticita (Turin, 1894), p. 193, is familiar in Eleotrodynamios. Cf. H.Lamb, PW2. Trans. Boy. Soc. vol. 178 (1888), p. 150, or J. J. Thomson, Recent Researches in Electricity and Magnetism, Oxford, 1893, p. 367. The underlying physical notion is, of course, identical with the relation of 'circulation ' to ' vortex strength ' brought to light in Lord Kelvin's memoir ' On Vortex Motion,' Edinburgh, Soy. Soc. Trans., vol. 25 (1869). 56 CUBVILINEAR COORDINATES [CH. I 22. Cylindrical and polar coordinates. In the case of cylindrical coordinates r, 6, z we have the line-element {{drf-^r^{defA-{dzf\^, and the displacements u„ m«, «.. The general formulae take the following forms :- (1) for the strains 1 aw, 3m9 tillr 'i>}h ^ _?!!»_ !^ , 1 ^- . (2) for the cubical dilatation (3) for the components of rotation 1 3 , . , 1 3Me , 8w«. 1 3m, Smb ZUr 3m, 13, , 1 3w,. In the case of polar coordinates r, 6, , we have the line-element {(^dry + r^ {def+r^ sin^ 6 (#)2}i, and the displacements «„ ue,u^. The general formulae take the following forms r— (1) for the strains 3r ?• 35 r '^'^ ?• sin 5 80 ?• »■ ' 1 fdu^ A 1 3tt8 1 duy 3tt« tt0 Bmb ue 1 3Mr (2) for the cubical dilatation 1 / pi o 3 1 ^ = r-^^I^ 137 ('■'''- ''" ^) +35 (™« ''" ^^ + 3^ (™*^/ = (3) for the components of rotation The verification of these formulae may serve as exercises for the student. APPENDIX TO CHAPTER I. GENERAL THEORY OF STRAIN. 23. The preceding part of this Chapter contains all the results, relating to strains, which are of importance in the mathematical theory of Elasticity, as at present developed. The discussion of strains that correspond with dis- placements in general, as opposed to small displacements, is an interesting branch of kinematics; and some account of it will now be given*. It may be premised that the developments here described will not be required in the remainder of this treatise. It is customary, in recent books on Kinematics, to base the theory of strains in general on the result, stated in Article 7, that the strain about a point is sensibly homogeneous, and to develop the theory of finite strain in the case of homogeneous strain only. From the point of view of a rigorous analysis, it appears to be desirable to establish the theory of strains in general on an independent basis. We shall begin with an account of the theory of the strain corresponding with any displacement, and shall afterwards in- vestigate homogeneous strain in some detail. 24. Strain corresponding with any displacement. We consider the effect of the displacement on aggregates of particles forming given curves in the unstrained state. Any chosen particle occupies, in the unstrained state, a point (x, y, z). The same particle occupies, in the strained state, a point {x -^u, y + v, z + w). The particles which lie on a given curve in the first state lie in general on a different curve in the second state. If ds is the differential element of arc of a curve in the first state, the * Beference may be made to Cauehy, Exercices de mathematique, Ann^e 1827, the Article ' Sur la condensation et la dilatation des corps solides ' ; Green's memoir on the reflexion of light quoted in the Introduction (footnote 42) ; Saint- Venant, ' M^moire sur I'^quilibre des corps solides. . . quand les d6placements..,ne sont pas tr^s petits,' Paris, C. R., t. 24 (1847); Kelvin and Tait, Nat. Phil., Part i. pp. 115—144 ; Todhunter and Pearson, History, vol. 1, Articles 1619—1622 ; J. Hadamard, Legons sur la propagation des ondes, Paris 1903, Chapter vi. An interesting ex- tension of the theory, involving the introduction of secondary elements of strain, has been made by J. Le Eoux, Paris, Ann. Ec. norm., t. 28, 1911, p. 523 and t. 30, 1913, p. 193. The secondary elements of strain are the curvature and twist of slender filaments of the material, and the curva- ture of thin sheets of the material, the filaments and sheets being straight and plane in the unstrained state. 58 GENERAL THEORY [CH. 1. APP. direction-cosines of the tangent to this curve at any point are dx dy dz ds ' ds' If dsi is the differential element of arc of the corresponding curve in the second state, the direction-cosines of the tangent to this curve are d {x +u) d{y + v) d(z + w) dsi ' dsi ' dsi Herein, for example. d (x + u) ds /dx du dx du dy du dz )■ •(1) dsj dsi\ds ' dxds dyds dz with similar formulae for the other two. Let I, m, n be the direction-cosines of a line in the unstrained state, li, wii, Til the direction-cosines of the corresponding line in the strained state, ds, ds-i the differential elements of arc of corresponding curves having these lines respectively as tangents. In the notation used above i = u = dx Ts' d {x + u) in-,= dy ds " d(y + v) ""^ds' dz ds' d{z+ w) dsi and the equations of type (1) may be written in such forms as h = ds ^^^ + 3x) + ^Ty + '^dz (2) On squaring and adding the right-hand and left-hand members, and remem- bering the equations Z^ -I- mM n» = 1, • I,- + m,' + n,^ = 1, we find an equation can be written where e ds) " ^^ '*' ^^^^^ ^' "•" "^-^ "•■ ^^'"'^ »i' + (1 + 26^^) ri" + 2e,j, mn + ^e^^ nl + ^e^y Im, . , (3) . are given by the formulae ''dx^H[dx dv dy dw '=yy ~ ;^, ,+ "2" -f-' -I- dwV' \ ^- = a^ + _dw dv dudu 3i;3?; dwdw "' dy dz dydz dydz dydz _du dw dudu dvdv dwdw dz dx dzdx dzdx dzdx _dv du dudu 9i) 3« dwdw ^^ dx dy dxdy dxdy dxdy' . .(4) 24, 25] OF STRAIN 59 The state of strain is entirely determined when we know the lengths in the strained and unstrained states of corresponding lines*. The quantity -5 1 is the extension of the linear element ds. This is determined by the formula (3). We observe that the extensions of linear elements which, in the unstrained state, are parallel to the axes of coordinates are respectively V(H-26^,.)-l. V(l + 2eyi,)-l, V(l+26,,)-l, where the positive values of the square roots are taken. We thus obtain an interpretation of the quantities Sxx, eyy, e^z- We shall presently obtain an interpretation of the quantities ey^, e^x, e^y, in terms of the angles, in the strained state, between linear elements, which, in the unstrained state, are parallel to the axes of coordinates. In the meantime, we observe that the strain at any point is entirely determined by the six quantities e^x, ^yy> ^zt> ^yz> ^zxi Sxy- These quantities will be called the components of strain. The quantities exx, • ■ ■ which were called " components of strain '' in previous Articles are suflSciently exact equivalents of exx, ••• when the squares and products of such quantities as du/dx are neglected. 25. Cubical Dilatation. The ratio of a differential element of volume in the strained state to the corresponding differential element of volume in the unstrained state is equal to the functional determinant d(x + u, y+v, z + w) d (x, y, z) or it is 9m du du ^^di' dy' Tz dv ^ dv dv d^' '^dy' dz dw dw -1 , Sw dx' dy' dz This will be denoted by 1 + A. Then A is the increment of volume per unit volume at a point, or it is the cubical dilatation. The quantity exx + ^yy + ^zz is a sufficiently exact equivalent of A when the displacement is small. We may express A in terms of the components of strain. We find by the process of squaring the determinant that (1 + A)^ = (1 +2exx) (1 + '^eyy) (1 + 2e2,) + ^ey;,e,xexy - (1 + ^^x^) ^y. - (1 + 2€yy)ezx- - (1 + 2622)63; .(5) * Lord Kelvin's method (Article 10, footnote) is applicable, as he points out, to strains of unrestricted magnitude. (iO GENERAL THEORY [CH. I. APP. 26. Reciprocal strain ellipsoid. The ratio dsi : ds, on which the extension of a linear element issuing from a point depends, is expressed in the formula (3) in terms of the direction- cosines of the element, in the unstrained state, and the components of strain at the point. The formula shows that, for any direction, the ratio in question is inversely proportional to the central radius vector, in that direction, of an ellipsoid which is given by the equation (1 + 2e^;,) x^ + (l + 2eyy) 2/' + (1 + 26j2) z'' + ley^yz + 'ie^^zx + ^e^yXy = const. (6) This is the reciprocal strain ellipsoid already defined (Article 6) in the case of homogeneous strains. Its axes are called the principal axes of the strain; they are in the directions of those linear elements in the unstrained state which undergo stationary (maximum or minimum or minimax) extension. The extensions of linear elements in these directions are called the principal extensions, e^, e^, 63. The values of 1 + ej, 1 + 62, 1 + 63 are the positive square roots of the three values of k, which satisfy the equation €xy, l+2eyy—K, Syz =0 (7) ^ZX' ^yz> 1 + ^^zz ~~ 1^ The invariant relation of the reciprocal strain ellipsoid to the state of strain may be utilized for the purpose of transforming the components of strain from one set of rectangular axes to another, in the same way as the strain quadric was transformed in Article 12. It would thus appear that the quantities exx, ■■■ ^x,, are components of a " tensor- triad." Three invariants would thus be found, viz. : fxx + e,i,i + e,^, eyy€^^ + e^^exx + e^xSyy - | (e^^^^ -I- e%x + e'^)\ ,q-. ^xi^mi^zz + 4 (^yz^zx^xy ~ ^xx^ yz — ^yy^ zx — ^zz^ xy)- ) 27. Angle between two curves altered by strain. The effect of the strain on the angle between any two linear elements, issuing from the point (x, y, z), can be calculated. Let I, m, n and I', m', n be the direction-cosines of the two lines in the unstrained state, and the angle between them : let k, m„ n^ and l^, to/, n^' be the direction-cosines of the corresponding lines in the strained state, and 6^ the angle between them, From the formulas such as (2) we find „ ds ds , n c, / cos ''i = ^ ^' Icos t' -I- 2 {exxU + eyymm + ezznn) -f Sy^ {mn + m'n) + e^x {nV + n'l) -t- e^y {Im' + I'm)}, (9) where ds^/ds and ds^jds' are the ratios of the lengths, after and before strain, of corresponding linear elements ia the two directions. 26-28] OF STRAIN 61 We observe that, if the two given directions are the positive directions of the axes of y and g, the formula becomes ey^ = >^{{\+2e^^){\+2eyy)]Qoae„ (10) and we thus obtain an interpretation of the quantity e^^. Similar inter- pretations can be found for e^^ and e^y From the above formula it appears also that, if the axes of x, y, z are parallel to the principal axes of the strain at a point, linear elements, issuing from the point, in the direction of these axes continue to cut each other at right angles after the strain. We may show that, in general, this is the only set of three orthogonal linear elements, issuing from a point, which remain orthogonal after the strain. For the condition that linear elements which cut at right angles in the unstrained state should also cut at right angles in the strained state is obtained by putting cos 6 and cos 6^ both equal to zero in equation (9). We thus find the equation ((1 + 26a;a;) I + e^ym + i^^.^] I' + {63^^ + (1 + 26j;j,) JM + e^jw) w! + {^zJ- + ej/zm + (1 + l^i^n] n' = 0, wherein IV + mm! + nn' = 0. This equation shows that each of two such linear elements, (besides being at right angles to the other), is parallel to the plane which is conjugate to the other with respect to the reciprocal strain ellipsoid. Any set of three such elements must therefore, (besides being at right angles to each other), be parallel to conjugate diameters of this ellipsoid. The formulae so far obtained may be interpreted in the sense that a small element of the body, which has, in the unstrained state, the shape and orienta- tion of the reciprocal strain ellipsoid, corresponding with that point which is at the centre of the element, will, after strain, have the shape of a sphere, and that any set of conjugate diameters of the ellipsoid will become three orthogonal diameters of the sphere. 28. Strain ellipsoid. We might express the ratio dsi : ds in terms of the direction of the linear element in the strained state instead of the unstrained. If we solved the equations of type (2) for I, m, n we should find that these are linear functions of li, mi, J7i with coefficients containing ds^/ds as a factor ; and, on squaring and adding and replacing l^ + m"-^ v? by unity, we should find an equation of the form f^y = (ai^i -(- &imi -I- Ci»ii)2 -I- (aa^i -I- h^m^ + c^n:^ ->r {a^li + hm^ + c^n^'', \dsj , , , du du dw where a^, ... depend only on ^, ^, ... ^ . 62 GENERAL THEORY [CH. I. APP. The ellipsoid represented by the equation (ois; + 6,2/ + Cizf + {a^x + h^y + CiZf + (os* + hy + c^zf = const. would have the property that its central radius vector, in any direction, is proportional to the ratio ds^, : ds for the linear element which, in the strained state, lies along that direction. This ellipsoifl is called the strain ellipsoid. The lengths of the principal axes of this ellipsoid and of the reciprocal strain ellipsoid are inverse to each other, so that, as regards shape, the ellipsoids are reciprocal to each other; but their principal axes are not in general in the same directions. In fact the principal axes of the strain ellipsoid are in the directions of those linear elements in the strained state which have undergone stationary (maximum or minimum or minimax) extension. The simplest way of finding these directions is to observe that the corresponding linear elements in the unstrained state are parallel to the principal axes of the strain, so that their directions are known. The formulse of type (2) express the direction-cosines, in the strained state, of any linear element of which the direction-cosines, in the unstrained state, are given. The direction-cosines of the principal axes of the' strain ellipsoid can thus be found from these formulse. 29. Alteration of direction by the strain. The correspondence of directions of linear elements in the strained and unstrained states can be made clearer by reference to the principal axes of the strain. When the axes of coordinates are parallel to the principal axes, the equation of the reciprocal strain ellipsoid is of the form (1 + e.fa:' + (l + e^fy^ + (1 -|- e.,;fz^ = const, where e,, e^, ej are the principal extensions. In the formula (9) for the cosine of the angle between the strained positions of two linear elements we have to put l+26,, = (l-f6,n l + 26,, = (H-e,y^ l + 26,, = (l+f3)^ e,,= 6,, = 6,, = 0. Let the line Q', m, n') of the formula (9) take successively the positions of the three principal axes, and let the line {I, m, n) be any chosen line in the unstrained state. We have to equate ds'jds,' in turn to (1 + e,)-\ (1 + e^)-\ (1 + e,)-\ and we have to put for dsjdsi the expression [(1 + e,f l' + (l+ e,ym' + (1 + e,)%^]-*. The formula then gives the cosines of the angles which the corresponding linear element in the strained state makes with the principal axes of the strain ellipsoid. Denoting these cosines by X, y,, v, we find (X, M, I') = [d + e.fl'' + (1 -f e,Ym'' + {\ + e^fn^y^ {(1 + e,)l, (1 + e,) m, {l+e,)n]. (11) 28-31] OF STRAIN 63 By solving these for I, m,n -vie find {I, m, n) = + -. l(l + e,y (1+6,)^ ^(1 + 63)^ Here I, m, n are the direction-cosines of a line in the unstrained state referred to the principal axes of the strain, and X, ix, v are the direction- cosines of the corresponding line in the strained state referred to the principal axes of the strain ellipsoid. The operation of deriving the second of these directions from the first may therefore be made in two steps. The first step* is the operation of deriving a set of direction-cosines (\, /tt, v) from the set {I, m, n) ; and the second step is a rotation of the principal axes of the strain into the positions of the principal axes of the strain ellipsoid. The formulae also admit of interpretation in the sense that any small element of the body, which is spherical in the unstrained state, and has a given point as centre, assumes after strain the shape and orientation of the strain ellipsoid with its centre at the corresponding point, and any set of three orthogonal diameters of the sphere becomes a set of conjugate diameters of the ellipsoid. 30. Application to cartography. The methods of this Chapter would admit of application to the problem of constructing maps. The surface to be mapped and the plane map of it are the analogues of a body in the unstrained and strained states. The theorem that the strain about any point is sensibly homogeneous is the theorem that any small portion of the map is similar to one of the orthographic projections of the corresponding portion of the original surface. The analogue of the properties of the strain -ellipsoid is found in the theorem that with any small circle on the original surface there corresponds a small ellipse on the map ; the dimensions and orientation of the ellipse, with its centre at any point, being known, the scale of the map near the point, and all distortions of length, area and angle are deter- minate. These theorems form the foundation of the theory of cartography. [Cf Tissot, M^moire sur la representation des surfaces et les projections des cartes geographiques, Paris, 1881.] 31. Conditions satisfied by the displacement. The components of displacement u, v, w are not absolutely arbitrary functions of x, y, z. In the foregoing discussion it has been assumed that they are subject to such conditions of differentiability and continuity as will secure the validity of the "theorem of the total differential f." For our purpose this theorem is expressed by such equations as Au, du dx du dy du dz ds dx ds dy ds dz ds ' Besides this analytical restriction, there are others imposed by the assumed condition that the displacement must be such as can be conceived to take place in a continuous body. Thus, for example, a displacement, by * This operation is one of homogeneous pure strain. See Article 33, infra. t Cf. Hamack, Introduction to the Calculus, London, 1891, p. 92. (;4 GENERAL THEORY OF STRAIN [CH. I. APP. which every point is replaced by its optical image in a plane, would be excluded. The expression of any component displacement by functions, which become infinite at any point within the region of space occupied by the body, is also excluded. Any analytically possible displacement, by which the length of any line would be reduced to zero, is also to be excluded. We are thus concerned with real transformations which, within a certain region of space, have the following properties : (i) The new coordinates (x + u, y + V, z + w) are continuous functions of the old coordinates (oc, y, z) which obey the theorem of the total differential, (ii) The real functions u, v, w are such that the quadratic function (1 + 26^^) l'' + {l+ 2e„,) vi^ + (1 + 2€,,) n" + 2e,„mn + 2e,^nl + 2e^lm is definite and positive, (iii) The functional determinant denoted by 1 + A is positive and does not vanish. The condition (iii) secures that the strained state is such as can be produced from the unstrained state, by a continuous series of small real displacements. It can be shown that it includes the condition (ii) when the transformation is real. From a geometrical point of view, this amounts to the observation that, if the volume of a variable tetrahedron is never reduced to zero, none of its edges can ever be reduced to zero. In the particular case of homogeneous strain, the displacements are linear functions of the coordinates. Thus all homogeneous strains are included among linear homogeneous transformations. The condition (iii) then excludes such transformations as involve the operation of reflexion in a plane in addition to transformations which can be produced by a continuous series of small displacements. Some linear homogeneous transformations, which obey the condition (iii), express rotations about axes passing through the origin. All others involve the strain of some line. In discussing homogeneous strains and rotations it will be convenient to replace (x + u, y + v, z + 'w)hj (x^, y^, z^). 32. Finite homogeneous strain. We shall take the equations by which the coordinates in the strained state are connected with the coordinates in the unstrained state to be .-i'l = (1 + Oil) « + (Xia^/ + aj3 z, yi=aaX+{\+a^)y+ a^z, ■ (13) z-i, = a^^x + a^^y + (1 + a^.^ z. The corresponding components of strain are given by the equations era; = Oil + ^ (au^ 4 a^ + as,'), ■=1/2 — Cf32 + 0^23 + ttjjais + £122*28 + (^SiOHi '\ (14) 31-33] FINITE HOMOGENEOUS STRAIN 65 The quantities 6^.^;, ..., defined in Article 8, do not lose their importance when the displacements are not small. The notation used here ma}- be identified with that of Article 8 by writing, for the expressions Oij, a^, Oss, 0-23 + Oaa, Osi + <*is, «i2 + O-n, 0^2 — ttas, «i3 — «3i, dii — Oi^. the expressions e^, e^y, e^, e^^, e^, e^y, 2tD-^, 2or^, 2Br-. Denoting the radius vector from the origin to any point P, or (x, y, z), by /■, we may resolve the displacement of P in the direction of r, and consider the ratio of the component displacement to the length /■. Let E be this ratio. We may define £ to be the elongation of the material in the direction of r. We find :J|(a^-^)^ + (y,-y)^ + (0,-z)^|; (1-5) E = j and this is the same as ■Er^ = e-cxH^ + Byyy^ + e^z^ + Cy^yz + e^zii: + e^ayy (16) A quadric surface obtained by equating the right-hand member of this equation to a constant may be called an elongation quadric. It has the property that the elongation in any direction is inversely proportional to the square of the central radius vector iu that direction. In the case of very small displacements, the elongation quadric becomes the strain quadric pre- viously discussed (Article 11). The invariant expressions noted in Article 13 (c) do not cease to be invariant when the displacements are not small. The displacement expressed by (13) can be analysed into two constituent displacements. One constituent is derived from a potential, equal to half the right-hand member of (16); this displacement is directed, at each point, along the normal to the elongation quadric which passes through the point. The other constituent may be derived firom a vector potential -U-^xif'+^'l n7y{z^ + ar'), ^.(x' + f)] (17) by the operation curl. 33. HomogeneoTis pure strain. The direction of a line passing through the origin is unaltered by the strain if the coordinates x, y, z of any point on the line satisfy the equations (1 + ajj)x + a^^y + a^^ _ a^a^ 4- (1 -I- a^y -I- a^z X ~ y ^ Oaia; -t- a^y -I- (1 -h flgs) ^ If each of these quantities is put equal to \, then X is a root of the cubic equation I 1 -I- a,! — \ Oia Ojs I I I aji H-cUs-X Ob 1 = (19) ' Oai fltss 1 -t- ass ~ ^ ; L. E. ^ 66 GENERAL THEORY OP [CH. I. APP. The cubic has always one real root, so that there is always one line of which the direction is unaltered by the strain, and if the root is positive the sense of the line also is unaltered. When there are three such lines, they are not necessarily orthogonal ; but, if they are orthogonal, they are by definition the principal axes of the strain. In this case the strain is said to be pure. It is worth while to give a formal definition, as follows : — Pure strain is such that the set of three orthogonal lines which remain orthogonal retain their directions and senses. We may prove that the sufiicient and necessary conditions that the strain corresponding with the equations (13), may be pure, are (i) that the quad- ratic form on the left-hand side of (20) below is definite and positive, (ii) that tj-a;, CTy, ■BT^ vanish. That these conditions are sufficient may be proved as follows: — When zy^, ^y, ■^z vanish, or aas = ctsa, ..., the equation (19) is the discriminating cubic of the quadric (1 -h Oil) a;'' -f- (1 + aa) 2/' + (1 + "ss) ^"^ + '^o.^iyz -|- ^a^^zx + la^^xy = const. ; (20) the left-hand member being positive, the cubic has three real positive roots, which determine three real directions according to equations (18) ; and these directions are orthogonal for they are the directions of the principal axes of the surface (20). Further they are the principal axes of the elongation quadric rtjiiZ^ -I- ttiil/^ + a^gZ^ + 2a2syz + la^^zx + ia^^xy — const., (21) for this surface and (20) have their principal axes in the same directions. The vanishing of ra-a,, ■uTy and OTj are necessary conditions in order that the strain may be pure. To prove this we suppose that equations (13) represent a pure strain, and that the principal axes of the strain are a set of axes of coordinates f, t?, f. The effect of the strain is to transform any point (^, r}, f ) into (^1, 77i, fi) in such a way that when, for example, 77 and f vanish, j?j and fi also vanish. Referred to principal axes, the equations (13) must be equivalent to three equations of the form fi = (l + ei)?, ^,=(H-6.)'?. ri = (l + e3)?, (22) where e,, 63, 63 are the principal extensions. We may express the coordinates ^, 7), i; in terms of x, y, z by means of an orthogonal scheme of substitution. We take this scheme to be X y Z 1 h Ml «1 V h WI2 «2 i h ™3 % Then we have = (1 + 61) i, {l^x + m^y + n^z) + {1 + e^) k {hos + m^y + n^z) + {l + e,) L, (Igx + m^y -I- n^z). 33-35] HOMOGENEOUS STEAIN 67 Hence Oja = (1 + ej) l^m^ + (1 + e^) krrh + (1 + €,) km^. We should find the same expression for osai , and in the same way we should find identical expressions for the pairs of coefficients a^, a,^ and a^^, a^^. It appears from this discussion that a homogeneous pure strain is equiva- lent to three simple extensions, in three directions mutually at right angles. These directions are those of the principal axes of the strain. 34. Analysis of any homogeneous strain into a pure strain and a rotation. It is geometrically obvious that any homogeneous strain may be produced in a body by a suitable pure strain followed by a suitable rotation. To deter- mine these we may proceed as follows: — When we have found the strain- components corresponding with the given strain, we can find the equation of the reciprocal strain ellipsoid. The lengths of the principal axes determine the principal extensions, and the directions of these axes are those of the principal axes of the strain. The required pure strain has these principal extensions and principal axes, and it is therefore completely determined. The required rotation is that by which the principal axes of the given strain are brought into coincidence with the principal axes of the strain ellipsoid. According to Article 28, this rotation turns three orthogonal lines of known position respectively into three other orthogonal lines of known position. The required angle and axis of rotation can therefore be determined by a well- known geometrical construction. [Cf Kelvin and Tait, Nat. Phil. Part i. p. 69.] 35. Rotation*. When the components of strain vanish, the displacement expressed by (13) of Article 32 is a rotation about an axis passing through the origin. We shall take 6 to be the angle of rotation and shall suppose the direction-cosines I, m, n of the axis to be taken so that the rotation is right- handed. Any point P, or {x, y, z), moves on a circle having its centre {G) on the axis, and comes into a position Pi, or (^'i, yj, z-ij. Let X, /x, •■ be the direc- tion-cosines of GP in the sense from C to P, and let Xi, j«i, vt be those of CP-^ in the sense from C to Pj. Prom Pi let fall PjiV perpendicular to CP. The di- rection-cosines of yVPi in the sense from iVto Pi aret mv — nfi, nK — Iv, Ifi — mX. Let f, 77, f be the coordinates of G. Then these satisfy the equations 1 = 1=^, l{^-x) + m{r,-y) + n{C-z) = 0, so that ^=l(lx + my + nz) with similar expressions for r), f. * Cf. Kelvin and Tait, Nat. Phil. Part i. p. 69, and Minchin, Statics, Third Edn. , Oxford 1886, vol. 2, p. 103. t The coordinate axes are taken to be a right-handed system. 5—2 68 GENERAL THEORY OF [CH. I. APP. The coordinates of Pj are obtained by equating the projection of CPi on any coordinate axis to the sums of the projections of CN and KP^. Projecting on the axis of x we find, taking p for the length of CI' or CP,, Xjp = Xp cos 0-\-{mv — nfi.) p sin 6, or ^i-^ = (.r-^)cose + {«i(0-f)-«(y-'7)}sin5, or xi = x-^{mz-ny)s,\.n6-{x-l{lx + my + m)}{\-co8 6) (23) Similar expressions for yi and z, can be written down by symmetry. The coefficients of the linear transformation (13) become jn this case aii=-(l-Z2)(l-cose), ^ ai2= — rasin 6 + lin(\-cosff), ai3= m,sm6-\-ln{\ — oos6). ,.(24) and it appears, on calculation, that the components of strain vanish, as they ought to do. 36. Simple extension. In the example of simple extension given by the equations a;i = (l+e).r, y-i,=y, 0i = 2, the components of strain, with the exception of f^x vanish, and fxx=e + ke'^. The invariant property of the reciprocal strain ellipsoid may be applied to find the components of a strain which is a simple extension of amount e and direction I, m, n. We should find ^-=... = ...=^ = ..=...=e + ie2. l^ "imn ^ The same property may be applied to determine the conditions that a strain specified by six components may be a simple extension. These conditions are that the invariants .:jj,f„ + .., + ...-i(eV + ••• + •••), ^'a:a: *^Vy "^22 "•" I \^y2 *ot ^x]1 ~ ^xx ^ yz~ ••• • • ■ / vani.sh. The amount of the extension is expressed in terms of the remaining invariant by the formula ^ {1+2 (f«; + f w + fuz)} - 1> tbe positive value of the square root being taken. Two roots of the cubic in n, (7) of Article 26, are equal to unity, and the third is equal to l + 2(fia;+e„ + fjj). The direction of the extension is the direction (?, m, n) that is given by the equations •if^ + ^xtm + e^n _ fj, + 2f „„TO + ty^n _ ej, + fy^m + 2c^to 22 2»l 2m -fxa:+fw + €M. 37. Simple shear. In the example of simple shear given by the equations 3Ci=x-irsy, yi=y, Zi=z, the components of strain are given by the equations " ^ ^vv — 5 *■ ^xy" By putting « = 2tana we may prove that the two principal extensions which are not zero are given, as in Article 3, by the equations l+e]=8eca — tana, 1 + €2 = 8eca + tana. 35-39] HOMOGENEOUS STRAIN 69 We may prove that the area of a figure in the plane of x, y is unaltered by the shear and that the difference of the two principal extensions is equal to the amount of the shear. Further we may show that the directions of the principal axes of the strain are the bisectors of the angle AOx in Fig. 2 of Article 5, and that the angle through which the principal axes are turned is the angle a. So that the simple shear is equivalent to a "pure shear" followed by a rotation through an angle a, as was explained before. By using the invariants noted in Article 26, we may prove that the conditions that a strain with given components (xxi--- may be a shearing strain are 2 (f a;3c + Hv + f aa) + 4 (f ji!/ ^zz + ^zz ^xx. + «xi ^wu) " (« V + ^'''ix + f \)/) =■ 0) ^e-ca: €yy fgg "T^yz ^zx ^xy ~ ^xx ^ yz ^yy ^ sx~ ^sz^ txy^^^t and that the amount of the shear is V {2 (exx + ^yy + ^zz)}- 38. Additional results relating to shear. A good example of shear* is presented by a sphere built up of circular cards in parallel planes. If each card is shifted in its own plane, so that the line of centres becomes a straight line inclined obliquely to the planes of the cards, the sphere become^ an ellipsoid, and the cards coincide with one set of circular sections of the ellipsoid. It is an instructive exercise to determine the principal axes of the strain and the principal extensions. We may notice the following methods t of producing any homogeneous strain by a sequence of operations : — (a) Any such strain can be produced by a simple shear parallel to one axis of planes perpendicular to another, a simple extension in the direction at right angles to both axes, an uniform dilatation and a, rotation. (6) Any such strain can be produced by three simple shears each of which is a shear parallel to one axis of planes at right angles to another, the three axes being at right angles to each other, an uniform dilatation and a rotation. 39. Composition of strains. After a body has been subjected to a homogeneous strain, it may again be subjected to a homogeneous strain; and the result is a displacement of the body, which, in general, could be effected by a single homogeneous strain. More generally, when any aggregate of points is transformed by two homo- geneous linear transformations successively, the resulting displacement is equivalent to the effect of a single linear homogeneous transformation. This statement may be expressed by saying that linear homogeneous transforma- tions form a group. The particular linear homogeneous transformations with which we are concerned are subjected to the conditions stated in Article 31, and they form a continuous group. The transformations of rotation, described in Article 35, also form a group; and this group is a sub-group included in the linear homogeneous group. The latter group also includes all homo- geneous strains; but these do not by themselves form a group, for two successive homogeneous strains J may be equivalent to a rotation. * Suggested by Mr B. E. Webb. Of. Kelvin and Tait, Nat. Phil. Part i. p. 122. t Cf. Kelvin and Tait, Nat. Phil. Part i. §§ 178 et seq. J A transformation such as (13) of Article 32, supposed to satisfy oonditioii (iii) of Article 31, expresses a rotation if all the components of strain (14) vanish. In any other case it expresses a homogeneous strain. 70 GENERAL THEORY OF [CH. I. APP. The result of two successive linear homogeneous transformations may be expressed conveniently in the notation of matrices. In this notation the equations of transformation (13) would be written («i, 2/1, ^i) = ( l + «u «i2 «i3 )(«. y> 4 (25) (^31 OS32 J- "I" '^aa and the equations of a second such transformation could in the same way be written Ct2, Vi, z-i) = { 1+^11 ^12 ^13 )(^i, 2/1. ^1) (26) 621 1 + 62a ^23 By the first transformation a point (a;, y, z) is replaced by {x^, y„ «i), and by the second {x^, y^, z,) is replaced by {x^, y^, z^). The result of the two operations is that {x, y, z) is replaced by (x^, y^, z^\ and we have (iBa, 2/2, ^j) = ( 1 +c„ C12 Ci3 )(«, y, ^), (27) C21 A + C22 C23 C31 C32 1 + C33 Cii = 611 + C^u 4- ^iiaii -f ftjaCtai + 6i3»3i> C12 = O12 + 0^12 4" t^n'^12 * ^^12^22 "T ^]3^32j where .(28) In regard to this result, we notice (i) that the transformations are not in general commutative; (ii) that the result of two successive pure strains is not in general a pure strain; (iii) that the result of two successive trans- formations, involving very small displacements, is obtained by simple super- position, that is by the addition of corresponding coefficients. The result (ii) may be otherwise expressed by the statement that pure strains do not form a group. 40. Additional results relating to the composition of strains. When the transformation (26) is equivalent to a rotation about an axis, so that its- coefficients are those given in Article .35, we may show that the components of strain •corresponding with the transformation (27) are the same as those corresponding with the transformation (25), as it is geometrically evident they ought to be. In the particular case where the transformation (25) is a pure strain referred to its principal axes, [so that aii = €i, a!22 = f2> <^33 = f3i ^.nd the remaining coefficients vanish], and the transformation (26) is a rotation about an axis, [so that its coefficients are those given in Article 35], the coefficients of the resultant strain are given by such equations as l+C,i = (H-.i){l-(l-?2)(l-C0sfl)}, Ci2 = (l+f2){- Wsin5 + ?OT(1 -COS 5)}, 39, 40] HOMOGENEOUS STRAIN 71 The quantities w^, ■my, m^, corresponding with this strain are not components of rotation, the displacement not being small. We should find for example 2a7a;= C32 - "23 = 2? sin 5+ (f 2 + es) ? sin 5 + (^2 - €3) mn (1 — cos 6). We may deduce the result that, if the components of strain corresponding with the transformation (27) vanish, and the condition (iii) of Article 31 is satisfied, the rotation expressed by (27) is of amount 6 about an axis (I, m, n) determined by the equations "32-^23 _ O13— £31 _ Oil— €12 _ 2 sin d I m n We may show that the transformation expressed by the equations Xi = x-'ni^i/ + ziryZ, i/i=i/-w^z+w^3!, Zi = z--iSyX + 'u!^y represents a homogeneous strain compounded of uniform extension of all lines which are at right angles to the direction {■uHx '■ 'Sy '■ afj) and rotation about a line in this direction. The amount of the extension is v'( 1 + 073.2 + or/ + ra'3^) — l, and the tangent of the angle of rotation is ^{■m^ + 'ajy^ + vj^). In the general case of the composition of strains, we may seek expressions for the resultant strain-components in terms of the strain-components of the constituent strains and the coefScients of the transformations. If we denote the components of strain corresponding with (25), (26), (27) respectively by {fxxia, ■•• ^xxxi, ■■■ {^xxX, ■■■ , we find such formulae as - (^xx)c = {^xx)a + ( 1 + <^u)'^xixi + «^2lf l/iyi + ^l^Slf ai^i • +aS2ia53lfiri8, + (l+'«n)ffl3lf2ia;i + (l+«n)a2lfa;,!,,, (f!/z)c=(fi/<,)<. + 2ai2ai3Em«i + 2 (1 +022) 023%,!,, + 2 (l+a33) asi^z^z, + {(l+a22)(l+«33) + a23«32}e!/,2, + {(l+«33)«12 + «32«13}fz,a^, + {(l+«22)«13 + »12«23}fx,l„- CHAPTER II. ANALYSIS OF STRESS. 41. The notion of stress in general is simply that of balancing internal action and reaction between two parts of a body, the force which either part exerts on the other being one aspect of a stress*. A familiar example is that of tension in a bar ; the part of the bar on one side of any normal section exerts tension on the other part across the section. Another familiar example is that of hydrostatic pressure. At any point within a fluid, pressure is exerted across any plane drawn through the point, and this pressure is esti- mated as a force per unit of area. For the complete specification of the stress at any point of a body we should require to know the force per unit of area across every plane drawn through the point, and the direction of the force as well as its magnitude would be part of the specification. For a complete specification of the state of stress within a body we should require to know the stress at every point of the body. The object of an analysis of stress is to determine the n.ature of the quantities by which the stress at a point can be specified!. In this Chapter we shall develope also those consequences in regard to the theory of the equilibrium and motion of a body which follow directly from the analysis of stress. 42. Traction across a plane at a point. We consider any area S in a given plane, and containing a point within a body. We denote the normal to the plane drawn in a specified sense by v, and we think of the portion of the body, which is on the side of the plane towards which v is drawn, as exerting force on the remaining portion across the plane, this force being one aspect of a stress. We suppose that the force, which is 'thus exerted across the particular area S, is statically equivalent to a force R, acting at in a definite direction, and a couple G, about a definite axis. If we contract the area S by any continuous process, keeping the point always within it, the force R and the couple G tend towards zero limits, and the direction of the force tends to a limiting direction (I, to, n). We assume ' For a discussion of the notion of stress from the point of view of Eational Mechanics, see Note B at the enJ of tliis book. t The theoiy of the specification of stress was given by Cauchy in the Article 'De la pression ou tension dans un corps solide ' in the volume for 1827 of the Exercices de matMmatiques. ^1~43] TRACTION ACROSS A PLANE 73 that the number obtained by dividing the number of units of force in the force R by the number of units of area in the area S (say R/S) tends to a limit F, which is not zero, and that on the other hand G/S tends to zero as a hmit. We define a vector quantity by the direction (I, m, n), the numerical measure F, and the dimension symbol (mass) (length)-' (time)-^ This quantity is a force per unit of area ; we call it the traction across the plane v at the point 0. We write X,, Y,, Z, for the projections of this vector on the axes of coordinates. The projection on the normal v is X, cos (*', v) + F„ cos (i/, v) + Z^ cos {z, v). If this component traction is positive it is a tension ; if it is negative it is a pressure. If dS is a very small area of the plane normal to v at the point 0, the portion of the body, which is on the side of the plane towards which v is drawn, acts upon the portion on the other side with a force at the point 0, specified by . , ^ (X^dS, Y^dS, Z^dS); this is the traction upon the element of area dS. In the case of pressure in a fluid at rest, the direction {I, m, n) of the vector (Z„, F„, Z„) is always exactly opposite to the direction v. In the cases of viscous fluids in motion and elastic solids, this direction is in general obliquely inclined to v. 43. Surface Tractions and Body Forces. When two bodies are in contact, the nature of the action between them over the surfaces in contact is assumed to be the same as the nature of the action between two portions of the same body, separated by an imagined surface. If we begin with any point within a body, and any direction for v, and allow to move up to a point 0' on the bounding surface, and v to coincide with the outward drawn normal to this surface at 0', then X,., Y„, Z, tend to limiting values, which are the components of the surface traction at 0'; and Z„ bS, Y^ 8S. Z, 8S are the forces exerted across the element S*Sf of the bounding surface by some other body having contact with the body in question in the neighbourhood of the point 0'. In general other forces act upon a body, or upon each part of the body, in addition to the tractions on its surface. The type of such forces is the force of gravitation, and such forces are in general proportional to the masses of particles on which they act, and, further, they are determined as to magnitude and direction by the positions of these particles in the field of force. liX, Y, Z ' are the components of the intensity of the field at any point, m the mass of a particle at the point, then mX, niY, mZ are the forces of the field that act on the particle. The forces of the field may arise from the action of particles 74 EQUATIONS OF MOTION OF BODIES [CH. II forming part of the body, as in the case of a body subject to its own gravi- tation, or of particles outside the body, as in the case of a body subject to the gravitational attraction of another body. In either case we call them body forces. 44. Equations of Motion. The body forces, applied to any portion of a body, are satically equivalent to a single force, applied at one point, together with a couple. The components, parallel to the axes, of the single force are p ^-^ jlpXdiJcdydz, lllp^^ 'J-f' dydz, \\\pZdxdydz, where p is the density of the body at the point («, y, z), and the integration is taken through the volume of the portion of the body. In like manner, the tractions on the elements of area of the surface of the portion are equivalent to a resultant force and a couple, and the components of the former are llx^dS, fjy.dS, JJZJS, where the integration is taken over the surface of the portion. The centre of mass of the portion moves like a particle under the action of these two sets of forces, for they are all the external forces acting on the portion. If then (fx'fy>.fz) is the acceleration of the particle which is at the point (x, y, z) at time t, the equations of motion of the portion are three of the type * jljpf^dxdydz=jjjpXdxdydz+jjx,dS, (1) where the volume-integrations are taken through the volume of the portion, and the surface-integration is -taken over its surface. Again the equations, which determine the changes of moment of momentum of the portion of the body, are three of the type jjjP (yfz - rfy) dj-dydz = jjjp (yZ - zY) dxdydz + j[(yZ, - zY,) dS ; -• (2) and, in accordance with the theorem f of the independence of the motion of the centre of mass and the motion relative to the centre of mass, the origin of the coordinates x, y, z may be taken to be at the centre of mass of the portion. The above equations (1) and (2) are the types of the general equations of motion of all bodies for which the notion of stress is valid. * The equation (1) is the form assumed by the equations of the type Smi" = SZ, of my Theoretical Mechanics, Chapter VI. ; and the equation (2) is the form assumed by the equations of the type Sm (yi -zy) = -Z(yZ~zY)ot the same Chapter. t Theoretical Mechanics, Chapter VI. 43-47] SPECIFICATION OF STRESS 75 45. Equilibrium. When a body is at rest under the action of body forces and surface tractions, these are subject to.the conditions of equilibrium, which are obtained from equations (1) and (2) by omission of the terms containing /,, /„, /^. We have thus six equations, viz. : three of the type n\pXdxdydz+ j jX,dS=0, (3) and three of the type \^jp{yZ-zr)dxdydz+U{yZ^-zr;)dS=o (4) It follows that if the body forces and surface tractions are given arbitrarily, there will not be equilibrium. In the particular case where there are no body forces, equilibrium cannot be maintained unless the surface tractions satisfy six equations of the types IJx^dS=0, and j l(yZ^-zr^)dS=0. 46. Law of equilibrium of surface tractions on small volumes. From the forms alone of equations (1) and (2) we can deduce a result of great importance. Let the volume of integration be very, small in all its dimensions, and let l^ denote this volume. If we divide both members of equation (1) by l^, and then pass to a limit by diminishing I indefinitely, we find the equation 1=0 Again, if we take the origin within the volume of integration, we obtain by a similar process from (2) the equation lim. 1-' ff(yZ, - z Y,) dS = 0. 1=0 The equations of which these are types can be interpreted in the statement : The tractions on the elements of area of the surface of any portion of a body, which is very small in all its dimensions, are ultimately, to a first approocimation, a system of forces in equilibrium. 47. Specification of stress at a point. Through any point in a body, there passes a doubly infinite system of planes, and the complete specification of the stress at involves the knowledge of the traction at across all these planes. We may use the results obtained in the last Article to express all these tractions in terms of the component tractions across planes parallel to the coordinate planes, and to obtain relations between these components. We denote the traction across a plane 07 = const, by its vector components (X,., Y^, Z^) and use a similar notation for the tractions across planes y = const, and z = const. The capital letters show the directions of the component tractions, and the suffixes the planes across which they act. The sense is such that X^ is positive when it is a tension, negative when it is a pressure. If the axis of x is supposed drawn 76 SPECIFICATION OF STRESS [CH. n Fig. 5. upwards from the paper (cf. Fig. 5), and the paper is placed so as to pass through 0, the traction in question is exerted by the part of the body above the paper upon the part below. We consider the equilibrium of a tetrahedral portion of the body, having one vertex at 0, and the three edges that meet at this vertex parallel to the axes of coordinates. The remaining vertices are the intersections of these edges with a plane near to 0. We denote the direction of the normal to this plane, drawn away from the interior of the tetra- hedron, by V, so that its direction-cosines are cos («, v), cos (2/, v), cos (^, v). Let A be the area of the face of the tetrahedron that is in this plane ; the areas of the remaining faces are A cos («, v), A cos (y, v), A cos {z, v). For a first approximation, when all the edges of the tetrahedron are small, we may take the resultant tractions across the face V to be Z;,A, ..., and those on the remain- ing faces to be — Xa;Acos(a;, i;), The sum of the tractions parallel to .'■ on all the faces of the tetrahedron can be taken to be X^A-Zj;Acos(«, v)-Xy^cQs{y, i')-X2Acos(5, v). By dividing by A, in accordance with the process of the last Article, we obtain the first of equations (5), and the other equations of this set are obtained by similar processes : we thus find the three equations X„ = Xa;Cos (x, v) +Xy cos (y, v) +Xz cos {z, v), \ F„ = Fj; cos {x, v) + Yy cos (y, v) + Y^cos(z, v), [ (5) Z^ = Zx cos (,v, v) + Zy cos (y, v) + Z^ cos {z, v). ) By these I'quations the traction across any plane through is expressed in terms of the tractions across planes parallel to the coordinate planes. By these equations also the component tractions across planes, parallel to the coordinate planes, at any point on the bounding surface of a body, are connected with the tractions exerted upon the body, across the surface, by any other body in contact with it. Again, consider a very small cube (Fig. 6) of the material with its edges parallel to the coordinate axes. To a first approximation, the resultant tractions exerted upon the cube across the faces perpendicular to the axis of x are AXj;, C^Yx, ^Z^, for thf face for which x is greater, and -AZ^;, - AF^;,- A.?j;, for the opposite face, A being the area of any face. Similar expressions hold for the other faces. The value of JJ{yZ,- zY^)dS for the cube can be 47, 48] BY MEANS OF SIX COMPONENTS 77 taken to be l^iZ^-Y^), where I is the length of any edge. By the process of the last Article we obtain the first of equations (6), and the other equations of this set are obtained by similar processes ; we thus find the three equations ^,= 7„ X, = Z^, Y, = X, (6) Fig. 6. By equations (6) the number of quantities which must be specified, in order that the stress at a point may be determined, is reduced to- six, viz. three normal component tractions X^, Yy, Z^, and three tangential tractions Fz, Zx, Xy. These six quantities are called the components of stress* at the point. The six components of stress are sometimes written xx, yy, zz, ye, zx, xy. A notation of this kind is especially convenient when use is made of the orthogonal curvilinear coordinates of Article 19. The six components of stress referred to the normals to the surfaces a, /8, 7 at a point will hereafter be denoted by aa, /S/3, 77, /S7, ') + 2;cos(y, v) + Z^cos{z, v)} -z{Yx cos {a!,v)+ Ty cos (y, v) + Y, cos {z, v)}] dS, by substituting for f^, ... from the equations of type (13), and for Y^, Z^ from (5). By help of Green's transformation, this equation becomes /// (Z„- Y^) dxdydz=0 ; and thus the equations of moments are satisfied identically in virtue of equations (6). It will be observed that, equations (6) might be proved by the above analysis instead of that in Article 47. (o) When the equations (14) are satisfied at all points of a body, the conditions of equilibrium of the body as a whole (Article 45) are necessarily satisfied, and the resultant of all the body forces, acting upon elements of volume of the body, is balanced by the resultant of all the tractions, acting upon elements of its surface. The hke statement is true of the resultant moments of the body forces and surface tractions. {d) An example of the application of this remark is afforded by Maxwell's stress-system described in (vi) of Article 53. We should find for example dx dy dz 47r dx ' where V^ stands for d^/dx^ + d''ldy^ + d^ldz^. It follows that, in any region throughout which ^727-^0, this stress-system is self-equilibrating, and that, in general, this stress-system is 1 /dV dV dV\ in equilibrium with body force specified by "7" ^^^ ( 5~> 3- i "3- j P^r unit volume. Hence the tractions over any closed surface, which would be deduced from the formulae for 1 /dV dV dV\ Xx, ..., are statically equivalent to body forces, specified by j— ^^^ ( 3" i -5- , ^ I per unit volume of the volume within the surface. (e) Stress-functions. In the development of the theory we shall be much occupied with bodies in equilibrium under forces applied over their surfaces only. In this case there are no body forces and no accelerations, and the equations of equilibrium are dX.dx^dZx dx dY dY^ ^_Zx dj^ dz^ W^'di^ SF" ' W^ dy '^ dz ~"' dx^ dy^ 'dz ~^ ' ^^^> while the surface tractions are equal to the values of {Xy, Yy, ZJ) at the surface of the body. The differential equations (16) are three independent relations between the six components 86 EQUATIONS OF MOTION AND OF EQUILIBRIUM [CH. II (if stress at any point ; by means of them we might express these six quantities in terms of three independent functions of position. Such functions would be called "stress-functions."' So long as we have no information about the state of the body, besides that contained in equations (16), such functions are arbitrary functions. One way of expressing the stress-components in terms of stress-functions is to assume* and then it is clear that the equations (16) are satisfied if ""^W "*" "3^ ' "" 3^2 "*■ 9^2 ' ^' 3^ -^ 3^,2 • Another way is to assume t ^^^^^9^90' ""923^' '~dxdi/' 13/ djfi (H^j ^\ y 1 9 fb'^x S^/fa . '^'^x '~~l^x\ dx'^dy clz)' 'i,dy\dx dy ^ dz 11 /3■) + ^ cos (/3, x) + f/ cos (7, x)} -^ + l{ya cos (a, x) + /37 cos (/8, x) + 77 cos (7, x)\ -^ . When we apply Green's transformation to this expression we find [ I X^dS = jljdad^dy |^ ^-^ {«« cos (a, «) + «^ cos (^, «) + 7a cos (7, x)} + + a/3 97 /^S^l fa/3 cos (a, «) + yS/3 cos (/3, «) + /37 cos (7, x)} r-r {ya COS (a, x) + ^y cos (0, x) + 77 cos (7, x)] and, since (h-^hji^ydad^dy is the element of volume, we deduce from (1) the equation 9 -=—r- [aa cos (a, x) + a/3 cos (/3, «) + 7a cos (7, x)] + 9/3 97 hshj [a^coa (a, x) + $^ cos (/3, «) + /S7 cos (7, «)) 9 r 1 --^ '-^ ^~- "11 + — f-r {7a cos (a, «) + 0y cos (;S, «) + 77 cos (7, x)} .(18) The angles denoted by (a, x), ... are variable with a, 0, 7 because the normals to the surfaces a = const., ... vary from point to point. It maybe shown* that for any fixed direction of x the differential coefficients of ♦ cos (a, x), ... are given by nine equations of the type icos(«,^) = -/i,|(l).cos(/3,*')-/^3^(^).cos(7,^), -^ cos (a, «) = /^i ^ ( y- ) . cos (j8, x), ^ cos (a, x) = hi :^ ( j-j . cos (7, x). We now take the direction of the axis of x to be that of the normal to the surface a = const, which passes through the point (a, /3, 7). After the differentiations have been performed we put cos (o, x)= 1, cos (/3, x) = 0, cos (7, x) = 0. * See the Note on applications of moving axes at the end of this book. In the special case of cylindrical coordinates the corresponding equations can be proved directly without any difficulty. 58, 59] EXPRESSED IN TERMS OP STRESS-COMPONENTS 89 We take /„ for the component acceleration along the normal to the surface a = const., and F^ for the component of body force in the same direction. Equation (18) then becomes The two similar equations containing components of acceleration and body force in the directions of the normals to /3 = const, and 7 = const, can be written down by symmetry. 59. Special cases of stress-equations referred to curvilinear co- ordinates. (1) In the case of cylindrical coordinates r, 6, z (cf. Article 22) the stress-equations are drr Idrd drz rr^-BQ „ . d7e Idee dTz %7d- „ ^ drz 1 ddz dzz rz dt j: (ii) In the case of plane stress referred to cylindrical coordinates, when there is equi- librium under surface tractions only, the stress-components, when expressed in terms of the stress-function x^ of equations (17), are given by the equations* (iii) In the case of polar coordinates r, 6, the stress-equations are (iv) When the surfaces a, /3, y are isostatic so that ^y=ya=a^ = 0, the equations can be written in such forms t as , daa aa-fifi aa-yy „ «1 -3— + — +P-l'a = p/a, Oa Pis P12 where pjj and pis are the principal radii of curvature of the surface a = const, which cor- respond respectively with the curves of intersection of that surface and the surfaces — const, and 7= const. * J. H. Miohell, London Math. Soc. Proc, vol. 31 (1899), p. 100. t Lame, Coordonnees curvilignes, p. 274. CHAPTER III. THE ELASTICITY OF SOLID BODIES. 60. In the preceding Chapters we have developed certain kinematical and djmamical notions, which are necessary for the theoretical discussion of the physical behaviour of material bodies in general. We have now to explain how these notions are adapted to elastic solid bodies in particular. An ordinary solid body is constantly subjected to forces of gravitation, and, if it is in equilibrium, it is supported by other forces. We have no experience of a body which is free from the action of all external forces. From the equations of Article 54 we know that the application of forces to a body necessitates the existence of stress within the body. Again, solid bodies are not absolutely rigid. By the application of suitable forces they can be made to change both in size and shape. When the induced changes of size and shape are considerable, the body does not, in general, return to its original size and shape after the forces which induced the change have ceased to act. On the other hand, when the changes are not too great the recovery may be apparently complete. The property of recovery of an original size and shape is the property that is termed elasticity. The changes of size and shape are expressed by specifying strains. The " unstrained state '' (Article 4), with reference to which strains are specified, is, as it were, an arbitrary zero of reckoning, and the choice of it is in our power. When the unstrained state is chosen, and the strain is specified, the internal configuration of the body is known. We shall suppose that the differential coefficients of the displacement {u, V, w), by which the body could pass from the unstrained state to the strained state, are sufficiently small to admit of the calculation of the strain by the simplified methods of Article 9 ; and we shall regard the configuration as specified by this displacement. For the complete specification of any state of the body, it is necessary to know the temperature of every part, as well as the configuration. A change of configuration may, or may not, be accompanied by changes of temperature. 61. Work and energy. Unless the body is in equilibrium under the action of the external forces, it will be moving through the configuration that is specified by the displace- ment, towards a new configuration which could be specified by a slightly 60, 61] ENERGY OF ELASTIC SOLID BODY 91 diiferent displacement. As the body moves from one configuration to another, the external forces (body forces and surface tractions) in general do some work ; and we can estimate the quantity of work done per unit of time, that is to say the rate at which work is done. Any body, or -any portion of a body, can possess energy in various ways. If it is in motion, it possesses kinetic energy, which depends on the distribution of mass and velocity. In the case of small displacements, to which we are restricting the discussion, the kinetic energy per unit of volume is expressed with sufficient approximation by the formula in which p denotes the density in the unstrained state. In addition to the molar kinetic energy, possessed by the body in bulk, the body possesses energy which depends upon its state, i.e. upon its configuration and the temperatures of its parts. This energy is called " intrinsic energy " ; it is to be calculated by reference to a standard state of chosen uniform temperature and zero displacement. The total energy of any portion of the body is the sum of the kinetic energy of the portion and the intrinsic energy of the portion. The total energy of the body is the sum of the total energies of any parts*, into which it can be imagined to be divided. As the body passes from one state to another, the total energy, in general, is altered ; but the change in the total energy is not, in general, equal to the work done by the external forces. To produce the change of state it is, in general, necessary that heat should be supplied to the body or withdrawn from it. The quantity of heat is measured by its equivalent in work. The First Law of Thermodynamics states that the increment of the total energy of the body is equal to the sum of the work done by the external forces and the quantity of heat supplied. We may calculate the rate at which work is done by the external forces. The rate at which work is done by the body forces is expressed by the formula ///K^l^^^^s)^*"^ « where the integration is taken through the volume of the body in the un- strained state. The rate at which work is done by the surface tractions is expressed by the formula * For the validity of the analysis of the energy into molar kinetic energy and intrinsic energy it is necessary that the dimensions of the parts in question should be large compared with mole- cular dimensions. (JO ELASTICITY OF SOLID BODIES [CH. Ill where the integration is taken over the surface of the body in the unstrained state. This expression may be transformed into an integral taken through the volume of the body, by the use of Green's transformation and of the formulse of the type X, = Z-. cos {x, v) + Xy cos (y, v) + X^ cos {z, v), We use also the results of the type Y^ = Zy, and the notation for strain- components e„, .... We find that the rate at which work is done by the surface tractions is expressed by the formula {{{\(dX, dXydZAdu(dX,dYyJJz\t ]]] [\d^ ^^^ dzJdt^Kdx '^ dy ^ dzjdt \dx dy dz J dt dxdydz dxdydz. ...(2) We may calculate also the rate at which the kinetic energy increases. This rate is expressed with sufficient approximation by the formula where the integration is taken through the volume of the' body in the un- strained state. If we use the equations of motion, (15) of Article 54, we can express this in the form !L ^^+t+t+t)s+---]^*'^- It appears hence that the expression ''df^ "^'^ ' dt ^^' dt +^" dt "*■ " dt dxdydz (4) represents the excess of the rate at which work is done by the external forces above the rate of increase of the kinetic energy. 62. Existence of the strain-energy-function. Now let S2\ denote the increment of kinetic energy per unit of volume, which is acquired in a short interval of time Bt. Let BU he the increment of intrinsic energy per unit of volume, which is acquired in the same interval. Let SWi be the work done by the external forces in the interval, and let BQ be the mechanical value of the heat supplied in the interval. Then the First Law of Thermodynamics is expressed by the formula i^{BT, + BU) dxdydz = BW, + BQ (5) 61, 62] EXPRESSED BY STRAIN-ENBRGY-FUNCTION 93 Now, according to the final result (4) obtained in Article 61, we have ^ J J J ^■^"^^''^ ■'" ^y^^m + ■^2^^22 + ^i^^ye + Z^he^^Jt Xyhexy)dxdydz, . . .(6) where ^e^x, ■■■ represent the increments of the components of strain in the interval of time St. Hence we have jfjsUdxdydz = 8Q+jjj{XJe^:,+ ...)da}dydz (7) The differential quantity BU is the differential of a function U, which is an one-valued function of the temperature and the quantities that determine the configuration. The value of this function U, corresponding with any state, is the measure of the intrinsic energy in that state. In the standard state, the value of U is zero. If the change of state takes place adiabatically, that is to say in such a way that no heat is gained or lost by any element of the body, SQ vanishes, and we have 8U = XxBexai+ YySeyy+ Z!iSez;i+ Y^Beyg-^- ZxSe^3, + XySe^-y (8) Thus the expression on the right-hand side is, in this case, an exact differen- tial ; and there exists a function W, which has the properties expressed by the equations dW dW VKxx 'Jf'yz The function W represents potential energy, per unit of volume, stored up in the body by the strain ; and its variations, when the body is strained adia- batically, are identical with those of the intrinsic energy of the body. It is probable that the changes that actually take place in bodies executing small and rapid vibrations are practically adiabatic. A function which has the properties expressed by equations (9) is called a " strain-energy-function." If the changes of state take place isothermally, i.e. so that the temperature of every element of the body remains constant, a function W having the properties expressed by equations (9) exists. To prove this we utilise the Second Law of Thermodynamics in the form that, in any reversible cycle of changes of state performed without variation of temperature, the sum of the elements BQ vanishes* The sum of the elements BU also vanishes; and it follows that the sum of the elements expressed by the formula 2 {XJe^ + YyBeyy + Z^Be^, + Y.Bey^ + ZJ>e^ -I- XyBe^) * Cf. Kelvin, Math, and Phys. Papers, vol. 1, p. 291. 94 ELASTICITY OP SOLID BODIES [CH. Ill also vanishes in a reversible cycle of changes of state without variation of temperature. Hence the differential expression ^X ^^aXC + Yy Beyy + Z^ ^JJ + Y^ SBj,; + Z^ BB^x + Xy Sfi^y is an exact differential, and the strain-energy-function W exists. When a body is strained slowly by gradual increase of the load, and is in continual equilibrium of temperature with surrounding bodies, the changes of state are practically isothermal. 63. Indirectness of experimental results. The object of experimental investigations of the behaviour of elastic bodies may be said to be the discovery of numerical relations between the quantities that can be measured, which shall be sufficiently varied and sufficiently numerous to serve as a basis for the inductive determination of the form of the intrinsic energy-function, viz. the function U of Article 62, This object has not been achieved, except in the case of gases in states that are far removed from critical states. In the case of elastic solids, the con- ditions are much more complex, and the results of experiment are much less complete ; and the indications which we have at present are not sufficient for the formation of a theory of the physical behaviour of a solid body in any circumstances other than those in which a strain-energy-function exists. When such a function exists, and its form is known, we can deduce from it the relations between the components of stress and the components of strain; and, conversely, if, from any experimental results, we are able to infer such relations, we acquire thereby data which can serve for the construction of the function. The components of stress or of strain within a solid body can never, from the nature of the case, be measured directly. If their values can be found, it must always be by a process of inference from measurements of quantities that are not, in general, components of stress or of strain. Instruments can be devised for measuring average strains in bodies of ordinary size, and others for measuring particular strains of small superficial parts. For example, the average cubical compression can be measured by means of a piezometer; the extension of a short length of a longitudinal filament on the outside of a bar can be measured by means of an extenso- meter. Sometimes, as for example in experiments on torsion and flexure, a displacement is measured*. External forces applied to a body can often be measured with great exactness, e.g. when a bar is extended or bent by hanging a weight at one end. In such cases it is a resultant force that is measured directly, not * For an account of experimental methods, which are commonly used, reference may be made to J. H. Poynting and J. J. Thomson, Properties of matter, London 1902, and G. F. C. Searle, Experimental Elasticity, Cambridge 1908. 62-64] EXPRESSED BY GENERALIZED HOOKe's LAW 95 the component tractions per unit of area that are applied to the surface of the body. In the case of a body under normal pressure, as in the experiments with the piezometer, the pressure per unit of area can be measured. In any experiment designed to determine a relation between stress and strain, some displacement is brought about, in a body partially fixed, by the application of definite forces which can be varied in amount. We call these forces collectively " the load." 64. Hooka's Law. Most hard solids show the same type of relation between load and measurable strain. It is found that, over a wide range of load, the measured strain is proportional to the load. This statement may be expressed more fully by saying that (1) when the load increases the measured strain increases in the same ratio, (2) when the load diminishes the measured strain diminishes in the same ratio, (3) when the load is reduced to zero no strain can be measured. The most striking exception to this statement is found in the behaviour of cast metals. It appears to be impossible to assign any finite range of load, within which the measurable strains of such metals increase and diminish in the same pjroportion as the load. The experimental results which hold for most hard solids, other than cast metals, lead by a process of inductive reasoning to the Generalized Hooke's Law of the proportionality of stress and strain. The general form of the law is expressed by the statement : — Each of the six components of stress at any point of a body is a linear function of the six components of strain at the point. It i.s necessary to pay some attention to the way in which this law represents the experimental results. In most experiments the load that is increased, or diminished, or reduced to zero consists of part only of the external forces. The weight of the body subjected to experiment must be balanced ; and neither the weight, nor the force employed to balance it, is, in general, included in the load. At the beginning and end of the experi- ment the body is in a state of stress ; but there is no measured strain. For the strain that is measured is reckoned from the state of the body at the beginning of the experiment as standard state. The strain referred to in the statement of the law must be reckoned from a different state as standard or "unstrained" state. This state is that in which the body would be if it were freed from the action of all external forces, and if there were no internal stress at any point of it. We call this state of the body the "unstressed state." Reckoned from this state as standard, the body is in a state of strain at the beginning of the experi- ment; it is also in a state of stress. When the load is applied, the stress is altered in amount and distribution ; and the strain also is altered. After the application of the load, the stress consists of two stress-systems: the stress-system in the initial state, and a stress-system by which the load would be balanced all through the body. The strain, reckoned from the unstressed state, is likewise compounded of two strains : the strain from 96 ELASTICITY OF SOLID BODIES [CH. Ill the unstressed state to the initial state, and the strain from the initial state to the state assumed uuder the load. The only things, about which the experiments can tell us any- thing, are the second stress-system and the second strain ; and it is consonant with the result of the experiments to assume that the law of proportionality holds for this stress and strain. The general statement of the law of proportionality implies that the stress in the initial state also is proportional to the strain in that state. It also implies that both the initial state, and the state assumed under the load, are derivable from the unstressed state by displacements, of amount sufficiently small to admit of the calculation of the strains by the simplified methods of Article 9. If this .were not the case, the strains would not be compounded by simple superposition ; and the proportionality of load and measured strain would not imply the proportionality of stress-components and strain-components. 65. Form of the strain-energy-function. The experiments which lead to the enunciation of Hooka's Law do not constitute a proof of the truth of the law. The law formulates in abstract terms the results of many observations and experiments, but it is much more precise than these results. The mathematical consequences which can be deduced by assuming the law to be true are sometimes capable of experimental verification ; and, whenever this verification can be made, fresh evidence of the truth of the law is obtained. We shall be occupied in sub- sequent chapters with the deduction of these consequences; here we note some results which can be deduced immediately. When a body is slightly strained by gradual application of a load, and the temperature remains constant, the stress-components are linear functions of the strain-components, and they are also partial differential coefficients of a function ( W) of the strain-components. The strain-energy-function, W, is therefore a homogeneous quadratic function of the strain-components. The known theory of sound waves* leads us to expect that, when a body is executing small vibrations, the motion takes place too quickly for any portion of the body to lose or gain any sensible quantity of heat. In this case also there is a strain-energy-function ; and, if we assume that Hooke's Law holds, the function is a homogeneous quadratic function of the strain- components. When the stress-components are eliminated from the equations of motion (15) of Article 54, these equations become linear equations for the determination of the displacement. The linearity of them, and the way in which the time enters into them, make it possible for them to possess solutions which represent isochronous vibrati«^s. The fact that all solid bodies admit of being thrown into states of isochronous vibration has been emphasized by Stokesf as a peremptory proof of the truth of Hooke's Law for the very small strains involved. The proof of the existence of W given in Article 62 points to different coefficients for the terms of W expressed as a quadratic function of strain-com- ponents, in the two cases of isothermal and adiabatic changes of state. These * See Bayleigh, Theory of Sound, Chapter XI. t See Introduction, footnote 37. 64-66] EXPRESSED BY GENERALIZED HOOKE's LAW 97 coefficients are the " elastic coastants," and discrepancies have actually been found in experimental determinations of the constants by statical methods, involving isothermal changes of state, and dynamical methods, involving adiabatic changes of state*. The discrepancies are not, however, very serious. To secure the stability of the body it is necessary that the coefficients of the terms in the homogeneous quadratic function W should be adjusted so that the function is always positive f. This condition involves certain relations of inequality among the elastic constants. If Hooke's Law is regarded as a first approximation, valid in the case of very small strains, it is natural to assume that the terms of the second order in the strain-energy-function constitute likewise a first approximation. If terms of higher order could be taken into account an extension of the theory might be made to circumstances which are at present excluded from its scope. Such extensions have been suggested and partially worked out by several writers}. 66. Elastic constants. According to the generalized Hooke's Law, the six components of stress at any point of an elastic solid body are connected with the six components of strain at the point by equations of the form y z — ''4i^a;a! T f^ii^yy + ^43^22 + Cf^eyz + Ci^Bzx + Ci^Bxy, .(10) The coefficients in these equations, c„, ... are the elastic constants of the substance. They are the coefficients of a homogeneous quadratic function 2W, where W is the strain-energy-function; and they are therefore con- nected by the relations which ensure the existence of the function. These relations are of the form Cm^Csr, (r, s = l,2, ... 6), (11) and the number of constants is reduced by these equations from 36 to 21. * The discrepancies appear to have been noticed first by P. Lagerhjelm in 1827, see Todhunter and Pearson's History, vol. 1, p. 189. They were made the subject of extensive experiments by G. Wertheim, Ann. de Chimie, t. 12 (1844). Information concerning the results of more recent experimental researches is given by Lord Kelvin (Sir W. Thomson) in the Article 'Elasticity' in Ency. Brit., 9th edition, reprinted in Math, and Phys. Papers, vol. 3. See also W. Voigt, Ann. Phys. Chem. (Wiedemann), Bd. 52 (1894). t Kirchhoff, Vorlesungen Uher . . .Mechanik, Vorlesung 27. For a discussion of the theory of stability reference may be made to a paper by R. Lipschitz, J. f. Math. (Crelle), Bd. 78 (1874). J Reference may he made, in particular, to W. Voigt, Ann. Phys. Chem. (Wiedemann), Bd. 52, 1894, p. 536, and Berlin Beriphte, 1901. L. E. 7 98 DETERMINATION OF STRESS [CH. Ill We write the expression for 2 W in the form 1. It ^ CiiB'xx + ^^ii^xx^yy + ^Cii^xx^zz + ^^nCxx^yz + ^Ci^^xx^zx ' ^^le^xx^xy + C226 yi, + ^^C^eyyBzZ + ^C^SyyCyz + ^C^eyyBix T ^(^iS^yy^Xy + Cjje 00 + ^CgiBiz^yz + ^Cs^^zz^zx I •^^3862062;^ + 04462/0 + ■'''456^0603; + ^C4e 6^0 Ca^y ' ^85 6 2a; I '^Cse &zx^xy (12) The theory of Elasticity has sometimes been based on that hypothesis concerning the constitution of matter, according to which bodies are regarded as made up of material points, and these points are supposed to act on each other at a distance, the law of force between a pair of points being that the force is a function of the distance between the points, and acts in the line joining the points. It is a consequence of this hypothesis* that the co- efficients in the function W are connected by six additional relations, whereby their number is reduced to 15. These relations are .(13) C23 — C44, C31 — Cjjj, C]2 — Cbs, I ^14 = C58 , C25 = C48 , C45 = C36 . J We shall refer to these as " Cauchy's relations " ; but we shall not assume that they hold good. 67. Methods of determining the stress in a body. If we wish to know the state of stress in a body to which given forces are applied, either as body forces or as surface tractions, we have to solve the stress-equations of equilibrium (14) of Article 54, viz. ^^^^fl + ^Zx^ ox dy dz '^ .(14) ox dy oz '^ dZx dY dZz _^ 7 .. and the solutions must be of such forms that they give rise to the right expressions for the surface tractions, when the latter are calculated from the formuliE (5) of Article 47, viz. A% == Xx cos (x, v) + Xy cos (y, v) H- Zx cos {z, v)]^ , , The equations (14) with the conditions (15) are not sufficient to determine the stress, and a stress-system may satisfy these equations and conditions and yet fail to be the correct solution of the problem ; for the stress-components are * See Note B at the end of this book. 66-68] IN AN ELASTIC SOLID BODY 99 functions of the strain- components, and the latter satisfy the six equations of compatibility (25) of Article 17, viz. three equations of the type d^eyy _^ 3^^ _ d'ey^ dz^ By' dydz' ^ . , > and three of the type J 2 ^'gja; = Af'_?^,?!^, ???y .(16) dydz dx\ dx dy dz When account is taken of these relations, there are sufficient equations to determine the stress. Whenever the forces are such that the stress-components are either con- stants or linear functions of the coordinates, the same is true of the strain- components, and the equations of compatibility are satisfied identically. We shall consider such cases in the sequel. In the general case, the problem may in various ways be reduced to that of solving certain systems of differential equations. One way is to form, by the method described above, a system of equations for the stress-components in which account is taken of the identical relations between strain-components. Another way is to eliminate the stress-components and express the strain- components in terms of displacements by using the formula _du _dv dw \ ex.-^, eyy-~, ^zz^Yz' _dw dv _ du dw dv du ^""'d^^di' ^'"'di'^dx' ^'^^dx'^Fy- Both these methods will be illustrated in the sequel. If the displacement can be obtained, the strain-components can be found by differentiation, and the stress-components can be deduced. If, on the other hand, the stress can be determined, the strains can be deduced, and the displacement can be found by the method indicated in Article 18. It will be proved in Chapter VII. that the solution of any problem of the kind considered here is effectively unique. We shall assume for the present that any solution, which satisfies all the conditions, is the solution. 68. Form of the strain-energy-function for isotropic solids. If we refer the stress-component^s and strain-components to a new system of axes of coordinates x, y', z instead of x, y, z, the stress-components must be transformed according to the formulae of Article 49, and the strain- components must be transformed according to the formulae of Article 12. When we substi- tute for Xx, ... and e^x, ... in the equations of the types (10) we find that the stress-components XV, ■■• and the strain-components e^jv, ■■■ are connected by linear equations. These may be solved for the X'^; . . . and the result will be that the X'^, . . . are expressed as linear functions of e^'x', ■ ■ ■ with coefiicients, 7—2 100 ELASTICITY OF ISOTROPIC SOLIDS [CH. Ill which depend on the coefficients Cn, ... in the formula (12), and also on the quantities by which the relative situations of the old and new axes are determined. The results might be found more rapidly by transforming the expression 2W according to the formulae of Article 12. The general result is that the elastic behaviour of a material has reference to certain directions fixed relatively to the material. If, however, the elastic constants are connected by certain relations, the formulae connecting stress-components with strain com- ponents are independent of direction. The material is then said to be isotropic as regards elasticity. In this case the function W is invariant for all trans- formations from one set of orthogonal axes to another. If we knew that there were no invariants of the strain, of the first or second degrees, independent of the two which were found in Article 13 (c), we could conclude that the strain- energy-function for an isotropic solid must be of the form 2 -^ \^xx ~r &yy + &zz) + 2 "^ \^ 2/z "I" ^'zx 'T~ ^ xy ^ ^^yy^zz ^^zz^xx ^^xx^yy)' We shall hereafter (Chapter vi.) perform the transformation, and verify that this is the actual form of W. At present we shall assume this form and deduce some of the simpler consequences of it. It will be convenient to write X -(- 2/i in place of A and fj. in place of B. We shall suppose the material to be homogeneous, so that X and /J, are the same at all points. 69. Elastic constants and modiiluses of isotropic solids. When W is expressed by the equation 2W = {\ + 2fi) (e^^ + eyy + ezzT + /J- (e%z + e%x 4- e\y - ieyyCzz - 4 " 2/x(3X + 2/i)- We write ^ = ^^f+^^ \ (20) (21) 2(X + /i) '"'fT Then E is the quantity obtained by dividing the measure of a simple longi- tudinal tension by the measure of the extension produced by it. It is known as Youngs modulus. The number o- is the ratio of lateral contraction to longitudinal extension of a bar under terminal tension. It is known as Poisson's ratio. Whatever the stress-system may be, the extensions in the directions of the axes and the normal tractions across planes at right angles to the axes are connected by the equations e^ = S-^{X,-cr(¥y+Z,)},] eyy = E-^{Yy-a{Z,+X,)},\ (22) e^ = E-^{Z,-a{X,+ Yy)].) Whatever the stress-system may be, the shearing strain corresponding with a pair of rectangular axes and the shearing stress on the pair of planes at right angles to those axes are connected by an equation of the form Xy = Mxy (23) This relation is independent of the directions of the axes. The quantity jx is called the rigidity. 70. Observations conceming the stress-strain relations in isotropic solids. (a) We may note the relations * E(T _ E , E ,„., ^ = (r+ ^, /fc would be negative, or the material would expand under pressure. If o- were< — 1, fi would be negative, and the function W would not be a positive quadratic function. We may show that this would also be the case if k were negative*. Negative values for a- are not excluded by the condition of stability, but such values have not been found for any isotropic material. (c) The constant k is usually determined by experiments on .compression, the constant E sometimes directly by experiments on stretching, and sometimes by experi- ments on bending, the constant ^ usually by experiments on torsion. The value of the constant a- is usually inferred from a knowledge of two among the quantities E, k, jtit. (rf) If Cauchy's relations (13) of Article 66 are true, X-=fi and 0-= J. (e) Instead of assuming the form of the strain-energy-function, we might assume some of the relations between stress-Components and strain-components and deduce the relations (18). For example J we may assume (i) that the mean tension and the cubical dilatation are connected by the equation ^(Xx+ Vy+Zi,) = i:A, (ii) that the relation A"„/ = fiex'y' holds for all pairs of rectangular axes of .t/ and y'. From the second assumption we should find, by taking the axes of x, y, z to be the principal axes of strain, that the principal planes of stress are at right angles to these axes. With the same choice of axes we should then find, by means of the formulae of transformation of Articles 12 and 49, that the relation Xx hh + Fy iriinii + Z^n^n^ = /j. {2exxhh + ^e^ym^m,^ + 'ie^jHin^) holds for all values oili, ... which satisfy the equation I1I2 + rriym^ + n^ti^ = 0. It follows that we must have Xx - ^iiCxx =Yv- Sjueji, = Z^- 2/:ie„. Then the first assumption shows that each of these quantities is equal to (k — |^) A. The relations (18) are thus found to hold for principal axes of strain, and, by a fresh application of the formulae of transformation, we may prove that they hold for any axes. (/) Instead of making the assumptions jxist described we might assume that the principal planes of stress are at right angles to the principal axes of strain and that the relations (22) hold for principal axes, and we might deduce the relations (18) for any axes. The working out of this assumption may serve as an exercise for the student. [g) We may show that, in the problem of the compression of a body by pressure uniform over its surface which was associated with the definition of ^, the displacement is expressed by the equations § 11, _v w p X y z~ Zk' (h) We may show that, in the problem of the bar stretched by simple tension T which was associated with the definitions of E and a-, the displacement is expressed by the equations v_w_ a-T_ \r u T_ (\ + iJ.)T y~ z~ E^ 2/i(3X-l-2^)' x~ E~ t).{Z\ + '2.iJ.)' * 2 \V may be written 0^ + llJ-) {exx + em + O^ + 1^ { («ct - « J^ + i^zz - ^xxV + Kx - ''yyf'} + A (V + <^J + e^^) ■ t Experiments for the direct determination of Poisson's ratio have been made by P. Cardani, Phys. Zeitschr., Bd. 4, 1903, and J. Morrow, Phil. Mag. (Ser. 6), vol. 6 (1903). M. A. Comu, Paris, G. R., t. 69 (1869), and A. Mallock, Proc. Boy. Soc, vol. 29 (1879), determined o- by experiments on bending. t This is the method of Stokes. See Introduction, footnote 37. § A displacement which would be possible in a rigid body may be" superposed on that given in the text. A like remark applies to the Observation (h). Of. Article 18, supra. 70-72] EXPRESSED BY TWO CONSTANTS 103 71 . Magnitude of elastic constants and moduluses of some isotropic solids. To give an idea of the order of magnitude of the elastic constants and moduluses of some of the materials in everyday use a few of the results of experiments are tabulated here. The table gives the density (p) of the material as well as the elastic constants, the constants being expressed as multiples of an unit stress of one dyne per square centimetre. Poisson's ratio is also given. The results marked "E" are taken from J. D. Everett's Illus- trations of the G.G.S. system of units, London, 1891, where the authorities for them will be found. Those marked "A" are reduced from results of more recent researches recorded in a paper by Amagat in the Journal de Physique (S6r. 2), t. 8 (1889). It must be understood that considerable differences are found in the elastic constants of different samples of nominally the same substance, and that such a designation as "steel," for example, is far from being precise. Material P E k /^ tT Eeference Steel 7-849 2-139 X 1012 1-841x1012 8-19x10" -310 E 51 2-041 X 1012 1-43 X10I2 -268 A Iron (wrought) 7-677 1-963x1012 1-456x1012 7-69 X 10" ■275 E Brass (drawn) 8-471 1-075x1012 3-66 X 10" E Brass 1-085x1012 1-05 X1012 -327 A Copper 8-843 1-234x1012 1-684x1012 4-47 X 10" -378 E j> 1-215x1012 1-166 X 1012 -327 A Lead 1-57 xlO" 3-62 XlO" •428 A Glass 2-942 6-03 xlO" 4-15 xlO" 2-40 X 10" ■258 E J5 6-77 XlO" 4-54 XlO" -245 A 72. Elastic constants in general. Materials such as natural crystals or wood which are not isotropic are said to be ceolotropic. The analytical expression of Hooke's Law in an seolotropic solid body is effected by the equations (10) of Article 66. In matrix notation we may write the equations ^yyj ^zz> ^yz) ^zxt ^ay)> {X^,Yy,Z„Y„Z^,Xy) = { Cn C31 C41 C61 C]2 C22 C32 C42 C52 Cl3 C33 C43 C53 C24 C34 C44 C54 ^64 Cl5 C36 C45 C16 ) \^x C26 C36 C46 ^66 .(25) where c,., = Cgr, (r, s = 1, 2, . . . 6). 104 ELASTICITY OF SOLIDS [CH. HI These equations may be solved, so as to express the strain-components in terms of the stress-components. If 11 denotes the determinant of the quantities c„, and Grs denotes the minor determinant that corresponds with Cm, so that n =CnGn + CnGn + CnG„ + GnCrt + Cr^Grs + CrsGre, (26) the equations that give the strain-components in terms of the stress-compo- nents can be written 11 \6xXi ^yyt ^zzi ^yzt ^zzj ^xy) = ( C7i] C/12 C7i3 C]4 0]5 G18 ) {^X! ^y> ■^Z) -'z; ^Xl -^y)) C (27) where G^ = G^^ (r, s = 1, 2, . . . 6). The quantities ^Cu, . . . Cjs, ... are the coefficients of a homogeneous quadratic function of e^x, ■■■■ This function is the strain-energy-function expressed in terms of strain-components. The quantities ^(7i,/n, ... Cia/II, ... are the coefficients of a homogeneous quadratic function of X^,.... This function is the strain-energy-function expressed in terms of stress-components. 73. Moduluses of elasticity. We may in various ways define types of stress and types of strain. For example, simple tension [XJ, shearing stress [F^], mean tension [J (Xx + Yy + Zz)\ are types of stress. The corresponding types of strain are simple extension [e^x], shearing strain [e^j] , cubical dilatation [e^x + eyy -y e^^. We may express the strain of any of these types that accompanies a stress of the corresponding type, when there is no other stress, by an equation of the form stress = M X (corresponding strain). Then M is called a "modulus of elasticity." The quantities n/C,i, H/G^ are examples of such moduluses. The modulus that corresponds with simple tension is known as Young's modulus for the direction of the related tension. The modulus that corre- sponds with shearing stress on a pair of orthogonal planes is known as the rigidity for the related pair of directions (the normals to the planes). The modulus that corresponds with mean tension or pressure is known as the modulus of compression. We shall give some examples of the calculation of moduluses. {a) Modulus of compression. We have to assume that Xx=Yy = Z^, and the remaining stress-components vanish; the corresponding strain is cubical dilatation, and we must therefore calculate exx-\-eyy + izz- We find for the modulus the expression n/(6'„ + C22-l-C33-^2<723 + 2C3,-l-2Ci2) (28) 72, 73] WHICH ARE NOT ISOTROPIC 105 As in Article 68, we see that the cubical compression produced in a body of any form by the application of uniform normal pressure, p, to its surface is p/zt, where h now denotes the above expression (28). (i) Rigidity. AVe may suppose that all the stress-components vanish except Y^, and then we have nfiy,= CnY^, so that II/C44 is tbe rigidity corresponding with the pair of directions y, z. If the shearing stress is related to the two orthogonal directions {I, m, n) and {I', m', n'), the rigidity can be shown to be expressed by nH-(Cii, C22, ... C12, ...){W, 2min', 2nn', mn' + m\ nl' + n'l, lm' + l'm)% ...(29) where the denominator is a complete quadratic function of the six arguments 2U', ... with coefficients Cn, C22, (c) Young's modulus and Poisson's ratio. We may suppose that all the stress-components vanish except X^, and then we have nej3,= CiiXc, so that n/Cn is the Young's modulus corresponding with the direction x. In the same case the Poisson's ratio of the contraction in the direction of the axis of y to the extension in the direction of the axis of ic is - Cia/Cn . The value of Poisson's ratio depends on the direction of the contracted transverse linear elements as well as on that of the extended longitudinal ones. In the general case we may take the stress to be tension X'x' across the planes 3/ = const., of which the normal is in the direction {I, m, n). Then we have X, = l^X'x,, Yy=m?X'x,, 4,=»2ZV, Y^ = mnX'x', Zx=nlX'x', Xy = lmX'x', and we have also Bxv = exxl^ + Byyin^ + e^n^ -t- By^in + e^^nl + exylm ; it follows that the Young's modulus E corresponding with this direction is n-^(Cii, (722, — C'i2, ...) (l\ m?, n% mn, nl, Imf, (30) where the denominator is a complete quadratic function of the six arguments 1^, ... with coefficients Cii, If {V, m', n') is any direction at right angles to of, the contraction, -ey,y,, in this direction is given by the equation %'!/' = Bxxl'^ + eyym'^ + fisX^ + /'.ygm'n' + e^n'l' + Sxyl'm', and the corresponding Poisson's ratio o- is expressible in the form "-"a-^L 3(^2) a(m2)+" 9(Mi')+"*'*8(m«) + '''9(«Z) + ""3(im)J' -^^^1 where ^ is the above-mentioned quadratic function of the arguments l^,..., and the differential coefficients are formed as if these arguments were independent. It may be observed that a-jE is related symmetrically to the two directions in which the corresponding contraction and extension occur. If we construct the surface of the fourth order of which the equation is (Cii, C22, ... C12, —){x\ y\ z\ yz, zx, xyY=cowt., (32) then the radius vector of this surface in any direction is proportional to the positive fourth root of the Young's modulus of the material corresponding with that direction*. * The result is due to Cauohy, Exercices de mathgmatique, t. 4 (1829), p. 30. 106 STRESS DUE TO VARIATION OF TEMPERATURE [CH. Ill 74. Thermo-elastic equations. The application of the two fundamental laws of Thermodynamics to the problem of determining the stress and strain in elastic solid bodies when variations of temperature occur has been discussed by Lord Kelvin* The results at which he arrived do not permit of the formulation of a system of differential equations to determine the state of stress in the body in the manner explained in Article 67. At an earlier date Duhamelf had obtained a set of equations of the required kind by developing the theory of an elastic solid regarded as a system of material points, and F. E. Neumann, starting from certain assump- tions;]:, had arrived at the same system of equations. These assumptions may, when the body is isotropic, be expressed in the following form : — The stress-system at any point of a body strained by variation of temperature consists of two superposed stress- systems. One of these is equivalent to uniform pressure, the same in all directions round a point, and proportional to the change of temperature ; the other depends upon the strain at the point in the same way as it would do if the temperature were constant. These assumptions lead to equations of the form dX^ dXy dZ^ d9 where /3 is a constant coefficient and 6 is the excess of temperature above that in the unstrained state. The stress-system at any point has com- ponents Y Z X ' ^^^^ in which A'^., ... are expressed in terms of displacements by the formulae (18) of Article 69. The equations are adequate to determine the displacements when 6 is given. When 6 is not given an additional equation is required, and this equation may be deduced from the theory of conduction of heat, as was done by Duhamel and Neumann. The theory thus arrived at has not been very much developed. Attention has been directed especially to the fact that a plate of glass strained by unequal heating becomes doubly refracting, and to the explanation of this effect by the inequality of the stresses in different directions. The reader who wishes to pursue the subject is referred to the following memoirs in addition to those already cited: — C. W. Borchardt, Berlin MonatsbericMe, 1873; J. Hopkinson, Messenger of Math. vol. 8 (1879); Lord Rayleigh, Phil. Mag. * See Introduction, footnote 43. t Paris, Mi>m....par divers savam, t. 5 (1838). t See his Vorlesungen Uber die Theorie der Elasticitat der festm KOrper, Leipzig, 1885, and cf. the memoir by Maxwell cited in Article 57 footnote. 74, 75] INITIAL STRESS 107 (Ser. 6) vol. 1 (1901), or Scientific Papers, vol. 4, p. 502; E. Almansi, Torino AUi, t. 32 (1897); P. Alibrandi, Qiornale di matem. t. 38 (1900). It must be observed that the elastic "constants" themselves are functions of the temperature. In general, they are diminished by a rise of temperature ; this result has been established by the experiments of Wertheim*, Kohlrauschf and Macleod and Clarke^:. 75. Initial stress. The initial state of a body may be too far removed from the unstressed state to permit of the stress and strain being calculated by the principle of superposition as explained in Article 64. Such initial states may be induced by processes of preparation, or of manufacture, or by the action of body forces. In cast iron the exterior parts cool more rapidly than the interior, and the unequal contractions that accompany the unequally rapid rates of cooling give rise to considerable initial stress in the iron when cold. If a sheet of metal is rolled up into a cylinder and the edges welded together the body so formed is in a state of initial stress, and the unstressed state cannot be attained without cutting the cylinder open. A body in equilibrium under the mutual gravitation of its parts is in a state of stress, and when the body is large the stress may be enormous. The Earth is an example of a body which must be regarded as being in a state of initial stress, for the stress that must exist in the interior is much too great to permit of the calculation, by the ordinary methods, of strains reckoned from the unstressed state as un- strained state. If a body is given in a state of initial stress, and is subjected to forces, changes of volume and shape will be produced which can be specified by a displacement reckoned from the given initial state as unstrained state. We may specify the initial stress at a point by the components and we may specify the stress at the point when the forces are in action by X^fi +Xx', .... In like manner we may specify the density in the initial state by /Jo and that in the strained state by p^ + p, and we may specify the body force in the initial state by (Xo, Y^, Z^ and that in the strained state by (Zj+X', Yo + Y', Zo+Z'). Then the conditions of equilibrium in the initial state are three equations of the type aZ,t«) 8X/) 8Z.i») , . -dx +-ar+-a7-+''°^«-^ ^^^^ * Ann. de Chimie, t. 12 (1844). t Ann. Phys. Chem. {Poggendorff), Bd. 141 (1870). J A result obtained by these writers is explained in the sense stated in the text by Lord Kelvin in the Article 'Elasticity' in Ency. Brit, quoted in the footnote to Article 65. 108 INITIAL STRESS [CH. Ill and three boundary conditions of the type X^w cos («, vo) +Zj,'°' cos (y, V,) +^^'»i cos(z,v,) = (36) in which i^o denotes the direction of the normal to the initial boundary. The conditions of equilibrium in the strained state are three equations of the form + ~iZ,m + zj) + {p, + p'){X, + X') = (37) and three boundary conditions of the type (Z^i»t +XJ) cos (*■, v) + (Xy'»' + Xy') cos (y, v) + (Z^^'>^+Z^')co8iz,v)=X,, (38) in which {X^, Yy, Z^) is the surface traction at any point of the displaced boundary. These equations may be transformed, when the displacement is small, by using the results (35) and (36), so as to become three equations of the type ■p>x ' TiX ' ?Z ' yv ^ + ^ + ^+ X' + p'X„ = (39 dx dy dz and three boundary conditions of the type Xx cos {x, v) +Xy' cos {y, v) +ZJ cos {z, v) = X^ — Xa;™ {cos (iC, v) — cos {x, Vn)] -Xy''"'* [cos {y, v) - cos (2/, z^o)} — Za;"" (cos (^;, I')— COS (2:, Vo)]. If the initial stress is not known the equations (35) and conditions (36) are not sufficient to determine it, and no progress can be made. If the initial stress is known the determination of the additional stress (X^;', ...) cannot be effected by means of equation (39) and conditions (40), without knowledge of the relations between these stress-components and the displace- ment. To obtain such knowledge recourse must be had either to experiment or to some more general theory. Experimental evidence appears to be entirely wanting*. Cauchyf worked out the consequences of applying that theory of material points to which reference has been made in Article 66. He found for Xx, ■ • • expressions of the form .(40) \ ...(41) * Reference may be made to a paper by F. H. Cilley, Am^. J. of Science (SilUman), (Ser. 4), vol. 11 (1901). t See Introduction and cf . Note B at the end of this book. 75] INITIAL STRESS 109 where (u, v, w) is the displacement reckoned from the initial state, and (Xx", . . ■) is a stress-system related to this displacement by the same equations as would hold if there were no initial stress. In the case of isotropy these equations would be (IS) of Article 69 with X put equal to /j.. It may be observed that the terms of X^', ■ - ■ that contain Xa,™, . . . arise from the changes in the distances between Cauchy's material points, and from changes in the directions of the lines joining them in pairs, and these changes are expressed by means of the displacement (m, v, w). Saint- Venant* has obtained Cauchy's result by adapting the method of Green, that is to say by the use of the energy-function. His deduction has been criticized by K. Pearsonf, and it cannot be accepted as valid. Green's original discussion | appears to be restricted to the case of uniform initial stress in an unlimited elastic medium, and the same restriction characterizes Lord Kelvin's discussion of Green's theory§. ♦ J. de Math. (Liouville), (S6r. 2), t. 8 (1863). + Todhunter and Pearson's History, vol. 2, pp. 84, 85. J See the paper quoted in the Introduction, footnote 81. § Baltimore Lectures on Molecular Dynamics and the Wave Theory of Light, London, 1904, pp. 228 et seq. CHAPTER IV. THE RELATION BETWEEN THE MATHEMATICAL THEORY OF ELASTICITY AND TECHNICAL MECHANICS. 76. Limitations of the mathematical theory. The object of this Chapter is to present as clear an idea as possible of the scope and limitations of the mathematical theory in its application to practical questions. The theory is worked out for bodies strained gradually at a constant temperature, from an initial state of no stress to a final state which differs so little from the unstressed state that squares and products of the displacements can be neglected ; and further it is worked out on the basis of Hooke's Law, as generalized in the statements in Article 64. It is known that many materials used in engineering structures, e.g. cast iron, building stone, cement, do not obey Hooke's Law for any strains that are large enough to be observed. It is known also that those materials which do obey the law for small measurable strains do not obey it for larger ones. The statement of the law in Article 64 included the statement that the strain disappears on removal of the load, and this part of it is absolutely necessary to the mathe- matical theory ; but it is known that the limits of strain, or of load, in which this condition holds good are relatively narrow. Although there exists much experimental knowledge* in regard to the behaviour of bodies which are not in the conditions to which the mathematical theory is applicable, yet it appears that the appropriate extensions of the theory which would be needed in order to incorporate such knowledge within it cannot be made until much fuller experimental knowledge has been obtained. * Information in regard to experimental results will be found in treatises on Applied Mechanics. The following may be mentioned : — W. J. M. Eankine, Applied Mechanics, 1st edition, London, 1858, (there have been numerous later editions); W. C. Unwin, The testing of materials of cojjstruction, London, 1888 ; J. A. Ewing, The Strength of Materials, Cambridge, 1899 ; Flamant, Stabilite des constructioTis, Resistance des materiaux, Paris, 1896; C. Bach, Elasticitdt und Festigkeit, 2nd edition, Berlin, 1894; A. Foppl, Vorlesungen iiber technische Mechanik, Bd. 3, Fe> 2660 2960 Steel (Bessemer) 1780 2650 79. Time-eflFects. Plasticity. The length of time that a body has been subjected to considerable load generally affects the strain produced, and the length of time that a strained body has been free from load generally affects the extent of the elastic recovery. The latter effect was discovered by W. Weberf in 1835 and has been called Elastische Nachwirkung or elastic after-working ; the former appears to have been first noted by Vicatj in 1834. When a body has been strained by a load surpassing the limit of perfect elasticity, and is set free, the set gradually diminishes. The body never returns to its primitive condition, and the ultimate deformation is the "permanent set," the part of the strain that gradually disappears is called "elastic after-strain." To produce the effect noted by Vicat very considerable stress is generally required. He found that wires held stretched, with a tension equal to one quarter of the breaking stress, retained the length to which this tension brought them throughout the whole time of his experiments (33 months), while similar wires stretched with a tension equal to half the breaking stress exhibited a notable gradual increase of extension. The gradual flow of solids under great stress, indicated by these experiments, has been made the subject of exhaustive investigation by H. Tresca§. He found, in his experiments on the punching and crushing of metals, results which point to the conclusion that all solids when subjected to very great pressure ultimately flow, i.e. take a set which increases with the time. This capacity of solids to flow under * Extracted from results given by Bauschinger, Mittheilungen, xiii. We may take 1000 atmospheres =6'56 tons per square inch = 1-0136 y 10" c.G.s. units of stress. t De fill Bcrmbycini vi Elastica. Gottingen, 1841. An off-print of a paper communicated to the Konigliche Gegellschaft der Wigsenschaften zu Gottingen, 1835, and practically translated in Ann. Phijs. Cham. (Poggendorff), Bde. 84 (1835) and 54 (1841). J \ote sur I'allongement progressif du fil de fer soumis a diverges tengiong. Annaleg del pontg et chaussges, ler semegtre, 1834. § Paris, M4m....par diverg savants, tt. 18 (1868), and 20 (1872). An account of some of Tresca's experiments is given by Unwin, loc. cit., pp. 46 et seq. 78-80] VISCOSITY OF SOLIDS 115 great stress is called plasticity. A solid is said to be " hard " when the force required to produce considerable set is great, " soft " or " plastic " when it is small. A substance must be termed " fluid " if considerable set can be pro- duced by any force, however small, provided it is applied for a sufficient time. In experiments on extension some plasticity of the material is shown as soon as the limit of linear elasticity is exceeded * If the load exceediag this limit is removed some set can be observed, but this set diminishes at a rate which itself diminishes. If the load is maintained the strain gradually increases and reaches a constant value after the lapse of some time. If the load is removed and reapplied several times, both the set and the elastic strain increase. None of these effecte is observed when the load is below the limit of linear elasticity. The possibility of these plastic effects tends to complicate the results of testing, for if two like specimens are loaded at different rates, the one which is loaded more rapidly will show a greater breaking stress and a smaller ultimate extension than the other. Such dif- ferences have in fact been observedf, but it has been shownj that under ordinary conditions of testing the variations in the rate of loading do not affect the results appreciably. 79 A. Momentary stress. A time-effect of the opposite kind has been observed in connexion with impact. It appears that very great stress acting for a very short time, of the order one-thousandth of a second, or less, does not necessarily cause any dangerous set, or fracture, when the stress exceeds that answering to statical limits of elasticity, or even when it exceeds the statical yield-point. Such stresses may, without sensible error, be calculated by the Mathematical Theory of Elasticity, as if there were no such things as limits of elasticity. This result has been established by B. Hopkinson and J. E. Sears §. 80. Viscosity of solids. "Viscosity" is a general term for all those properties of matter in virtue of which the resistance, which a body offers to any change, depends upon the rate at which the change is effected. The existence of viscous resistances involves a'dissipation of the energy of the substance, the kinetic energy of molar motion being transformed, as is generally supposed, into kinetic energy of molecular agitation. The most marked effect of this property, if it exists in the case of elastic solids, would be the subsidence of vibrations set up in the solid. Suppose a solid of any form to be struck, or otherwise suddenly disturbed. It will be thrown into more or less rapid vibration, and the stresses * Bausohinger, Mittheilungen, xni. (1886). t Cf. TJnwin, loc. cit., p. 89. J Bausohinger, Mittheilungen, xx. (1891). § B. Hopkinson, London, Proc. R. Sac, vol. 74, 1905, p. 498, and London, Phil. Trans. R. Soc. (Ser. A), vol. 213, 1914, p. 437 ; J. B. Sears, Cambridge, Proc. Phil. Soc, vol. 14, 1908, p. 257, and Cambridge, Trans. Phil. Soc, vol. 21, 1912, p. 49. 8—2 116 CHANGES OF QUALITY [CH. IV developed in it would, if there is genuine viscosity, depend partly on the dis- placements, and partly on the rates at which they are effected. The parts of the stresses depending on the rates of change would be viscous resistances, and they would ultimately destroy the vibratory motion. Now the vibratory motion of elastic solid bodies is actually destroyed, but the decay appears not to be the effect of viscous resistances of the ordinary type, that is to say such as are proportional to the rates of strain. It has been pointed out by Lord Kelvin * that, if this type of resistance alone were involved, the proportionate diminution of the amplitude of the oscillations per unit of time would be inversely proportional to the square of the period; but a series of experiments on the torsional oscillations of wires showed that this law does not hold good. Lord Kelvin pointed out that the decay of vibrations could be accounted for by supposing that, even for the very small strains involved in vibratory motions, the effects of elastic after-working and plasticity are not wholly absent. These effects, as well as viscous resistances of the ordinary type, are included in the class of hysteresis phenomena. All of them show that the state of the body concerned depends at any instant on its previous states as well as on the external conditions (forces, temperature, &c.) which obtain at the instant. Hysteresis always implies irreversibility in the sequence of states through which a body passes, and is generally traced to the molecular structure of matter. Accordingly, theories of molecular action have been devised by various investigators f to account for viscosity and elastic ^fter- working. 81. .ffiolotropy induced by permanent set. One of the changes produced in a solid, which has received a permanent set, may be that the material, previously isotropic, becomes aeolotropic. The best known example is that of a bar rendered seolotropic by permanent torsion. Warburgj found that, in a copper wire to which a permanent twist had been given, the elastic phenomena observed could all be explained on the supposition that the substance of the wire was rendered seolotropic like a rhombic crystal. When a weight was hung on the wire it produced, in addition to extension, a small shear, equivalent to a partial untjyistingg of the wire; this was an elastic strain, and disappeared on the removal of the * Sir W. Thomson, Article ' Elasticity,' Ency. Brit, or Math, and Phys. Papers, vol. d, Cambridge, 1890, p. 27. t The following may be mentioned :— J. C. Maxwell, Article ' Constitution of Bodies,' Ency. Brit, or Scientific Papers, vol. 2, Cambridge, 1890; J. G. Butcher, London Math. Soc. Proc, vol. 8 (1877) ; 0. E. Meyer, J.f. Math. {Crelle),BdL. 78 (1874) ; L. Boltzmann, Ann. Phys. Ghem. (Poggendorff), Ergzgsbd. 7 (1878). For a good account of the theories the reader may be referred to the Article byF. Braun in Winkelmann's ffandimcft der PAj/siA;, Bd. 1 (Breslau, 1891), pp. 321 — 342. For a more recent discussion of the viscosity of metals and crystals, see W. Voigt, Ann. Phys. Ghem. (Wiedemann), Bd. 47 (1892). t Ann. Phys. Ghem. (Wiedemann), Bd. 10 (1880). § Cf. Lord Kelvin, loc. cit.. Math, and Phys. Papers, vol. 3, p. 82. 80-82 aJ produced by overstrain 117 load. This experiment is important as showing that processes of manufac- ture may induce considerable seolotropy in materials which in the unworked stage are isotropic, and consequently that estimates of strength, founded on the employment of the equations of isotropic elasticity, cannot be strictly interpreted *- 82. Repeated loading. A body strained within its elastic limits may be strained again and again without receiving any injury; thus a watch-spring may be coiled and uncoiled one hundred and twenty millions of times a year for several years without deterioration. But it is different when a body is strained repeatedly by rapidly varying loads which exceed the Limits of elasticity. Wohler'sf ex- periments on this point have been held to show that the resistance of a body to any kind of deformation can be seriously diminished, by rapidly repeated applications of a load. The result appears to point to a gradual deterioration;]: of the quality of the material subjected to repeated loading, which can be verified by the observation that after a large number of applications and removals of the load, bars may be broken by a stress much below the statical breaking stress. Bauschinger§ made several independent series of experiments on the same subject. In these the load was reversed 100 times a minute, and the specimens which endured so long were submitted to some millions of repe- titions of alternating stress. In some cases these severe tests revealed the existence of flaws in the material, but the general result obtained was that the strength of a piece is not diminished by repeated loading, provided that the load always lies within the limits of linear elasticity. An analogous property of bodies is that to which Lord Kelvin || has called attention under the name "fatigue of elasticity." He observed that the torsional vibrations of wires subsided much more rapidly when the wires had been kept vibrating for several hours or days, than when, after being at rest for some days, they were set in vibration and immediately left to themselves. Experimental results of this kind point to the importance of taking into account the manner and frequency of the application of force to a structure in estimating its strength. 82 A. Elastic Hysteresis. It has been known for a long time that there may be a defect of Hooke's Law (Article 64) even within the limits of perfect elasticity. For example, in * Cf. Unwin, loc. cit., p. 25. f Ueber Festigkeitsversuche mit Eisen und StaM, Berlin, 1870. An acconnt of Wohler's experi- ments is given by tlnwin, loc. cit., pp. 356 et seq. X A different explanation has been proposed by K. Pearson, Messenger of Math. vol. 20 (1890). § Mittheilungen, xx. (1891) and xxv. (1897) edited by Fdppl. 11 Loc. eit.. Math, and Phys. Papers, vol. 3, p. 22. 118 ELASTIC [CH. IV tensile tests, a specimen which has been loaded gradually, and then unloaded gradually, may show no measurable subsequent extension, that is to say, it may be perfectly elastic within the limits of the applied load; and, never- theless, the extension under a given load, during the process of unloading, may be appreciably different from the extension, under the same load, during the process of loading. Thus the limits of sensibly linear elasticity may be narrower than those of sensibly perfect elasticity. When this is the case it is found that the strain during unloading is greater, not less, than that, at the same load, during loading. The earliest account of this phenomenon, which I have found, is in a paper by Ewing*, describing experiments made on steel wire. They showed that the effect in question, described by the author as "hysteresis," is more marked when the loading and unloading are rapid than when they are slow, but it is sensible when they are performed very slowly, so that there appears to be a true statical hysteresis. It appeared later that with very rapid alternations of stress the observable amount of hysteresis is less than the statical amountf. For hard metals it seems that there is no hysteresis with moderate load, but for rocks, such as granite and marble J, there is hysteresis at quite moderate loads. The effect in metals is especially important in torsion tests§. The general nature of the effect may be described in the words — The stress-strain diagram is a closed curve. It appears to follow that some energy is dissipated in putting a specimen through a cycle of stress-changes, and Ewing (loc. cit.) calls attention to the bearing of this con- clusion upon Wohler's experiments on repeated loading and alternating stress. The subject, however, is still rather obscure ||. It is perhaps a little unfortunate that the term "elastic hysteresis'' should have been appropriated to describe the special phenomenon noticed by Ewing, because, as has been already observed, elastic after-working, plasticity, and viscosity are also properties which indicate hysteresis, in the general sense of the word, and the same may be said of the property called fatigue of elasticity, and of those brought to light in experiments on repeated loading and alter- nating stress. They all imply a dependence of the instantaneous state of a Dody upon its previous states as well as upon the instantaneous conditions. A rather promising beginning of a mathematical theory of elastic after- working, possibly applicable also to other phenomena involving hysteresis, in the general sense of the word, has been made by Volterra, starting with the physical theory developed by Boltzmann, cited in Article 80. He describes * J. A. Ewing, Brit. Assoc. Rep., 1889, p. 502. + B. Hopkinson and G. T. Williams, London, Proc. E. Soc. (Ser. A.), vol. 87, 1912, p. 502. J F. D. Adams and E. G. Coker, An investigation into the elastic constants of rocks, more especially with reference to cubic compressibility (Washington, 1906). § Cf. Searle, Experimental Elasticity, p. 152. II Cf. the ' Report on Alternating Stress ' by W. Mason in Brit. Assoc. Rep., 1913, p. 183, and the Eeport by the same author ' On the Hysteresis of Steel under repeated Torsion ' in Brit. Assoc. Rep., 1916, p. 285. 82 a, 83] HYSTEBESIS 119 the physical circumstances by the epithet "ereditario" (hereditary) and shows that the theory leads to equations of a type, which he names "integro-dif- ferential," and by aid of the theory of integral equations he has obtained some special solutions of. his integro-differential equations. We shall not pursue the matter, but refer the reader to the papers cited below*. 83. Hypotheses concerning the conditions of rupture. Various hypotheses have been advanced as to the conditions under which a body is ruptured, or a structure becomes unsafe. Thus Lamef supposed it to be necessary that the greatest tension should be less than a certain limit. Ponceletj, followed by Saint-Venant§, assumed that the greatest extension must be less than a certain limit. These measures of tendency to rupture agree for a bar under extension, but in general they lead to different limits of safe loading 1 1. Again, Tresca followed by G. H. Darwin IT makes the maxi- mum difference of the greatest and least principal stresses the measure of tendency to rupture, and not a very different limit would be found by follow- ing Coulomb's** suggestion, that the greatest shear produced in the material IS a measure of this tendency. An interesting modification of this view has been suggested and worked out geometrically by 0. Mohrft- I* would enable us to take account of the possible dependence of the condition of safety upon the nature of the load, i.e. upon the kind of stress which is developed within the body. The manner and frequency of application of the load are matters which ought also to be taken into account. The conditions of rupture are but vaguely understood, and may depend largely on these and other acci- dental circumstances. At the same time the question is very important, as a satisfactory answer to it might suggest in many cases causes of weakness previously unsuspected, and, in others, methods of economizing material that would be consistent with safety. * V. Volterra, JSomo, Ace. Line. Rend. (Ser. 5), t. 18 (2), 1909, pp. 295 and 577, t. 19 (1), 1910, pp. 107 and 239, t. 21 (2), 1912, p. 3, t. 22 (1), 1913, p. 529; G. Laurieella, Roma, Ace. Line. Rend. (Ser. 5), t. 21 (1), p. 165. t See e.g. the memoir of Lam4 and Clapeyron, quoted in the Introduction (footnote 39). X See Todhunter and Pearson's History, vol. 1, art. 995. § See especially the Historique Abrege in Saint- Venant's edition of the Lemons de Navier, pp. oxeix — ccv. II For examples see Todhunter and Pearson's History, vol. 1, p. 550 footnote. IT ' On the stresses produced in the interior of the Earth by the weight of Continents and Mountains,' Phil. Trans. Roy. Sac, vol. 173 (1882). The same measure is adopted in the account of Prof. Darwin's work in Kelvin and Tait's Nat. Phil. Part ii. art 882'. ** ' Essai sur une application des regies de Maximia &c.,' Mem,... par divers savans, 1776, Introduction. tt Zeitschr. des Vereines Deutscher Ingenieure, Bd. 44 (1900). A discussion by Voigt of the views of Mohr and other writers will be found in Ann. Phys. (Ser. 4), Bd. 4 (1901). The subject is dis- cussed further in the light of new experimental researches by Th. v. K4rmAn, ' Pestigkeitsversuche unter allseitigem Druok,' Zeitschr. d. Vereines Deutscher Ingenieure, 1911, also by W. A. Scoble, ' Report on Combined Stress.' Brit. Assoc. Rep., 1913. 120 HYPOTHESES CONCERNING THE [CH. IV In all these hypotheses it is supposed that the stress or strain actually produced in a body of given form, by a given load, is somehow calculable. The only known method of calculating these effects is by the use of the mathematical theory of Elasticity, or by some more or less rough and ready rule obtained from some result of this theory. Suppose the body to be subject to a given system of load, and suppose that we know how to solve the equations of elastic equilibrium with the given boundary-conditions. Then the stress and strain at every point of the body can be determined, and the principal stresses and principal extensions can be found. Let 7" be the greatest principal tension, S the greatest difiference of two principal tensions at the same point, e the greatest principal extension. Let To be the breaking stress as determined by tensile tests. On the greatest tension hypothesis T must not exceed a certain fraction of T^. On the stress-difference hypothesis S must not exceed a certain fraction of To- On the greatest extension hypo- thesis e must not exceed a certain fraction of T„/E, where E is Young's modulus for the material. These conditions may be written T which occurs in them is called the "factor of safety." Most English and American engineers adopt the first of these hypotheses, but take to depend on the kind of strain to which the body is likely to be subjected in use. A factor 6 is allowed for boilers, 10 for pillars, 6 for axles, 6 to 10 for railway-bridges, and 12 for screw-propeller-shafts and parts of other machines subjected to sudden reversals of load. In France and Germany the greatest extension hypothesis is often adopted. Recently attempts have been made to determine which of these hypo- theses best represents the results of experiments. The fact that short pillars can be crushed by longitudinal pressure excludes the greatest tension hypo- thesis. If it were proposed to replace this by a greatest stress hypothesis, according to which rupture would occur when any principal stress (tension or pressure) exceeds a certain limit, then the experiments of A. Foppl* on bodies subjected to very great pressures uniform over their surfaces would be very important, as it appeared that rupture is not produced by such pressures as he could apply. These experiments would also forbid us to replace the greatest extension hypothesis by a greatest strain hypothesis. There remain for examination the greatest extension hypothesis and the stress-difference hypothesis. Wehage's experimentsf on specimens of wrought iron subjected to equal tensions (or pressures) in two directions at right angles to each other have thrown doubt on the greatest extension hypothesis. From experiments on metal tubes subjected to various systems of combined stress J. J. GuestJ * Mittheilungen (Munchen), xxvii. (1899). + Mitthciluvgen der medmnigch-technischen Versuchsanstalt zu Berlin, 1888. + Phil. Mag. (Ser. .5), vol. 48 (1900). Mohr {loc. cit.) has criticized Guest. 83, 84] COXDITIONS OP SAFETY 121 has concluded that the stress-difference hypothesis is the one which accords best with observed results. The general tendency of modern technical writings seems to be to attach more importance to the limits of linear elasticity and the yield-point than to the limits of perfect elasticity and the breaking stress, and to emphasize the importance of dynamical tests in addition to the usual statical tests of tensile and bending strength. 84. Scope of the nxathematical theory of elasticity. Numerical values of the quantities that can be involved in practical problems may serve to show the smallness of the strains that occur in struc- tures which are found to be safe. Examples of such values have been given in Articles 1, 48, 71, 78. A piece of iron or steel with a limit of linear elasticity equal to 10| tons per square inch, a yield-point equal to 14 tons per square inch and a Young's modulus equal to 13000 tons per square inch would take, under a load of 6 tons per square inch, an extension 0'00046. Even if loaded nearly up to the yield-point the extension would be small enough to require very refined means of observation. The neglect of squares and products of the strains in iron and steel structures within safe limits of loading cannot be the cause of any serious error. The fact that for loads much below the limit of linear elasticity the elasticity of metals is very iniperfect may perhaps be a more serious cause of error, since set and elastic after- working are unrepresented in the mathematical theory; but the sets that occur within the limit of linear elasticity are always extremely small. The effects produced by unequal heating, with which the theory cannot deal satis- factorily, are very important in practice. Some examples of the application of the theory to questions of strength may be cited here: — By Saint-Venant's theory of the torsion of prisms, it can be predicted that a shaft transmitting a couple by torsion is seriously weakened by the existence of a dent having a curvature approaching to that in a reentrant angle, or by the existence of a flaw parallel to the axis of the shaft. By the theory of equilibrium of a mass with a spherical boundary, it can be predicted that the shear in the neighbourhood of a flaw of spherical form may be as great as twice that at a distance. The result of such theories would be that the factor of safety should be doubled for shafts transmitting a couple when such flaws may occur. Again it can be shown that, in certain cases, a load suddenly applied may cause a strain twice* as great as that produced by a gradual application of the same load, and that a load suddenly reversed may cause a strain three times as great as that produced by the gradual application of the same load. These results lead us to expect that additional factors of safety will be required for sudden applications and sudden reversals, and they suggest that these extra factors may be 2 and 3. Again, a source of weakness in structures, * This point appears to have been first expressly noted by Ponoelet in his Introduction a la Mecanique indmtrielle, physique et experimentale of 1839, see Todhunter and Pearson's History, vol. 1, art. 988. 122 SCOPE OF THE MATHEMATICAL THEORY OF ELASTICITY [CH. IV some parts of which are very thin bars or plates subjected to thrust, is a possible buckling of the parts. The conditions of buckling can sometimes be determined from the theory of Elastic Stability, and this theory can then be made to suggest some method of supporting the parts by stays, and the best places for them, so as to secure the greatest strength with the least expendi- ture of materials; but the result, at any rate in structures that may receive small permanent sets, is only a suggestion and requires to be verified by experiment. Further, as has been pointed out before, all calculations of the strength of structures rest on some result or other deduced from the mathe- matical theory. More precise indications as to the behaviour of solid bodies can be deduced ' from the theory when applied to obtain corrections to very exact physical measurements*. For example, it is customary to specify the temperature at which standards of length are correct ; but it appears that the effects of such changes of atmospheric pressure as actually occur are not too small to have a practical significance. As more and more accurate instruments come to be devised for measuring lengths the time is probably not far distant when the effects produced in the length of a standard by different modes of support will have to be taken into account. Another example is afforded by the result that the cubic capacity of a vessel intended to contain liquid is in- creased when the liquid is put into it in consequence of the excess of pressure in the parts of the liquid near the bottom of the vessel. Again, the bending of the deflexion-bars of magnetometers affects the measurement of mag- netic force. Many of the simpler results of the mathematical theory are likely to find important applications in connexion with the improvement of measuring apparatus. * Cf. C. Chree, Phil. Mag. (Ser. 6), vol. 2 (1901). CHAPTER V. EQUILIBRIUM OF ISOTROPIC ELASTIC SOLID BODIES. 85. Recapitulation of the general theory. As a preliminary to the furtlier study of the theory of elasticity some parts of the general theory will here be recapitulated briefly. (a) Stress. The state of stress at a point of a body is determined when the traction across every plane through the point is known. The traction is estimated as a force per unit of area. If v denotes the direction of the normal to a plane the traction across the plane is specified by means of rectangular components X^, Yp, Z^ parallel to the axes of coordinates. The traction across the plane that is normal to v is expressed in terms of the tractions across planes that are normal to the axes of coordinates by the equations X,=Xx', v: are functions of x, y, z. (d) Strain. The strain at a point is determined when the extension of every linear element issuing from the point is known. If the relative displacement is small, the extension of a linear element in direction (l, to, n) is exxl' + ei/yin'^ + ei^n^ + ey^mn + eixnl + e^ylm, (4) "^ where e^: denote the following : — dw dv _ dti dw dw _dv 9m " dx dy,' .(5) Zw dv ^'^y~lz .(6) The quantities exx, ■■■ exy are the "components of strain." The quantities ■zsx, '^y, ■oi'z determined by the equations du dw dv du dx'' '"dx dy are the components of a vector quantity, the "rotation." The quantity A determined by the equation 9m 9;; dw ,_, ^=?:^+9-^ + & '(^) is the "dilatation." (e) Stress-strain relations. In an elastic solid slightly strained from the unstressed state the components of stress are linear functions of the components of strain. When the material is isotropic we have Y^=liey2, Zx=iiezx, ^v=l'^xy< > and by solving these we have .(8) ^XX — "cT { ^ I ■_ ?^o, 32-. dx ' 9z 9^ ' and therefore we must have where m„ and vo are functions of x and y. The equations du _dv _ a-gpz 3i~9y~~ E give 9Mb_. ?^_n S^_ The state of curvature expressed by i?i and B^ is maintained by couples applied to the edges. The couple per unit of length, applied to that edge X = const, for which m has the greater value, has its axis parallel to the axis of y, and its amount is r'' -rr , , • L • 2 Eh? (I a An equal and opposite couple must be applied to the opposite edge. The corresponding couple for the other pair of edges is given by J_^-.F,d.,which-iS3^-— ,^^ + -j^J. - I'V-^- The value of the strain-energy-function at any point can be shown without difficulty to be and the potential energy of the bent plate per unit of area is IS[(i-i)'-'--'>«]- It is noteworthy that this expression contains the sum and the product of the principal curvatures. 91. Equations of equilibriTom in terms of displacements. In the equations of type dx dy dz "^ we substitute for the normal stress-components Za,, ... such expressions as XA -f 2fidulda;, and for the tangential stress-components Y^,... such expressions as /j,{dw/dy + dv/dz) ; and we thus obtain three equations of the type {X + ,^)^^ + f^V'u + pX = 0, .'....(19) ^ du dv dw ^,_^,^.^ where A = ^ + ^^+-^-, ^ - 9^,. + g^. + a^^ • These equations may be written in a compact form ....(20) 9-2 (7\ 7) 7^ \ doo' dy' W ^ ~ ^ ^^^^ ^'^'" '^"' '^^' 132 SYSTEM OF EQUATIONS [CH. V If we introduce the rotation {vs^, tsy, OTj) = \ curl (if, V, w), _ 1 /9w 'bv dii dw dv du\ ~2[dy~dz' dz~dx' doo~dy)' and make use of the identity 1 1 dy ' dz. the above equations (20) take the form (\ + 2/x) (^ , ^ , I) A - 2/. curl (^,, t^„ tiT,) + p {X, Y,Z) = 0. .. .(21) We may note that the equations of small motion (Article 54) can be expressed in either of the forms or / 7\ 7i 7^ \ ^2 •••(22) The traction (X„, F„, Z„) across a plane of which the normal is in the direction v, is given by formulae of the type X, = cos {.r, v) \\^ +2fi^j + cos (y, ") /" (g^ + g^j + cos (^, v)M'[^ + fdu dw\ By J ' ^""^^'"^'"Ife + S^j' Xy = XA cos {x, v) + fj, \:^ + cos («, ") g- + cos (y, v) ^ + cos {z, v)-^ ■ , (23) or X„ = XAcos(a;, j/) + 2^-^^ •sj-y cos (^;, i^) + tn-^ cos (y, j/) [ , ...(24) , 9m , . 3m , , , 9m , , 9m where — = cos (x, v)^^ + cos (y, v)^ ■{- cos (^^, i/) ^r ■ dv ^ 'ox ^^ ' dy ^ 'dz If V is the normal to the bounding surface drawn outwards from the body, and the values of A, du/dx, ... are calculated at a point on the surface, the right-hand members of (23) and the similar expressions represent the com- ponent tractions per unit area exerted upon the body across the surface. 92. Equilibrium under surface tractions only. We record here some results deducible from the displacement-equations ^'^^^Kl' i- a-> + /^^^(«'-'^) = o (25) (i) By differentiating the left-hand members of these equations with respect to x, y, e, and adding the results, we find (X-(-2^)v2A = 0, (26) so that A is an harmonic function, i.e. a function satisfying Laplace's equation, at all points within the body. 91-93] SATISFIED BY THE STRESS-COMPONENTS 133 (ii) It follows from this and (25) that each of u, v, w satisfies the equation V«<^ = (27) at all points within the body. All components of strain and of stress also satisfy this equation. (iii) Again, by differentiating the left-hand member of the third of equations (25) with respect to y, and that of the second with respect to 2, and subtracting the results, we find V'm^=0 (28) Similar equations are satisfied by zny and m^, so that each of the components of the rotation is an harmonic function at all points within the body. (iv) The stress-components satisfy a system of partial differential equations. In order to obtain them it is convenient to introduce a quantity @, the sum of the principal stresses at any point ; we have = (3X, + 2/i)A; (29) thus @ is an harmonic function at all points within the body. Further we find * dx ^^ dx- In like manner we find -■-'-'-m's-o ■•« -•^l^^'a^l-'' <-> Similar formulse can be obtained for V^Fj,, V%, V'Z^, V^Z^. The coefficient 2 (\ + /.t)/(3\ -I- 2/i) is 1/(1 -f- o-). (v) As an example of the application of these formulse, we may observe that Maxwell's stress-system, described in (vi) of Article 53, cannot occur in an isotropic solid body free from the action of body forces and slightly strained from a state of no stresst. This appears at once on observing that Xx+Ty-^Z^, as given for that system, is not an harmonic function. 93. Various methods and results. (i) The equations of types (30) and (31) may also be deduced J from the stress-equations (3) and the equations of compatibility of the strain-components (Article 17). We have, for example. Thus the equation Q ^yy I P ^zz ^ ^yz dz^ dy^ dydz becomes J {(If and three equations of the type p-J(i+-)v^xi-e]=o. (33) where e is written for It may be shown also that the stress-functions ^jrl, i//-2, V's "^ ^^^ same Article satisfy three equations of the type (^+^)v^|fe-S-' (-) and three equations of the type •'-'"iC^'-t-t)-^!!- « where 6 is written for ?!ii+9N:?+?!i3 (37) dydz dzdx dxdy " (iii) It may be shown t also that, when there are body forces, the stress-components Sjitisfy equations of the types _2j._^ 1 9^0 0- /9A' 9r_^9^\ dX .... . „2T^ , 1 9^© 9^ dV ,.., ""•^ ^'^^+l-+7 9^.= -''9l,-''^ ('') t The equations of these two types with the equations (3) are a complete system of equations satisfied by the stress-components. * Ibbetson, Mathematical Theory of Elasticity, London, 1887. + Michell, loc. cit. 93, 94] PLANE STRAIN AND PLANE STRESS 135 94. Plane strain and plane stress. States of plane strain and of plane stress can be maintained in bodies of cylindrical form by suitable forces. We take the generators of the cylin- drical bounding surface to be parallel to the axis of z, and suppose that the terminal sections are at right angles to this axis. The body forces, if any, must be at right angles to this axis. When the lengths of the generators are small in comparison with the linear dimensions of the cross-section the body becomes a flaie and the terminal sections are its faces. In a state of plane strain, the displacements u, v are functions of a;, y only and the displacement w vanishes (Article 1-5). All the components of strain and of stress are independent of z; the stress-components Z^, T^ vanish, and the strain-components e-ix, eyz, e^ vanish. The stress-component Z^ does not in general vanish. Thus the maintenance of a state of plane strain requires the application of tension or pressure, over the terminal sections, adjusted so as to keep constant the lengths of all the longitudinal filaments. Without introducing any additional complication, we may allow for an uniform extension or contraction of all longitudinal filaments, by taking w to be equal to ez, where e is constant. The stress-components are then expressed by the equations X. = (X + 2M)| + xg-f.), F. = 0, 9a; \dt/ The functions u, v are -to be determined by solving the equations of equi- librium. We shall discuss the theory of plane strain more fully in Chapter ix. In a state of plane stress parallel to the plane of (x, y) the stress- components Z^, Tz, Z^ vanish, but the displacements w, ■», w are not in general independent of z. In particular the strain-component 632 does not vanish, and in general it is not constant, but we have , 3w \ /^^^* I g^^-_AA (40) ^'^'■dz~ X-t-2^W 9W 2/i The maintenance, in a plate, of a state of plane stress does not require the application of traction to the faces of the plate, but it requires the body forces and tractions at the edge to be distributed in certain special ways. We shall discuss the theory more fully in Chapter ix. An important generalization* can be made by supposing that the normal * Cf. L. N. G. Filon, FUl. Trans. Roy. Soc. (Ser. A.), vol. 201 (1903). 136 BENDING OF NARROW BEAM [CH. traction Z^ vanishes throughout the plate, but that the tangential tractions Z^, F, vanish at the faces z=±h only. If the plate is thin the deter- mination of the average values of the components of displacement, strain and stress taken over the thickness of the plate may lead to knowledge nearly as useful as that of the actual values at each point. We denote these average values by M, ... e^^, ... Z^, ... so that we have for example w = (2A)-i[' udz (41) We integrate both members of the equations of equilibrium over the thickness of the plate, and observe that Z^ and F^ vanish at the faces. We thus find that, if there are no body forces, the average stress-components X^, Xy, Yy satisfy the equations ax,_^az,^ ?|_^ + ^^/ = o (42) dx dy ox dy Since Z^ vanishes equations (40) hold, and it follows that the average dis- placements li, V are connected with the average stress^components Z^,, Xy, Yy by the equations X^ = ~ — ^ 5- -f 5- U- 2/X ;^ , X + 2ij,\.dx dyi dx ^ (dv du States of stress such as are here described will be termed states of" generalized plane stress." 95. Bending of narrow rectangular beam by terminal load. A simple example of the generalized type of plane stress, described in Article 94, is afforded by a beam of rectangular section and small breadth (2/i), bent by forces which act in directions parallel to the plane containing the length and the depth. We shall take the plane of (a?, y) to be the mid-plane of the beam (parallel to length and depth) ; and, to fix ideas, we shall regard the beam as horizontal in the unstressed state. The top and bottom surfaces of the beam will be given by 2/ = + c, so that 2c is the depth of the beam, and we shall denote the length of the beam by I. We shall take the origin at one end, and consider that end to be fixed. From the investigation in Article 87, we know a state of stress in the beam, given by Xx = — EyjR ; and we know that the beam can be held in this state by terminal couples of moment ^hcfE/R about axes parallel to the axis of z. The central-line of the beam is bent into an arc of a circle of radius R. The traction across any section of the beam is then statically 94, 95] BY TERMINAL LOAD 137 equivalent to a couple, the same for all sections, and equal to the terminal couple, or bending moment. w Let us now suppose that the beam is bent by a load W applied at the end a; = Z as in Fig. 13. This force cannot be balanced by a couple at any section, but the traction across any section is equivalent to a force W and a couple of moment W{1 — x). The stress-system is therefore not so simple as in the case of bending by couples. The couple of moment W {I — x) could be balanced by tractions X^, given by the equation w Fig. 13. X.= ■4A^3^(^-)^; .(44) and the average traction X^ across the breadth would be the same as X^. We seek to combine with this traction X^ a tangential traction Xy, so that the load W may be equilibrated. The conditions to be satisfied by Xy are the following : — (i) Xy must satisfy the equations of equilibrium -- = 0, dXx dXy _ - dX, dx dy ' dx (ii) Xy must vanish when y=±c, (iii) 2h I Xydy must be equal to W. J -e These are all satisfied by putting W{c^-y^) (45) X -^ It follows that the load W can be equilibrated by tractions X^ and Xy, with- out Yy, provided that the terminal tractions, of which W is the resultant, are distributed over the end so as to be proportional to c^ - y\ As in Article 89, the distribution of the load is important near the ends only, if the length of the beam is great in comparison with its depth. We may show that a system of average displacements which would correspond with this system of average stresses is given by the equations 2^M = W (Zc^y-f)- 3X+-2fi 4Ac3 {Qlxy +y^ -Zx^y), .(46) Since these are deduced from known stress-components a displacement possible in a rigid body might be added, so as to satisfy conditions of fixity at the origin. 138 GENERAL EQUATIONS [CH. V These conclusions may be compared with those found in the case of bending by couples (Article 88). We note the following i-esults : — (i) The teusion per unit area across the normal sections {X^) is connected with the bending moment, W {l-x), by the equation tension = - (bending moment) (y//), whore // is distance from the neutral plane, and / is the appropriate moment of inertia. (ii) The curvature {oPvidx^)ii^a is , ^ /qx ^.n \ ' ®° ^^^^ ^® '^^^'^ *^^ equation curvature = (bending moment)/(£^/). (iii) The surface of particles which, in the unstressed state, is a normal section does not continue to cut at right angles the line of particles which, in the same state, is the line of centroids of normal sections. The cosine of the angle at which they cut when the beam is bent is (3S/3.j; + 3M/3y)j^5, and this is 3 WjSixhc. (iv) The normal sections do not remain plane, but are distorted into curved surfaces. A line of particles which, in the unstressed state, is vertical becomes a curved line, of [normal of central line tangent of central line central tangent Fig. 14. which the equation is determined by the expression for m as a function of y when x is constant. This equation is of the form u=:ay + ^y\ and the corresponding displacement consists of a part ay which does not alter the planeness of the section combined with a part which does. If we construct the curve x = fy^ and place it with its origin {x = 0, y = 0) on the strained central-line, and its tangent at the origin along the tangent to the line of particles which, in the unstressed state, is vertical, the curve will be the locus of these particles in the strained state. Fig. 14 shows the form into which an initially vertical filament is bent and the relative situation of the central tangent of this line and the normal of the strained central-line. 96. Equations referred to orthogonal curvilinear coordinates. The equations such as (21) expressed in terms of dilatation and rotation can be transformed immediately by noticing the vectorial character of the terms. In fact the terms f — , g- , ^ J A may be read as " the gradient of A," and then the equations (22) may be read (\ + 2/i) (gradient of A) - 2/i (curl of w) + p (body force) = p (acceleration), (47) 95-98] IN CURVILINEAR COORDINATES 139 where in- stands temporarily for the rotation (0-3,, btj,, la-^), and the factors such as \ + 2/i are scalar. Now the gradient of A is the vector of which the component, in any direction, is the rate of increase of A per unit of length in that direction ; and the components of this vector, in the directions of the normals to three orthogonal surfaces a, /3, 7 (Article 19), are accordingly SA 3A aA We have already transformed the operation curl, and the components of rotation, as well as the dilatation (Article 21) ; and we may therefore regard A and ot-„, zTp, ot^ as known in terms of the displacement. The equation (47) is then equivalent to three of the form where F,,, F^, Fy are, as in Article 58, the components of the body force in the directions of the normals to the three surfaces. 97. Polar coordinates. As an example of the equations (48) we may show that the equations of equilibrium under no body forces when referred to polar coordinates take the forms (X + 2;.)sinfl|^-2^{^--|.(m^sin5)} = 0, ^ (^ + ^'^)sil-.|-^''{|(-.)-'l'}=°- (^«) (X + 2^) r sin 5 g^ - 2;. jg-^ (^^ sm 6) - g^«| =0- ^ We may show also that the radial components of displacement and rotation and the dilatation satisfy the equations f,V^ru;) + (\ + lx)r~-2f.A=0, v2a=0, vHrmr)=0; but that some solutions of these equations correspond with states of stress that would require body force for their maintenance*. 98. Radial displaeementf- The simplest applications of polar coordinates relate to problems involving purely radial displacements. We suppose that the displacements ug, u,j, vanish, and we write U in place of Ur. Then we find from the formulae of Articles 22 and 96 the following results :— (i) The strain-components are given by do _U _ _ _n * Miohell, London Math. Soc. Proc, vol. 32 (1901), p. 24. t Most of the regults given in this Article are due to Lam^, Legons sur la thgm-ie...de Velasti- cite, Paris 1852. 140 RADIAL STRAIN IN [CH. V (ii) The dilatation and rotation are given by (ill) The stress-components are given by (\v\ The general equation of equilibrium, under radial body force R, is (v) If ^ = 0, the complete primitive of the equation just written is U=Ar + Br-\ where A and B are arbitrary constants. The first term corresponds with the problem of compression by uniform normal pressure [Article 70 (jr)]. The complete primitive cannot represent a displacement in a solid body containing the origin of r. .The origin must either be outside the body or inside a cavity within the body. (vi) The solution in (v) may be adapted to the case of a shell bounded by concentric spherical surfaces, and held strained by internal and external pressure. We must have a {7 U _ f-po whenr=?-o. (x.2,)^^^.xf={:j; vhen r = ri, where /j„ is the pressure at the external boundary (r = r„), and pi is the pressure at the internal boundary (r—rj). We should find 3X + 2,i V-n^ ¥ ro^-ns j-2" The radial pressure at any point is Ti^ r^ — 'fi r^ 1^ — ri^ P^ '^ ,^J^j3 +Po^ rj^3 ' and the tension in any direction at right angles to the radius is 1 rj^ r^%r^ _ 1 j-q^ 2r3 +r-y^ 2^' r^ 77^7i3 ~ 2^° 73" ra^-ri^ ' In case pu = 0, the greatest tension is the superficial tension at the inner surface, of amount \P\ (V + 2n^)/(''o^-»'i') ; and the greatest extension is the extension at right angles to the radius at the inner surface, of amount Pi 'V3X + 2,.'^V/- ( vii) If in the general equation of (iv) R= - grjr^ , where g is constant, the surface )• = »•„ is free from traction, and the sphere is complete up to the centre, we find U= --- SPI^ /5X + 6/i _ r2\ 10X + 2/iV3X + 2/i ro^/' This corresponds with the problem of a sphere held strained by the mutual gravitation of its parts. It is noteworthy that the radial strain is contraction within the surface *' = »'ov/{(3- '•> '•') The additional stress when the compound tube is subjected to internal pressure p may be taken to be given by the equations The diminution of the hoop tension 66 at the inner surface r=ri may be taken as an index of the increased strength of the compound tube. 102. Rotating cylinder*. An example of equations of motion is afforded by a rotating cylinder. In equations (52) we have to put fr= - ay'r, where a is the angular velocity. The equations for the displacements are ,. „ X 3 /3f U dw\ d fdU dxo\ (^ + ^'')a-. (3;^ + 7 + 3ij+ '^9^ (^ " 3?)= "" '"' with the conditions rr=rz=(J when r=a or r = a', rz=zz==0 when z= +1. The cylindrical bounding surface is here taken to be r=a, and it is supposed that there is an axle-hole given by r=a'; the terminal sections are taken to be given by z=+l, so that the cylinder is a shaft of length il, or a disk of thickness il. Case (a). Rotating shaft. An approximate solution can be obtained in the case of a long shaft, by treating the problem as one of plane strain, with an allowance for uniform longitudinal extension, e. We regard the cylinder as complete, i.e. without an axle-hole ; and then the approximate solution satisfies the equations rz = throughout, rr = when r=a, but it does not satisfy 20= when z= ±1. The uniform longitudinal extension e can be adjusted so that the tractions zz on the ends shall have no statical resultant, i.e. .(61) /: zzrdr = ; ' and then the solution represents the state of the shaft with sufficient exactness over the greater part of the length, but is defective near the ends. [Cf Article 89.J We shall state the results in terms of E and er. We should find ^=^'— 8^ 1^^ -' '" = ^^' (^2) * See papers by C. Chree in Cambridge Phil. Sue. Proc, vol. 7 (1892), pp. 201, 283. The problem had been dlHcussed previously by several writers among whom Maxwell {loc. cit. Article 57), and Hopkinson, Mess, of Math. (Ser. 2), vol. 2 (1871), may be mentioned. 101,102] AND ROTATING DISK 145 where the constants A and e are given by the equations . a^pa' 3 — 5cr ay^pa^a- ,„„, ^ = WT3^' '^--^E- ^^'^ The stress-components are given by the equations — a)V(a2-»-^) 3-2• • • •(!) ■^-exyilim^+km^-] If the material possesses, at each point, a centre of symmetry, a figure consisting of equivalent rays going out from the point allows the operation of central perversion. The corresponding substitution is given by the equations x =-x, y' = -y, z' = -z. 150 EFFECT OF SYMMETRY [CH. VI This substitution does not affect any component of strain, and we may con- clude that the elastic behaviour of a material is in no way dependent upon the presence or absence of central symmetry. The absence of such symmetry in a material could not be detected by experiments on the relation between stress and strain. It remains to determine the conditions which must hold if the strain- energy-function is unaltered, when the strain-components are transformed by the substitutions that correspond with the following operations : — (1) reflexion in a plane, (2) rotation about an axis, (3) rotation about an axis combined with reflexion in the plane'at right angles to the axis. We shall take the plane of S3rmmetry to be the plane of x, y, and the axis of symmetry, or of alternating symmetry, to be the axis of z. The angle of rotation will be taken to be a given angle 6, which will not in the first instance be thought of as subject to any restrictions. The conditions that the strain-energy-function may be unaltered, by any of the substitutions to be considered, are obtained by substituting for Bx'id, ■■■, in the form CueVa;' + •••> their values in terms of e^x, ■■■, and equating the coefficients of the several terms to their coefiicients in the form The substitution which corresponds with reflexion in the plane of (x, y) is given by the equations x' = x, y' = y, z =-z; and the formulae connecting the components of strain referred to the two systems of axes are 6y'z! = 6yz> ^ix! = ^zxt ^x/y' — ^xy The conditions that the strain-energy-function may be unaltered by this sub- stitution are ^14 ^^ <^15 ^^24^ C25 ^ C34 = C36 ^ C46 = Cge ^ " \^) The substitution which corresponds with rotation through an angle 6 about the axis of z is given by the equations x = X cos 6 + y sin 9 , y'= — xsind + y cos 0, z' — z; (3) and the formulae that connect the components of strain referred to the two systems of axes are Bz'x' = Sxx cos'' 6 + Byy sin^ + Bxy sin cos 0, By^ = Bxx sin^' + Byy cos^ — Bxy sin cos 0, ^z'z' ^ Bzz> ey'z' — Syz cos — Czx sin 0, ' ^ ^ Bia! = fiyz sin -f ezx COS 0, e^y' = — 2exx sin cos ^ -)- 2eyy sin cos ^ -I- Cxy (cos'' ^ - sin" 0). I 105] ON STRAIN-ENERGY-PDNCTION 151 The algebraic work required to determine the conditions that the strain-energy-function may be unaltered by this substitution is more complicated than in the cases of central perversion and reflexion in a plane. The equations fall into sets connecting a small number of coefficients, and the relations between the coefficients involved in a set of equations can be obtained without much difficulty. We proceed to sketch the process. We have the set of equations Cii = Cn cos* 5 + 2ci2 sin^ 6 cos^ 6 + c^ sin* 6 - 4ci6 cos' 6 sin 6- 4c26 sin^ 6 cos 6+ 4c68 sin^ B cos^ 6, e22 = Cn sin* d -t- 2ci2 sin^ 6 cos^ B + C22 cos* B + 4ci6 sin' 5 cos 5 + 4c26 cos' fl sin 5 -I- ic^a sin^ B cos^ B, "n = On sin^ 6 cos^ 6 + C12 (cos* 6 + sin* i9) -I- C22 siu^ 6 cos^ 5 -(- 2 (c,8 - cje) sin fl cos B (cos^ tf - sin^ B) — ic^ain'^ B cos^ 6, "66 = Cn sin^ 6 cos^ 6-ic■^2 sin^ B cos^ fl -f C32 sin^ 5 cos'^ 5-1-2 (cjo - C26) sin 6 cos 5 (cos^ fl - sin^ 5) -{■Cm{oos^B-sm^6f, "16= Cii cos' 5 sin 5 - Cj2 sin 6 cos 5 (cos^ 6 - sin^ fl) — C22 sin' 6 cos 5 -f Cje cos^ 5 (cos^ B-Z a\v? B) + C26 sin2 5 (3 cos2 B-s\rfiB)- ^c^ sin (9 cos B (cos^ 5 - sin^ i9), C26=Cn sin' B cos 5 -)- O12 sin fl cos 5 (oos^ B — sin^ 5) - C22 cos' B sin fi-l-Cie sin^ 5 (3 cos'^ 6 - sin^ 5) -I- C26 cos^ 5 (cos^ 5 — 3 sin^ 6) + 2cei3 sin 5 cos B (cos^ 5 — sin^ 5). The equations in this set are not independent, as is seen by adding the first four. We form the following combinations : — cie + "26 = (cii - «22) sin 5 cos 5 -Kc,e -I- c^) (cos* 6 - sin* B), Oh - C22= (cii - C22) (cos* B - sin* 5) - 4 (cie -1- C26) sin 5 cos B, from which it follows that, unless sin 5 = 0, we must have Oil = 022, 026=— C16. When we use these results in any of the first four equations of the set of six we find (ci, - C12 - 2ce6) sin^ B cos^ B + 2cig sin 6 cos B (cos^ 5 - sin^ 5) =0, and when we use them in either of the last two equations of the same set we find - 8ci6 sin^ B cos2 5 +{cn - c,2 - 2cee) sin B cos B (cos^ B - sin^ 5) = ; and it follows that, if neither sin B nor cos 6 vanishes, we must have 0m = i{''n-0i2), Ci6 = 0. Again we have the set of equations «i3 = "13 cos^ B + C33 sin^ B - 2c36 sin 5 cos B, C23 — 013 sin^ 5-1-C23 cos^ 5-l-2c36sin B cos B, C35= (ci3 - C23) sin B cos B + Css (oos^ 5 - sin^ 6) ; from which it follows that, unless sin 5=0, we must have 013 = 023, C3e=0. In like manner we have the set of equations C44= On cos^ 5 -I- C66 siu^ 5 -t- 2046 sin 5 cos 5, 055=044 sin'^ 5-1-065 cos^ 5 — 2045 sin 5 cos 5, C45 = - (C44 — 055) sin 5 cos 5 -1- O45 (cos^ 5 — sin^ 5) ; from which it follows that, unless sin 5 = 0, we must have 044 = «65, 046 = 0. In like manner we have the set of equations O34 = C34 cos 5 -I- C36 sin 5, C36 = - C34 sin 5 +C36 cos 6 ; 152 EFFECT OF SYMMETRY [CH. VI from which it follows, since cos 5^1, that we must have 034=^36 = 0. Finally we have the set of equations (,'„ = Ci4 cos' d + Ci6 cos^ 6 sin 6 + C24 sin^ 6 00s 5 + C2j sin' 6 - 2045 cos^ 6 sin 6 - 2c^ sin^ 6 cos ^, (■].-, = - Ci4 cos^ dam 8 + Cjs cos' 5 — c^ sin' ^+025 sin^ 5 cos 6 + 2c4c sin^ 5 cos 5 - 2ci^ cos^ 5 sin 6, Ci4 = r,4sin2 fl cosfl+c,5sin'fl + C24Cos' 5 + C26Cos2 S sin5 + 2c46coa2 5sia5 + 2c6osin2 5cosd, Cjs = - C]4 sin' 5 + Ci5 sin^ 5 cos d - c^ cos^ 5 sin 5 + Cjs cos' 5 - 2045 sin^ 6 cos 5 + 2C50 cos" 6 sin 5, '■40 = Ci4 cos^ fl sin 5 + C16 sin^ d cos fl — C24 cos^ 6 sin 5 - C26 sin^ 5 cos B + (C48 cos 6 + C68 sin 6) (cos^ 5 - sin^ 6), <■.-,; = - f ■, 4 sin^ 5 cos d + c,5 cos'-* fl sin tf + C24 sin- 6 cos 5 - C26 cos^ 5 sin fl - (C46 sin S-cjo cos fl) (cos^ 5 - sin^ 6). From these we form the combinations Ci4 + C24 = (ci4 + C24) COS 5 + (ci5 + C25) sin 8, «ir, + C25 = - (ci4 -(- C24) sin 5 + (ci6 + 025) cos d ; uiid it follows, since cos 6=^1, that we must have Cl4 + C24 = 0, C,6 + C25 = 0. Assuming these results, we form the combinations (ci4 - cse) = (ci4 - «56) cos 6 - (ci6 + c^e} sin 5, ('-'15 + C46) = (Ci4 - cse) sin 5 + (Ci6 + C45) cos 6 ; from which it follows that Cl4 = ''66j ''if,^— <'46- Assuming these results, we express all the coefficients in the above set of equations in terms of c^s and c^, and the equations are equivalent to two : — C46 (1 - cos' + 3 sin2 d cos 6) - c^ (3 cos^ 5 sin 5 - sin' 6) = 0, C46 (3 oos2 19 sin 5 - sin' 6)-\-Ci^ (1 - cos' 5+3 sin^ 6 cos 5) = 0. The condition that these may be compatible is found to reduce to (1 - cos fl) (1 + 2 cos 5)^=0 ; so that, unless cos 5= -\, we must have «46 = C66 = 0- We have thus found that, if the strain-energy-function is unaltered by a substitution which corresponds with rotation about the axis z, through any angle other than tt, \it, |7r, the following coefficients must vanish : — ^\%t *^26» ^38, C4fl, Cgg, C45 , C14, C24 , C15, C25, C34, C35 J \b) and the following equations must hold among the remaining coefficients : — "u = C22, Ci3 = Cjs, C44=C{5, Cos = ^ (Cji — C42) (6) When the angle of rotation is tt, the following coefficients vanish : — Ci4i C24, C15, C25, C48, Cje, C34, C35 ; (7) no relations between the remaining coefficients are involved. When the angle of rotation is \it, the following coefficients vanish : — C30) C48, C58, C45, Ci4, C24, Ci5, C25, C34, C3B ; (o) and the following equations connect the remaining coefficients : — Cll ^ C22, Ci3=C2j, 044=655, 025= — Cij (y) 105-107] ON STEAIN-ENEEGY-FUNCTION 153 When the angle of rotation is | tt, the following coefficients vanish : — Cl6, C28, C3„, C45, C34, C35; (10) and the following equations connect the remaining coefficients : — Cll = C22, Cis = C23, C44 = Cb5, Cfis = 1(^11 " Cjs), | ,jj> Ci4 = ~ C24 = C58 , — Ciis = C25 = Cjj . ) In like manner, when the axis of z is an axis of alternating symmetry, and the angle of rotation is not one of the angles vr, ^tt, \tv, the same coefficients vanish as in the general case of an axis of symmetry, and the same relations connect the remaining coefficients. When the angle is tt, we have the case of central perversion, which has been discussed already. When the angle is \ir, the results are the same as for direct symmetry. When the angle is ^vr, the results are the same as for an axis of direct symmetry with angle of rotation |7r. 106. Isotropic solid. In the case of an isotropic solid every plane is a plane of symmetry, and every axis is an axis of symmetry, and the corresponding rotation may be of any amount. The following coefficients must vanish : — Cl4) Ci5, C16, C24, Cj5, Cjs, C34, C35, Cjg, C4S, C46, C56...(12) , and the following relations must hold between the remaining coefficients : — Cn = C22 = C33 , C23 = C31 = C]2, C44 = C55 = Cgs = ■5- (Cii — C12) (13) Thus the strain-energy-fiinction is reduced to the form ^ Cii (e a;a; + e yy + e zz) + C]2 \fiyy&zz + ^zz^ma "I" ^xx^yy) + i (cn - C12) (eV + e%r + e\y\ (14) which is the same as that assumed in Article 68. 107. Symmetry of crystals. Among seolotropic materials, some of the most important are recognized as crystalline. The structural symmetries of crystalline materials have been studied chiefly by examining the shapes of the crystals. This examination has led to the construction, in each case, of a figure, bounded by planes, and having the same symmetry as is possessed in common by the figures of all crystals, formed naturally in the crystallization of a material. The figure in question is the " crystallographic form " corresponding with the material. F. Neumann* propounded a fundamental principle in regard to the physical behaviour of crystalline materials. It may be stated as follows : — Any kind of symmetry, which is possessed by the crystallographic form of a material, is possessed by the material in respect of every physical quality. In * See his Vorlesungen iiber die Theorie der Elasticitdt, Leipzig, 1885. 154 NATURE OF THE SYMMETRY [CH. VI other words we may say that a figure consisting of a system of rays, going oufc from a point, and having the same symmetry as the crystallographic form, is a set of equivalent rays for the material. The law is an induction from experience, and the evidence for it consists partly in a posteriori verifications. It is to be noted that a crystal may, and generally does, possess, in respect of some physical qualities, kinds of symmetry which are not possessed by the crystallographic form. For example, cubic crystals are optically isotropic. Other examples are afforded by results obtained in Article 105. The laws of the symmetry of crystals are laws which have been observed to be obeyed by crystallographic forms. They may be expressed most simply in terms of equivalent rays, as follows : — (1) The number of rays, equivalent to a chosen ray, is finite. (2) The number of rays, equivalent to a chosen ray, is, in general, the same for all positions of the chosen ray. "We take this number to be iV- 1, so that there is a set of N equivalent rays. For special positions, e.g. when one of the rays is an axis of symmetry, the number of rays in a set of equivalent rays can be less than N. (3) A figure, formed of N equivalent rays, is a symmetrical figure, allowing all the covering operations of a certain group. By these operations, the N equivalent rays are inter- changed, so that each ray comes at least once into the position of any equivalent ray. Any figure formed of equivalent rays allows all the covering operations of the same group. (4) When a figure, formed of N equivalent rays, possesses an axis of symmetry, or an axis of alternating symmetry, the corresponding angle of rotation is one of the angles 77, |?r, ^ff, \t! *. It can be shown that there are 32 groups of covering operations, and no more, which obey the laws of the symmetry of crystals. With each of these groups there corresponds a class of crystals. The strain-energy -function corresponding with each class may be written down by making use of the results of Article 105 ; but each of the forms which the function can take corresponds with more than one class of crystals. It is necessary to describe briefly the symmetries of the classes. For this purpose we shall now introduce a few definitions and geometrical theorems relating to axes of symmetry : — The angle of rotation about an axis of symmetry, or of alternating symmetry, is iw/n, where n is one of the numbers : 2, 3, 4, 6. The axis is described as " w-gonal." For »i = 2, 3, 4, 6 respectively, the axis is described as "digonal," "trigonal," "tetragonal," "hexagonal." Unless otherwise stated it is to be understood that the n-gonal axis is an axis of symmetry, not of alternating symmetry. The existence of a digonal axis, at right angles to an w-gonal axis, implies the existence of n such axes ; e.g. if the axis z is tetragonal, and the axis x; digonal, then the axis i/ and the Unes that bisect the angles between the axes of x and y also are digonal axes. The existence of a plane of symmetry, passing through an m-gonal axis, implies the existence of n such planes; e.g. if the axis 2 is digonal, and the plane x=0 is a plane of symmetry, then the plane y=0 also is a plane of symmetry. If the H-gonal axis is an axis of alternating symmetry, the two results just stated still hold if n is uneven ; but, if n is even, the number of axes or planes implied is ^n. * The restriction to these angles is the expression of the "law of rational indices." 107, 108] OF CRYSTALLINE MATERIALS 155 108. Classification of crystals. The symmetries of the classes of crystals may now be described by reference to the groups of covering operations which correspond with them severally : — One group consists of the identical operation alone ; the corresponding figure has no symmetry; it will be described as "asymmetric." The identical operation is one of the operations contained in all the groups. A second group contains, besides the identical operation, the operation of central perversion only ; the symmetry of the corresponding figure will be described as "central." A third group contains, besides the identical operation, the operation of reflexion in a plane only ; the symmetry of the corresponding figure will be described as "equatorial." Besides these three groups, there are 24 groups for which there is a. "principal axis"; that is to say, every axis of symmetry, other than the principal axis, is at right angles to the principal axis ; and every plane of symmetry either passes through the principal axis or is at right angles to that axis. The five remaining groups are characterized by the presence of four axes of .trigonal symmetry equally inclined to one another, like the diagonals of a cube. When there is an w-gonal principal axis, and no plane of symmetry through it, the symmetry is described as "n-gonal" ; in case there are digonal axes at right angles to the principal axis, the symmetry is further described as " holoaxial " ; in case there is a plane of symmetry at right angles to the principal axis, the symmetry is further described as " equatorial " ; when the symmetry is neither holoaxial nor equatorial it is further described as "polar." When there is a plane of symmetry through the n-gonal principal axis, the symmetry is described as "di-w-gonal"; it is further described as "equatorial" or "polar," according as there is, or is not, a plane of symmetry at right angles to the principal axis. When the principal axis is an axis of alternating symmetry, the symmetry is described as "di-/i-gonal alternating," or "re-gonal alternating," according as there is, or is not, a plane of symmetry through the principal axis. The appended table shows the names* of the classes of crystals so far described, the symbols t of the corresponding groups of covering operations, and the numbers of the classes as given by Voigt|:. It shows also the grouping of the classes in systems and the names of the classes as given by Lewis §. The remaining groups, for which there is not a principal axis, may be described by reference to a cube ; and the corresponding crystals are frequently called "cubic," or "tesseral," crystals. All such crystals possess, at any point, axes of symmetry which are distributed like the diagonals of a cube, having its centre at the point, and others, which are parallel to the edges of the cube. The latter may be called the " cubic axes. " The symmetry about the diagonals is trigonal, so that the cubic axes are equivalent. The symmetry with respect to the cubic axes is of one of the types previously named. There are five classes of cubic crystals, which may be distinguished by their symmetries with respect to these axes. The table shows the names of the classes (Miers, Lewis); the symbols of the corresponding groups (Schoenflies), the numbers of the classes (Voigt), and the character of the symmetry with respect to the cubic axes. * The names are those adopted by H. A. Miers, Mineralogy, Oxford, 1902. t The symbols are those used by Schoenflies in his book Krystallsysteme und KrystalUtructur . X Bapports prSsenteei au Gongres International de Physique, t. 1, Paris, 1900. § W. J. Lewis, Treatise on Crystallography, Cambridge, 1899. The older classification in six (sometimes seven) "systems" as opposed to the 32 "classes" is supported by some modem authorities. See V. Goldschmldt, Zeitschr. f. Krystallographie, Bde. 31 and 32 (1899). 156 CLASSES OF CRYSTALS [CH. VI System Name of olass [Miers] Symbol of group [Sohoenflies] Number of olass [Voigt] Name of class [Lewis] Triclinic ( or \ Anorthic 1, Asymmetric Central 0, S2 2 1 Anorthic I Anorthic II Monoclinic or Oblique Equatorial Digonal polar Digonal equatorial s 4 5 3 Oblique II Oblique I Oblique III Rhombic [ or ] Prismatic 1 Digonal holoaxial Didigonal polar Didigonal equatorial V v 7 8 6 Prismatic I Prismatic III Prismatic II Hexagonal and J Rhombohedral Trigonal polar Trigonal holoaxial Trigonal equatorial Ditrigonal polar Ditrigonal equatorial Hexagonal polar Hexagonal alternating Hexagonal holoaxial Hexagonal equatorial Dihexagonal polar Dihexagonal alternating Dihexagonal equatorial 13 10 27 11 26 25 12 23 24 22 9 21 Rhombohedral I Rhombohedral IV Rhombohedral VI Rhombohedral V Rhombohedral VII Hexagonal I Rhombohedral 11 Hexagonal V Hexagonal II Hexagonal III Rhombohedral III Hexagonal IV Tetragonal ■ Tetragonal polar Tetragonal alternating Tetragonal holoaxial Tetragonal equatorial Ditetragonal polar Ditetragonal alternating Ditetragonal equatorial 0, A (74" A" 18 20 15 17 16 19 14 Tetragonal III Tetragonal VII Tetragonal V Tetragonal IV Tetragonal VI Tetragonal I Tetragonal II Name of class Symbol of group Number Symmetry with respect [Miers] [Lewis] [Sohoenflies] [Voigt] to the cubic axes tesseral polar Cubic III T 32 digonal tesseral holoaxial Cubic I 29 tetragonal tesseral central Cubic IV xm 31 digonal equatorial ditesseral polar Cubic V TW 30 tetragonal alternating ditesseral central Cubic II ow 28 tetragonal equatorial 108,109] ELASTICITY OF CRYSTALS 157 109. Elasticity of crystals. We can now put down the forms of the strain-energy-function for the different classes of crystals. For the classes which have a principal axis we shall take this axis as axis of z ; when there is a plane of symmetry through the principal axis we shall take this plane as the plane {x, z) ; when there is no such plane of symmetry but there is a digonal axis at right angles to the principal axis we shall take this axis as axis of y. For the crystals of the cubic system we shall take the cubic axes as coordinate axes. The classes will be described by their group symbols as in the tables of Article 108 ; we shall first write down the symbol or symbols, and then the corresponding strain-energy-function ; the omitted terms have zero coefiScients, and the con- stants with different suffixes are independent. The results* are as follows : — Groups (7i, Sa — (21 constants) ^Cufi XX I" Gi2^xx^yy "t" ^is^zx^zz ~r Cu&xx^yz "^ ^is^xx^zx *• ^\6^xx^zy "T" 2 ^22^ yy •" ^2S^yy^zz i" ^Si^yy^yz ' ^zn^yy^zx * ^2J5^yy^xy "^ i CsaS ;!!! -f- C3i6iz^yz "t" 0356221622; "f" C^iBz^Bxy "^i^^i^yz "r ^4S^yz^zx "r CisSyiexy '^2^65^ zx I' C^e^x^xy '^i^^ee^ xy Groups S, Os, O/ — (13 constants) h "ne^xa; + Cn^xxByy + C^exx^zz + Cie^xx^xy + \c^e\y +C^eyyezz + Cs6 Syy 6xy + 2 "336 zz -|- Cs^S^exiy + 2''446 !/z + ^K^yz^zx + 2 "55? zx + iCfsB xy Groups V, G/, F''— (9 constants) 2*^11^ XX ~(" ^I2^xx^yy ~T C^^Bxx^zz -^ -^C^e yy -r Ci^SyySzz + 2^336 zz + 2^44® yz + ji^m^ zx + iCm^ xy Groups C's, (Sg— (7 constants) \Cn^'xx + (^w^xx^yy + ^^w^xx^zz + CuBxx^yz + Oi^^xx^zx + •^011 e\y -I- C]3 Byy ^ZZ — 0)4 Byy Byg — C15 Byy B^x + J C33e''zz + 2 ^44^ yz ~ OisByuBxy + \Cue\x +Cuezxe^ + i{Cn-Ci^e'^xy * The results are due to Voigt. 158 EFFECT OF SYMMETRY [CH. VI Groups -D3, Ci, Se" — (6 constants) + i \^n C12) 6 xy- Groups Cs*, A*, C^e, A, C'e*, Os", A*— (5 constants) 2 CjiC aa + ^^xx^yy + CisCaixCzz + ^CiiB yy + CiS^yy^zz Groups C4, (84, C4* — (7 constants) jCjiB xx + <6* for which the constants are cii = 2746, 033 = 2409, Ci2 = 980, 013 = 674, 044=666. For a bar whose axis is in the direction of the principal axis of symmetry ^=2100, For a bar whose axis is in the direction of a secondary axis of symmetry £!= 2300. The first of these is about the same as that for steel, and the second is rather greater. The principal rigidities are 666 and 883, of which the first is less and the second considerably greater than the rigidity of steel. Cauchy's relations are approximately verified. Quartz is a rhombohedral crystal of the class specified by the group 1)3. The constants are o„ = 868, 033 = 1074, Ci3=143, Ci2 = 70, 044=582, 015= -171, and £! in the direction of the principal axis is 1030. * For references see Introduction, footnote 55. t See table, Article 71. i It has been suggested that these somewhat paradoxical results may be due to "twinning" of the crystals. L. E. 11 162 CURVILINEAR [CH. VI Topaz is a rhombic crystal (of the class specified by the group I'*) whose principal Young's moduluses and rigidities are greater than those of ordinary steel. The constants of formula ( 15) are for this mineral ^ = 2870, 5=3560, C=3000, ^'=900, (? = 860, B=1'280, i = 1100, J/^=1350, i\^=1330. The principal Voung's moduluses are 2300, 2890, 2650. Barytes is a crystal of the same class, and its constants are ^=907, 5 = 800, C=U)74, i^=273, = 275, ir=468, Z=122, M=293, iV^=283. These results show that for these materials Cauchy's reduction is not valid. 114. Curvilinear seolotropy. As examples of curvilinear aeolotropy (Article 110) we may take the problems of a tube (Article 100) and a spherical shell (Article 98) under pressure, when there is transverse isotropy about the radius vector*. (a) In the case of the tube we should have dr .(25) r dr ' or r where H is written for A - iN. The displacement Urn given by the equation ^d'U GdU AU {F-H)e „ ^3;:2.+ 7 3;:--^- + — 7^=0, (26) of which the complete primitive is F- H U=ar"'+fir-'' + -j~~,er, (27) n being written for V(4/(7), and a and /S being arbitrary constants. The constants can be adjusted so that rr has the value -p^ at the outer surface r = r(„ and -p^ at the inner surface r=?-,. The constant e can be adjusted so as to make the resultant of the longi- tudinal tension S over the annulus ro>r>ri balance the pressure tt {p^r-i^ - p^r^') on an end of the cylinder. (i) In the case of the sphere we should find in like manner that the radial displacement U satisfies the equation ^^ + T ^-2(^+^-^7.=0; (28) so that 5 + /3r where n2 = |^Jl+f * Saint-Venant, J. de Math. (Liouville), (S^r. 2), t. 10 (1865). 113, 114] ^OLOTROPY 163 and we can find the formula which agrees with the result obtained in (vi) of Article 98 in the case of isotropy. The cubical dilatation of the spherical cavity is the value of 3U/r when r=ri, and this is ?-o^-?-i2"l {n-i)0+2F '^ " (n + i)C-%F J ^ ' This result has been applied by Saint-Venant to the theory of piezometer experiments, in which a discrepancy appears to have been observed between the results obtained and the dilatation that should theoretically be found to occur if the material were isotropic. The solution in (30) contains 3 independent constants and Saint-Venant held that these could be adjusted so as to accord with bhe experiments in question. CHAPTER YII GENERAL THEOREMS. 115. The variational equation of motion*. Whenever a strain-energy-function, W, exists, we may deduce the equa- tions of motion from the Hamiltonian principle. For the expression of this principle, we take T to be the total kinetic energy of the body, and V to be the potential energy of deformation, so that V is the volume-integral of W. We form, by the rules of the Calculus of Variations, the variation of the integral 1{T — V)dt, taken between fixed initial and final values (to and t-,) for t. In varying the integral we assume that the displacement alone is subject to variation, and that its values at the initial and final instants are given. We denote the variation so formed by iJ(r-F) dt. We denote by SWi the work done by the external forces when the displace- ment is varied. Then the principle is expressed by the equation BJ{T-V)dt + lsW,dt = (1) We may carry out the variation of Tdt. We have and therefore BJTdt=jdtjjjp g ^ -H ... + ...) d^dyd. = III SI p S^" + 1 ^' + ^ ^^) ^''^y^' Here t^ and t^ are the initial and final values of t, and Sm, . . . vanish for both these values. The first term may therefore be omitted ; and the equation (1) * Cf. Kirehhoff, Vorleaungen Uber..,Mechanik, Vorlesung 11. 115,116] VARIATIONAL EQUATION OF MOTION 165 is then transformed into a variational equation of motion. Further, 87 is \\\^Wdmdydz, and hW^ is given by the equation S Tf, = ^^^p {Xhu + Yhv + Zhw) doodydz + \{{X, Su + Y,Bv + Z, Sw) dS. Hence the variational equation of motion is of the form ///I d^u ^ d^ ^ a^i -flip (^Sm + YSv + ZSw) dxdydz - ll(XM + T^Bv + ZJw) dS = 0. ...(3) Again, 6 kK IS ^ — be^x + 5 — oByy + . . . + ^ — Se^y, where, for example, he^x is dhujdx. Hence / \\BW dxdydz may be transformed, by integration by parts, into the sum of a surface integral and a volume integral. We find f\\zWdxdydz=\\Y^^^os{x, ^) + ^cos(2,, ^) + gcos(., .)| 8« + ... + , -/// 9 dW 19;^,19Z^g„. + ' dxde^ dyde^ dzde^J dS dxdydz. ...(4) The coefficients of the variations Su, ... under the signs of volume integration and surface integration in equation (8), when transformed by means of (4), must vanish separately, and we thus deduce three differential equations of motion which hold at all points of the body, and three conditions which hold at the boundary. The equations of motion are of the type d^_ -^ d_dW ^dW ddW , Pdf'~P ^dxde^x^dyde^y^dzde,^' ^^' and the surface conditions are of the type dW , ^ dW , , , aif - — cos(a;, I') +5— r- cos(a;, v)+^^ cos(y, v) + ^cos(z, v) = X, (6) 116. Applications of the variational equation. (i) As an example* of the application of this method we may obtain the equations (19) of Article 58. We have * Cf. J. Larmor, Cambridge Phil. Soc. Trans., vol. 14 (1885). IQQ VARIATIONAL EQUATIONS [CH. VII and, by the formulae (36) of Article 20, we have also ««^v=||(^3K) + ||(^.S-^). Every term of ( ( Uw ° f ^ is now to be transformed by the aid of the formulse of the type ( ( (p- dad^dy= j jh^ks^ COS (a, r) dS, and the integral will then be transformed into the sum of a surface integral and a volume integral, in such a way that no differential coeflacients of Sua, Su^, 8uy occur. We may collect, for example, the terms containing SUa. in the volume integral. They are [ r fri /" J_ ^\ _ li (L\^_K^ _ 11 (L\ ^ ~ J J J [da y/hh deaj hgda\hj de^^ h^ da XhJ Be^y ,3/1 dW\ , a/ 1 dWV]. , ,,,, The equations in question can be deduced without difficulty. (ii) As another example, we may obtain equations (21) of Article 91 and the second forms of equations (22) of the same Article. For this purpose we observe that Hence the strain-energy-function in an isotropic body may be expressed in the form /■/■/■„ fbw dv dv dw\ , , , ^°^ ]]r\d^jdz-TyTzr'^y'^' f f [Vfdw dbv dw d8v\ (dv dSw dv dSw\~\ , , , = j}Jl{d^~d^-^w)^\^^W~^i/^)i''^ = nr|cos(2, r) g^-cos(!/, v)^j 8v+^coa(y, v)^^-ao%{z, v)-^^hw^dS; J^ J^ - =^ -^ ) in JFdo not contribute anything to the volume integral in the transformed expression for I I \bWdxdydz. Hence the eqiiations of motion or of equilibrium can be obtained by forming the variation of f f f [i e^ + 2/i) A2 -f 2^ (Br:,2 + ^^2 + ^^2)] ^^c?y «^2 instead of the variation of \A \ Wdxdydz. The equations (21) and the second forms of equations (22) of Article 91 are the equations that would be obtained by this process. The result here found is that the differential equations of vibration, or of equilibrium, of an isotropic solid are the same as those of a body possessing potential energy of deformation per unit of volume expressed by the formula i(X-f2^)A2 + 2,x(w/-fV+^/)- 116,117] or EQUILIBRIUM 167 The surface conditions are different in the two cases. In MacCuUagh's theory of optics* it was shown that, if the luminiferous sether is incompressible and possesses potential energy according to the formula 2fi.{7iTx' + T!jy^ + '!ii^^), the observed facts about reflexion and re- fraction of light are accounted for ; the surface conditions which are required to hold for the purposes of the optical theory are precisely those which arise from the variation of the volume integral of this expression. Larmort has described a medium, which possesses potential energy in the required manner, as "rotationally elastic." The equations of motion of a rotationally elastic medium are formally identical with those which govern the propagation of electric waves in free aether. 117. The general problem of equilibrium. We seek to determine the state of stress, and strain, in a body of given shape which is held strained by body forces and surface tractions. For this purpose we have to express the equations of the type d 7dW\ d /dW\ d^(dW\ ^^^ ^ dx \dexx' dy \dexy) dz \de^xJ as a system of equations to determine the components of displacement, u,v,iu; and the solutions of them must be adapted to satisfy certain conditions at the surface jS of the body. In general we shall take these conditions to be, either (a) that the displacement is given at all points of *S, or (6) that the surface tractions are given at all points of S. In case (a), the quantities u, V, w have given values at (S ; in case (6) the quantities of the type "^ dW~ dW dW X„ , = ^ — cos {x, v) + X — cos {y, v) + ^ — cos {z, v), J have given values'at S. It is clear that, if any displacement has been found, which satisfies the equations of type (7), and yields the prescribed values for the surface tractions, a small displacement which would be possible in a rigid body may be superposed and the equations will still be satisfied ; the strain and stress are not altered by the superposition of this displacement. It follows that, in case (6), the solution of the equations is indeterminate, in the sense that a small displacement which would be possible in a rigid body may be superposed upon any displacement that satisfies the equations. The question of the existence of solutions of the equations of type (7) which also satisfy the given boundary conditions will not be discussed here. It is of more importance to remark that, when the surface tractions are given, the equations and conditions are incompatible unless these tractions, with the body forces, are a system of forces which would keep a rigid body in equilibrium. Suppose in fact that m, v, w are a system of functions which satisfy the equations of type (7). If we integrate the left-hand member * Dublin, Trans. R. Irish Acad., vol. 21 (1839), or Collected Works of James MacCullagh, Dublin, 1880, p. 145. t Phil. Trans. Boy. Soc. (Ser. A), vol. 185 (1894). 168 GENERAL THEORY [CH. VU of (7) through the volume of the body, and transform the volume integrals 9 /dW\ of such terms as ^ ( r — I by Green's transformation, we find the equation 00) \dexxl llx^dS + \\\pXdxdydz = (8) If we multiply the equation of type (7) which contains Z by y, and that which contains F by z, and subtract, we obtain the equation )]] [^ \dx [dej "^ dy [deyj "*" dz \de^)\ ^ \dx [dej "^ dy \deyy) ^ dz \deyj\ + p{yZ — zY) dxdydz = 0; and, on transforming this by Green's transformation, we find the equation |[(yZ, - z F,) dS + \^^p (yZ - zY) dxdydz = (9) In this way all the conditions of statical equilibrium may be shown to hold. 118. Uniqueness of solution *- We shall prove the following theorem : — If either the surface displacements or the surface tractions are given the solution of the problem of equilibrium is unique, in the sense that the state of stress (and strain) is determinate without ambiguity. We observe in the first place that the function W, being a homogeneous quadratic function which is always positive for real values of its arguments, cannot vanish unless all its arguments vanish. These arguments are the six components of strain ; and, when they vanish, the displacement is one which would be possible in a rigid body. Thus, if W vanishes, the body is only moved as a whole. Now, if possible, let u', v', w' and u", v", w" be two systems of displace- ments which satisfy the equations of type (7), and also satisfy the given conditions at the surface S of the body. Then u' — u", v' — v" , w' — w" is a system of displacements which satisfies the equations of the type dxXdexx/ dy\dexyJ dz\dezx) throughout the body, and also satisfies conditions at the surface. Denote this displacement by (u, v, w). Then we can write down the equation j}}\_ [dx\dexxJ dyXdexyJ dz\de^xJ\ \fix\dexy/ oy\deyy/ dz\deyj] \d /dW\ ^(^\ ^(^\V]dxd dz = {dxKde^J dyKdeyJ dzKdezJjj * Cf. Kirchhoff, J.f. Math. (Crelle), Bd. 56 (1859). 117-119] OF EQUILIBRIUM 169 and this is the same as cos (.r, v) ^ h cos {y, v) ^ h cos {z, v) 5 — oexx oSxy oezx + two similar expressions dS 'dW dW dW dW dW dW g Sxa; + 5 Byy + gjj + ;r e,„ + ■ e^x + ^ I Lf^e^x deyy ■'■> de^s dcy^ ■" de^^ de^y dxdydz = 0. When the surface conditions are of displacement u, v, w vanish at all points of S ; and when they are of traction the tractions calculated from u, v, w vanish at all points of S. In either case, the surface integral in the above equation vanishes. The volume integral is I2W dxdydz; and since W is necessarily positive, this cannot vanish unless W vanishes. Hence («, v, w) is a displace- ment possible in a rigid body. When the surface conditions are of displacement W, v,w must vanish, for they vanish at all points of & 119. Theorem of minimum energy. The theorem of uniqueness of solution is associated with a theorem of minimum potential energy. We consider the case where there are no body forces, and the surface displacements are given. The potential energy of deformation of the body is the volume integral of the strain-energy-function taken through the volume of the body. We may state the theorem in the form : — The displacement which satisfies the differential equations of equilibrium, as well as the conditions at the bounding surface, yields a smaller value for the potential energy of deformation than any other displacement, which satisfies the same conditions at the bounding surface. Let {u, V, w) be the displacement which satisfies the equations of equilibrium throughout the body and the conditions at the bounding surface, and let any other displacement which satisfies the conditions at the surface be denoted by (m + u', V + v', w -I- w'). The quantities u', v', w vanish at the surface. We denote collectively by e the strain-components calculated from u, v, w, and by e' the strain-components calculated from u\ v', w' ; we denote by /(e) the strain- energy-function calculated from the displacements u, v, w, with a similar notation for the strain-energy-function calculated from the other displacements. We write V for the potential energy of deformation corresponding with the displacement (u, v, w), and V^ for the potential energy of deformation corre- sponding with the displacement {u + u,v + v', w + w'). Then we show that Fi — F must be positive. We have ^^-'^ = jjj{ /■(« + O - f(e)] dxdydz, 170 GENERAL THEORY [CH. VII and this is the same as r,-v- ie''^+m dxdydz, because f {e) is a homogeneous quadratic function of the arguments denoted collectively by e. Herein / {e) is necessarily positive, for it is the strain- energy-function calculated from the displacement {u, v, w'). Also we have, in the ordinary notation, -,9/'(e) du'dW dv'dW dw'dW 2.e • = I I de doo dexx ^y ^^yy d^ ^^zz /3«/ 8t/\SF /aw' du/\dW /; \dy dzlde^t \dz dx j de^x \ dv' du'\ dW .dy dzJ deyi \dz dx ) de^x xdso dy j de^y' We transform the volume integral of this expression into a surface integral and a volume integral, neither of which involves differential coefficients of u', v', w'. The surface integral vanishes because u',v', w' vanish at the surface. The coefficient of u' in the volume integral is dx\dexxi dy\dexy/ dzKde^xJ' and this vanishes in virtue of the equations of equilibrium. In like manner the coefficients of v and w' vanish. It follows that V,-V^^\^f{e') dxdydz, which is necessarily positive, and therefore F< F,. The converse of this theorem has heen employed to prove that there exists a solution of the equations of equilibrium which yields given values for the displacements at the boundary*. If we knew independently that among all the sets of functions u, v, w, which take the given values on the boundary, there must be one which gives a smaller value to Wdxdydz than any other gives, we could infer the truth of this converse theorem. /// The same difficulty occurs in the proof of the existence- theorem in the Theory of Potential t. In that theory it has been attempted to turn the difficulty by devising an explicit process for constructing the required function:]:. In the case of two-dimensional potential functions the existence of a minimum for the integral concerned has been proved by Hilbert§. ♦ Lord Kelvin (Sir W. Thomson), Fhil. Tram. Roy. Soc, vol. 153 (1863), or Math, and Phyi. Papers, vol. 3, p. 351. t The difficulty appears to have been pointed out first by Weierstrass in his lectures on the Calculus of Variations. See the Article ' Variation of an integral ' in Ency. Brit. Supplement, [Ency. Brit., 10th ed., vol. 33 (1902)]. t See, e.g., C. Neumann, Untersuchungen iiber das logarithmische und Newton'sche Potential, Leipzig, 1877. § ' Ueber das Dirichlet'sche Prineip,' {Festschrift zur Feier des 150 jahrigen Bestehens d. K&nigl. Ges. d. Wiss. zu Gottingen), Berlin, 1901. 1.19-121] OF EQUILIBRIUM ' 171 120. Theorem concerning the potential energy of deformation*. The potential energy of deformation of a body, which is in equilibrium under given load, is equal to half the work done by the external forces, acting through the displacements from the unstressed state to the state of equilibrium. The work in question is jlj p(uX + vY+ wZ) dxdydz+ jj (uX, + 1> F„ + wZ,) dS. The surface integral is the sum of three such terms as and the work in question is therefore equal to ([['' ii( X + -~ +- — + -— ] Xdxd d Jjj\ \ dxdexx dydexy dzde^xJ j ^ ^ ^ffff 5F , dW dW dW dW dW\,.. + J JJ I'- s^ + '^'' a^, + ^- 9i; + ^'^ a^ + ^- 8i + ^- a^.) '^"'^^^^- The first line of this expression vanishes in virtue of the equations of equi- librium, and the second line is equal to 2 1/ 1 Wdxdydz. Hence the theorem follows at once. 121. The reciprocal theorem f. Let u, V, w be any functions of x, y, z, t which are one- valued and free from discontinuity throughout the space occupied by a body ; and let us suppose that u, V, w are not too great at any point to admit of their being displacements within the range of " small displacements " contemplated ia the theory of elasticity founded on Hooke's Law. Then suitable forces could maintain the body in the state of displacement determined by u, v, w. The body forces and surface tractions that would be required can be determined by calculating the strain-components and strain-energy-function from the displacement (u, v, w) and substituting in the equations of the types ■^ d / dW \ d /dW\ ^(^\^ 9^ *" dx \dexcc) oy \dexy) dz Kde^:) ^ dff' ' X. = cos {X, V) (^j -F cos {y, V) (g- j + cos (., .) ^ . * In some books the potential energy of deformation is called the "resilience " of the body. + The theorem is due to E. Betti, H nuovo Cimento (Ser. 2), tt. 7 and 8 (1872). It is a special case of a more general theorem given by Lord Eayleigh, London Math. Soc. Proc, vol. 4 (1873), or Scientific Papers, vol. 1, p. 179. For a general discussion of reciprocal theorems in Dynamics reference may be made to a paper by H. Lamb, London Math. Soc. Proc, vol. 19 (1889), p. 144. 172 BETTl's THEOREM OF RECIPROCITY [CH. VII The displacement u, v, w is one that could be produced by these body forces and surface tractions. Now let (w, V, w), (u', v', w) be two sets of displacements, {X, Y, Z) and {X', Y', Z') the corresponding body forces, (Z„ Y„ Z,) and {X'„, Y\, Z\) the corresponding surface tractions. The reciprocal theorem is as follows : — The whole work done by the forces of the first set (including kinetic reactions), acting over the displacements produced by the second set, is equal to the whole work done by the forces of the second set, acting over the displacements produced by the first. The analytical statement of the theorem is expressed by the equation + jj{XX+Yy + ZM)dS + ii{X\u+ Y\v + Z'„w)dS. (11) In virtue of the equations of motion and the equations which connect the surface tractions with stress-components, we may express the left-hand member of (11) in terms of stress-components in the form of a sum of terms containing u , v', w' explicitly. The terms in u' are JJ] " \dx [de^J dy \de^y) dz Kde^J] + jJ u jcos (., V) (|£) + cos {y, v) g^) + cos (., .) (|£)| dS. It follows that the left-hand member of (11) may be expressed as a volume integral ; and it takes the form /// dxdydz. By a general property of quadratic functions, this expression is symmetrical in the components of strain of the two systems, e^x, ■■■ and e'xx,---- It is there- fore the same as the result of transforming the right-hand member of (11). 122. Determination of average strains*. We may use the reciprocal theorem to find the average values of the strains produced in a body by any system of forces by which equilibrium can be maintained. For this purpose we have only to suppose that u, v', w' are * The method is due to Betti, loc. cit. 121-123] AVERAGE VALUES OF STRAIN-COMPOJfENTS 173 displacements corresponding with a homogeneous strain. The stress-com- ponents calculated from u, v', w' are then constant throughout the body. Equation (11) can be expressed in the form I 1 I (ea;a;Z'a; -I- eyy Y'y + B^^Z';, + gj/j Y' ^ + ^Z^Z' X + ^xy^'y) dX CLy dZ = llip {Xu' + Yv' + Zw') dxdydz + lUxy + Y,v' + ZM) dS... .{12) If X'x is the only stress- component of the uniform stress that is different from zero the corresponding strain-components can be calculated from the stress- strain relations, and the displacements (u, v', w') can be found. Thus the quantity I 1 e^xdxdydz can be determined, and this quantity is the product of the volume of the body and the average value of the strain-component e^x taken through the body. In the same way the average of any other strain can be determined. To find the average value of the cubical dilatation we take the uniform stress-system to consist of uniform tension the same in all directions round a point. 123. Average strains in an isotropic solid body. In the case of an isotropic solid of volume V the average value of e^x is ^jjjp{Xx-a(Yy + Zz)}dxdydz + -^jj{X^x-<.iY,y + Z,z)}dS;...(U) the average value of Cyz is ^jjjp {Y^ + Zy) dxdydz + 2^|Jp {Y,z + Z,y) dS; (14) the average value of A is ^jjjp i^x + Yy + Z^) dxdydz + ^jj (X^ + Y,y + Z^z) dS... (15) The following results* may be obtained easily from these formulss : — (i) A solid cylinder of any form of section resting on one end on a horizontal plane is shorter than it would be in the unstressed state by a length Wl/2Eca, where W is its weight, I its length, m the area of its cross-section. The volume of the cylinder is less than it would be in the unstressed state by W7/6^. (ii) When the same cylinder hes on its side, it is longer than it would be in the unstressed state by t Wh/Fw, where A is the height of the centre of gravity above the plaae. The volume of the cylinder is less than it would be in the unstressed state by Whist (iii) A body of any form compressed between two parallel planes, at a distance c apart, will have its volume diminished by pcjSk, where p is the resultant pressure on either plane. * Numerous examples of the application of these formulae, and the corresponding formulse for an asolotropio body, have been given by C. Chree, Cambridge Phil. Soc. Trans., vol. 15 (1892), p. 313. 174 GENERAL THEORY [CH. VII If the body is a cylinder with plane ends at right angles to its generators, and these ends are in contact with the compressing planes, its length will be diminished by pcjEa, where m is the area of the cross-section. (iv) A vessel of any form, of internal volume Fj and external volume V^, when subjected to internal pressure joi and external pressure poi 'will be deformed so that the volume T^ - F, of the material of the vessel is diminished by the amount {poVt^— piVijjk. 124. The general problem of vibrations. Uniqueness of solution. When a solid body is held in a state of strain, and the forces that maintain the strain cease to act, internal relative motion is generally set up. Such motions can also be set up by the action of forces which vary with the time. In the latter case they may be described as " forced motions." In problems of forced motions the conditions at the surface may be conditions of displacement or conditions of traction. When there are no forces, and the surface of the body is free from traction, the motions that can take place are "free vibrations.'' They are to be determined by solving the equations of the type dx UJ 92/ KdeJ^dz [deJ'P dt'' ^^^^ in a form adapted to satisfy the conditions of the type dW dW dW cos (x, v)^— + cos (y, v)- — + cos (z, z/) r— = (17) OBxx OBxy OSzx I at the surface of the body. There is an infinite number of modes of free vibration, and we can adapt the solution of the equations to satisfy given conditions of displacement and velocity in the initial state. When there are variable body forces, and the surface is free from traction, free vibrations can coexist with forced motions, and the like holds good for forced motions produced by variable surface tractions. The methods of integration of the equations of free vibration will occupy us immediately. We shall prove here that a solution of the equations of free vibration which also satisfies given initial conditions of displacement and velocity is unique*. If possible, let there be two sets of displacements («', v', w') and {u", v", w") which both satisfy the equations of type (16) and the conditions of type (17), and, at a certain instant, t = to, let (u', v , w) = (m", v", w") and /9m' a?/ Sm/\ ^ /du" diT dw"\ \dt' dt" dt)~[~dt ' 'dt' ~di)- The difference (u' - u", v' - v", w - w") would be a displacement which would also satisfy the equations of type (16) and the conditions of type (17), and, * Cf. F. Neumann, Vorlesungen uber...Elasticitat, p. 125. 123-125] OF VIBRATIONS 175 at the instant * = io,this displacement and the corresponding velocity would Tanish. Let (u, v, w) denote this displacement. We form the equation Ml 'du( ^ _d_ /dW\ _ d_ /dW\ _ d_ idW\ \ _ a_ /dw\ _ d_ idW\] J ^y\^e^) dz\dej) dtY dt^ dxKdeJ dyXdeJ d2\dej] + dv Pdt^ dt dw { d^w di d_ fdW\ _ ^ /dW\ _ d_ fdW\ dos \de. ) dy\deyyj dz\deyj] {pw-wM)-h © - 1 ©}] '^''^y'^' = ^' • • -^^'^ in which the components of strain, e^x---, and the strain- energy- function, W, are to be calculated from the displacement {u, v, w). The terms containing p can be integrated with respect to t, and the result is that these terms are equal to the kinetic energy at time t calculated from dujdt, ..., for the kinetic energy at time 4 vanishes. The terms containing W can be transformed into a surface integral and a volume integral. The surface integral is the sum of three terms of the type and this vanishes because the surface tractions calculated from {u, v, w) vanish. The volume integral is />//. 'dWde, de. dWde^ dt deyy dt dWde^, dWdey^ de^z dt deyg dt + dWde,^ dWde. de,. + dt de^ jay dt dxdydz. and this is the value of 1 1 1 Wdxdydz at time t, for W vanishes at the instant < = ^o.ljecause the displacement vanishes throughout the body at that instant. Our equation (18) is therefore fduY . fdv^ /dw\ SmiTJ(-«+-+-«')+g— g^ + g^^+g^pr The right-hand member may be transformed into a volwiie integral and a surface integral. The terms of the volume integral which contain ii are and the terms of the surface integral which contain u are II'- •s — COS (^, i')+o — cos (y, i')+-^ — cos (z, v)^ «r!>T, + wZ^. This vector therefore may be used to calculate the flux of energy. 126. Free vibrations of elastic solid bodies. In the theory of the small oscillations of dynamical systems with a finite number of degrees of freedom, it is shown that the most general small motion of a system, which is slightly disturbed from a position of stable equilibrium, is capable of analysis into a number of small periodic motions, each of which could be executed independently of the others. The number of these special types of motion is equal to the number of degrees of freedom of the system. Each of them is characterized by the following properties : — (i) The motion of every particle of the system is simple harmonic. (ii) The period and phase of the simple harmonic motion are the same for all the particles. (iii) The displacement of any particle from its equilibrium position, estimated in any direction, bears a definite ratio to the displacement of any chosen particle in any specified direction. When the system is moving in one of these special ways it is said to be oscillating in a " principal " (or " normal ") mode. The motion consequent upon any small disturbance can be represented as the result of superposed motions in the different normal modes. 125, 126] FREE VIBRATIONS 177 When we attempt to generalize this theory, so as to apply it to systems with infinite freedom, we begin by seeking for normal modes of vibration*. Taking p/27r for the frequency of such a mode of motion, we assume for the displacement the formulae u = u' cos {pt + e), v = v' cos {pt + e), w = w' cos {pt + e), ... (22) in which u', v', w' are functions of x, y, z, but not of t, and p and e are constants. Now let W be what the strain-energy-function, -W, would become if u', v', w' were the displacement, and let X'a;, . . . be what the stress-components would become in the same case. The equations of motion under no body forces take such forms as ^f^'^^'-w*^-'-"-' '^^> and the boundary conditions, when the surface is free from traction, take such forms as cos (a;, v)X'x + (ios{y, v)X'y+ coa(z, v)Z'x = (24) These equations and conditions suffice to determine u', v', w as functions of X, y, z with an arbitrary constant multiplier, and these functions also involve p. The boundary conditions lead to an equation for p, in general transcendental and having an infinite number of roots. This equation is known as the " frequency-equation." It thus appears that an elastic solid body possesses an infinite number of normal modes of vibration. Let pi, P2, ... be the roots of the frequency-equation, and let the normal mode of vibration with period ^irjpr be expressed by the equations U = ArUy cos {prt + e^), V = ArVr COS {p^t + e,), W = ArWr COs{prt -(- 6,.), . . .(25) in which Ar is an arbitrary constant multiplier; the functions Ur, Vr, Wr are called " normal functions." The result of superposing motions in the different normal modes would be a motion expressed by equations of the type in which ^, stands for the function Ar cos (prt + Sr). The statement that every small motion of the system can be represented as the result of super- posed motions in normal modes is equivalent to a theorem, viz.: that any arbitrary displacement (or velocity) can be represented as the sum of a finite or infinite series of normal functions. Such theorems concerning the ex- pansions of functions are generalizations of Fourier's theorem, and, from the point of view of a rigorous analysis, they require independent proof. Every problem of free vibrations suggests such a theorem of expansion. * See Clebsoh, Elasticitat, or Lord Rayleigh, Theory of Sound, vol. 1. 12 178 GENERAL THEORY OF [CH. VII 127. General theorems relating to free vibrations*. (i) In the variational equation of motion jjjhWd^rdydz +jjjp (^ 8m + |^ 8v + ^ Swj dxdydz = ...(27) let u, V, w have the forms Wr^r, Vrr, Wrs, Vs(j)s, Wg^s, where <^r and <^s stand for A^ cos {p^t + e^) and Ag cos {p^t + €«), and the constants A^ and Ag may be as small as we please. Let W become ir, when Ur, Vr, Wr are substituted for u, v, w, and become Wg when Vg, Vg, Wg are substituted for u, v, w. Let e denote any one of the six strain-com- ponents, and let e, and eg denote what e becomes when Ur, Vr, w,- and Vg, Vg, Wg respectively are substituted for u, v, w. Then the variational equation takes the form \\\'^ i^^'' ®«) dxdydz = Pr^\\\p (UrUg + VrVg + WrWg) dxdydz. The left-hand member is unaltered when e^ and gg are interchanged, i.e. when u, V, w are taken to have the forms Ug<^g, ... and hu, Bv, Bw are taken to have the forms Urj>r, ■■■ and then the right-hand member contains p/ instead of p^^. Since p,- and pg are unequal it follows that ///' I p (UrUg + Vri>s + w^Wg) dxdydz = (28) This result is known as the "conjugate property" of the normal functions. (ii) We may write (^,. in the form A,, coa prt + B^ sin p^t, and then the conjugate property of the normal functions enables us to determine the con- stants Ar, Br in terms of the initial displacement and velocity. We assume that the displacement at any time can be represented in the form (26). Then initially we have U,i = %ArUr, Vo='^ArVr, Wq = S^rW^, (29) Ua = '^BrPrUr, ilf, — liBrPrVr, Wo = l.BrPr'Wr (30) where (u„, v^, w^) is the initial displacement and {do, v^, w^) is the initial velocity. On multiplying the three equations of (29) by pUr, pVr, pWr respectively, and integrating through the volume of the body, we obtain the equation Arj j \ p {Ur^ + Vr^ + w/) dxdy dz = 1 1 \p{UoUr+ v„Vr + WoW,.) dxdydz. ...(31) The other coefficients are determined by a similar process. (iii) The conjugate property of the normal functions may be used to show that the frequency-equation cannot have imaginary roots. If there * These theorems were given by Clebsch as a generalization of Poisson's theory of the vibra- tions of an elastic sphere. See Introduction. 127,128] FREE VIBRATIONS 179 were a root ^^^ of the form a + tyS, there would also be a root ja/ of the form a-t^. With these there would correspond two sets of normal functions Mr, '»r, Wr and Mj, Vg, Wg which also would be conjugate imaginaries. The equation \\\p{urUs + VrVg + WrWg) dxdy dz = could not then be satisfied, for the subject of integration would be the product of the positive quantity p and a sum of positive squares. It remains to show that p/ cannot be negative. For this purpose we consider the integral \\\p {ur^ + Vr^ f w^) dxdydz, which is equal to where Zj,"'', ... are what X^, ... become when Ur, Vr, w^ are substituted for u, V, w. The expression last written can be transformed into - Pr~"j[['>^T (cos (x, v) X^f' + cos (y, v) X/^ + cos {z, v)Z^''^)] + ... + ...]dS^ + pr-^ 'jj2 Wrdxdydz, in which the surface integral vanishes and the volume integral is necessarily- positive. It follows that p/ is positive. 128. Load suddenly applied or suddenly reversed. The theory of the vibrations of solids may be used to prove two theorems of great importance in regard to the strength of materials. The first of these is that the strain produced by a load suddenly applied may be twice as great as that produced by the gradual application of the same load ; the second is that, if the load is suddenly reversed, the strain may be trebled. To prove the first theorem, we observe that, if a load is suddenly applied to an elastic system, the system will be thrown into a state of vibration about a certain equilibrium configuration, viz. that which the system would take if the load were applied gradually. The initial state is one in which the energy is purely potential, and, as there is no elastic stress, this energy is due simply to the position of the elastic solid in the field of force constituting the load. If the initial position is a possible position of instantaneous res% in a normal mode of oscillation of the system, then the system will oscillate in that normal mode, and the configuration at the end of a quarter of a period will be the equilibrium configuration, i.e. the displacement from the equilibrium configuration will then be zero ; at the end of a half-period, it 12—2 180 SUDDEN APPLICATION OR REVERSAL OF LOAD [CH. VII will be equal and opposite to that in the initial position. The maximum displacement from the initial configuration will therefore be twice that in the equilibrium configuration. If the system, when left to itself under the .suddenly applied load, does not oscillate in a normal mode the strain will be less than twice that in the equilibrium configuration, since the system never passes into a configuration in which the energy is purely potential. The proof of the second theorem is similar. The system being held strained in a configuration of equilibrium, the load is suddenly reversed, and the new position of equilibrium is one in which all the displacements are reversed. This is the position about which the system oscillates. If it oscillates in a normal mode the maximum displacement from the equilibrium configuration is double the initial displacement from the configuration of no strain ; and, at the instant when the displacement from the equilibrium con- figuration is a maximum, the displacement from the configuration of no strain is three times that which would occur in the equilibrium configuration. A typical example of the first theorem is the case of an elastic string, to which a weight is suddenly attached. The greatest extension of the string is double that which it has, when statically supporting the weight. A typical example of the second theorem is the case of a cylindrical shaft held twisted. If the twisting couple is suddenly reversed the greatest shear can be three times that which originally accompanied the twist. CHAPTER VIII. THE TRANSMISSION OF FORCE. 129. In this Chapter we propose to investigate some special problems of the equilibrium of an isotropic solid body-~under no body forces. We shall take the equations of equilibrium in the forms ^'^ + ^)(a^' I' l,)^ + f^^Hu,v,^) = o, (1) and shall consider certain particular solutions which tend to become infinite in the neighbourhood of chosen points. These points must be outside the body, or in cavities within the body. We have a theory of the solution of the equations, by a synthesis of solutions having certain points as singular points, analogous to the theory of harmonic functions regarded as the poten- tials due to point masses. From the physical point of view the simplest singular point is a point at which a force acts on the body. 130. Force operative at a point*. When body forces (X, Y, Z) act on the body the equations of equi- librium are ^'^^"^Qx' Ty £)^ + '^'^'("' '' «') + M^. Y, Z) = o, ...(2) and the most general solution of these equations will be obtained by adding to any particular solution of them the general solution of equations (1). The effects of the body forces are represented by the particular solution. We seek such a solution in the case where {X, Y, Z) are different from zero within a finite volume T and vanish outside T. The volume T may be that of the body or that of a part of the body. For the purpose in hand we may think of the body as extended indefinitely in all directions and the volume T as a part of it. We pass to a limit by diminishing T indefinitely. We express the displacement by means of a scalar potential

* The results obtained in this Article are due to Lord Kelvin. Sse Introduction, footnote 66. 182 FORCE OPERATIVE [CH. VIII and we express the body force in like manner by means of formulse of the type .(4) dao dy dz ' Since A = V''^, ..., the equations (2) can be written in such forms as Kdy dz / \dx dy and particular solutions can be obtained by writing down particular solutions of the four equations fi^' G + pM=0, iiV^H+ pN= Now X, Y, Z can be expressed in forms of the type (4) by putting '=0, ) ^=0. I .(6) M = J. 47r 47r ///(^•^■-^'9^'*^''' 3^-1 Sf-ix (7) where X' , Y', Z' denote the values of X, Y, Z at any point {x, y', z) withio T, r is the distance of this point from x, y, z, and the integration extends through T. It is at once obvious that these forms yield the correct values for X, Y, Z at any point within T, and zero values at any point outside T. We now pass to a limit by diminishing all the linear dimensions of T indefinitely, but supposing that iljX'dx'dy'dz' has a finite limit. We pass in this way to the case of a force X,, acting at (x', y', /) in the direction of the axis of x. We have to put pf'Jlx'dx'dy'dz' = X„ (8) and then we have * = - 1 V ^~' r A ji^ 1 IT dr-' „ 1 „ dr-' ... Now V^{dr/dx) = 2dr-^/dx, and we may therefore put Stt (X. + 2/x) dx '■ ' Bir/ji. dz ' *- sll- -W 130, 131] AT A POINT 183 The corresponding forms for u, v, w are S-TTfi (X, + 2yi4) dx' 4nrfj,r ' (X + fi) Xq d^r '" 8'7rfi(\ + 2/u.)dxdy' \ ^^^^ (X + /i) Zo 3^r Stt/li (X + 2fj,) dxdz ' More generally, the displacement due to force (Z„, F„, ^„) acting at the point (x', y', z), is expressed by the equation ^ X + M fx-x' y-y' z-z' \ Xo{x-x')+Y,(y-y') + Z,(z-/) S-TTfi (X + 2/i) V ?■ ' r ' r / r^ (12) When the forces X, Y, Z act through a volume T of finite size, particular integrals of the equations (2) can be expressed in such forms as X + A^ STT/i (X + 2/^) /■/•/•JX + 3/i Z' , ,,Z'(«-a'')+F'(w-2/')+^'(i:-/)] , ,, ,, , ,,„, rnxV^'-r^f^''-^^— V^ ^ yx'dydz',...{lZ) where t,he integration extends through the volume T. It may be observed that the dilatation and rotation corresponding with the displacement (11) are given by the equations 131. First type of simple solutions * . When the force acts at the origin parallel to the axis of z we may write the expressions for the displacement in the forms xz _ . yz . 1^ X + 3/i 1 M=>i-5, v=A^, m; = 4 -5 + '"i) (1^) It may be verified immediately that these constitute a solution of equations (1) in all space except at the origin. We suppose that the origin is in a cavity within a body, and calculate the traction across the surface of the cavity. The tractions corresponding with (15) over any surfaces bounding a body are a system of forces in statical equilibrium when the origin is not a point of the body [cf. Article 117]. It follows that, in the case of the body with the cavity, the resultant and resultant moment of these tractions at the outer boundary of the body are equal and opposite to the resultant and resultant moment of the tractions at the surface of the cavity. The values of these tractions at the outer boundary do not depend upon the shape or size of the cavity, and they may therefore be calculated by * The solution expressed in equations (15) has received this title at the hands of Boussinesq, At^licatiom des Potentiels 184 EFFECT OF FORCE OPERATIVE AT A POINT [OH. VIII taking the cavity to be spherical and passing to a limit by diminishing the radius of the sphere indefinitely. In this way we may verify that the displacement expressed by (15) is produced by a single force of magnitude Stt/x (\ + 2/i) A/(K+h) applied at the origin in the direction of the axis of z. We write equations (15) in the form 2u 3r~i The cubical dilatation A corresponding with the displacement (16) is A -^ — , and A ~i~ u OZ the stress-components can be calculated readily in the forms „ _ dr dr dr~'^ ' ox cy OZ The tractions across any plane (of which the normal is in direction v) are given by the equations J,= 2H [3g- g^ ^ + xT;^ {cos (., .) ^-COS (^, .) ^| J, and, when v is the iuwards drawn normal to a spherical surface with its centre at the origin, these are _QfiAxz QfiAt/z 2/i4 / 02 ^ A-.= = 2^ A dr- dz ■HI)' X+J' n = = 'ifiA dr- dz ■Hl)'- X+J' 2^ = '2.^ A dr- dz ■HD- ^^\- y_e^Ayz 2,iA f z^ ^\ Whatever the radius of the cavity may be, this system of tractions is statically equivalent to a single force, applied at the origin, directed along the axis of z in the positive sense, and of magnitude SvixA (X+2/i)/(X+/i). Some additional results in regard to the state of stress set up in a body by the application of force at a point will be given in Article 141 infra. 132. Typical nuclei of strain. Various solutions which possess singular points can be derived from that discussed ifr Article 131. In particular, we may suppose two points at which forces act to coalesce, and obtain new solutions by a limiting process. It is convenient to denote the displacement due to force {Xq, Yu, Zq) applied at the origin by {XaUi+ Y0U2 + Z0U3, X0V1+ T(,V2 + Z„Vs, XoWi+YoW^+ZgWa), so that for example (mj, vi, ■!»,) is the displacement obtained by .replacing X^ by unity in equations (11). We consider some examples* of the synthesis of singularities : — (a) Let a force h~^P be applied at the origin in the direction of the axis of x, and let an equal and opposite force be applied at the point (A, 0, 0), and let us pass to a limit * In most of these the leading steps only of the analysis are given. The results (a') and (b') are due to J. Dougall, Edinlmrgh Math. Soc. Proc, vol. 16 (1898). 131,132] TYPICAL NUCLEI OF STRAIN 185 by supposing that h is diminished indefinitely while P remains constant. The displace- ment is We may describe the singularity as a "double force without moment." It is related to an axis, in this case the axis of x, and is specified as regards magnitude by the quantity P. {a!) We may combine three double forces without moment, having their axes parallel to the axes of coordinates, and specified by the same quantity P. The resultant displace- ment is Now the result (12) shows that we have V3 = W2, «'i = l«3, Ui = V^, (19) and thus (18) may be written P ( A, , Aj , A3), where Ai is the dilatation when the displacement is (Ml, v-^, Wi), and so on. Hence the displacement (18) is P fBr-i a?--^ 3?- ' I 4,r(X + 2;^)l Sa; ' dy ' dz ] ^ ' We may describe the singularity as a "centre of compression"; when Pis negative it may be called a "centre of dilatation." The point must be in a cavity within the body ; when the cavity is spherical and has its centre at the point, it may be verified that the traction across the cavity is normal tension of amount {^iP/(X+2;.)^}r- 3 (6) We may suppose a force h~^P to act at the origin in the positive direction of the axis of X, and an equal and opposite force to act at the point (0, h, 0), and we may pass to a limit as before. The resultant displacement is p /dui dvi dwi\ \dy ' dy^hyj' We may describe the singularity as a "double force with moment." The forces applied to the body in the neighbourhood of this point are statically equivalent to a couple of moment P about the axis of z. The singularity is related to this axis and also to the direction of the forces, in this case the axis of x. ih') We may combine two double forces with moment, the moments being about the same axis and of the same sign, and the directions of the forces being at right angles to each other. We take the forces to be A"' P and -h'^P parallel to the axes of x and y at the origin, - h-^P parallel to the axis of x at the point (0, h, 0), and h-'^P parallel to the axis of y at the point {h, 0, 0), and we pass to a limit as before. The resulting displace- ment is J/Smi dui\ (dv^ 3^\ /8wi gwgM ^\\dy'dx)' \dy~dx)' \dy dxJS' or it is 47r/i \ dy ' dx ' J We may describe the singularity as a "centre of rotation about the axis of z." The forces applied to the body in the neighbourhood of this point are statically equivalent to a couple of moment 2P about the axis ofz; the singularity is not related to the directions of the 186 DISPLACEMENT DUE TO [CH. VIII forces. In like manner we may have singularities which are centres of rotation about the axes of X and y, for which the displacements have the forms aud (c) We suppose that centres of dilatation are distributed uniformly along a semi-infinite line. The line may be taken to be the portion of the axis of z on which z is negative. The displacement is given by equations of the form where 5 is a constant, and R^ = x'+2/'^ + {z + ^y. „ fd^ 1 /"p + g'l 1 A z\ 1 "^ Jo R^~x^+2/^/ ol B.'j r'-z'\ r) r{z + r)' and the displacement is given by the equations M=S-.^, v = B-^,, w = - (24) These displacements constitute the "simple solutions of the second type*." The result may be expressed in the form {u,v,w)^b{1^, I-, ^\\og{z+r) (25) ' y 3^ ' dy' '6z j A singularity of the type here described might be called a "line of dilatation," and B might be called its " strength." If B is negative, the singularity might be called a "line of compression." {d) A line of dilatation may be terminated at both ends, and its strength may be variable. If its extremities are the origin and the point (0, 0, —V), and its strength is proportional to the distance from the origin, we have _, p/rfz' _, [^!!di „, [^{z+z')z!d^ ,_., "=^^j„^' '' = ^-^J„-l^' ^=^Jo-^^— ' ^''^ where C" is constant. Now we have fk ^dz' _ f'" I'z + z' z\ 1 1 z (z + h z\ jo^P~}o\B' "R'j r'S^'^+fK^^r)' where R^=x'^+y'^ + {z+kY. The integral remains finite when h is increased indefinitely, and we have r z^_l z (, z\_ \ Jo R^ ~ r r^-z^X fj~z+', ["{z + z')^., k f^dz' k . z + k + R, 5 + '' Again we have This does not tend to a limit when k is increased indefinitely. Let C" ((/, V, W) denote the displacement (26) ; and, in addition to the line of dilatation which gives rise to the * Boussinesq, loc. cit. 132,133] TYPICAL NUCLEI OF STRAIN 187 displacement {U, V, W), let there be a line of compression, with the same law of strength, extending from the point {h, 0, 0) to the point (A, 0, — k). We pass to a limit by taking h to diminish indefinitely and C to increase indefinitely, in such a way that G'h has a finite limit, C say. The displacement is given by the equations ^dV „9F ^dW OX ox ox „ dW kx ~ Now = + Zx M^^ Ri{z+k + £i) r{e+r)' and this has a finite limit when k is increased indefinitely, viz. —x/r (z+r). The displace- ment due to such a semi-infinite double line of singularities as we have described here is expressed by the equations n=c(^ T^X v=-C-^^, w=-C-j^^, (27) or, as they may be written, {U,v,v>)=-C(^^„ g|^, ^){.log(. + r)-r} (28) In like manner we may have (e) Instead of a line-distribution of centres of dilatation, we may take a line-distribution of centres of rotation. Prom the result of example (6') we should find M = 0, ■""fZ-w"'' -=^/„"l^^"' where Z) is a constant, and the axes of the centres of rotation are parallel to the axis of x. This gives In like manner we may have u=0, v=-~, w = D-/-^ (30) ---, v=0, w=-D-r^^ (31) or, as they may be written, {u,v,w)^dQ-^, 0, _^){log(0-fr)} (32) Other formulae of the same kind might be obtained by taking the line of singularities in directions other than the axis of z. The reader will observe that, in all the examples of this Article, except (a) and (5), the components of displacement are harmonic functions, and the cubical dilatation vanishes. The only strains involved are shearing strains, and the displacements are independent of the ratio of elastic constants X : /». 133. Local Perturbations. Examples (a) and (a') of the last Article show in particular instances how the application of equilibrating forces to a small portion of a body sets up strains which are unimportant at a distance from the portion. The displace- ment due to a distribution of force having a finite resultant for a small volume varies inversely as the distance ; that due to forces having zero resultant for 188 EFFECT OF PRESSURE APPLIED [CH. VIII the small volume varies inversely as the square of the distance, and directly as the linear dimension of the small volume. We may conclude that the strain produced at a distance, by forces applied locally, depends upon the resultant of the forces, and is practically independent of the mode of distribu- tion of the forces which are statically equivalent to this resultant. The effect of the mode of distribution of the forces is practically confined to a compara- tively small portion of the body near to the place of application of the forces. Such local effects are called by Boussinesq " perturbations locales*." The statement that the mode of distribution of forces applied locally gives rise to local perturbations only, includes Saint- Venant's "Principle of the elastic equivalence of statically equipollent systems of load," which is used in problems relating to bars and plates. In these cases, the falling off of the local perturbations, as the distance from the place of application of the load increases, is much more rapid than in the case of a solid body of which all the dimensions are large compared with those of the part subjected to the direct action of the forces. We may cite the example of a very thin rect- angular plate under uniform torsional couple along its edges. The local perturbations diminish according to an exponential function of the distance from the edgef. 134. Second type of simple solutions. The displacement is expressed by the equations given in Article 132 (c), viz. : — u = B Z' v = B - ^, . , w = -, (24fe) r{z + r) r{z + r) r ^ ' or, as they may be written, ^_^ aiog(^-fr) ^_-g d\o%{z^T ) „^._^ aiog(^ + r) 8« ' 9y ' "dz ' It may be verified immediately that these expressions are solutions of the equations (1) at all points except the origin and points on the axis of z at which z is negative. There is no dilatation, and the stress-components are given by the equations rs y I- T^iz-^rJ * Boussinesq, loc. cit. t Kelvin and Tait, Nat. Phil., Part ii. pp. 267 et seq. Cf . Articles 226 b. and 249 a. infra. 133-135] TO PART OF A BOUNDING SURFACE 189 At the surface of a hemisphere, for which r is constant and z is positive, these give rise to tractions the normal {v) being drawn towards the centre. 135. Pressure at a point on a plane boundary. We consider an elastic solid body to which forces are applied in the neighbourhood of a single point on the surface. If all the linear dimensions of the body are large compared with those of the area subjected to the load, we may regard the body as bounded by an infinite plane. We take the origin to be the point at which the load is applied, the plane 2^ = to be the bounding surface of the body, and the positive direction of the axis of z to be that which goes into the interior of the body. The local effect of force applied at the origin being very great, we suppose the origin to be excluded by a hemispherical surface. The displacement expressed by (15) could be maintained in the body by tractions over the plane boundary, which are expressed by the equations ^ \ + lx r'' ^ \ + /i r'' and by tractions over the hemispherical boundary, which are expressed by the equations (17). The resultant of the latter for the hemispherical surface is a force in the positive direction of the axis of z of amount ^iTfiA (X + 2/tt)/(X + fi). The displacement expressed by (24) could be maintained in the body by tractions over the plane boundary, which are expressed by the equations Z. = -2^5^„ F, = -2^£^, Z, = 0. (34) and by tractions over the hemispherical boundary, which are expressed by the equations (33). The resultant of the latter is a force in the positive direction of the axis of z of amount iT/jbB. If we put B= -A/j-KX + fi), the state of displacement expressed by the sum of the displacements (15) and (24) will be maintained by forces applied to the hemispherical surface only ; and, if the resultant of these forces is P, the displacement is given by the equations P wz P a> ] " 4!7rtJ,r' 4,-7r{\+fi)r{z + r)' y_^ P y^ P y 4nrfi r' 4iir {X + /j.) r(z + r)' _ _P_ z^ P{\ + 2fj-) 1 477 /u r'^ 47r/x (A. + ft,) r ,(35) 190 EFFECT OF LOOAL PRESSURE [CH. VIII At all points not too near to the origin, these equations express the displace- ment due to a pressure of magnitude P applied at the origin. For the discussion of this solution, it is convenient to regard the plane boundary as horizontal, and the body as supporting a weight P at the origin. We observe that the tractions across a horizontal plane are so that the resultant traction per unit area exerted from the upper side across the plane at any point is a force directed along the radius vector drawn from the origin and of magnitude f (P/ttt-^) cos^ 6, where d is the angle which the radius vector drawn from the origin makes with the vertical drawn downwards. The tractions across horizontal planes are the same at all points of any sphere which touches the bounding plane at the origin, and their magnitude is fP/TrZ)^ where D is the diameter of the sphere. These expressions for the tractions across horizontal planes are independent of the elastic constants. The displacement may be resolved into a horizontal component and a vertical com- ponent. The former is iTTfir \_ X+/i (H-cos5)J' it is directed towards or away from the line of action of the weight according as the radius vector is without or within the cone which is given by the equation (X + /i) cos 5(1 + cos d) = li. When Poisson's ratio for the material is ^ the angle of the cone is about 68° 32'. At any point on the bounding plane the horizontal displacement is directed towards the axis and is of amount ^Pjnr (X + /i). The vertical displacement at any point is P /X + 2^ ^ , A it is always directed downwards. Its magnitude at a point on the bounding plane is ji'(X + 2fi)/7r?-/i(X + |i). The initially plane boundary is deformed into a curved surface. The parts which are not too near the origin come to lie on the surface formed by the revolution of the hyperbola xz = lP {\ + '2,x)lnix{'k+jx) about the axis of z. 136. Distributed pressure. Instead of supposing the pressure to be applied at one point, we may suppose it to be distributed over an area on the bounding plane. Let («', y', 0) be any point of this plane, P' the pressure per unit of area at this point, r the distance of a point {x, y, z) within the body from the point («', y', 0). Let i/r denote the direct potential of a distribution P' over the area, x the logarithmic potential of the same distribution, so that ^lr = \^\^P'rdxdy', x=\\P'^°E{^ + r)dx'dy', (36) where the integrations are taken over the area subjected to pressure. We observe that V^X = 0, V^t = 2^ = 2ffycZ^W = 2<^>say, (37) 135-137] - PRESSURE BETWEEN TWO BODIES 191 where We observe that these expressions are finite and determinate for all values of (w, y, z), provided z is positive ; and that, as the point (x, y, z) approaches any point (a;', y', 0), they tend to definite finite limits. They represent the displacement at all points of the body, bounded by the infinite plane = 0, to which pressure is applied over any areaf. The normal com- ponent, w, of the displacement at any point on the surface of the body is (\ + 2/i) ^/47r/x (X + /i). 137. Pressure between two bodies in contact. — Geometrical Pre- liminaries. Let two bodies be pressed together so that the resultant pressure between them is P. The parts of the bodies near the points of contact will be com- pressed, so that there is contact over a small area of the surface of each. This common area will be called the compressed area, and the curve that bounds it the curve of compression. We propose to determine the curve of compression and the distribution of pressure over the compressed areaj. The shapes, in the unstressed state, of the two bodies near the parts that come into contact can be determined, with sufficient approximation, by equations of the form z^ = A,!)i? + B,f + 2H,xy,\ ^^^^ z^ = A^x'-+B^y^ + 2H^xy,] * These fonnulffi are due to Hertz, J. f. Math. {Crelle), Bd. 92 (1881), reprinted in Ges. Werke von Heiiirich Hertz, Bd. 1, Leipzig 1895, p. 155. t A number of special oases are worked out by Boussinesq, loc. cit. i The theory is due to Hertz, loc. cit. 192 PRESSURE BETWEEN TWO [CH. VIII the axes of z^ and z^ being directed along the normals drawn towards the interiors of the bodies respectively. In the unstressed state, the bodies are in contact at the origin of {x, y), they have a common tangent plane there, and the distance apart of two points of them, estimated along the common normal, is expressed with sufficient approximation by the quadratic form (J-i + A^) a? + (5, + 5„) ?/^ + 2 (^1 + Hi) xy. This expression must be positive in whatever way the axes of x and y are chosen, and we may choose these axes so that H^ + H^ vanishes. Then Ai + A, and 5j + B.^ must be positive. We may therefore write A, + A, = A, B, + B, = B, H, = -H„ (41) A and B being positive. If i^i, iJi' are the principal radii of curvature at the point of contact for the body (1), and R^, R^ those for the body (2), and if these have positive signs when the corresponding centres of curvature are inside the bodies respectively, we have 2(4 + 5) = l/iJi + l/i?/+l/i?, + Vi?/ (42) The angle (aj) between those normal sections of the two surfaces in which the radii of curvature are iij, Eg is given by the equation The angle (&>') between the {x, z) plane, chosen so that H^^ — Hi, and the normal section in which the radius of curvature is R^ is given by the equation If we introduce an angle t by the equation ^-^ cosT = -g-j-^, (45) so that 2A cosec^ |t = 25 sec^ ^t = 1/E, + 1/i?/ + \IR^ + 1/R^', (46) the shape of the " relative indicatrix," Ax^ + By^ = const., depends on the angle t only. When the bodies are pressed together there will be displacement of both. We take the displacement of the body (1) to he (ui, v^, Wj) relative to the axes of {x, y, z^), and that of the body (2) to be (u^, v^, w^) relative to the axes of (x, y, z^. Since the parts within the compressed area are in contact after the compression, we must have, at all points of this area, ■Si + Wi = - {z^ + Wa) + a. 137, 138] BODIES IN CONTACT 193 where a is the value of w^ + w^ at the origin*. Hence within the compressed area we have Wi+ Wi = a-Ax^-By^, (47) and outside the compressed area we must have Wi + Wj >a- Ax^-By'^ (48) in order that the surfaces may be separated from each other. 138. Solution of the problem of the pressure betMreen two bodies in contact. We denote by Xj, /ti the elastic constants of the body (1), and by Xj, /t^ those of the body (2). The pressure P between the bodies is the resultant of a distributed pressure (P' per unit of area) over the compressed area. We may form functions (/)i, ^i, Oj for the body (1) in the same way as ^, ;^, O were formed in Article 136, and we may form corresponding functions for the body (2). The values of w^ and w^ at the common surface can then be written, Wl=^1^0. W2 = ^g<^o. (4-9) where ^i =(A,i + 2/;t,)/47r/ii(\i +/ii), ^2 = (X2+ 2/i,)/47r/^(X2 4-/^2), ...(50) and (^Q is the value of (^1 or <^2 at the surface, i.e. the value of the convergent integral \P'r~^dx'dy' at a point on the surface. The value of <^o at any point within the compressed area is determined in terms of the quantity a and the coordinates of the point by the equation '^°=^¥rT¥/""^*'~-^2/o (51) This result suggests the next step in the solution of the problem. The functions denoted by <^i and ^2 are the potentials, on the two sides of the plane = 0, of a superficial distribution of density P' within the compressed area, and the potential at a point of this area is a quadratic function of the coordinates of the point. We recall the result that the potential of a homo- geneous ellipsoid at an internal point is a quadratic function of the coordinates * If the points (ajj, y^, zj of the body (1) and (sj, y^, %) of the body (2) come Into contact, we must have « and in equation (47) we identify (rCi, yj with (x.^, y^. We may show that, without making this identification, we should have Wi + MJa = o - ylXi^ - Bj/jS - 2 [^2 J (xi + aja) K - ig + -82 4 (2/1 + ^3) ("i - "2) + -H^si {a:i («i - "2) + yi K - "2) }]• In the result we shall find for w^ + w^ an expression of the order Aa?, where a is the greatest diameter of the compressed area, and itj , u^ ... wUl be of the same order in a as w^ + w^; thus the terms neglected are of a higher order of small quantities than those retained. If the bodies are of the same material we have Ui=^ and v^ — v^ when x.^=x^ and 2/1 = 2/21 ^-nd thus the identifica- tion of (Xj , 2/1) with (I2 , 2/a) leads in this case to an exact result. L. E. 13 194 PRESSURE BETWEEN TWO [CH. VIII of the point. We therefore seek to satisfy the conditions of the problem by assuming that the compressed area is the area within the ellipse, regarded as an ellipsoid very much flattened, and that the pressure P' may be obtained by a limiting process, the whole mass of the ellipsoid remaining finite, and one of its principal axes being diminished indefinitely. In the case of an ellipsoid of density p, of which the equation referred to its principal axes is x^'/a'' + y^b" + g^'/c^ = 1, the mass would be ^irpabc ; the part of this mass that would be contained in a cylinder standing on the element of area dx'dy' would be 2pdx'dy'c V(l - x'^a' - y'"-/b% and the potential at any external point would be , r /-^ X' y^ z" \ d^ '^''""Ll a^ + t b■'^r^ C» + t/{(a^+i/r)(6= + i/r)(c^+^)}*' where v is the positive root of the equation x'lip? + i^) + 2/7(6'' + I/) + z^l{& +v) = \. At an internal point we should have the same form for the potential with written for v. We have now to pass to a limit by taking c to diminish in- definitely, and p to increase indefinitely, while a. and b remain finite, in such a way that (i) f TT {pc) ab = P, (ii) 2 ipc) V(l - ^'V«' - /V&O = -f". (iii) rf)„ = wab (pc) \ (l ; r - ,/ , ) — , , the third of these conditions being satisfied at all points within the compressed area. Hence we have 3P P' = 2'!rab ^/(-s-i") (-) and ^-^^{a-Ax'^-Bf) .(63) =ipr(i ^ y^) ^ JoV a^' + f b^ + f''[{a-' + f)(b^ + 'f)f}i The equation (52) determines the law of distribution of the pressure P' over the compressed area, when the dimensions of this area are known. The equation (53) must hold for all values of x and y within this area, aijd it is therefore equivalent to three equations, viz. a = ^P(% + %)r ^ ., J {{a' + a/t) (b' + yjr) -f }5 A = iP (% + %) f J^ 5 = |P(^i + ^,)[ dyjr .'o (62 + ^)l{(a2 + i^)i|^}i', .(54) 138] BODIES IN CONTACT 195 The second and third of these equations determine a and h, and the first of them determines a when a and b are known. If we express the results in terms of the eccentricity (e) of the ellipse, e will be determined by the equation ■'o(l + #{f(l-e^ + D!* •'«(1- dK (l-e^ + ?)*i?(H-r))*' di; (i + r)M?(i-e^ + r)!* a will be given by the equation and a will be given by the equation 3P ,e. . . . f " d^ ...(55) .(56) t=^(^l + a,) — 4a Jo {^(1. !*• .(57) {i:(i+0(i-e^+r)p We observe that e depends on the ratio A : B only. Hertz has tabulated the values of 6/a, = (1 — ^)^, in terms of the angle t, of which the cosine is {B — A)I{B + A). He found the following results : — 7- = 90° 80° 70° 60° 50° 40° 30° 20° 10° 0° hla= 1 0-79 0-62 0-47 0-36 0-26 0-18 0-10 0-05 At points on the plane z = which are outside the compressed area, ^o is the potential, at external points in this plane, due to the distribution P' over the compressed area. It follows from (49) that at points on the surfaces of the bodies, outside the compressed area and not far from it, we may write, with sufficient approximation ZP r f. <^ _ _t_ \ d'^ Wi + w, = (^, + ^, zp_r where v is the positive root of the equation «V(a^ + v) + 2/7(6'' + v) = 1 (58) Hence we have {w^^w^-{(x-Ax'-By'') SP = -(^. + ^.)^J^(i-^r^-^ d-lf .(59) Now, when -^ lies between and v, the point («, y), which is on the ellipse (58), is outside the ellipse a;V(a2 + i/r) + y7(6'' + i^r) = 1, and therefore the expression on the right-hand side of equation (59) is positive. The condition of inequality (48) is therefore satisfied. The assumptions that the compressed area is bounded by an ellipse x^ja? + y^lh'^=\, where a and b are determined by the second and third of equations (54), and that the pressure P' over this area is expressed by the 13—2 196 HERTZ S THEORY [CH. VIII formula (52), satisfy all the conditions of -the problem. When P' is known the functions (p, x> ^ ^^^ ^^^^ °^ ^^^ bodies can be calculated, and hence we may determine the displacement and the distribution of stress in each body. Hertz* has drawn the lines of principal stress in the {x, z) plane for the case in which X = 2f* (Poisson's ratio —\). His drawing was in part conjectural, as the difierential equation determining the directions of the lines of principal stress cannot be integrated exactly. A more exact result has been obtained by S. Fuchst, by a method of approximate inte- gration, in the case of a sphere resting on a plane. The lines of principal stress in the body with the spherical boundary are represented in Fig. 15, where the full curved lines Fig. 1.5. .■ire lines of principal stress along which the traction is pressure, and the dotted lines are lines of principal stress along which the traction is tension. It will be observed that near the compressed area both the principal stresses are pressures. A little further away one set of lines shows tension near the surface and pressure in the central portions. Still further away the same set of lines shows tension throughout. The other set of lines are always lines of pressure. Hertz made a series of experiments with the view of testing the theory. The result that the linear dimensions of the compressed area are proportional to the cube root of the pressure between the bodies was verified very exactly ; the dependence of the form of the compressed area upon the form of the relative indicatrix was also verified in cases in which the latter could be determined with fair accuracy. 139. Hertz's theory of impact. The results obtained in the last Article have been applied to the problem of the impact of two solid bodies^. The ordinary theory of impact, founded by Newton, divides bodies into two classes, " perfectly elastic " and " imper- fectly elastic." In the case of the former class there is no loss of kinetic energy in impact. In the other case energy is dissipated in impact. Many actual bodies are not very far from being perfectly elastic in the Newtonian sense. Hertz's theory of impact takes no account of the dissipation of energy; the compression at the place of contact is regarded as gradually produced and as subsiding completely by reversal of the process by which it is produced. The local compression is thus regarded as a statical effect. In order that such a theory may hold it is necessary that the duration of the impact should * Verhandlungen des Vereim zur Beforderung des Gewerbefleisses, 1882, reprinted in Hertz, Ges. Werke, Bd. 1, p. 174. t Physikalische Zdtschr., 1913, p. 1282. t Hertz, /./. Math. {Crelle), Bd. 92 (1881). 138, 139] OF IMPACT 197 be' a large multiple of the gravest period of free vibration of either body which involves compression at the place in question. A formula for the duration of the impact, which satisfies this requirement when the bodies impinge on each other with moderate velocities, has been given by Hertz, and the result has been verified experimentally *. At any instant during the impact, the quantity a is the relative displace- ment of the centres of mass of the two bodies, estimated from their relative positions at the instant when the impact commences, and resolved in the direction of the common normal. The pressure P between the bodies is the rate of destruction of the momentum of either. We therefore have the equation ^U^J^ = ^P, (60) where d stands for dajdt, and m^, m.^ are the masses of the bodies. Now P is a function of t, so that the principal semi-diameters a and h of the compressed area at any instant are also functions of t, determined in terms of P by the second and third of equations (54) ; in fact a and h are each of them pro- portional to P^. Equation (57) shows that a is proportional to P^, or that P is proportional to a^ ; we write P = hS (61) where {iyk^A{% + %f r dx 1"^ r Jo {f(i + f)(i_e=+f)}ij ■ ifl (l + r)*{r(l-e^ + DP (62) Equation (60) may now be written ■d = -hk^(t^, (63) where k^ — (mj 4- m^jmim^,. This equation may be integrated in the form i(d^-t)0 = -|X;Aa*, (64) where v is the initial value of d, i.e. the velocity of approach of the bodies before impact. The value of a at the instant of greatest compression is .(6.5) ' 5 Y f'vy hkj \2) ' and, if this quantity is denoted by «!, the duration of the impact is g /•'■' da. * Sohneebeli, Arch, des sci. phys., Geneva, t. 15 (1885). Investigations of the duration of impaet in the case of high velocities were made by Tait, Edinburgh Boy. Soc. Trans., vols. 36, 37 (1890, 1892), reprinted in P. G. Tait, Scimtific Papers, vol. 2, Cambridge 1900, pp. 222, 249. The theory will be discussed further in Chapter XX infra. 198 DISPLACEMENT DUE TO NUCLEI OF STRAIN [CH. VIII We may express Oj in terms of the shapes and masses of the bodies and the velocities of propagation of waves of compression in them ; let Fj and V^ be these velocities*, p^ and p^ the densities of the bodies, a^ and o-j the values of Poisson's ratio for the two materials ; then (1 - T.y ^ (1 - '^^y % = so that ■7rr-'p,{l-2;? .(70) Hence the duration of the impact and the radius of the (circular) compressed area are determined. In the particular case of equal spheres of the same material the duration of the impact is ,.^n..,{rm}'jn "■' where r is the radius of either sphere, o- is the Poisson's ratio of the material, and V is the velocity of propagation of waves of compression. The radii of the circular patches that come into contact are each equal to m _16 l-2(r These results have been verified experimentally +. .(72) * F,2 is (Xi + 2Mi)/ft and V,' is (Xs + 2^)/p2. See Chapter XIII infra. t Schneebeli, Rep. d. Phys., Bd. 22 (1886), and Hamburger, Tagehlatt d. Nat. Vers, in Wies- baden, 1887. 139-141] EXPRESSED IN TERMS OF POLAR COORDINATES 199 141. Effects of nuclei of strain referred to polar coordinates. We may seek solutions of the equations (1) in terms of polar cooi'dinatea, the displace- ment being taken to be inversely proportional to the radius vector r. The displacement must satisfy equations (49) of Article 97. If we take u^ and Ug to be proportional to cos n^, and ^ to be proportional to sian(/>, we may show that* ^^cosMj (^ ()i + cos 6) tan" ~+B (k- cos 6) cot" ^l , sin nd) („ , 6 _ , 6] S^'r ^ \C tan" ~-J) cot" -| , coswdi f X-l-2u j-^A ^^ 6 „ ,„fll «r= ^J-_!_^ ~+etan"-+i)cot''5L »• I /I cos»0 2 2J where A, B, C, D are arbitrary constants; and then we may show that «e= — ^^i--H-^smfl-T3( -l+cosfl (Ctan"jr+i>cot"5) r sm fl ( 2/x rffl \cos ?i(^/ \ 2 2/ + (?tan"| + fi^cot"^|, „,=^L^M J^_,„,g/(.tan»|-iJcot-f)-(?tan-§ + ^cot"a, where G' and fl^ are arbitrary constants. In the particular cases where «=0 or 1 some of the solutions require independent investigation. These cases include the first type of simple solutions for any direction of the applied force, the second type of simple solutions, and the solutions arrived at in Article 132, examples (d), (e). We give the expressions for the displacements and stress-components in a series of cases. * J. H. Miohell, London Math. Soc. Proc., vol. 32 (1900), p. 23. 200 FORCE APPLIED AT [CH. VIII (a) The first type of simple solutions, corresponding with a force F parallel to the axis of z, is expressed by the equations F cos 5 X + 3/i F ^\n6 the stress-components are expressed by the equations -— 3X-|-4u -^ cosfl S2 r> u F cose "■=-^+2)7 4^-^' ^^='^''' = XT2;:4^-^' The meridian planes (0 = const.) are principal planes of stress ; and the lines of principal stress, which are in any meridian plane, make with the radius vector at any point angles ^ determined by the equation tan 2^|^ = - {2/i/(3X + 5/i)} tan e. These lines have been traced by Michell, for the case where X = fi, with the result shown in Fig. 16, in which the central point is the point of application of the force. O) When the line of action of the force F' is parallel to the axis ofx, the displacement is expressed by the equations _ F' sin5cos<^ _ \ + 3fi. F' cos5cos0 _ X + 3/i F' sin0 "••"4^^ r ' "«~2(X-)-2;t.) i^ij. r ' ''*" ~ 2(X-|-2;ii) 'ii^ ~T' ' the stress-components are expressed by the equations •— 3X -I- 4/i F' sin 6 COS (b 'Ty. <--, iJ- F' sin 6 cos A "•=-XT%r4^---7^-' ^^='^<^=xf2M4^-^^' •;ri r, ^ IJ. F' sm -^ IJ- F' COS 6 cos -r- „ C sin <6 'tj „ C cosd) 141, 142] VERTEX OF CONE 201 (f) The solution (31) obtained in Article 132 (e) is expressed by the equations D sin 6 cos tb D , D . % = - 1 ,„„„/! ) Mo= — cosd, u*,= sin (A; r 1+cosS " r -ri V J, T- ; the stress-components are expressed by the equations '^ ^ - 2) sin ^ cos d) -n ^ '^r2 1 + C0SI9 ^^ ' 21 2) sin 5 sin d) r- 2) /„ 1 \ • ^ Oi -0 /„ 1 \ . e, r6=-u.—„ [2 — - ;: cosd). , f-^2 i+cos5 ^ ^r^V l + cosSy ^' '^ r^ \ l+cosfl/ ^ 142. Problems relating to the equilibrium of cones*. (i) We may combine the solutions expressed in (a) and (y) of the last Article so as to obtain the distribution of stress in a cone, subjected to a force at its vertex directed along its axis, when the parts at a great distance from the vertex are held fixed. If fl = a is the equation of the surface of the cone, the stress-components 66, 0^, rd must vanish when 6 = a, and we have therefore X + 2fi 4-77 1-1-cosa The resultant force at the vertex of the cone may be found by considering the traction in the direction of the axis of the cone across a spherical surface with its centre at the vertex ; it would be found that the force is F 2(\ + 2^) ^^^'^~ °°^^ «) + /* (1 - cos a) (1 4- cos^ a)}, and, when F is positive, it is directed towards the interior of the cone. By putting 0=^77 we obtain the solution for a point of pressure on a plane boundary (Article 135). (ii) We may combine the solutions expressed in (/3), (8), (e) of the last Article so as to obtain the distribution of stress in a cone, subjected to a force at its vertex directed at right angles to its axis. The conditions that the surface of the cone may be free from traction are 20 — -. DBma=0, sm a F' 2C- 1) (1 + 2 00s a)- ■ cos a (1 -I- cos a) = 0, -2C-^. " + 2DBma + -. — ,. , - , sina(l-f-cosa) = 0, sma 47r(A + 2/i) _ i?"(l-|-cosa)2 F'{\+ coaa) S^"°S ^=- 8,r(X-l-2^) ' ^~- in(\ + 2^) ' The resultant force at the vertex is in the positive direction of the axis of a;, when F' is positive, and is of magnitude /"(2-t-cosa)X-|-2fi,, .„ — 5^ r — ~ (1 - cos a)^. 4 \ + 2^ "■ ' By combining the results of problems (i) and (ii) we may obtain the solution for force acting in a given direction at the vertex of a cone; and by putting 0=^77 we may obtain the solution for force acting in a given direction at a point of a plane boundary. * Miohell, loc. eit. CHAPTER IX. TWO-DIMENSIONAL ELASTIC SYSTEMS. 1 43. Methods of the kind considered in the last Chapter, depending upon simple solutions which tend to become infinite at a point, may be employed also in the case of two-dimensional elastic systems. We have already had occasion (Chapter v.) to remark that there are various ways in which such systems present themselves naturally for investigation. They are further useful for purposes of illustration. As in other departments of mathematical physics which have relations to the theory of potential, it frequently happens that the analogues, in two dimensions, of problems which cannot be solved in three dimensions are capable of exact solution ; so it will appear that in the theory of Elasticity a two-dimensional solution can often be found which throws light upon some wider problem that cannot be solved completely. 144. Displacement corresponding with plajie strain. In a state of plane strain parallel to the plane {x, y), the displacement w vanishes, and the displacements u, v are functions of the coordinates x, y only. The components of rotation ts^ and ■nry vanish, and we shall write 07 for 13-2. When there are no body forces, the stress-equations of equilibrium show that the stress-components Xx, Ty, Xy can be expressed in terms of a stress-function -x^, which is a function of x and y, but not of z, by the formulae ^'''df "'dx'^' y~ dxdy ^^ The identical relation between strain-components (Article 17) P ^xx , ^yy _ O ^xy /a\ 3/ dx' dxdy ^ ^ -X||+4(X + ^)||, = 0, = (3) We shall denote the operator d'/dx^ + d'/dy^ by Vj^, and then this equation is ^I'X — ^- -'■t shows that ^i'x i^ ^ plane harmonic function. takes the form If-^-'g- -^da-]^dx^X-^ + ^''^dx^ or ^'x.^x,^ ^'x dx* dy" "dofidy'' 143,144] DISPLACEMENTS CORRESPONDING WITH PLANE STRAIN 203 The equations of equilibrium in terms of dilatation and rotation are From these we deduce that A and m- are plane harmonic functions, and that (\ + 2/ji) A + i2fii!r is a function of the complex variable cc + ly. The plane harmonic function V^^i^ is equal to 2 (\ + ;tt) A. We introduce a new function |^ + 1?? of a; + ty by means of the equation f + t'>7= n(X+2/i)A+ t2/f5j-} d (a; + iy) (5) so that .(6) Then we have Also we have It follows that dy dx '^dx dy^ 2(\ + fj,) '^ dx'^dx' ■^9y dx'' 2(X4-^) '-^ dy^ dy' dy dxdy dxdy dy' 2u^ = -i!^ + 2»,iir = - -^ +^ . ^dx dxdy dxdy dx' TUfese equations enable us to express the displacement when the stress- function X is known. Again, when A and cr are known, we may find expressions for u, v. We have the equations ^u dv _ dv _du _ ,_, dx dy ' dx dy These, with (6), give _ ^ / yv \ ^ fy^\ , / " ~ a^ U (\ + 2^)r 92/ l2/.r ** ' _a/ yv \_l(yj\_^^, dy\2{\+2u,)J dx\2^J^ ' in which [Article 14 (d)] v' + m' is a function of x + ly. We may put dx' dy' 204 DISPLACEMENTS CORRESPONDING WITH [CH. IX where / is a plane harmonic function, and then u, v can be expressed in the forms , V \ + /i, dy df 1 2' a„ ■•" ; .(9) 2(\ + 2//,) 2/M(\+2/j.)-^dy dy' We may show without difficulty that the corresponding form of x is ^=-^''/+j!t$^'" (1°) and we may verify that the forms (7) for u, v are identical with the forms (9). 145. Displacement corresponding with plane stress. In the case of plane stress, when every plane parallel to the plane of w, y is free from traction, we have Xz = F^ = Zj = 0. We wish to determine the most general forms for the remaining stress-components, and for the corre- sponding displacement, when these conditions are satisfied and no body forces are in action. We recall the results of Article 92 (iv). It was there shown that, if @ = Xx +Yy + Z^, the function is harmonic, and that, besides satisfying the three equations of the type dXx dXy dZx_f. ,.,, 'd^'^^^Tz'^' ^"^ the stress-components also satisfy six equations of the types -^-rf.^-. "--rf.H- <-' Since X^, Tz, Z-^ are zero, 'b%l'dz is a constant, /3 say, and we have @ = ©o + /8^ (13) where @o is a function of x and y, which must be a plane harmonic function since @ is harmonic, or we have V,=e„ = (14) The stress-components X^, Yy, Xy sue derived from a stress-function x< svhich is a function of x, y, z, in accordance with the formulae (1), and we have Vi^X = 00 -I- /8^ (15) The first of equations (12) gives us 82/^ 1 + o- W ' or, in virtue of (14) and (1 5), dy^ Kdz'^ + **V i + or ay" \dz^ V l + a dy- '■^•.o, 144, 145] PLANE STRAIN AND PLANE STRESS 205 In like manner the remaining equations of (12) are da? W + IT^®»J-^' .fS+.4-_@o) = 0. dxdy Xdz^ 1 + o- It follows that -^ + cj @o is a linear function of x and y, and this function oz 1 + 0" may be taken to be zero without altering the values of X^, Yy, Xy. We therefore find the following form for ;\; : — %=%o+Xl^-il^®o^^ (16) where )(_„ and p(;i are independent of e and satisfy the equations V,=X„ = e„, V,^^, = /3 (17) We may introduce a pair of conjugate functions ^ and t] oi w and y which are such that 8.. + /* 2/t (\ + 2/*) \ + M A%^u,\ 2 (\ + 2/ii) To make v one-valued we must put A '" 2(\ + 2,Lt) The formulae for u, v then become (26) The corresponding (27) 0, A 2 (\ + 2/i,) logr. u = v = ■ A log r + 2/i(A, + 2/i) X + H ^sey 2/i,(\ + 2/i) r"' A r >- + /* .(28) The stress-components X^, Ty, Xy are given by the equations X( 2\ + ^,JL 2{X + ,JL) f\ 2(\ + /.) f\ \ + 2fi X + 2fi 2{X + fi} or 5- .(29) «~ 7^\X+2fj, ' X+2fi The origin must be in a cavity within the body ; and the statical resultant of the tractions at the surface of the cavity is independent of the shape of the cavity. The resultant may be found by taking the cavity to be bounded (in the plane) by a circle with its centre at the origin. The component in the direction of the axis of x is expressed by the integral j^ -{x4.X.f)r^. 208 EFFECT OF FORCE OPERATIVE [CH. IX which is equal to — IAtt. The component in the direction of the axis of y vanishes, and the moment of the tractions about the centre of the cavity also vanishes. It follows that the state of stress expressed by (29) is that produced by a single force, of magnitude lirA, acting at the origin in the negative sense of the axis of x. The effect of force at a point of a plate may be deduced by writing \' in place of X and replacing u, X^, ... by m, X^ 149. Force operative at a point of a boundary. If the origin is at a point on a boundary, the term of (27) which contains Q can be one-valued indepen- dently of any adjustment of v! , v'. It is merely necessary to fix the meaning of e. In Fig. 17, OX is the initial line, drawn into the plate, and the angle XOT=a. Then may be taken to lie in the interval We may seek the stress-system that would correspond with (27) if u' and v' were put equal to zero. We should find ^ 2(\ + /x ) ^ =" X + 2fi r*' Pig. 17. 2{X + ^) xf _ 2(\ + f,) x>y ^y- x+2f. ^^' ^'- x + 2/. ^^ ^^^) In polar coordinates the same stress-system is expressed by the equations p^2JX^)^c_os^ ^ = 0, Te^O (31) This distribution of stress is described by Michell* as a "simple radial distribution." Such a distribution about a point cannot exist if the point is within the body. When the origin is a point on the boundary, the state of stress expressed by (31) is that due to a single force at the point. We calculate the resultant traction across a semicircle with its centre at the origin. The a;-component of the resultant is or it is X-I-/X \ + 2fi. — I rr . cos 6 . rdO, J —IT+a. IT. The y-component of the resultant is -/ rr . sin .rd6. — TT+a London Math. Soc. Proc, vol. 32 (1900), p. 35. 148-151] AT A POINT OF A PLATE 209 or it is zero. Thus the resultant applied force acts along the initial line and its amount is ■7rA(X + fj,)/(X + 2fi); the sense is that of the continuation of the initial line outwards from the body when A is positive. This result gives us the solution of the problem of a plate with a straight boundary, to which force is applied at one point in a given direction. Taking that direction as initial line, and F as the amount of the force, the stress- system is expressed by the equations ;^=__i^£^!_ ,^ = 0, ^=0, (32) and these qiiantities are of course averages taken through the thickness of the plate. 150. Case of a straight boundary. In the particular case where the boundary is the axis of x, the axis of y penetrates into the plate, and the force at the origin is pressure F directed normally inwards, the average Stresses and displacements are expressed by the equations ^':---^-^' ^v=--^:pi' ^v=--J^^, (33) and ,7= ^ (e-"!^) + J^ 27r/x(X' + ^) ^ 2,r/i;-2 .(34) This solution* is the two-dimensional analogue of the solution of the problem of Boussinesq (Article 135). Since u, ?do not tend to zero at infinite distances, there is some difficulty in the application of the result to an infinite plate ; but it may be regarded as giving correctly the local eflfect of force applied at a point of the boundary. 151. Additional results. (i) The sti-ess-function corresponding with (32) of Article 149 is — 7r~'/Vflsin 6. (ii) The effect of pressure distributed uniformly over a finite length of a straight boundary can be obtained by integration. Itp is the pressure per unit of length, and the axis of X is the boundary, the axis of y being drawn into the body, the stress-function is found to be ^-ir'^p {('a^^a — '"i^^i)}. where rj, 6i and ;-2, 02 are polar coordinates with the axis of .r for initial line and the extremities of the part subject to pressure for origins. It may be shown that the lines of stress are confocal conies having these points as focit. (iii) Force at an angle. The results obtained in Article 149 may be generalized by supposing that the boundary is made up of two straight edges meeting at the origin. Working, as before, with the case of plane strain, we have to replace the Umit —ir+a of integration in the calculation of * Flamant, Paris, C. R., t. 114, 1892. For the verification by means of polarized Hght see Mesnager in Bapports presentes au emigres internationai de physique, t. 1, Paris 1900, p. 348. Cf. Cams Wilson, Phil. Mag. (Ser. 5), vol. 32 (1891), where an equivalent result obtained by Boussinesq is recorded. t MicheU, Londwi Math. Soc. Proc, vol. 34 (1902), p. 134. L. E. 14r !10 FORCE AT AN ANGLE [CH. IX the force by —y + a, where y is the angle between the two straight edges. We find for the .r-component of force at the origin the expression and, for the y-component of force at the origin, we find the expression X + fi tan (j}= The direction of maximum radial stress is not, in this case, that of the resultant force former of these is the initial line, making angles a and a-y witTi the edges ; the latter makes with the same edges angles and y-4>, where _ y sin a — sin y sin (a — -y) y COS a + sin y cos (a — y)' It follows that the angle a is given by the equation tan a = 7 ^ ^ <> - ^in y sin (y - 0) y cos (^ - sin y cos (y - 0) ' When a given force F is applied in a given direction, (p will be known, and a can be found from this equation ; and the constant A can be determined in terms of the re- The force initial line Fig. 18. sultant force F. The conditions that the radial stress may be pressure everywhere are "■^o' y~" ^n ^ _ « ^+M fo_ y^ d_ .^\ "~ ^ 2^ (X + 2;x) ay ^^^''' 2^. (X + 2;x) \dy r^ "^ S.t: r^ ) ' „ X + 3fi 8 X + /i /d x^ d xy\ "= ^27IX + 2;0 3S(*°S'-) + '^2MX + 2;r)(3S7^+9^^j' ("'^) = f(-|' i)l°g'- (38) This displacement is expressible in polar coordinates by the formulee M,=0, us = Blixr; (39) it involves no dilatation or rotation. The stress is expressed by the formulse '^=68=0, re=-2Br-% (40) so that the state of stress is that produced by a couple of magnitude inB applied at the origin. (c) We may take (\ + 2fij A + i2fiu; = C log (x + ty). Since tu is not one- valued in a region containing the origin, we shall suppose the origin to be on the boundary. Equation (5) becomes ^ + iri = C{x\ogr->/6-x) + iC{ylogr+xe-i/), and the displacement may be taken to be given by the formulse C , , , (2X-|-3f.)C , u = ^^{xlogr-x)-y-^^^!^0, The stress is then given by the formulse We may take n'^d'^O, the axis of x to be the boundary, and the axis of 7/ to be drawn into the body. Then the traction on the boundary is tangential traction on the part of the boundary for which x is negative; and the traction is of amount Cw (\ +/i)/(X + 2/i), and it acts towards the origin if G is positive, and away from the origin if G is negative. The most important parts of v, near the origin, are the terms containing log r and B, and if x is negative both these have the opposite sign to C, so that they are positive when G is negative. We learn from this example that tangential traction over a portion of a surface tends to depress the material on the side towards which it acts*. * Cf. L. N. G. Filon, London, Phil. Tram. R. Soc. (Ser. A), vol. 198 (1902). 152-154] TRANSFORMATION OF PLANE STRAIN 213 153. Transformation of plane strain. We have seen that states of plane strain are determined in terms of functions of a complex variable x + ly, and that the poles and logarithmic infinities of these functions correspond with points of application of force to the body which undergoes the plane strain. If the two-dimensional region occupied by the body is conformally represented upon a different two-dimen- sional region by means of a functional relation between complex variables x + ly' and x + ly, a new state of plane strain, in a body of a different shape from that originally treated, will be found by transforming the function (A, -I- 2|i) A + f 2fi-s7 into a function of os' + ty' by means of the same functional relation. Since poles and logarithmic infinities are conserved in such con- formal transformations, the points of application of isolated forces in the two states will be corresponding points. We have found in Article 149 the state of plane strain, in a body bounded by a straight edge and otherwise un- limited, which would be produced by isolated forces acting in given directions at given points of the edge. We may therefore determine a state of plane strain in a cylindrical body of any form of section, subjected to isolated forces at given points of its boundary, whenever we can effect a conformal repre- sentation of the cross-section of the body upon a half-plane. It will in general be found, however, that the isolated forces are not the only forces acting on the body; in fact, a boundary free from traction is not in general transformed into a boundary free from traction. This defect of correspondence is the main difficulty in the way of advance in the theory of two-dimensional elastic systems. We may approach the matter from a different point of view, by con- sidering the stress-function as a solution of Vj*;y; = 0. If we change the independent variables from x, y to x', y', where x' and y' are conjugate functions of x and y, the form of the equation is not conserved, and thus the form of the stress-function in the (x, y') region cannot be inferred from its form in the {x, y) region. 154. Inversion*. The transformation of inversion, x + ly' = {x + lyY^, constitutes an excep- tion to the statement at the end of Article 153. It will be more convenient in this case to avoid complex variables, and to change the independent variables by means of the equations x' = fe/r^, y = %/rS in which k is the constant of inversion, and r^ stands for x- + y^. We write in like manner r'"- for x"" + y'^ Expressed in polar coordinates the equation Vj^i^ = becomes "19/ a\ . 1 9^' ^-)+^i r dr \ dr) r" 36' r dr \ dr) r^dO'- * Miohell, loc. cit., p. 209. = 0; (41) 214 TRANSFORMATION OF INVERSION [CH. IX and, when the variables are changed from r, to r', 6, this equation may be sho^vn to become [7 ^- ('■.^) - 7. w] [r. |. {'■ 37 <-^)} - ;5-. |.<-x.] = 0. ...,«, It follows that, when x is expressed in terms of x, /, r\ satisfies the equation i&^^W-^^'^y'^y^^-'' (*-^> and therefore r'-x is a stress-function in the plane of {x', y). The stress-components derived from r'^% are given by the equations* ^ (44) where ^ is the same as Q; and we find where rr, ^^, r^ are the stress-components derived from %, expressed in terms of r, 6. Thus the stress in the (r', ff) system differs from that in the (r, 6) system by the factor r^, by the reversal of the shearing stress r6, and by the superposition of a normal traction 2 /^ — r ddxl^r)], the same in all directions round a point. It follows that lines of stress are trans- formed into lines of stress, and a boundary free from stress is transformed into a boundary under normal traction only. Further this normal traction is constant. To prove this, we observe that the conditions of zero traction across a boundary are cos (x, v) .-^ - cos {y, v) 5—^ =0, - cos ix, v) ,-^ -H cos(w, v) ^ = 0, 6y^ '' ' dxdy cxdy ^ da? and these are the same as OH \oy) ds \dxj where ds denotes an element of the boundary. Hence cj^iox and d^/^y are constant along the boundary, and we have ds\^ drJ dsV dx by J ds ds dx ds dy * See the theorem (ii) of Article 59. APPLIED TO PLANE STRAIN 215 154, 155] It follows that a boundary free from traction in the (r, 6) system is trans- formed into a boundary subject to normal tension in the (r', 0') system. This tension has the same value at all points of the transformed boundary, and its effect is known and can be allowed for. 155. Equilibrium of a circular disk under forces in its plane*. (i) We may now apply the transformation of inversion to the problem of Articles 149, 150. Let 0' be a point of a fixed straight line O'A (Fig. 19). If ffA were the boundary of the section of a body in which there was plane strain produced by a force F directed along Fig. 19. O&X, the stress-function at P would be -ir'^FrO sin 6, where r stands for (yP; and this may be written — ir~^F6y, where y is the ordinate of P referred to OX. When we invert the system with respect to 0, taking k=0O, P is transformed to P', and the new stress- function is -ir-^ri^F(ei + e^lc^y'lri^, where 5j and ^-6^ are the angles XOP, XffP', and we have written r^ for OP', and y' for the ordinate of P' referred to OX. Further the line O'A is transformed into a circle through 0, 0', and the angle 2a which 00' subtends at the centre is equal to twice the angle AO'A". Hence the function -7r~^F'y'{6^ + d2) is the stress-function corresponding with equal and opposite isolated forces, each of magnitude F', acting as thrust in the line 00', together with a certain constant normal tension round the bounding circle. To find the magnitude of this tension, we observe that, when P' is on the circle, ?•, cosec 02 = ^2 cosec 6^ — k cosec (^i -I- ^2) = 2i2, * The results of (i) and (ii) are due to Hertz, Zeit. f. Math. u. Physik, Bd. 28 (1883), or Ges. Werke, Bd. 1, p. 283, and Michell, London Math. Soc. Proc, vol. 32 (1900), p. 35, and vol. 34 (1902), p. 134. 216 EQUILIBRIUM OF CIRCULAR DISK [CH. IX where R is the radius of the circle. Further, the formulse (1) of Article 144 give for the stress-compoaents ^Ffaoa^Bx cosMgN „ 2F' / cos 6^ sin^ 6 i , cos 6.^ sin^ gg^ _ 2i^' / cos^ ^1 sin 6i cos^ gg sin gg N ^"-"^V »•. ~ '•2 /■ Also the angle (<^ in the figure) which the central radius vector {R) to F makes with the axis of x, when P is on the circle, is \-rr -a + 'HB-^, or ^ir + 61-62. Hence the normal tension across the circle is X^ sin2 {6i - 61) + 7y cos2 {6^ - 61) + 2Xy sin {82 - fli) cos (6^ - 6^), and this is - {F' sin ajlvR. If the circle is subjected to the two forces F' only there is stress compounded of mean tension, equal at all points to {F' sin a)l-n-R, and the simple radial distributions about the points and ff in which the radial components are - (2/" cos 5i)/ff?-i and -{^Fcoad^)/^^!- (ii) Circular plate subjected to forces acting on its rim. If the force F' is appUed at in the direction 00' (see Fig. 19) and suitable tractions are applied over the rest of the rim the stress-function may consist of the single term -n-'^F'y'Bi. Let r and 6 be polar coordinates with origin at the centre of the circle and initial line parallel to 00'. The angle (r, rj) between the radii vectores drawn from the centre and from to any point on the circumference is ^w — d^. The stress-system referred to (ri, 61} is given by the equations r^j =-(2F' cos 6i)l{nrj), 6^i = 0, ?^i = ; and therefore, when referred to (r, 6), it is given, at any point of the boundary, by the equations ^ iF' cos 61 sin^ 62 2fl _ 2-^' '^°^ ^1 °°^^ ^2 '^ _ ML' cos^ijjos 62 shi^ or we have at the boundary ^- /^' cos 61 sin 62 '~a_ ^' "^"^ ^1 '^°® ^2 "■""V " R ^' T ~li ' and this is the same as — F'sina F' ,. „. --i F'oosa _ F' ,a a\ where a, =61 + 62, is the acute angle subtended at a point on the circumference by the chord 00'. Hence the traction across the boundary can be regarded as compounded of (i) uniform tension - ^ {F' sin ajjirR in the direction of the normal, (ii) uniform tangential traction \ {F' cos a)lnR, (iii) uniform traction — ^F'j-rrR in the direction OC. Let any number of forces be applied to various points of the boundary. If they would keep a rigid body in equilibrium they satisfy the condition 2i^' cos a= 0, for iP'R cos a is the sum of their moments about the centre. Also the uniform tractions corresponding with (iii) in the above solution would have a zero resultant at every point of the' rim. Hence the result of superposing the stress-systems of type (32) belonging to each of the forces would be to give us the state of stress in the plate under the actual forces and a normal 155, 156] UNDER FORCES IN ITS PLANE 217 tension of amount - 2 {F' sin a)l^rrR at all points of the rim. The terms F' sin a of this summation are equal to the normal (inward) components of the applied forces. Mean tension, equal at all points to 2 {F' sin a)/2n-/J, could be superposed upon this distribution of stress, and then the plate would be subject to the action of the forces F' only. (iii) Heavy disk*. The state of stress in a heavy disk resting on a horizontal plane can also be found. Let iff be the weight per unit of area, and let r, 6 be polar coordinates with origin at the point of contact A and initial line drawn vertically upwards, as in Fig. 20. Fig. 20. The stress can be shown to be compounded of the systems (i) X^ = ^{y + R), Yy=-^w{y-R), Xy=-^x, (ii) ?T= -2M?^V-icosfl, ^ = 0, r6=Q. The traction across any horizontal section is pressure directed radially from A, and is of amount ^r~^ {'iR^coa'd—r^) ; the traction across anysection drawn through^ is hori- zontal tension of amount ^w{2Rcosd-r). 156. Examples of transformation. (i) The direct method of Article 153 wiU lead, by the substitution x+iy=ifl/(a/ + iy') in the formula (\ + 2/i) A+2/xtrar = ^(a; + iy-*)-i, (46) to a stress-system in the plane of (x', y'), in which simple radial stress at the point [k, 0) is superposed upon a constant simple tension (X,.) in the direction of the axis n/. If the boundary in the {so, y) plane is given by the equation y = {a! — k)i3Xia, the boundary in the {a/, y) plane will be a circle, and the results given in (i) and (ii) of Article 155 can be deduced. * The solution is due to Michell, loc. cit., p. 208. Figures showing the distribution of stress in this case and in several other cases, some of which have been discussed in this Chapter, are drawn by Michell. 218 TRANSFORMATION OF PLANE STRAIN [CH. IX (ii) By the transformation ^ + ty = (y + ty')" the wedge-shaped region between 3/'=0 and y/y = tan tt/k is conformally represented on the half-plane y>0. If we substitute for x + iy YD. (46) we shall obtain a. state of stress in the wedge-shaped region bounded by the above two lines in the plane of [x', y'j, which would be due to a single force applied at (/;''", 0), and certain tractions distributed over the boundaries. When to=2 the traction over y' = Q vanishes and that on y = becomes tension of amount proportional to (iii) By the transformation z=(e'' - l)/(e^-H), where z=j: + Ly and z'=x' + iy', the strip between y' = and y' = 7r is conformally represented upon the half-plane y>0, so that the origins in the two planes are corresponding points, and the points (±1, 0) in the plane of {x, y) correspond with the infinitely distant points of the strip. Let a single force F act at the origin in the {x, y) plane in the positive direction of the axis of y. Then the sohition is given by the equation (X-1-2;.)A + 2M.r.= -i:?^ -^— . ^ 1^' ^ '^ n- X-|-/i x + iy Transforming to {x\ y') we find „ . „ F \ + 2ui sin y' ■+ 1 sinh x' X-|-2fi ^ + a^-s,= - — ^— ^ — f5 y , w \ + ii coahaf -cosy and ^ + ir] = —- -J(2tan~i -, — ^— ^^^ —y'j -ilog (cosh a;' -cosy) V -)- const. This solution represents the effect of a single force 2i^, acting at the origin in the positive direction of the axis of y, and purely normal pressure of amount Fj{\-\-oos\i x') per unit of length, acting on the edge y' = -n- of the strip, together with certain tangential tractions on the edges of the strip. The latter can be annulled by superposing a displacement (a', V) upon the displacement V2^ ^ 2;i (X -I- %ii) " dy" %{\ + ip.) 'iji (X + 2^i) -^ Zy' provided that and this additional displacement does not affect the normal tractions on the boundary. "'+"'-27^S^(''-'^)' APPENDIX TO CHAPTERS VIII AND IX. VOLTERRA'S THEORY OF DISLOCATIONS. 156 a. The analytical possibility that the stress-function of Article 144 may be many- valued was explicitly treated by J. H. Mich ell*, under the condition that the displacement must be expressed by one-valued functions. The analytical possibility of displacements expressed by many- valued functions is implicitly present in Article 148, where the ambiguity was removed for the reason that an actual physical displacement would appear to be necessarily one-valued. In a body, such as a hollow cylinder, occupying a multiply- connected region of space, there is, however, a physical possibility of many- valued displacements. Suppose, for example, that a thin slice of material, bounded by two axial planes, is removed from such a cylinder, and the new surfaces thus formed are brought together and joined. The body so formed will be in a state of initial stress (Article 75), which may be determined by assuming the displacement of a point in the hollow cylinder, resolved at right angles to the axial plane through the point, or rather the ratio of this displacement to distance from the axis, to be many-valued with a cyclic con- stant, and adjusting the constant so that the corresponding points of the two axial planes, after displacement, may coincide. The possibility of such inter- pretations of many-valued displacements appears to have been first indicated by G. Weingartenf . The subject was discussed with more detail by A. Timpej for two-dimensional systems, such as a plane circular ring. A more general theory was afterwards developed by V. Volterra in a series of notes, and a comprehensive account of the theory with some improvements by E. Cesaro was published by the same author|. He describes the kind of deformations to which the theory is applicable by the name " distorsioni." I have ventured to call them " dislocations." The multiply-connected region occupied by the body can be reduced to a simply-connected region by means of a system of barriers||. For example, * London, Proc. Math. Soc, vol. 30 (1900), p. 103. t Rama, Ace. Line. Rend. (Ser. 5), t. 10 (1 Sem.), 1901, p. 57. + ProUeme d. Spannungsverteilung in ebmen Systemen einfach gelost mit Hilfe d. Airyschen Funktion, Gottingen Diss., Leipzig 1905. § V. Volterra, " Sur I'^quUibre des corps ^lastiques multiplement coiinexes," Paris, Ann. Ec. ■norm. (Ser. 3), t. 24, 1907, pp. 401—517. II In regard to the theory of multiple connectivity reference may be made to J. C. Maxwell, A treatise on Electricity and Magnetism, vol. 1, 2nd Edn., Oxford 1881, pp. 16—24, or H. Lamb, Hydrodynamics, 4th Edn., Cambridge 1916, pp. 47 — 55. 220 APPENDIX TO CHAPTERS VIII AND IX the region between the bounding cylindrical surfaces of a hollow cylinder can be rendered simply-connected by a plane barrier passing through the axis of the cylinder, and having that axis for an edge. The stress in the body, and therefore also the strain, must be one-valued and continuous, but the dis- placement may be discontinuous in crossing the barrier. With a view to determining the nature of the possible discontinuities we shall (a) prove a general theorem for the expression of a displacement answering to given strain-components, and then (6) use this theorem to determine the nature of the discontinuities. We shall then, (c) and (d), apply the theory to two par- ticular examples. (a) Displacement answering to given strain. The displacement {u, v, w) and the rotation {r^x, '^y, '"'z) are not necessarily one- valued, but the strain-components e^x, ■ • ■ have definite values at any point (w, y, z). Let («», v^, Wq) denote one of the values of the displacement at a point M^ or (xo, y„, z^). Then one of the values {ui, v^, w^) of the dis- placement at another point M^ or {ici, y^, z-^) can be obtained by evaluating the line-integral / ^^' du -, du , du , ^- ax + :^ dy + ~ az M,ox dy " dz taken along any path joining the points M^, M^. But different values may be obtained by choosing different paths of integration. Now we have in general du du . du ^ Hence u-^ — u^—l e^idx + ^e^ydy + \ezxdz+ \ •usydz — 'Oi^dy (1) ■' M^ J M, Let (tD-a;™, OTj,'"', -oj-^'"') denote one of the values of the rotation at the point Jf„, or the value if there is only one. Then rM, rM, tsydz-is^dy^] 'ST^d{yi-y)-'!i!yd{z-^- z) J jr„ .' M^ = ^2/"" («i - ^o) - W^ (2/a - yo) - (yi - y) d-s^^ - (z, - z) d^y, J M, where, for example, d-sr^ =-^dx + ~ dy + ^ dz. ox dy ^ dz Now we have identically 2 ^z ^ 9^ _ 2 8e^ 2 d^ ^ 2 ^ _ ^ 9a! dx dy ' dx dz dx ' 2 ?^ = 2 ^^ - ?^ 2 ^^ — ^^ — ^' dy dx dy ' dy dz dx 5ar2 _ 9e^ ^ 9e^ ^ 9^ ^ g^ ^ ge^ dz dx 9y ' dz dz dx 156 aJ dislocations 221 and we thus obtain the equation ,=u, + i!T,/^{2,-z,)-WHy^-yo)+l ' ^dx + vdy + ^dz, ...(2) ■I 31, where ?-^ + (3/.-.)(^-i'^^) + (^:-)(^-4^^), .9y dy Similar equations can be obtained for v^ and Wj. The proof is due to Cesd,ro. (6) Discontinuity at a harrier. Now suppose the multiply-connected region to be reduced to a simply- connected one by means of a system of barriers. We are going to apply equation (1) to a circuit, so that Jfj and M„ coincide, taking for the path of integration a non-evanescihle circuit, or one which cannot be contracted to a point without passing out of the region. We shall take the circuit to cut a particular barrier fi once, at a point M, and not to cut any of the other barriers. To fix ideas we may, if we like, think of the region as the doubly- connected region between two coaxial cjdinders, having the axis of z as their common axis, and the barrier O as formed by that part of the plane a; = 0, lying between the two cylinders, on which y is positive. But the result is general, not restricted to this particular example. Then we take M^ and M^ to be close to M on opposite sides of 12, and treat the path of integration as not cutting fl. It will be observed that |, r), ^, given by (3), are such that the equations dy dz ' dz dx ' dw dy are satisfied identically in virtue of equations (25) of Article 17, and therefore the value of the integral ^dx + ridy + ^dz (4) / is the same for all reconcilable circuits, that is to say such as can be deformed, the one into the other, without passing out pf the region. For it is possible to construct a surface S, having any two such circuits for edges, and lying entirely in the region, and then the difference between the values of the integral (4) taken along, the two circuits is equal to the integral //|(|-I)'-(M)-(M)"}-' taken over 8, where I, m, n> denote the direction-cosines of the normal to S drawn in a determinate sense; and this surface integral vanishes. Hence, 222 APPENDIX TO CHAPTERS VIII AND IX whatever point on Xl we take for M, the integral (4) has the same value for all circuits beginning and ending at M, provided they cut the barrier fl nowhere except at M and do not cut any other barrier. It follows that tfi — Mo has, at each point ilf on fi, a definite value which may depend upon the position of M. The like holds for v^ — v„ and w^ — w^. We now consider the variation of Mj — u„ as M moves on fl. Let M and M' be any two points on fl, if„ and Jfj points close to M on opposite sides of fl, il/o' and My points close to M' on opposite sides of 11, (m„', d/, w^) the dis- placement at M^, (w/, Vi, Wi) that at if/. The theorem expressed by equa- tion (2) may be applied to the path from Mo to Jf„', coinciding with a curve traced on fi and joining M to M'. It gives rM' uo' - M„ = ■=Ty<»t (z„' - z„) - W^ (2/„' - 2/„) + ^dx + 7)dy + ^dz. ■ ' M The same theorem may be applied to the path from M^ to M^, coinciding with the same curve, and gives rM' «/ - u, = -syyi" (V - z,) - tD-^w (y,' - 2/0 + ^dx + rjdy + ^dz. •I M Since {x^, y^, Zj) is the same as (.r,,, y^, z^) and («/, y/, ^'i') is the same as («(,', y„', ^„'), these equations give u,' - u; = (tt, - M„) -I- (^V" - ^^'"0 (V - ^o) - (^.'" - WViy: - yo), where the coefficients of ^„' - ^^o and y^ — y^ are independent of {x^, y^, z„'). Similar equations hold for v^' — vj and w/ — Wq'. It thus appears that the discontinuities of u, v, w at D, are expressed by equations of the form v,-u„=l,+p^z -psyA ^1 - t'o = k+P3«:-pi^, > (^) Wi - w„ = k+piy —piX, ] where l^, U, I, and pi, p^, ps are constant over CI. This is Weingarten's result. It may be interpreted in the statement that the displacement of the matter on one side of a barrier relative to the matter on the other side is a displace- ment which would be possible in a rigid body. Let the multiply-connected region occupied by the body be reduced to a simply-connected region by a system of barriers, and suppose each barrier to be the seat of an actual physical breach of continuity, such as could be effected by cutting the material along the barrier. After such dissection the body will not be divided into parts, but will still be a coherent body. The effect of the dissection will be that the region of space occupied by the body will become a simply-connected region. Let one of the faces of the dissected body formed by any barrier be displaced relatively to the other by a small dis- placement, which would be possible for a rigid body, that is to say let the 156 aJ dislocations 2ii3 face be moved as a whole by translation and rotation. Let this be done for each pair of faces, and thereafter let opposing faces be joined, by removal or insertion of a thin sheet of matter of the same kind as that forming the original body. The new body so formed will, in general, be in a state of initial stress. The operation of deriving the new body from the original body is a dislocation. It appears that, if dislocations are permitted, the theorem of Article 118 requires modification. To secure uniqueness of solution, it is necessary that the six constants of each barrier, occurring in equations (5), should be given. It is open to us either to regard the displacement in the body, occupying the multiply-connected region, as one-valued and discontinuous at the barriers, or as many-valued and continuous in the region, supposed without barriers. In the latter case the sum of the increments of it taken along any such circuit as has been described is the same for all reconcilable circuits passing through a given point. Thus the position of the barriers is, to a great extent, immaterial. In a body, which has suffered a dislocation, and is consequently in a state of initial stress, there is, in general, nothing to show the seat of the dislocation. A similar result occurs in the theory of electric currents. The magnetic potential due to a unit current flowing in a closed circuit is many-valued and continuous in the multiply- connected region surrounding the linear conductor, which carries the current. One of its values at any point is the solid angle subtended at the point by any surface having the line of the conductor for an edge, the solid angle being specified as regards sign by a certain conventional rule. The soUd angle thus specified is one- valued and discontinuous at the surface. But the magnetic field of the current is in no way dependent upon the choice made among the possible surfaces. (c) Hollow cylinder deformed by removal of a slice of uniform, thickness. Volterra has given a very complete discussion of the problem for a hollow cylinder and for some systems of thin rods. We shall select for detailed treatment one of his examples of dislocation in a hollow cylinder. A possible formula for a many-valued two-dimensional displacement is o-iven in equations (27) of Article 148 supra. In a hollow circular cylinder whose axis is the axis of ^^ a possible displacement is given by th.e equations A , \+ a .y- -^ a '^ + 1^ A^y n (6) Here r, 6 are cylindrical polar coordinates in the plane of a cross-section, and .4 is a constant. The displacement is many-valued and continuous ; but, if we restrict by a convention such as lir +a>6 >a, it becomes one-valued and discontinuous, the y-component decreasing suddenly by J.7r/(X -I- 2^) as 6 changes from 27r -h a to «. In particular we may take a = 0. Then the 224 APPENDIX TO CHAPTERS VIII AND IX displacement at the axial section y = 0, a; > is greater on the negative side {y < 0) than on the positive side {y > 0) by this amount, and there is a dis- location, equivalent to the removal of a thin slice of thickness Airl{\ + ^fi), bounded by the parts of two planes y = ± \Airj(}^ + 2;a) on which x>Q, and subsequent joining of the plane faces so formed. As in the general theory, so here, the seat of dislocation need not be y = 0, it may be any plane y = const, which meets the inner boundary in real points. The state of stress answering to the displacement (6) is expressed by the equations ^^~ X + 2^ ^r^' ^y~~X + 2^ r^' ^'~ \ + 2ti^ f^' y,-o, z^-0, ^y-^^:^-^^- ..(7) Cf equations (30) of Article 149. The state of the hollow cylinder with the dislocation can be maintained by suitable surface tractions on the inner and outer cylindrical boundaries and on the terminal faces. The traction across any cylindrical surface is normal pressure, or tension, expressed by the equation ^ 2 (\ + Ai) . cos rr = ^--^'A (8) The tractions on the terminal faces are expressed by the value of Z^. The tractions across the cylindrical boundaries can be nullified by super- posing on the displacement (6) a one-valued displacement. For this purpose we consider in the first place the displacement X+3/X X + fi f X + fi xy " 2^(X-f-2/.)^'^°S'^+2^(\-F2^)'^V^' ''-2/.(X + 2/.)^'^' *"-"• (9) Cf equations (28) of Article 148. This gives rise to tractions across any cj'lindrical surface r = const, expressed by the equations In the second place we consider a displacement expressed by equations of the form lYi^ __ yy2 u = A,[{X-ti)x'' + (:A\ + 5iJi)y^] + A^—-^ v = -2A2{X + 3fi)xy + 2A3^, w = 0, (11) which may easily be shown to satisfy the equations of equilibrium under no 156 aJ dislocations 225 body force. This displacement gives rise to tractions across any cylindrical surface r = const, expressed by the equations Z,.= - 4^ (X + /i) A^ - — ^ - ifiAs"^—^ , Let r = ri and r = r^ be the inner and outer cylindrical boundaries. When the displacements expressed by equations (6), (9), (11) are superposed, and the resulting tractions on these boundaries equated to zero, the following equations are found to hold : — \ + fJ, s A,=-A A,= A. + 2/i' A 4(\ + 2^)=(n^+r-„^)' (X + m) r-iVoM } (12) The composition of the displacements expressed by equations (6), (9), (11), and substitution of these values for A^, A^, A^, yields a displacement in the dislocated hollow cylinder free from traction over the inner and outer cylin- drical boundaries, the length being maintained constant by suitable tractions on the terminal faces. These tractions are normal tensions and pressures expressed by the equation ^/^ Ax U^ r^^+r,y ^^^^ ' (X + lixf By means of an additional one-valued displacement these tractions also could be nullified. We do not know how to determine this displacement in the general case, but it can be found in the case where (rj — r-^jr-^ is small. For this the reader may refer to Volterra's memoir, where will also be found photographs of a hollow cylinder under no external forces, but in a state of initial stress due to the formation of a dislocation of the type here discussed, and described as a " parallel fissure." [d) Hollow cylinder v>itk radial fissure. Another of Volterra's examples is that of a hollow cylinder with a dislocation due to the removal of a thin slice bounded by two axial planes. In polar coordinates the many- valued displacement is expressed by the equations ««r=4 j— ^rlogr, ue=Ar6 (14) It is a good exercise to obtain these formulae from those of Article 145 by assu.ming (\ + 2)1) A + i.2iixa = ^iiA log (x + ly), adjusting the subsidiary displacement v!, v', and superposing an additional displacement L. E. 15 226 APPENDIX TO CHAPTERS VIII AND IX which obviouslj- satisfies the equations of equilibrium under no body force. The displace- ment expressed by (14) can be shown to give rise to tensions or pressures on the cylindrical boundaries expressed by the equation and to terminal tensions or pressures expressed by the equation -2.= - A ^^^ {2,1. log »• + X + 3;x). The tractions on the cylindrical boundaries can be nullified by combining the displacement expressed by (14) with a suitable displacement of the type considered in Article 100, and the tractions on the terminal faces can also be nullified by superposing an additional dis- placement, which can be determined approximately when the wall of the hollow cylinder is thin. The question is discussed fully by Volterra. It wiU be observed that the problems (c) and (d) arise by putting in equations (5) either 1^ = 1^=0 and pi=p2=p3 = 0, or 1^ = 1.^ = 1^=0 &nd pi=p2=0. The nature of the dislocations expressed by l^, I3, pi, p^ in a hollow cylinder is also discussed by Volterra. CHAPTER X. THEOKY OP THE INTEGRATION OF THE EQUATIONS OP EQUILIBRIUM OF AN ISOTROPIC ELASTIC SOLID BODY. 157. Nature of the problem. The chief analytical problem of the theory of Elasticity is that of the solution of the equations of equilibrium of an isotropic body with a given boundary when the surface displacements or the surface tractions are given. , The case in which body forces act upon the body may be reduced, by means of the particular integral obtained in Article 130, to that in which the body is held strained by surface tractions only. Accordingly our problem is to determine functions u, v, w which within a given boundary are continuous and have continuous differential coefficients, which satisfy the system of partial differential equations (\ + ^i)~+tN'u = 0, (K + fi)y+fiV''v = 0, (X + iJ,)^^ + fiV'w = 0, ...(1) , . du , dv dw .„. "^^^^ ^=aS+ai; + a^' (2) and which also satisfy certain conditions at the boundary. When the surface displacements are given, the values of u, v, w at the boundary are prescribed. We know that the solution of the problem is unique if fi and 3X + 2/u, are positive. When the surface tractions are given the values taken at the surface by the three expressions of the type v XA cos (x, ^) + H']^ + ^ cos (oc, I/) + g^ cos (y, i/) + ^ cos {z, j/)J . . .(3) are prescribed, dv denoting an element of the normal to the boundary. We know that the problem has no solution unless the prescribed surface tractions satisfy the conditions of rigid -body-equilibrium (Article 117). We know also that, if these conditions are satisfied, and if fj, and 3X -I- 2fj, are positive, the solution of the problem is effectively unique, in the sense that the strain and stress are uniquely determinate, but the displacement may have superposed upon it an arbitrary small displacement which would be possible in a rigid body. 16—2 228 RiSUM]^ OF THE THEORY [CH. X 158. Resume of the theory of Potential. The methods which have been devised for solving these problems have a close analogy to the methods which have been devised for solving corre- sponding problems in the theory of Potential. In that theory we have the problem of determining a function U which, besides satisfying the usual conditions of continuity, shall satisfy the equation V=Cr=0 (4) at all points within a given bolindary*, and either (a) shall take an assigned value at every point of this boundary, or (6) shall be such that 9 Ujdv takes an assigned value at every point of this boundary. In case (6) the surface integral I -K-.dS taken over the boundary must vanish, and in this case the function U is determinate to an arbitrary constant prhs. There are two main lines of attack upon these problems, which may be described respectively as the method of series and the method of singularities. To illustrate the method of series we consider the case of a spherical boundary. There exists an infinite series of functions, each of them rational and integral and homogeneous in x, y, z and satisfying equation (4). Let the origin be the centre of the sphere, let a be the radius of the sphere, and let r denote the distance of any point from the origin. Any one of these functions can be expressed in the form V^Sn, where n is an integer, and 8n, which is inde- pendent of r, is a function of position on the sphere. Then the functions Sn have the property that an arbitrary function of position on the sphere can be oo expressed by an infinite series of the form 2 -4^„. The possibility of the expansion is bound up with the possession by the functions Sn of the conjugate property expressed by the equation SnS,ndS=0 (5) The function U which satisfies equation (4) within a sphere r = a, and takes on the sphere the values of an arbitrary function, is expressible in the form If the surface integral of the arbitrary function over the sphere vanishes there is no term of degree zero (constant term) in the expansion. The function U which satisfies equation (4) when r< a, and is such that 9 U/dv has assigned values on the sphere r = a, is expressed by an equation of the form >/i. „=i wa" - '\ * A function which has these properties is said to be "harmonic " in the region within the given boundary. 158] OP THE NEWTONIAN POTENTIAL 229 The application of the method of series to the theory of Elasticity will be considered in the next Chapter. The method of singularities depends essentially upon the reciprocal theorem, known as Green's equation, viz. : [[[( UV^V- VV^U) dxdydz = [[(CT ^ - F ^) dS (6) in which U and V are any two functions which satisf;f the usual conditions of continuity in a region of space ; the volume-integration is taken through this region (or part of it), and the surface-integration is taken over the boundary of the region (or the part). The normal v is drawn away from the region (or the part). The method depends also on the existence of a solution of (4) having a simple infinity (pole) at an assigned point ; such a solution is 1/r, where r denotes distance from the point. By taking for V the function 1/r, and, for the region of space, that bounded externally by a given surface 8 and internally by a sphere S with its centre at the origin of r, and by passing to a limit when the radius of % is indefinitely diminished, we obtain from (6) the equation -"^-m-^^' « so that U is expressed explicitly in terms of the surface values of U and 3 Ujdv. The term that contains 3 TJjdv explicitly is the potential of a " simple sheet," and that which contains U explicitly is the potential of a' "double sheet." In general the surface values of U and d Ujdv cannot both be pre- scribed, and the next step is to eliminate either U ovd U/dv — the one that is not given. This is effected by the introduction of certain functions known as " Green's functions." Let a function G be defined by the following con- ditions : — (1) the condition of being harmonic at all points within S except the origin of r, (2) the possession of a simple pole at this point with residue unity, (3) the condition of vanishing at all points of S. The function G may be called " Green's function for the surface and the point." The function G-l/r is harmonic within 8 and equal to — 1/r at all points on 8, and we have the equation jj dv\ r) \ r) dv d8 = 0. Since G vanishes at all points on 8 we find that (7) may be written i^U = -jju^^dS. (8) Hence IT can be expressed in terms of its surface values if G can be found. When the values of dU/dv are given at the boundary we introduce a function T defined by the following conditions:— (1) the condition of being 230 METHOD OF INTEGRATING THE EQUATIONS [CH. X harmonic at all points within S except the origin of r and a chosen point A, (2) the possession of simple poles at these points with residues + 1 and — 1, (3) the condition that dT/dv vanishes at all points of S. We find for U the equation 47r(fr- U^)^jJT^-^dS (9) Hence U can be expressed effectively in terms of the surface values of 9 U/dv when r is known. The function F is sometimes called the " second Green's function." Green's function G for a surface and a point may be interpreted as the electric potential due to a point charge in presence of an uninsulated conducting surface. The second Green's function r for the surface, a point P and a chosen point A may be interpreted as the velocity potential of incompressible fluid due to a source and sink at P and A within a rigid boundary. The functions G and r are known for a few surfaces of which the plane and the sphere* are the most important. The existence of Green's functions for any surface, and the existence of functions which are harmonic within a surface and take prescribed values, or have prescribed normal rates of variation at all points on the surface are not obvious without proof. The eflfbrts that have been made to prove these existence-theorems have given rise to a mathematical theory of great interest. Methods have been devised for constructing the functions by convergent processes t ; and these methods, although very complicated, have been successful for certain classes of surfaces (e.g. such as are everywhere convex) when some restrictions are imposed upon the degree of arbitrariness of the prescribed surface values. Similar existence-theorems are involved in the theory of Elasticity. The subject will not be pursued here, but the reader who wishes for further information in regard to the problem of given surface displacements is referred to the following memoirs : — G. Laurioella, Boma, Ace. Line. Rend. (Ser. 5), t. 15, Sem. 1, 1906, p. 426, and Sem. 2, 1906, p. 75, and t. 16, Sem. 2, 1907, p. 373, also A. Korn, Munehen, Ber., Bd. 36, 1906, p. 37, Paris, Ann. Ec. Norm., t. 24, 1907, p. 9, Palermo, Cire. Mat. Rend., t. 30, 1910, pp. 138, 336. For the problem of given surface tractions reference may be made to A. Korn, Toulouse, Ann. (S^r. 2), t. 10, 1908, p. 165. References to writings on another method of attacking these problems will be given in the next Chapter. There is a corresponding theory for vibrations, in regard to which reference may be made to A. Korn, Munehen, Ber., Bd. 26, 1906, p. 351, and Palermo, Cire. Mat. Rend., t. 30, 1910, p. 153. For a general survey of such questions, and the methods proposed for deahng with them up to 1906, reference may be made to the Article by 0. Tedone "Allgemeine Theorems d. math. Elastizitatslehre," Ency. d. math. Wiss., Bd. IV., Art. 24, Leipzig, 1907. 159. Description of Betti's method of integration. The adaptation of the method of singularities to the theory of Elasticity was made by BettiJ, who showed how to express the dilatation A and the * See e.g. Maxwell, Electricity and ilagnetism, 2nd edition, Oxford 1881, and W. M. Hicks, London, Phil. Tram. R. Soc, vol. 171 (1880). t See e.g. Poincar^, Theorie dupotentiel Newtonien, Paris 1899. X See Introduction, footnote 65. A general account of the extension of the theory to seolotropio solid bodies is given by I. Fredholm, Acta Math., t. 23, 1900, p. 1. 158-160] OF EQUILIBRIUM OF ISOTEOPIC SOLIDS 231 rotation (wa;, sTy, -sr^) by means of formulae analogous to (7) and containing explicitly the surface tractions and surface displacements. These formulae involve special systems of displacements which have been given in Chapter viil. Since A is harmonic the equations (1) can be written in such forms as V»[m + -^(1 + X/;x)«A] = (10) and thus the determination of u, v, w when A is known and the surface values of u, V, w are prescribed is reduced to a problem in the theory of Potential. If the surface tractions (Z„, F„, Z^) are prescribed, we oljserve that the boundary conditions can be written in such forms as g- = „- -X"^ - „- A COS {x, v) + ■sjy cos {z, v) - 1ST;, COS (y, v), . . .(11) so that, when A and •Era,, ra-^, ot^ are found, the surface values of dujdv, dv/dv, dw/dv are known, and the problem is again reduced to a problem in the theory of Potential. Accordingly Betti's method of integration involves the determination of A, and of w^, tsy, OTj, in terms of the prescribed surface displacements or surface tractions, by the aid of subsidiary special solutions which are analogous to Green's functions. 160. Formula for the dilatation. The formula analogous to (7) is to be obtained by means of the reciprocal theorem proved in Article 121. When no body forces are in action the theorem takes the form f [ (Z,m' ■+ Y.,v'. + Zlv/) dS = \\{X,Ju + YJv + ZJw) dS, (12) in which (w, v, w) is a displacement satisfying equations (1) and X^, T^, Z, are the corresponding surface tractions, and also (u , v', w') is a second dis- placement and Xy, YJ, ZJ are the corresponding surface tractions. Further, the integration is taken over the boundary of any region within which u, v, w and II , v', w satisfy the usual conditions of continuity and the equations (1). We take for m', v' , w' the expressions given in (20) of Article 132. It will be convenient to denote these, omitting a factor, by m„, ■Uj, Wo, and the corre- sponding surface tractions by Z,"", 7^'°', Z^"». We write («„,.„, «;o) = (-g^, ^, -g^-J (13) and then the region in question must be bounded internally by a closed surface surrounding the origin of r. The surface will be taken to be a sphere 2, and we shall pass to a limit by diminishing the radius of this sphere indefinitely. The external boundary of the region will be taken to be the surface S of the body. 232 METHOD OF INTEGRATING THE EQUATIONS [CH. X Since the values of cos {x, v),...a.tl, are - x/r, — yjr, — z/r, the contribu- tion of S to the left-hand member of (12) is x /du dw\ , y fdw dv\ z / . „ du dw dz dx dw dv dy dz. dw\ dr2 dx dy which is /•rr, A „ /x'^du y^dv , /■/■ [yz (dw dv\ zx (du dw\ acy /dv_ du\ JJ ^ ''i^Kd^'^dz)^'^ \dz + dx) + r* \dx ^ dy) dl. All the integrals of type lyzdl, vanish, and each of those of type j j«^a!2 is equal to i4nrr*, and therefore the limit of the above expression when the radius of S is diminished indefinitely is 47r (X + f /u,) (A)o, where (A)o denotes the value of A at the origin of r. Again, since the values of Z^"», F„i°', Z„<»' are expressed by formulae of the type Z„"'>=2/i cos(a;, j/)g^-l-cos(y, i^)g^ + cos(^:, v)g^ -^, the contribution of X to the right-hand side of (12) is 2f.jf-2 '^ + 'y' dt. Now such integrals as xdX vanish, and we therefore expand the functions u, V, w in the neighbourhood of the origin of r in such forms as {du\ (du\ (du\ and retain first powers of x, y, z. Then in the limit, when the radius of S is diminished indefinitely, the above contribution becomes or — -i^'rrytt(A)i,. Equation (12) therefore yields the result 47r(X -l-2ya)(A)o= [[[(Z, (»)«■+ 7,^<'H + Z,^'^w)-{X,u,+ Y,v, + Z,iu,)]dS. (14) The formula (14) is the analogue of (7) in regard to the dilatation. 160, 161] OF EQUILIBRIUM OF ISOTEOPIC SOLIDS 233 This formula has been obtained here by a strictly analytical process, but it may also be arrived at synthetically* by an interpretation of the displacement (uq, Vo, Wq)- This displacement could be produced in a body (held by suitable forces at the boundary) by certain forces applied near the origin of r. Betti's reciprocal theorem shows that the work done by the tractions JT^, ... on the surface S, acting through the displacement (m^, vg, Wg), is equal to the work done by certain forces applied at, and near to, the origin, 9,cting through the displacement (m, v, w), together with the work done by the tractions X,,^, ... on the surface S, acting through the same displacement. Let forces, each of magnitude P, be applied at the origin in the positive directions of the axes of coordinates, and let equal and opposite forces be applied in the negative directions of the axes of x, y, z respectively at the points {h, 0, 0), (0, A, 0), (0, 0, A). Let us pass to a limit by increasing P in- definitely and diminishing h indefinitely in such a way that limPA = 47r (X + 2^). We know from Article 132 that the displacement {u^, i^q, w^ will be produced, and it is clear that the work done by the above system of forces, Applied at, and near to, the origin, acting through the displacement (m, d, w) is -47r (\ + 2/x) (A);,. 161 . Calculation of the dilatation from surface data. (a.) When the surface displacements are given u, v, w are given at all points of S but X„, F„, Z^ are not given. In this case we seek a displace- ment which shall satisfy the usual conditions of continuity and the equations (1) at all points within 8, and shall become equal to (m„, «„. '"'o) at all points on S. Let this displacement be denoted by (mq', v^ ,'w^), and let the corre- sponding surface tractions be denoted by X/'"', 7/'°', ZJ^''\ Then we may apply the reciprocal theorem to the displacements (u, v, w) and {u^, V, Wo') which have no singularities within 8, and obtain the result [ |(X;(») u + F/c' V + ZJ^o) w) d8 = f ((X.wo' + TX + ^.W) d8 = (XyUo + Y^Vo + Z,Wa) d8. We may therefore write equation (14) in the form 47r(\ + 2/i)(A)„= jJ[(Z,'°' -Z;(«')m+(F,(«) - 7/ <»')?; + (^„ <»' -^/ <»!)«;] cZ^. (15) The quantities X„"" — X/'"', ... are the surface tractions calculated from displacements «„ — «o'. • ■ ■ and they are therefore the tractions required to hold the surface fixed when there is a "centre of compression'' at the origin of r. To find the dilatation at any point we must therefore calculate the surface tractions required to hold the surface fixed when there is a centre of compression at the point; and for this we must find a displacement which (1) satisfies the usual conditions of continuity and the equations of equili- brium everywhere except at the point, (2) in the neighbourhood of the point tends to become infinite, as if there were a centre of compression at the point, (3) vanishes at the surface. The latter displacement is analogous to Green's function. * J. Dougall, Edinbwrgh Math. Soc. Proc, vol. 16 (1898). 234 METHOD OF INTEGRATING THE EQUATIONS [OH. X (6) When the surface tractions are given, we begin by observing that ^f"", F„"", ZJ"* are a system of surface tractions which satisfy the conditions of rigid-body-equilibrium. Let (mo", V. Wo") be the displacement produced in the body by the application of these surface tractions. We may apply the reciprocal theorem to the displacements (w, v, w) and («„", v^', w^'), which have no singularities within S, and obtain the result and then we may write equation (14) in the form 47r (X + 2/.) (A)„ = [[{Z, («„" - Mo)+ F. W - V,) + Z, (w„"-w„)! (i>Sf.. . . (16) To find the dilatation at any point we must therefore find the displacement produced in the body when the surface is free from traction and there is a centre of dilatation at the point. This displacement is (mq" — Mq, v,^' — v^, W - Wo); it is an analogue of Green's function. The dilatation can be determined if the displacement (wo", v",Wo") can be found. The corresponding surface tractions being given, this displacement is indeterminate in the sense that any small displacement possible in a rigid body may be superposed upon it. It is easily seen from equation (16) that this indeterminateness does not affect the value of the dilatation. 162. Formulae for the components of rotation. In applying the formula (12) to a region bounded externally by the surface S of the body, and internally by the surface 2 of a small sphere surrounding the origin of r, we take for («', v', w') the displacement given in (22) of Article 132. It will be convenient to denote this displacement, omitting a factor, by (Ui, v^, w^,* and the corresponding surface tractions by X,(«), r,(<), ZJ-'>'\ We write {Ui,Vi,w,) = [0, -^, --g^j (17) The contributions of 2 to the left-hand and righi-hand members of (12) may be calculated by the analytical process of Article 160. We should find that the contribution to the left- hand member vanishes, and that the contribution to the right-hand member is 87r^(nrj.)o, where (ctj;),, denotes the value of m^ at the origin of r. We should therefore have the formula 8,r;u (ra-^)„= I [{(X.Ui + Y^Vi + Z^Wi)-{X^mu+ YJ^*)v + Z^mw)}dS, (18) which is analogous to (7). The same result maybe arrived at by observing that {u^,Vi,w^) is the displacement due to forces Anjijh applied at the origin in the positive and negative directions of the axes of y and z respectively, and to equal and opposite forces applied respectively at the points (0, 0, h) and (0, h, 0), in the limiting condition when h is diminished indefinitely. It is clear that the work done by these forces acting over the displacement {u, V, w) is in the limit equal to iirn ( ■= -k-) ■ Formulae of the same type as (18) for TZy and Bjj can be written down. * This notation is adopted. in accordance with the notation (itj, tij, Wj), ... of Article 132 for the displacement due to unit forces. 161-164] OF EQUILIBRIUM OF ISOTROPIC SOLIDS 235 163. Calculation of the rotation from surface data. (a) When the surface displacements are given, we introduce a displacement (m/, d/, w^) which satisfies the usual conditions of continuity and the equations of equilibrium (1), and takes at the surface the value (M4, vi, Wi); and we denote by X/(^), ?„'(*), ZJi*) the corre- sponding surface tractions. Then equation (18) can be written Stt^i (ti7:,)o = f f {(x;w - z,(«)) u + ( r;(4) - r^m) v + (^;w - .?,w) w}ds, (i9) in which the quantities Z^^'W- X^W, ... are the surface tractions required to hold the surface fixed when a couple of moment Stt/i about the axis of x is applied at the origin in such a way that this point becomes "a centre of rotation" about the axis of x. The corresponding displacement (m/ — M4, d/ — ■1)4, wi — w^ is an analogue of Green's function. (6) When the surface tractions are given we observe that the tractions X^t*), F^W, ^^(*), being Statically equivalent to a couple, do not satisfy the conditions of rigid-body-equi- librium, and that, therefore, no displacement exists which, besides satisfying the usual conditions of continuity and the equations of equilibrium, gives rise to surface tractions equal to X„(*), ...*. We must introduce a second centre of rotation at a chosen point A, so that the couple at A is equal and opposite to that at the origin of n Let ■u,i'^\ v,^-^), w^-^) be the displacement due to a centre of rotation about an axis at A parallel to the axis of x, so that {u,W,v,(A)^,c,W^=(o,^Ijp-, -^"jp), (20) where r^ denotes distance from A. Let X„"(*), F/'(*), ZJ'W denote the surface tractions calculated from the displacement {u^ — u^-^), v^ — v^^), w^ — w^^)). The conditions of rigid body equilibrium are satisfied by these tractions. Let (M4", ^4", W4") be the displacement which, besides satisfying the usual conditions of continuity and the equations of equi- librium, gives rise to the surface tractions X„"(*) Then, denoting by (or^)^ the value of CTj. at the point A, we find by the process already used to obtain (18) the equation Stt/. {(^,)o - (^.)^} = J j"[{Z„ («4 - u^W) + ...}- {Z;'(*> u+...)-\dS; and from this again we obtain the equation STTfi {(ro-^)o - {.■b!^)a] = \\{X^ {Ui - u^W - %") -I- Y, (Vi - v^i^) - <) +Z^{Wi-WiW-w^')}dS. (21) The quantities U4, - UiW — ui', ... are the components of displacement produced in the body by equal and opposite centres of rotation about the axis of x at the origin of r and a parallel axis at the point A when the surface is free from traction. This displacement is an analogue of the second Green's function. The rotation can be determined if such a displacement as (M4", vl\ W4") can be found. The indeterminateness of this displacement, which is to be found from surface conditions of traction, does not afiect the rotation, but the indeterminateness of vix which arises from the additive constant (crj;)^ is of the kind already noted in Article 157. 164. Body bounded by plane — Formulae for the dilatation. The difficulty of proceeding with the integration of the equations in any particular case is the difficulty of discovering the functions which have been denoted above by <, Wo", V, •••. These functions can be obtained when the * J. Dougall, loc. cit., p. 233. 236 SOLUTION OP THE [CH. X boundary of the body is a plane*. As already remarked (Article 135) the local effects of forces applied to a small part of the surface of a body are deducible from the solution of the problem of the plane boundary. Let the bounding plane be z = 0, and let the body be on that side of it on which z>0. Let («', y', z) be any point of the body, {x, y', — z) the optical image of this point in the plane z = 0, and let r, R denote the distances of any point {x, y, z) from these two points respectively. For the determination of the dilatation when the surface displacements are given we require a displacement (mq', V; '^o') which, besides satisfying the usual conditions of continuity and the equations of equilibrium (1) in the region z>0, shall at the plane z = Q have the value (m,,, Vq, Wq), i.e. (9r~'/9a;, dr~'^/dy, dr~'^/dz), or, what is the same thing, {dR~^/dx, dR~^/dy, —dR~^/dz). It can be shown without diffieultyf that the functions Ug, Vo, w^ are given by the equations 37?- % = w„ =-■ dx dR-' dy dR-^ l^^X+^^S^R- X + 3/ct - X + w + 2 , . .r .g 9a; 9^ ' dz ' X + 3/A ~ dydz ^ ■ " dz'' .(22) X + 3/U, , The surface tractions Z,'»), F,'<", Z,'»' on the plane ^ = calculated from the displacement (m„, Vo, Wo) are, since cos(z, v) = -l, given by the equations Z ("I = - ■ 2 — = dzdx 2/i 9'E-i 929a; ' dm-'- 9V-1 £Ji, — dz^ •2/--5;r = -2/^ d'R- dz^ .(23) * The application of Betti's method to the problem of the plane was made by Cerruti. Introduction, footnote 68.) t If in fact we assume for Uq, vq', wq such forms as the following :— SB-} ■■^0 = — 5r-+2«'. {Sie zv', Wn'= ■ dz we find for u', v', w' the equations ^ 1'^-'^) I (£ - % - %) ---4 - (^-^) 'i-^^'^-H-^.) dz :{,. , J/■^ , \ 9 fSu' dv' Sw'\ „„ ,1 which are all satisfied by gsjj- _ 2(\+jn) 32j;-i \ + 3/i dxdz ' ' X + 3/i 3i/3«' \ + 3/i 3z' ' for these functions are harmonic and are such that ~ + ~ +~=0. ox ay oz 164, 165] PROBLEM OF THE PLANE 237 and the surface tractions X/'"', ... on the plane z = calculated from the displacement (uq, »„', w,,') are given by the equations /9mo' dWo'\ „ \ + u d'R-' \ J. V — dx J ^'\ + 3/j, dxdz ' /9wo' 9y«'\ „ X + ii d'R- ) = -2/. ^'"--ffe-i-l?)-^'^ \ + 3/x 92/9^ ' 9wo' = 2/t X+/1 9^i2-' ..(24) ' X + 3/u. 9^^ We observe that X/'"', F/"", Z„'"" are equal respectively to the products of XJ"', 7„"", ^^<») and the numerical factor - (\ + ytt)/(X + Sfi), and hence that «', Vo", <') = - {(X + 3/^)/(X + /.)} «, i;„', w„'). It follows that, when the surface displacements are given, the value of A at the point («', y', z") is given by the equation ^'^-^(X + 3;.)JJfe^ + pi^+^'^)'^^'^y' .(25) the integration extending over the plane of {x, y). When the surface tractions are given the value of A at the point {x', y, z') is 27r (X + n) ^^ 1 St*""*" 9?" ^\ .(26) 165. Body bounded by plane — Given surface displacements. The formula (25) for the dilatation at («', y', z) can be written A = - /* [^'//^^^^-97//^'^^^^-^l#^^^^}--(^^) .(28) 7r(X + 3/i)9/ If we introduce four functions L, M, N, 4> by the definitions - L = lj^dxdy, M=jj^dxdy, N={f^dxdy, ._dL dM M '^~dx'^ dy'^dz" these functions of x\ y', z' are harmonic on either side of the plane z' = 0, ,-,.,,,. ,• 1 9i ,. 1 9ilf and at this plane the values oi u, v, w are limj'=+(, ~ h" 3^ ' •'i°i2'-+o ~ 2~ 9^ ' limj'=+o - s^ ^577 . The value of A at («', y , z') is - ; lirdz equations of equilibrium can be written X + yU. V'2 " 27r(X + 3/i) 9< _ ^ + M / 9^ ^~27r(X + 3^) dy' X + /M ,d(}) 27r(X + 3;a)^ 9< w — TT (X + 3/a) 0^' and the = 0, = 0, y .(29) ,* 238 SOLUTION OF THE [CH. X where V'^ = d'-/dx"- + d^/dy"- + d'/dz'\ The three functions such as u - {(X + /x)/27r (A, + 3/x)) z' (9^/9*') are harmonic in the region z' > 0, and, at the plane z' = 0, they take the values — ^7r~'^{dL/dz'), ..., which are themselves harmonic in the same region. It follows that the values of u, v, w at {x, y', sf) are given by the equations* __J^9Z 1 \ + fjL ,9^ ^~ ¥ird7 ^ 2^\ + %x^ dy" __ IdN J^ \ + /x , 9^ "'~ 2^97'^2^X + 3;u,^ a?' The simplest example of these formulae is afforded by the case in which u and v vanish at all points of the surface, and w vanishes at all such points except those in a very small area near the origin. In this case the only points {x, y, z) that are included in the integration are close to the origin, and is the potential of a mass at the origin. We may suppress the accents on a/, y', z' and obtain the solution } (30) .xz «z , A+3u 1 22\ which was considered in Article 131. In the problem of the plane this solution gives the displacement due to pressure of amount — 4w/i ^- — -A exerted at the origin when the plane 2 = is held fixed at all points that are not quite close to the origin. In an unlimited solid it is the displacement due to a single force acting at the origin and directed along the axis of 2. A second example is afforded by the ease in which v and w vanish at all points of the surface, and u vanishes at all such points except those in a very small area near the origin. The values of u, v, w at any point {x, y, z) are given by equations of the form B_'d^l\ ^ X+/i x-^\ , B \ + fi. 9?-i u= — :\r X + 3^ r^J^ '. ■2n dz\t-\ + 3li r^ J ^ 2n\ + 3^^ dz ' __B^ X + li d_ (xy\ ''~ 2jr X + 3^i92 Vr3y' '^~ 27r X + 3^ 82 Vr^J 27rX + 3)I~8F' where B= \ [ udxdy taken over the area in question. In an unlimited solid this would be the displacement due to (i) a " double force with moment," the forces of the double force being parallel to the axis of x, and the axis of the equivalent couple being parallel to the axis of y [Article 132 (6)], and (ii) a "centre of rotation about the axis of y" [Article 132 (6')]. The solution of the example in which u and w vanish at all points of the surface, and v vanishes at all such points except those in a very small area near the origin, may be written down and interpreted in the same way ; and the solution expressed by (30) may be built up by synthesis of the solutions of these three examples. * The results are due to Boussinesq. See Introduction, footnote 67. 165, 166] PROBLEM OF THE PLANE 239 166. Body bounded by plane — Given surface tractions* It is unnecessary to go through the work of calculating the rotations by the general method. The formula (26) for A can be expressed in the form 2ir (\ + (li) dz' ■ To effect this we introduce a function ;)(; such that dx/Sz' = 1/r at ^^ = 0, The required function is expressed by the formula X = log(z + z' + R); (31) it is harmonic in the space considered and has the property expressed by the equations 1-14 (^« Now at the surface z = we have dr2 _ dR-^ dR^ _ _ d\ ^__ o'x 9^~' _ ^'X dx ~ dx ~ dx' ~ dz'dx" dy ~ dy'dz" dz dz'^' If therefore we write F = Jlx,X<^dy. Q=jJY^xTrfjL d/ ^ 47r (X + fi) It follows that w is given by the equation '^ ~ 47r/^ 93" "*" 47r (X + m) "^ 47r/x^ 9^'" " Again the first of the equations of equilibrium is 47rytt dx'_ and the first of the boundary conditions is /du dw\ f .(35) = 0. "[dF^dx'J^^- 47r/A 9«' 1 d^F 1 d'H 9i|r Hence at / = _9^r _ ^ _ dz' \_ 4>Trfi doc'j 2-77 fi dz'^ iTr/j, dx'dz' "^ ^tt/m (X + fi) doc' ' and it follows that u is given by the equation 1 dF I dH X dH _9>h_J_y9i (36) 27r/i 9^:' 47r/i 9a;' 47rya (X + fi) dx 47rya 9a;" where a/tj is an harmonic function which has the property d^Jfi/dz' = yjr. Such a function can be obtained by introducing a function il by the equation D,:={z + z')log(z+/ + R)- R (37) Then fl is harmonic in the space considered and has the property 9fl 9fi ,_. 97=9? = ^ (^^^ If we write F,= llx,n,dxdy, 0^ = jiY,n,dxdy, H,= iiz^ndxdy/ ._dJ\d_G.dH, ^' ~ dx' + dy' + dz' ' I... (39) then all the functions Fi,Gi,Hi, yfr^ are harmonic in the space considered and dF, P 9Gi ^ dHi „ 9-\/r, In the same way as we found u we may find v in the form I dG 1 dH + - d^r. 1 ,dy!r — z .(40) .(41) 27r/i 9/ 47r/i dy' 4nrfi, (X + /a) dy' 4nrf^ dy' ' The simplest example of these formulae is aflforded by the case in which X„, Yy vanish at all points of the surface, and Z^ vanishes at all such points except those in a very small area near the origin, but I \Z^dxdy taken over this area =P. The values of m, v, w at 166, 167] PROBLEM OF THE PLANE 241 (x,y, z) are thea given by equations (35) of Article 135. In an unlimited solid this solution represents, as we know, the displacement due to (i) a single force acting at the origin and directed along the axis of z, (ii) a line of centres of dilatation along the axis of z from the origin to — oo [Article 132 (c)]. A second example is afforded by the case in which F„, Z, vanish at all points of the surface, and Xy vanishes at all such points except those in a very small area near the origin, but I I X^dxdy taken over this area =& The values of m, w, xo at {x, y,z) are given by the equations ^_ ^ A + 3^ 1 x'^\ s \ s r 1 • + - 2n-(\ + /j,)r 47r(X-f/i) \z + r r {z+rf _ S xy S xy ""4^ 1^~ 4:n-{\ + iJ.) r{z+rf' S X S X 27r{\+ix)r{z+r) Air (X + fi) r {z + r)' In the solid with a plane boundary these equations express the displacement due to tangential force S applied at the origin, the rest of the boundary being free from traction. In an unlimited solid they express the displacement due to (i) a single force acting at the origin, and directed along the axis of x, (ii) a line of centres of rotation along the axis of z from the origin to — oo , the axes of the equivalent couples being parallel to the axis of y [Article 132 (e)], (iii) a double line of centres of dilatation along the axis of z from the origin to — 00 , the axes of the doublets being parallel to the axis of x [Article 132 (rf)]. The solution of the example in which Xy, Z, vanish at all points of the surface, and Yy vanishes at all such points except those in a very small area near the origin, may be written down and interpreted in the same way ; and the solution expressed by equations (35)j (36), (41) may be built up by synthesis of the solutions in these three examples. 167. Historical Note. The problem of the plane — sometimes also called the "problem of Boussinesq and Cerruti" — ^has been the object of numerous researches. In addition to those mentioned in the Introduction, pp. 15, 16, we may cite the following : — J. Boussinesq, Paris, C. R., 1. 106 (1888), gave the solutions for a more general type of boundary-conditions, viz. : the normal traction and tangential displacements or normal displacement and tangential tractions are given. These solutions were obtained by other methods by V. Cerruti, Roma, Ace. Line. Rend. (Ser. 4), t. 4 (1888), and by J. H. Michell, London Math. Soc. Froc, vol. 31 (1900), p. 183. The theory was extended by J. H. Michell, London Math. Soe. Froc, vol. 32 (1901), p. 247, to seolotropio solid bodies which are transversely isotropic in planes parallel to the boundary. The solutions given in Articles 165 and 166 were obtained by a new method by C. Somigliana in 11 Nuovo Cimento (Ser. 3), tt. 17—20 (1885—1886), and this was followed up by G. Lauricella in 11 Nuovo Cimento (Ser. 3), t. 36 (1894). Other methods of arriving at these solutions have been given by H. Weber, Part. Diff.-Gleiohungen d. math. Physik, Bd. 2, Brunswick 1901, by H. Lamb, London Math. Soc. Froc., vol. 34 (1902), by 0. Tedone, Ann. di mat. (Ser. 3), t. 8 (1903), and by E. Marcolongo, Teoria .matematica dello equilibrio dei corpi elastici, Milan 1904. The extension of the theory to the case of a body bounded by two parallel planes has been discussed briefly by H. Lamb, loe. cit., and more fuUy by J. Dougall, Edinburgh Roy. Soc. Trans., vol. 41 (1904), and also by 0. Tedone, Palermo, Circ. Mat. Rend., t. 18 (1904), and by L. Orlando, Palermo, Circ. Mat. Rend., t. 19 (1905). L. E. 16 242 THE PKOBLEM OF THE PLANE [CH. X 1 68. Body bounded by plane — Additional results. (a) In the calculation of the rotations when the surface tractions are given we may take the point A of Article 163 (6) to be atan infinite distance, and omit Ui^--^\... altogether. We should find for M4", V4", a'4" the forms u"- 2: ^'^ ^^ ^'^ * dxdydz X+iid.vdi/' "^ ~ dfdz x+^i df^ di^' and we may deduce the formula 47r/i L ^ + M 9y' S*' In like manner we may prove that '-ini\_ X + 2/X dy^ _3_ (do _ dF For the calculation of ra^we should require a subsidiary displacement which would give rise to the same surface tractions as the displacement {dr-'^jdy, —dr-'^jdx, 0), and this displacement is clearly {-dR~'^jdy, dR'^jdx, 0) and we can deduce the formula 1 8_/3^_8^\ " inji dzf \dx' dy' ) ' (b) As an example of mixed boundary-conditions we may take the case where u, v, Z^ are given at z=0. To calculate A we require a displacement (it', «', w') which at 2 = shall satisfy the conditions where {XJ, TJ, ZJ) is the surface traction calculated from (m', v', vi). Then we may show that the value of A at the origin of r is given by the equation 4,r (X + 2,x) A = f f {(X.W - X;) u + ( r„(0) - TJ) v-Z, (wo - vS)) dxdy. We may show further that and then that 1 8 fazT „ /az a¥\] 2,r (X-t-2fi) Bz- jaz' '" \dx' "^ ayVJ ' and we may deduce the value of {u, v, w) at (si, y', 2') in the form 1 'cL \+^ a fag /3^,9^\1 271- 5/ 4ff;i (^ + 2m) ay loz' '^ W c^VJ ' 1 ai/ x+/i a [9^ /sz a^\| * 2^80- 47^^(x+2/x) ayjaz' '^Vay + syyf' _ j_ eg 1 \dH (^.^_MyX '^ In II hz' "'' 47r (X + 2^) 1 82' '" l^ay "*■ ay // 47r^(X-t-2/i) , a lag „ /8i ai/\] ^aZJay-^'^Vsy+ayJr 168, 169] somigliana's integrals 243 (c) As a second example we may take the case where X^, Y^, w are given at 2 = 0. To calculate A we require a displacement (m", v", w") which at 2 = shall satisfy the conditions XJ'=XJ!>\ 7;'=r,(o), «;"=w„, where XJ', TJ', Z^' denote the surface tractions calculated from (m", h", w"). "We can prove that the value of A at the origin of r is given by the equation and that « = — ^5 — , V = = — , w =—- 3a; ' 8y ' 32 and then we can find for A the formula 1 9 fdF . dG „ dNs _ ' + ^-2/* and for {u, v, w) the formulas _ 1 dF I dJV X+^ 3 fdFi dOi ^ \ ■" 27r,i g/ "•■ Stt dxf "*■ 4,r/.(X+2fi) 3y V3^' 3y / "47rix (\ + 2m) 3a^ VS^ ¥ '^ d^J' " 2,r/i 32' "^ 2t 3y "^ 47r^ (X + 2ii) df V 3^ "^ 3^' ** / X+jot d fdF dG d^\ iTTfj. (X + 2/i) 3y \dx' ■*■ 3/ '^ 3/ y ' 1 3i\^ X+/^ 3 /3i?^ 3^_ 3i^\ '^ 2v d/ 47r^t (X + 2;ii) 32' V dx' "•" 3y '* 82' y ' 169. Pormulse for the displacement and strain. By means of the special solutions which represent the effect of force at a point we may obtain formulae analogous to (7) for the components of displacement. Thus let (mj, Vi, Wi) represent the displacement due to unit force acting at {a/, y', 2') in the direction of the axis of X, so that iU„V„W,)=-^--^^~^-^^[^^,-^^-, g-g-, gJg^J, (42) and let -T^W, F„(i), Z^(^) be the surface tractions calculated from (mi, v^, Wi). We apply the reciprocal theorem to the displacements {u, v, w) and (mi, tJi, Wj), with a boundary consisting of the surface 8 of the body and of the surface 2 of a small sphere surrounding {x, y\ 2'), and we proceed to a limit as before. The contribution of 2 can be evaluated as before by finding the work done by the unit force, acting over the displacement (m, v, w), and the same ' result would be arrived at analytically. If the body is subjected to body forces {X, T, Z) as well as surface tractions Xy, Y^, Z^, we find the formulse* (tt)o= \ \ \p (Xwi+Fvi+Zwi) dxdydz + /"f[(.l>i+ r„«;i + .?,Wi)- (Z„Wm+ Y,mv-^ZiS)w)'\dS, (43) * The formulsB of this type are due to C. Somigllana, 11 Nuovo Cimento (Ser. 3), tt. 17—20 (1885, 1886), and Ann. di mat. (Ser. 2), t. 17 (1889). ' 16—2 244 somigliana's integrals [ch. X where the volume integration is to be taken (in the sense of a convergent integral) throughout the volume within S. We should find in the same way and {w)o= j j Ip [Xv^ + Yv^ + Zw^) dx dy dz + I ll{XyU,+ Y,V3 + Z,Ws)-{A\^^)u+ r,i:')v + ZJ.^)w)]dS. A method of integration similar to that of Betti has been founded upon these formulae*. It should be noted that no displacement exists which, besides satisfying the usual con- ditions of continuity and the equations of equilibrium (1), gives rise to surface tractions equal to Z^(i), F^O, ZJ^^), or to the similar systems of tractions X^i^), ... and A'^l^), ..., for none of these satisfies the conditions of rigid-body-equilibrium t. When the surface tractions are given we must introduce, in addition to the unit forces at {a/, y', 0'), equal and opposite unit forces at a chosen point A, together with such couples at A as will, with the unit forces, yield a system in equilibrium. Let (mj', h/, Wj^) be the displacement due to unit force parallel to x at (x', ij, z') and the balancing system of force and couple at A, and let X^'O, F„'('), Z/(') be the surface tractions calculated from (mi', v^, wi). Also let (Ml", vi", wi') be the displacement which, besides satisfying the usual conditions of continuity and the equations of equilibrium (1), gives rise to surface tractions equal to AV'I, jy', ZJ''^). We make the displacement precise by supposing that it and the corre- sponding rotation vanish at A. Then we have (m)o= I I \p{Xu{+Tvi+Zwi)dxdydz + jj{XA''h'-u,") + Y,{W-vi') + Z,(w,'-wn}dS. (44) The problem of determining u is reduced to that of determining {ui', vi', wi'). The dis- placement (mj' - ui', vi - vi', wi - wi') is an analogue of the second Green's function. If, instead of taking the displacement and rotation to vanish at A, we assign to ^ a series of positions very near to {x', y', z'), and proceed to a limit by moving A up to coin- cidence with this point, we can obtain expressions for the components of strain in terms of the given surface tractions J. In the first place let us apply two forces, each of magnitude A~' at the point (x', y', z') and at the point ix' -\-h, y', z'), in the positive and negative directions respectively of the axis of x. In the limit when h is diminished indefinitely the displacement due to these forces is ( -^ , ^ , 5-7 ) . Let (wu, Vn, Wii) be the displace- ment produced in the body by surface tractions equal to those calculated from the displacement i-^, -^ , -^ ) . Then the value of (dujdx) at the point {x', y', z') is given by the formula In like manner formulae may be obtained for dvjdy and dwjdz. ' G. Lauricella, Pirn Ann., t. 7 (1895), attributes the method to Volterra. It was applied by C. Somigliana to the problem of the plane in II Nuovo Cimmto (1885, 1886). t J. Dougall, loc. cit. p. 233. J G. Lauricella, loc. cit. 169, 170] VARIOUS METHODS OF INTEGRATION 245 Again, let us apply forces of magnitude ^-i in the positive directions of the axes of y and z at the origin of r, and equal forces in the negative directions of these axes at the points (*■', y, «' + A) and {id, y' + h, z') respectively, and proceed to a Kmit as before. This system of forces satisfies the conditions of rigid-body-equilibrium, and the displacement due to it is / 3m3 8^2 dvi dvi &W3 3w2\ \dy^ dz' dy'^ dz' dy^dz)' Let (itas) "23) '"'23) be the displacement produced in the body by surface tractions equal to those calculated from the displacement (-3-^ + -3^1 ■••) ■•• ) • Proceeding as before we obtain the equation -//[^■{(¥*^)-""}*''-{(l*&)-"4*^-{(¥ + ^}-'-}]'*-(«' In like manner formulae may be obtained for du/dz+dw/dx and dv/dx + dujdy. 170. Outlines of various methods of integration. One method which has been adopted sets out from the observation that, when there are no body forces, ra-j., nr^, nr^, as well as A, are harmonic functions within the surface of the body, and that the vector (cr^, nr^, cj^) satisfies the circuital condition dziTa. d'u^ti 3^z ^ 5 -I i J ?=0. dx dy dz From this condition it appears that cTj., ruy, sr^ should be expressible in terms of two in- dependent harmonic functions, and we may in fact write* dd) dy dv " dy dx dz ' 9d) 9y 9y ' dz tiy " dx' where (^ and x ^^"^ harmonic functions. The equations of equilibrium, when there are no body forces, can be written in such forms as '^~y'dx^'° W^y dxdy + ^ dxdz) and it follows that =-£(^+^1+4-^^1; Cf. Lamb, Hydrodynamics, Chapter xi 246 VARIOUS METHODS OF INTEGRATION [CH. X This expression ■ represents, as it should, an harmonic function ; and the quantities A, cjj., ■Sly, cTj are thus expressible in terms of two arbitrary harmonic functions ^ and )^. If now these functions can be adjusted so that the boundary-conditions are satisfied A and (tETj., TUy, -01^ will be determined. This method has been applied successfully to the problem of the sphere by C. W. Borchardt* and V. Cerruti f . Another method % depends upon the observation that, in the notation of Article 132, %=«!, U3=Wi, w^=V3, and therefore the surface traction J!„(') can be expressed in the form where I, m, n are written for cos (.r, v), cos (y, v), cos (z, v). The surface tractions -ly^), AV^) can be written down by putting v and w respectively everywhere instead of u in the expression for 2^). It follows that (Z„('), X^P', Xv<^') is the displacement produced by certain double forces. In like manner (7„(i), I'^P)^ y_^(3)) and (Z^m, Z,'^\ ZJ.^)) are systems of displacements which satisfy the equations (1) everywhere except at the origin of r%. On this result has been founded a method (analogous to that of 0. Neumann || in the theory of Potential) for solving the problem of given surface displacements by means of series. The equations of equilibrium, when there are no body forces, can also be written in the forms v^(«+^.a)=o, v^(.+^,a)=o, .^(.+^.a)=o, showing that the three expressions of the type M+J/i~i (X + /i.);j;A are harmonic functions. These three harmonic functions must be adjusted so that the relation 3m ov dw dx dy dz ' where A also is an harmonic function, may be satisfied, and they must also be adjusted so as to satisfy the boundary-conditions. This method has been developed by 0. Tedonet and applied by him to the problems of a solid bounded by a plane, by two parallel planes, by a sphere, by two concentric spheres, by an ellipsoid of revolution, and by a right circular cone. * Berlin Monatsber., 1873, reprinted in C. W. Borohardt's Ges. Werke, Berlin, 1888, p. 245. t Comptes rendus de V Association Fran^aise pour I'avancementde Science, 1885, and Roma, Ace. Line. Rend. (Ser. 4), t. 2 (1886). X G. Laurloella, Pisa Ann., t. 7 (1895), and Ann. di mat. (Ser. 2), t. 23 (1895), and II Nitovo Cimento (Ser. 4), tt. 9, 10 (1899). § The result is due to C. Somigliana, Ann. di mat. (Ser. 2), t. 17 (1889). II Untersuchungen ilber das logarithmische und Newton'sche Potential, Leipzig, 1877. Of. Poinoar^, loc. cit. p. 230. H Ann. di mat. (Ser. 3), t. 8, 1903, p. 129; (Ser. 3) t. 10, 1904, p. 13; Roma, Ace. Line. Bend. (Ser. 5), t. 14, 1905, pp. 76 and 316. CHAPTEE XI THE EQUILIBRIUM OF AN ELASTIC SPHERE AND RELATED PROBLEMS 171. In this Chapter will be given examples of the application of the method of series (Article 158) to the problem of the integration of the equations of equilibrium of an isotropic elastic solid body. Of all the problems which have been solved by this method the one that has attracted the most attention has been the problem of the sphere. In our treatment of this problem we shall follow the procedure of Lord Kelvin*, retaining the equations referred to Cartesian coordinates instead of transforming to polar coordinates, and we shall give his solution of the problem. The solution is expressed by means of infinite series, the terms of which involve spherical harmonics. We shall begin with a general form of solution involving such functions. 172. Solution in spherical harmonics of positive degrees. We propose to solve the equations ('^+'^)(l^' af t)+/^^^K".-) = o. (1) where A = 5- +5- +3-, (2) dec dy oz subject to the conditions that u, v, w have no singularities in the neighbour- hood of the origin. Since A is an harmonic function, we may express it as a sum of spherical solid harmonics of positive degrees, which may be infinite in number. Let Am be a spherical solid harmonic of degree n, that is to say a rational integral homogeneous function of x, y, z of degree n which satisfies Laplace's equation ; then A is of the form A = SA„, * See Introduction, footnote 61. Eeferences will be given in the course of the Chapter to other solutions of the problem of the sphere, and additional references are given by B. Marcolongo, Teoria matematica dello equilibria dei corpi elastici (Milan, 1904), pp. 280, 281. 248 SOLUTION OF THE [OH. XI the summation referring to different values of n. Take one term A„ of the series, and observe that 9A„/9a; is a spherical solid harmonic of degree n—l, and that, if r denotes the distance of the point (x, y, z) from the origin. We see that particular integrals of equations (1) could be written in such forms as \ + /^ r' aA„ "~ jx. 2(2?i + l) da; ' and more general integrals can be obtained by adding to these expressions for w, . . . any functions which satisfy Laplace's equation in the neighbourhood of the origin, provided that the complete expressions for u, ... yield the right value for A. The equations (1) and (2) are accordingly integrated in the forms A = SA„, \ in which [/"„, F,i, W-n are spherical solid harmonics of degree n, provided that these harmonics satisfy the equation ^-— ^-r'-5^i--^©+v"-'i') (*) Introduce the notation ^" = "^^ + -37"'^^^ ^^^ then T|r„ is a spherical solid harmonic of degree n, and equation (4) requires that A„ and i/rn should be connected by the equation The harmonic function A„ is thus expressed in terms of the comple- mentary functions Z/n+i, . . . ; and the integrals (3) may be expressed as sums of homogeneous functions of degree n in the forms («,.,^) = -2ilf.r^f-^, ^-|^\ ^-^^) + S(fr„,F„,Tf „).... (7) where P„, F„, 1F„ are spherical solid harmonics of degree n, M^ is the constant expressed by the equation M ^ ^ + /"■ ""2(?i-1)\ + (3m-2)^= W and ^n-\ is a spherical solid harmonic of degree n—\ expressed by the equation ^"-'-■9^ + ^ + 17" (^) 172, 173] PROBLEM OF THE SPHERE 249 It may be observed that equations (7) also give us a solution of the equations of equi- librium when n is negative, but such a solution is, of course, valid only in regions of space which exclude the origin. As an example, we may put »= — 1, and take £^„=0, 7„=0, TF„=l/r. We should thus obtain the solution which was discussed in Article 131. 1 73. The sphere with given surface displacements. In any region of space containing the origin of coordinates, equations (7) constitute a system of integrals of the equations of equilibrium of an isotropic solid body which is free from the action of body forces. We may adapt these integrals to satisfy given conditions at the surface of a sphere of radius a. When the surface displacements are prescribed, we may suppose that the given values of m, ?;, w at r = a are expressed as sums of surface harmonics of degree n in the forms (m, D, wV=„ = S (4„, 5„, (7„) (10) Then r".4„, r"5„, r"C„ are given spherical solid harmonics of degree re. Now select from (7) the terms that contain spherical surface harmonics of degree n. We see that when r = a the following equations hold : — (11) -A --M ^ aP. '^T't+x , Tj \ — B --M ^ a'^-^^^+V The right-hand and left-hand members of these equations are expressed as spherical solid harmonics of degree n, which are equal respectively at the surface r= a. It follows that they are equal for all values of x, y, z. We may accordingly use equations (11) to determine ?7„, Fm, TTm in terms of ■^m Jjni ^n- For this purpose we differentiate the left-hand and right-hand members of equations (11) with respect to x, y, z respectively and add the results. Utilizing equation (9) we find the equation ^»-4.S^») + a-yG^«)4.(J^^») (^^) Thus all the functions i/r^ are determined in terms of the corresponding An, Bn, On, and then ?/„,... are given by such equations as U -^A +M ^ a^^izit? a" dx ' The integrals (7) may now be written in the forms 250 SOLUTION OF THE [OH. XI in which if„+, = 2 (^+i);, + (3« + 4)^ •• and V^„+, = g^ (^^^, .4„+,J + g^ (^^, i^„«j + g^ ^^;^, 0„ By equations (13) the displacement at any point is expressed in terms of the prescribed displacements at the surface of the sphere. 174. Generalization of the foregoing solution. (i) The expressions (7) are general integrals of the equations of equilibrium arranged as sums of homogeneous functions of x, y, z of various integral degrees. By selecting a few of the terms of lowest orders and providing theni with undetermined coefficients we may obtain solutions of a number of special problems. The displacement in an ellipsoid due to rotation about an axis has been found by this method*. (ii) If we omit the terms such as AnirjaY from the right-hand members of equations (1.3) we arrive at a displacement expressed by the equation di'Ty'Wzr^*' ^^'^^ This displacement would require body force for its maintenance, and we may show easily that the requisite body force is derivable from a potential equal to ?[(»i + 1)X + (3to + 4);i]x/a„ + i, and that the corresponding dilatation is — 2(?i.+ l)-v^„+i. We observe that, if X and fi, could be connected by an equation of the form (TO-l-l),X + (3?l-|-4);u = 0, (15) the sphere could be held in the displaced configuration indicated by equation (14) without any body forces, and there would be no displacement of the surface. This result is in apparent contradiction with the theorem of Article 118 ; but it is impossible for X and /i to be connected by such an equation as (15) for any positive integral value of w, since the strain-energy-function would not then be positive for all values of the strains. (iii) The results just obtained have suggested the following generalization t ; — Denote (X+li)lli by T. Then the equations of equilibrium are of the form ox We may suppose that, answering to any given bounding surface, there exists a sequence of numbers, say tj, t^, ..., which are such that the system of equations of the type ^"3^(1^+1^ + ^)+^'^' = °' ('= = 1'2,...) * C. Chree, Quart. J. of Math., vol. 23 (1888). A number of other applications of the method were made by Chree in this paper and in an earlier paper in the same Journal, vol. 22 (1886). t E. and F. Cosserat, Paris, C. R., tt. 126 (1898), 133 (1901). The generalization here indicated is connected with researches on the problem of the sphere by E. Almansi, Boma, Ace. Line. Rend. (Ser, 5), t. 6 (1897), and on the general equations by G. Laurioella, Ann. di mat. (Ser. 2), t. 23 (1895), and II Nuovo Cimento (Ser. 4), tt. 9, 10 (1899). The theory of the solution of the equations of equilibrium by this method is discussed further by I. Fredhohn, Arkiv fdr mat.,fys. och astr., Bd. 2 (1905), Nr. 28, and A. Korn, Acta Math., t. 32 (1909), p. 81. 173-175] PROBLEM OF THE SPHERE 251 possess solutions which vanish at the surface. Denote dU„ldx + dVJd7/ + dW^/dz by A«. Then A^ is an harmonic function, and we may prove that, if k' is different from k, j I U^A,,dx:d^dz = 0, (16) where the integration is extended through the volume within the bounding surface. We may suppose accordingly that the harmonic functions A, are such that an arbitrary harmonic function may be expressed, within the given surface, in the form of a series of the functions A» with constant coefficients, as is the case with the functions ^„ + i when the surface is a sphere. Assuming the existence of the functions C,, ... and the corresponding numbers t,, we should have the following method of solving the equations of equilibrium with prescribed displacements at the surface of the body : — Let functions «„, wq, Wq be determined so as to be harmonic within the given surface and to take, at that surface, the values of the given components of displacement. The function Uq, for example, would be the analogue of S — J„ in the case of a sphere. Calculate from uo, «o, Wq the harmonic function Aq deter- mined by the equation , " da dy oz Assume for u, v, w within the body the expressions {u,v,w) = {uo,Vg,Wo)-TS — —iU,, F„ W^), (17) T-Ti; where the A's are constants. It may be shown easily that these expressions satisfy the equations of equilibrium provided that S4,A, = Ao. The conjugate property (16) of the functions A^ enables us to express the constants A by the formula -^K / / \{ii'K)^dxdydz= j I j ^o^ndaidydz (18) the integi-ations being extended through the volume of the body. The problem is there- fore solved when the functions U^, ... having the assumed properties are found*. 1 75. The sphere with given surface tractions. When the surface tractions are prescribed, we may suppose that the tractions Xr, Y^, Z^ at r = a are expressed as sums of surface harmonics of various degrees in the forms {Xr, Yr, Zr)r=a'=^{Xn, Y^, Z^), (19) so that r^Xn, ^Yn, r'^Z^ are given spherical solid harmonics of degree n. Now Xn ... are expressed in terms of strain-components by formulae of the type * E. and F, Cosserat, Paris, C. R., t. 126 (1898), have shown how to determine the funotions in question when the surface is an ellipsoid. Some solutions of problems relating to ellipsoidal boundaries have been found by C. Chree, loc. cit. p. 250, and by D. Edwardes, Quart. J. of Math., vols. 26 and 27 (1893, 1894). 252 THE PROBLEM OF THE SPHERE [CH. XI and these are equivalent to formulae of the type rXr ^^ * , 9? , du ,„., -y^r^+i^'-di^-^ ^2^) in which f = ux + vy + wz, (21) so that ^Ir is the radial component of the displacement. We have now to calculate X^, ... by means of the formulae of type (20) from the displacement expressed by the equations (7). We know already that this displacement can be expressed by such formulae as ?t = 2 ;^"9+^-«'^-'*^'] (^^> We proceed to calculate Xr, Yr, Zr from these formulae. In the result we shall find that An, Bn, On can be expressed in terms of X„, F„, Zn- When these expressions are obtained the problem is solved. We have at once ?=2 (wAn + yBn + zGn) ^ + Mn^-^a'in + l)i|r„+, - MnT^ (n - l)>/r„_i The terms such as xAnr^/a^ are products of solid harmonics, and we transform them into sums of terms each containing a single surface harmonic by means of such identities as «/(«. 2/. ^) = ; 'df _ r^ _9 /^^ dx a2i+i9a;Vr-™+i-^, (23) 2n+l We obtain in this way the equation {xAn + yBn + zCn) -„ = g^^^ i^n-. " -^, ),...(31) 1 X(w + 2)-/^(?i-3) , where Jiin^n — tttt tt^ — Tq o\ V"-'^ 2n + l\{n—l) + fi{3n — 2) .(34) 254 THE PROBLEM OF THE SPHERE [CH. XI From the sum of all the terms in the expression for rX^/fi we select those which contain spherical surface harmonics of degree n. The value of the sum of these terms at the surface r = a must be the same as the value of r''Z„/a"~'/i. at this surface. We have therefore the equation (n-l)A„--^„r--g^(^^-2-^_^-yg-^^Si^,_„_2 and constant factors. On substituting in the equations of type (33) we have J.„, ii„, C'„ expressed in terms of X„, F„, Z^. The problem is then solved. 176. Conditions restricting the prescribed surface tractions. The prescribed surface tractions must, of course, be subject to the conditions that are necessary to secure the equilibrium of a rigid body. These conditions show immediately that there can be no constant terms in the expansions such as 2X„. They show also that the terms such as Xj, Y^, Z^ cannot be taken to be arbitrary surface harmonics of the first degree. "We must have, in fact, three such equations as //' (y2Z„-02r„)rf5=O, 175-177] WITH GIVEN SURFACE TRACTIONS 255 where the integration is extended over the surface of the, sphere. Writing this equation in the form and transforming it by mejins of identities of the type (23), we find the equation For any positive integral value of n, the subject of integration in the second of these integrals is the product of a power of r (which is equal to a) and a spherical surface harmonic, and the integral therefore vanishes, and the like statement holds concerning the first integral except in the case m = l. In this case we must have three such equa- tions as |(.^,)=gi(,-r,), and these equations show that rX-^, rY-y, rZi are the partial differential coefficients with respect to a;,y, a of a homogeneous quadratic function of these variables. Let XJ>-), ... be the stress-components that correspond with the surface tractions Xi, .... Then we have such equations as rX^=xX,m+yXym+zX,m, It thus appears that X^W, ... are constants, and the corresponding solution of the equations of equilibrium represents the displacement in the sphere when the material is in a state of uniform stress. 177. Surface tractions directed normally to the boundary. When the surface traction consists of tension or pressure at every point of the surface we may take the normal traction to be expressed as a sum of surface harmonics in the form TiRn- Then we have at the surface rXr=x^-Rn, r7.,=y^-Rn, rZr=z^—R^. Now the first of these equations gives for rX^ at r=a the formula rXr H- The right-hand member of this equation must therefore be the same as the left-hand member of equation (33), or it must be the same as By the processes already employed we deduce the two equations [(..Hl) + (. + 2)(2»+5)^„.J^„..J^^±|S±^©''^^^tl, and - „„,, - 0-n-i- '-' 2n + l\aJ ^-""' 2n + 3\aJ ^i ' and then we can easily find the ^'s, B'a and C's. In the case where 2^„ reduces to a 256 THE PEOBLEM OF THE SPHERE [CH. XI single term Rn+i, the only A's, ... which occur have suffixes n and n + 2, and we may show that with like expressions for the B's and C's. 178. Solution in spherical harmonics of negative degrees. When the space occupied by the body is bounded by two concentric spheres* solutions can be obtained in the same way as in Article 172 by the introduction of spherical harmonics of negative degrees in addition to those of positive degrees. To illustrate the use of harmonics of negative degrees we take the case where there is a spherical cavity in an indefinitely extended mass. Using, as before, U„, F„, Tf„ to denote spherical solid harmonics of positive integral degree n, we can write down a solution of the equations of equilibrium in the form where ^n + 1 - ?• ^g^ \^^2n ^ij^di/ Vr^" + V ^ 3^ \r'"' + : and Kn — 2(re+2)\ + (3n + 5)/x' The function i//-„+i is a spherical solid harmonic of degree n + 1, and the dilatation calculated from the above expression for the displacement is given by the formula 2_(2m+3)^ r^ X+^ '^V^-' + a- The solution expressed by a sum of particular solutions of the above type can be adapted to satisfy conditions of displacement or traction at the surface of a cavity r=a. An example of some interest is afforded by a body in which there is a distribution of shearing strain t. At a great distance from the cavity we may take the displacement to be given by the equation {u,v,w) = {sy, 0,0), where s is constant. In this example we may show that, if the cavity is free from traction, the displacement at any point is expressed by equations of the form where A, B, C are constants, and we may find the following values for A, B, C : — 3X + 8;x ^. 3(X+^) 3(X + ^) * Lord Kelvin's solution is worked out for the case of a shell bounded by concentric spheres, and includes the solution of this Article as well as that of Articles 172, 173, 175. + See Vhil. Mag. (Ser. 5), vol. 38 (1892), p. 77. 177-179] DEFOEMED BY BODY FORCES 257 The value of the shearing strain =- + =- can be calculated. It will be found that, at the point 3:=0, y = 0, r=a, it is equal to =rr — 7t^«. The result shows that the shear in the neighbourhood of the cavity can be nearly equal to twice the shear at a distance from the cavity. The existence of a flaw in the form of a spherical cavity may cause a serious diminution of strength in a body subjected to shearing forces*. 179. Sphere subjected to forces acting through its volume. Par- ticular solution. When the sphere is subjected to body forces we seek in the first place a particular solution of the equations of equilibrium of the type (X + /x)|^ + /jiV^u + pX = 0, and then, on combining this solution with that given in (13), we obtain expressions for the displacement which are sufficiently general to enable us to satisfy conditions of displacement or traction at the surface of the sphere. If the body force (X, Y, Z) is the gradient of a potential V which satisfies Laplace's equation, the particular integral can be obtained in a simple form ; for, within the sphere, V can be expressed as a sum of spherical solid harmonics of positive degrees. Let F= 2F„, where Vn is such an harmonic function, and consider the equations of the type (X + ;.)g + ;.V^. + P^^» = (37) Particular integrals of these equations can be obtained by putting 3d) 9(i dd> dx dy oz if (X + 2At)V^i + 3) + 2yL.(>i + 1) ,„+>!( Ill p, X + 2/x L (2k + l)fj, dx (2n + 1) (2n + 3) /x da;W"+\ (39) The formulie for Y^ and Zr can be written down by substituting d/dy and d/dz successively for d/dx in the right-hand member of (39). 180. Sphere deformed by body force only. When the surface is free from traction the displacement is obtained by adding the right-hand members of equations (13) and (38), in the former of which the functions An, ■•• are to be determined in terms of F„, ... by the conditions that the sum of the expressions for rX^jfi, in the left-hand member of (33) and the right-hand member of (39) must vanish. We take the potential SF„ to consist of a single term F^+i, in which n > 1, and then we have three equations of the type (n + l)^„+3^„^, i4„+,r ^^\:^^,) 2^7Tia^l^^^^-"-' ^ p r \ + (» + 2)/x djn^ _ X(2n-f5) + 2/^(w + 2) d_ fV^\ \ + 2fil (2?i-f3)/i dx {2n + Z){2n + 5)ix dx\r^+'J^ (40) which hold at the surface of the sphere, and therefore, in accordance with an argument already employed, hold everjnvhere. We notice that, if the material is incompressible so that the ratio fi/X vanishes, the particular integrals expressed by (.38) vanish, but the surface tractions depending upon the particular integrals do not vanish. The right-hand member of (39) becomes, in fact, ^2n + 3 ' In this case the equations by which A„, ... are to be determined are the same as those which were used in Article 177, provided that, in the latter, (rla)"*^ B„ + i is replaced by p 'i. + i- It follows that the displacement produced in an incompressible sphere by body force derived from a potential F„+i is the same as that produced by purely normal surface traction of amount p ['„ + j «" + '/?•'' "*"'*. Returning to the general case, we find, as in Article 177, that yp-n+i and ^_„_^ are the only yjr and (f) functions that occur, and that the only ^'s, ... * Chree, Cambridge Phil. Soc. Train., vol. 14 (1889), p. 250. 179-1 80 aJ deformed by body forces 259 which occur have suffixes n and ?j + 2. By the processes already employed we obtain the equations 2n + I {a) "P-"-^ - J::^ {2n + 3)f, ^^ + ^> ^"+^- The value of f at r = a is 2 (2« + 5) X + 2/. "* "^"^^ + 2^rr5 '''»+^ ~ 2^m UJ '^^-^' and, since ilr^+i and -n-i are determined by the method of Article 180, it will be found that the complementary solution is of the form (u, V, ^") = Fn.(J^,^, ^) Vn+ Gn.{oo, y, z) Vn, (42 A) where the coefficient F„ is of the form a + /3r^, a and /3 being constants, and the coefficient (?„ is constant. For a shell bounded by concentric spherical surfaces, as well as for a solid sphere, we might begin by assuming formulae of this type, in which F^ and Gn are functions of r, and determine these functions directly from the equations of equilibrium under no body force. We should find ^ = {^S' + ''^+('* + ^)^^»' (^2^) and thence with similar expressions for d^/dy and dA/dz, and we should also find " ~ 1 rfr= '^ r dr^ ^^"j 5* ^ { dr' ^ r 'W\ "^ ^'" with similar expressions for V''v and V^w. The equations of equilibrium under no body force are then satisfied if Fn and Gn satisfy the two equations and (42D) From these two equations we find the equation 1 I ilK ,2ndt\ I (d^Gn ^2(n + 2)dGJ „ r dr I dr^ ^ r dr ^ ^^"| {"d^ + r ~d^] = "' which, by means of the identity 1 d^ (d'Fn 2n dJ\l ^{d^ 2(n + l) d) /I dP^N rdr\dT-' r dr ] \dr' r drjWdr)' can be written iy-^im-'-h" <-^> * other applications of the method will be found in the Essay by the Author Some ProUemi of Geodytianiics, Cambridge, 1911. 180 a] under body foeces 261 The complete primitive of this equation is 1 dFn p _. , Bn r dr """ ~ ^" "•" ^2»+i ' where An and 5„ are arbitrary constants. This equation gives ^» = r(?„ + 4„r+^:, (42F) and '^^r^^O +A -^-^ whence we obtain d'Fn , 2ndFn -^ dGn /„ „x ^ .„ ,x . ^^ +7" *^+ ^^" = '" V ^^^'^ + ^^ ^" + ^^''^ -^^^^ so that equation (42 C) becomes iX + 2^)\r^ + (2n + 3)Gn} + A„{nX + (3n + l)f.} + '^^^^±^-^^0. The complete primitive of this equation is p _ n\ + (Sn+l),j. njX + fi) Bn C» ,„ p. (2« + 3)(\ + 2/a) " 2(X,+2jc<,)r»+i"^r^"+" •••^*^^^ where Cn is an arbitrary constant, and, when this form is substituted for (?„, equation (42 F) becomes dFn^ (n + 3)\ + (n + 5)/j, (w- 2)X + («- 4)/^ ^„ g„ dr " (2n + 3)(X + 2fi) "^ 2(A,+ 2/i) ' r'n + ^m+2' from which we find (n + 3)X + (w + 5)/^ (w-2)\ + (w-4)^ ^„ C„ "~ 2(2n + 3)(A, + 2/i) " "^ 2 (2re - 1) (X, + 2^) r^"-' (2m+1)?^"+i "' (42 H) where Z)„ is an arbitrary constant. Equations (42 H) and (42 G) express the general forms of Fn and Gn- In the case of a solid sphere the terms in Bn and Gn are to be omitted. We may now calculate the component tractions across any spherical surface concentric with the boundary by means of the formulae of the type (20). The formula (42 B) for A gives "^ - t X+2f. ^" + X + 2^ r^"+4 "■ Again we have ^=(nFn + r'Gn)rn, (421) so .w ^I.i.,.,^a4'^-,{^^^rf-,2G.}.r.. 262 EQUILIBRIUM OF A SPHERE [CH. XI and this is dx 2 (2ft + 3) (X. + 2/i) n + 1 AnT'- + 2ft + 1 ?•«'+! Also we have du + dFn n{(n + l)\ + (n + S)fi] B^ 2 (2n - 1) (X + 2/x) r-^-^ (w + l){n\ + (n-2)^} (2n + 3)(X + 2;L4) ft{(w + l)X + (»+3)M} ^n _ (W + 1) On ■2{\ + 2/j,) r"'+' r^+^ a; Vji. ]dVn , f dG„ ^'_« = l,.^ + ^,^_2)^4^'+ r!9^ + »iG„UF„, ...(42 J) ( dr die dr and this is du r= u = or {(n + B)\ + {n + 5)fj,} ., {n + \){{n-2)-K + {n-^),x] Bn 2(27i + 3)(X-+2ia) "'^ " 2(2n-l)(X + 2/x) r«'-' + (2?i m+ 3 2?i + l r^* (-'n - + {n-2)Dn dx + n{nX + (Sn + 1) fi} . (2n + 3)(\+2/^) " + w(w + l)(\ + At) Bn 2{\ + 2fi) r^-t ■(n + 3) w iZJ Ktj. Hence we find rXr n (n + 2) X + (n' + 27i - 1) fj, ^ (n ■l)\ + (n'-2)^ B„ (2k + 3) (X + 2/i) + 2(w + 2 ) Cn 2m + 1 r'"+^ + 2{n-l)Dn dVn + + dx n (X - fi) B„ (2?i-l)(X + 2/i) r^~' (2n' + 4ft + 3)X+ 2('n? + n+l)fi (2n + 3) (X + 2yii) 2(n + 2)(7„ -^n ^ ' n- .(42 K) X + 2/A r^' To obtain the complete expressions for the surface tractions in the shell under body force we have to add to the expressions of this type those of the type expressed in (39), which may be written rXr P n + 1 ^dVn X + 2/Li, 2w + 3 '^ 'dx |^^?i!L±i)Ur„; ..(42 L) \ + 2fji.[fi ' 2n + 3 and, when this is done, the coefficients oidVn/dx and xVn must vanish for the values of r answering to the bounding surfaces. We thus find four equations to determine the constants A„, B^, G^, Dn- An example of this method is afforded by a rotating sphere or spherical shell*. If the axis of rotation is the axis of z, and a is the angular velocity, the equations of motion of the rotating solid are identical with the equations of equilibrium of the same solid under body force per unit of mass, having as components parallel to the axes of x, y, z a^x, (o^y, 0. The complete solution, expressed in terms of polar coordinates, for both the sphere and the shell, is given by Chree in the memoir cited on p. 258, and further discussed by him in Cambridge, Trans. Phil. Soc, vol. 14 (1889), p. 467. 180a, 181] UNDER BODY FORCES 263 These forces are derived from a potential V, which is given by the equation or, as it may be written, where the second term is a spherical solid harmonic, while the first term gives rise to purely radial forces. The displacement due to the radial force | a^r may be found by the method of Article 98. The displacement due to the remaining forces may be found by the method of this Article, by putting n = 2 and substituting the expression for Fa- The solution of the problem is to be obtained by combining the two displacements thus determined. For a solid sphere the result is M _ y _ pa' x~y~ 3 1 / 5X + 6;t 5(X + 2/i)l,3\ + 2^i 2_^5^^+^„._,.)], 181. Gravitating incompressible sphere. The chief interest of problems of the kind considered in Article 179 arises from the possibility of applying the solutions to the discussion of problems relating to the Earth. Among such problems are the question of the. dependence of the ellipticity of the figure of the Earth upon the diurnal rotation, and the question of the effects produced by the disturbing attractions of the Sun and Moon. All such applications are beset by the diflSculty which has been noted in Article 75, viz. : that, even when the effects of rotation and disturbing forces are left out of account, the Earth is in a condition of stress, and the internal stress is much too great to permit of the direct application of the mathematical theory of superposable small strains*. One way of evading this diflSculty is to treat the material of which the Earth is composed as homogeneous and incompressible. When the homogeneous incompressible sphere is at rest under the mutual gravitation of its parts the state of stress existing in it may be -taken to be of the nature of hydrostatic pressure f; and, ii po is the amount of this pressure at a distance r from the centre, the condition of equi- librium is dp„/dr = -gpr/a, (43) where g is the acceleration due to gravity at the bounding surface r= a. Since ^o vanishes at this surface, we have p, = yp(a'-r')/a (44) * The difficulty has been emphasized by Chree, Phil. Mag. (Ser. 5), vol. 32 (1891). t Cf. J. Larmor ' On the period of the Earth's free Eulerian precession,' Cambridge Phil. Sac. Proc, vol. 9 (1898), especially § 13. 264 GRAVITATING INCOMPRESSIBLE SPHERE [OH. XI When the sphere is strained by the action of external forces we may- measure the strain from the initial state as " unstrained " state, and we may suppose that the strain at any point is accompanied by additional stress superposed upon the initial stress po- We may assume further that the com- ponents of the additional stress are connected with the strain by equations of the ordinary form in which we pass to a limit by taking X to be very great compared with /x, and A to be very small compared with the greatest linear extension, in such a way that X,A is of the same order of magnitude as /xexx,---- We may put lim. XA = — p, and then ^o + P is the mean pressure at any point of the body in the strained state. Let V be the potential of the disturbing forces. The equations of equi- librium are of the form The terms containing — po and —gp cancel each other, and this equation takes the form -^-'•" + '5 = »- The equations of equilibrium of the homogeneous incompressible sphere, deformed from the state of initial stress expressed by (44) by the action of external forces, are of the same form as the ordinary equations of equilibrium of a sphere subjected to disturbing forces, provided that, in the latter equations, XA is replaced by — ^J arid /itA is neglected. The existence of the initial stress jOq has no influence on these equations, but it has an influence on the special conditions which hold at the surface. These conditions are that the deformed surface is free from traction. Let the equation of the deformed surface be r = a-V eS, where e is a small constant and S is some function of position on the sphere r = a. The " inequality" eS must be such that the volume is unaltered. We may calculate the traction {X,, Yy, Z^) across the surface r = a + eS. Let I', in', n be the direction cosines of the outward drawn normal v to this surface. Then Xy = I' (Xx -po) + m'Xy + ri'Zz. In the terms X^, Xy, X^, which are linear in the strain-components, we may replace I', m, n' by nc/a, y/a, zja, for the true values differ from these values by quantities of the order e ; but we must calculate the value of the term — I'pa at the surface r = a + eiS correctly to the order e. This is easily done because 181,182] DEFORMED BY BODY FORCES 265 Pf, vanishes at r = a, and therefore at r = a + €S it may be taken to be eSf-^j ,OT—gpeS. Neglecting e^ we may write -l'po = ^9peS. Hence the condition that X^ vanishes at the surface r = a + eS can be written {Xr)r^, + %peS=0 (45) Cb The conditions that Y^, Z„ vanish at this surface can be expressed in similar forms and the results may be interpreted in the statement: Account can be taken of the initial stress by assuming that the mean sphere, instead of being free from traction, is subjected to pressure which is equal to the weight per unit of area of the material heaped up to form the in- equality* 182. Deformation of gravitating incompressible sphere by external forces. Let the external disturbing forces be derived from a potential satisfying Laplace's equation ; and, within the sphere, let this potential be expressed as a sum of spherical solid harmonics of positive degrees in the form STr„. Let the surface of the sphere be deformed, and let the height of the inequality be expressed as a sum of spherical surface harmonics in the form 2e„)Sn, e„ being a small quantity which is at most of the order of magnitude of the inequality. The attraction of the inequality is a body force acting on the matter within the sphere, and at points within the sphere this force is derived from a potential of amount 47r7pa2 (2n + 1)-^ e„ (r/a)»>S„, where 7 is the constant of gravitation. When the potential of all the disturbing forces is expressed, as in Article 179, in the form 2F„, we have ^»=^"+2^l'"5'^»' ^*^^ in which 47ry/3a has been replaced by the equivalent expression Zg. The displacement within the sphere is expressed by formulae of the type 1 p ^ ^( iY \ 2 (2?i + 3) \ + 2/i a* ^'' "^ dso .(47) * This result is often assumed without proof. It appears to involve implicitly some such argument as that given in the text. 266 GRAVITATING INCOMPRESSIBLE SPHERE [CH. XI where An,... are unknown surface harmonics, and Mn and i|^«_i are expressed by means of equations (9) and (12). To complete the solution we must determine the harmonics A,i, Bn, On, Sn in terms of the known harmonics W,j, .... In the process we make such simplifications as arise from the assumption that the material is incompressible. The boundary-conditions which hold at the surface ?■ = a are of two kinds. We have, in the first place, the kinematical condition that the radial displacement at this surface is that which has been denoted by 2e„iS„, and, in the second place, the condition that the surface traction, calculated from the displacements of type (47), is equi- valent to a pressure equal to the weight of the inequality. The kinematical condition is expressed by the equation n + 2 2 (2?i -t- 3) X + 2/i a2 Wn + ^9 2ft -f-1 + 2 2n + l "'fn-i - -^^^ - (49) which are obtained by simplifying the expressions in (33) and (39) in accordance with the condition of incompressibility. The conditions in addition to (48) which hold at the surface are obtained by equating the expression on the right-hand side of (49) to - ix-^gpx^enS^. We thus find the equation 2n+\ d_(r-„\ a» dx\r'^+\ dx I ^n+i 2n + l\ dx dx W,. + ^ (n-l)4„-- — ^ "a" (n (n-2) l)(2w + l) 1 dx\r' an—ij 2n + \dx \d^+^ ^-"-y 0, .(50) 182] DEFORMED BY BODY FORCES 267 which holds' at the surface r = a. When we select from this equation the terms that contain surface harmonics of order n we find the equation {2n + 3)iM2n + 3 ^"+' dx U"+' +V (2w -l)fi 2n-l ^"- a"-> di ~ /i 2n + 3 dx fji2n—\dx -H^/i. i;^«^„ (n-l)(2n + l) aa;U^-' 1 9 /r^i+s \ "2^^TT9Sl^^^'^-^-^j=^' ^-^^^ in which the left-hand member is a solid harmonic of order n. Since this harmonic function vanishes at the surface r = a, it vanishes for all values of X, y, z. There are two similar equations, which are obtained by considering the tractions in the directions of y and z. We differentiate the left-hand members of the three equations of type (51) with respect to x, y, z respectively and add the results. We thus obtain the equation gp2{n -2 )w(2w -H) r"-' _p w(2w-|-l) „ {2n-iy -)- (n, - 1 ) t„_i -I- ^— j^ i|r„_i = 0. This equation holds for all values of n. When we replace n by n + 2 it becomes gp 2n(n+2)(2n + 5) ?•"+> „ _p (n + 2) {2n -h 5) J {2n+Sy ^»+ict«+i'^"+> f^ 2/1 + 3 ^"+' , 2(n + 2y + l + ^^^ 'f«+i = (52) Again we multiply the left-hand members of the three equations of type (51) by x, y, z respectively and add the results. We thus obtain the equation gp(i? 2n (n + 1) r»+^ „ pa? w + 1 ^ 2% r^^, _ ^ '^ (2« + 3)» ^"+^ a«+' "+^ " /x 2»i + 3 '^"+' 2n + 1 a^+' ''""-^ ~ "• (53) The equations (48), (52), (53) determine (S^+i, -»|^»+i, <^-h-2 in terms of TFn+i. Hence all the functions denoted by 8, -if, S'„+i is the only function of the type S. The equations of type (51) show that, of the functions An, ..., those which occur have 268 GRAVITATING INCOMPRESSIBLE SPHERE [CH. XI suffixes either ?i or n + 2. The result that Sn+^ is a multiple of Wn+i may be interpreted in the same way as the corresponding result noted in Article 180. 183. Gravitating body of nearly spherical form. The case of a nearly spherical body of gravitating incompressible material can be included in the foregoing analysis. The surface conditions as regards traction are still expressed by such equations as (50), but we have not now the kinematical condition expressed by (48). If the equation of the surface is of the form r=a + fn + iS„+i the values of \fr„^i and <^_„_2 are given by putting zero for W^+i in equations (52) and (53), and the harmonic functions such as An and A„ + 2 are determined by equations of the type of (51) from which the W'b are omitted. G. H. Darwin has applied analysis of this kind, without, however, restricting it to the case of incompressible material, to the problem of determining the stresses induced in the interior of the Earth by the weight of continents*. Apart from the difficulty concerning the initial stress in a gravitating body of the size of the Earth — a difficulty which it is troublesome to avoid without treating the material as incompressible — there is another difficulty in the application of such an analysis to problems concerning compressible gravitating bodies. In the analysis we take account of the attraction of the inequality at the surface, but we neglect the inequalities of the internal attraction which arise from the changes of density in the interior; yet these inequalities of attraction are of the same order of magnitude as the attraction of the surface inequality. To illustrate this matter it will be sufficient to consider the case where the density po in the initial state is uniform. In the strained state the density is expressed by po(l — A) correctly to the first order in the strains. The body force, apart from the attraction of the surface inequalities and other disturbing forces, has components per unit of mass equal to ffx/a, gyja, gzja. Hence the expressions for pX, ... in the equations of equilibrium ought to contain such terms as gpfjXa~'^{\ - A), and the terms of type —gp(,xAJa are of the same order as the attractions of the surface inequalities t. 184. Rotating sphere. In the case of the Earth the most interesting problems are those of the ellipticity of figure due to the diurnal rotation and of the tidal deformation produced by the attractions of the Sun and Moon. The effect of the rotation can be represented as due to body force of magnitude a>'^{x, y, 0), where w is the angular velocity, and the force at any point may be derived from a potential of magnitude \ ai- (x- + y^). This potential may be arranged as the sum of two terms J &)= {x- + y^ + z^)-^ w'' (2z^ -x^- y^), of which the former, equal to |&)^r^ gives rise to a radial force fwV. This term can be included in the term —gpr/a of equation (43) by writing 5f f 1 — o — ] instead of g. Since, in the case of the Earth co''a/g is a small fraction, equal to about 5^, we may, for the present purpose, disregard * London, Phil. Trans. R. Sac, vol. 173 (1882), reprinted in revised form in G. H. Darwin's Scientific Papers, vol. 2, p. 459. Darwin's results have been discussed critically by Chree, Cuvibridge Phil. Soc. Trans., vol. 14 (1889), and Phil. Mag. (Ser. 5), vol. 32 (1891). t See a paper by J. H. Jeans, London, Phil. Trans. R. Soe. (Ser. A), vol. 201 (1903). 182-184] DEFORMED BY BODY J-ORCES 269 this alteration of g. The term -^(o''{2z'-!c'-y'') when expressed in polar coordinates is - ■^&)V=' (f cos" - ^), so that it contains as factors r^ and a spherical surface harmonic of degree 2. We may determine the effect of the rotation from the results of Article 182 by putting n=\ and using this expression — ^ wV (f cos^ ^ - i) for W^. When n = l equations (52) and (53) become 19^ _ 21p 2 r^ 2 r" i2 ri ] 5-0 ^2 -2 -S^a = a and equation (48) becomes '2 n Hence we find the height of the harmonic inequality in the form It follows that the inequality is less for a solid incompressible sphere of rigidity /a than it would be for a sphere of incompressible fluid in the ratio 19 w 1 : 1 + -^ . If the sphere has the same size and mass as the Earth, this 2 gpa ^ ratio is approximately equal to ^ when the rigidity is the same as that of steel, and approximately equal to f when the rigidity is the same as that of glass. The ellipticity of the figure of the Earth is about ^^. The ellipticity* of a nearly spherical spheroid of the same size and mass as the Earth, con- sisting of homogeneous incompressible fluid, and rotating uniformly at the rate of one revolution in 24 hours, is about ^ku. The ellipticity which would be obtained by replacing the homogeneous incompressible fluid by homogeneous incompressible solid material of the rigidity of glass, to say nothing of steel, is too small ; in the case of glass it would be -g^-g nearly. The result that a solid of considerable rigidity takes, under the joint influence of rotation and its own gravitation, an oblate spheroidal figure appropriate to the rate of rotation, and having an ellipticity not incomparably less than if it were fluid, is important. It is difficult, however, to base an estimate of the rigidity of the Earth upon the above numerical results because the deformation of a sphere by rotation is very greatly affected by heterogeneity of the material. In the case of the Earth the average density of surface rock is about half the Earth's mean density. It is not difiicult to see that, in the case of an incompressible solid stratified in nearly spherical layers of equal density, * An eqiTation of the form r = a{l-|e(feos2fl-i)}, represents, when e is small, a nearly spherical spheroid of ellipticity e. 270 TIDAL EFFECTIVE RIGIDITY [CH. XI deficiency of density in the layers nearest the surface may tend to increase the ellipticity of figure due to rotation*. In our equations we have taken the density to be uniform, but we may take account of variations of density, in a roughly approximate fashion, by observing that the weight of the inequality, and the potential of it at internal points, must be proportional to the mean density of the surface layer. Let p denote this density. The rough approximation referred to would be made by writing p for p in the first two lines of equation (50). The result would be that, instead of the expression 19 u 1 + -r- -^~ in the denominator of the right-hand member of (64) we should 2 gpa have ~ + -^ -^. If p were Jo the numbers which were i for steel and p 2 gpa "^ -'^ f for glass would become | for steel and f for glass, and the ellipticity of the figure would, if this rough approximation could be trusted, be increased accordingly. 185. Tidal deformation. Tidal effective rigidity of the Earth. The tidal disturbing forces also are derived from a potential which is a spherical solid harmonic of the second degree. The potential of the Moon at any point within the Earth can be expanded in a series of spherical solid harmonics of positive degrees. With the terms of the first degree there correspond the forces by which the relative orbital motion of the two bodies is maintained, and with the terms of higher degrees there correspond forces which produce relative displacements within the Earth. By analogy to the tidal motion of the Sea relative to the Land these displacements may be called " tides." The most important term in the disturbing potential is the term of the second degree, and it may be written (ilf 77-^/!)') (|cos^^-|), where M denotes the mass of the Moon, D the distance between the centres of the Earth and Moon, 7 the constant of gravitation, and the axis from which 6 is measured is the line of centres f. This is the "tide-generating potential " referred to the line of centres. When it is referred to axes fixed in the Earth, it becomes a sum of spherical harmonics of the second degree, with coefficients which are periodic functions of the time. Like statements hold with reference to the attraction of the Sun. With each term in the tide-generating potential there coiTesponds a deformation of the mean surface of the Sea into an harmonic spheroid of the second order, and each of these deformations is called a " tide." There are diurnal and semi-diurnal tides * This result was noted by Chree, Phil. Mag. (Ser. 5), vol. 32 (1891), p. 249. lu the case of a. fluid, deficiency of density in the outer layers may tend to diminish the ellipticity of figure due to the rotation. In Laplace's "law of density in the interior of the Earth" the pressure and density are assumed to be connected by a certain law, and the density of the heterogeneous fluid is adjusted so as to make the ellipticity the same as that observed in the case of the Earth. See Kelvin and Tait, Nat. Phil., Part 11. p. 403. t See Lamb's Hydrodynamics, Appendix to Chapter YIII. 184,185] OF THE EARTH 271 depending on the rotation of the Earth, fortnightly and monthly tides de- pending on the motion of the Moon in her orbit, annual and semi-annual tides depending on the motion of the Earth in her orbit, and a nineteen- yearly tide depending on periodic changes in the orbit of the Moon which are characterized by the revolution of the nodes in the Ecliptic. The inequality which would be produced at the surface of a homogeneous incompressible fluid sphere, of the same size and mass as the Earth, or of an- ocean covering a perfectly rigid spherical nucleus, by the force that corre- sponds with any term of the tide-generating potential, is called the "true equilibrium height " of the corresponding tide. From the results given in Article 184 we learn that the inequalities of the surface of a homogeneous incompressible solid sphere, of the same size and mass as the Earth and as rigid as steel, that would be produced by the same forces, would be about J of the true equilibrium heights of the tides. They would be about § of these heights if the rigidity were the same as that of glass. It follows that the height of the ocean tides, as measured by the rise and fall of the Sea relative to the Land, would be reduced in consequence of the elastic yielding of the solid nucleus to about | of the true equilibrium height, if the rigidity were the same as that of steel, and to about | of this height if the rigidity were the same as that of glass. The name " tidal effective rigidity of the Earth " has been given by Lord Kelvin* to the rigidity which must be attributed to a homogeneous incompressible solid sphere, of the same size and mass as the Earth, in order that tides in a replica of the actual ocean resting upon it may be of the same height as the observed oceanic tides. If the tides followed the equilibrium law, the rigidity in question could be determined by observation of the actual tides and calculation of the true equilibrium height. It would be necessary to confine attention to tides of long period because those of short period are not likely to follow the equilibrium law even approximately. Of the tides of long period the nineteen-yearly tide is too minute to be detected with certainty. The annual and semi-annual tides are entirely masked by the fluctuations of ocean level that are due to the melting of ice in the polar regions. From observations of the fortnightly tides which were carried out in the Indian Oceanf it appeared that the heights of these tides are little, if anything, less than two-thirds of the true equilibrium heights. If the fortnightly tide followed the equilibrium law, we could infer that the tidal effective rigidity of the Earth is about equal to the rigidity of steel. The fact that there are observable tides at all, and the above cited results in reference to the fortnightly tides in the Indian Ocean, have been held by Lord Kelvin to disprove the geological hypothesis that the Earth has a * Sir W. Thomson, London, Phil. Trans. B. Soc, vol. 153 (1863), and Math, and Phys. Papers, vol. 3, p. 317. t Kelvin and Tait, Nat. Phil , Part n. pp. 442—460 (contributed by G. H. Darwin). 272 APPLICATION TO GEOPHYSICAL PKOBLEMS [CH. XI molten interior, upon which there rests a relatively thin solid crust, and, on this and other independent grounds, he has contended that the Earth is to he regarded as consisting mainly of solid matter of a high degree of rigidity. The dynamical theory of the tides of long period can be worked out for an ocean of uniform depth covering the whole globe, the' nucleus being treated as rigid* It is found that the heights of such tides, on oceans of such depths as actually exist, would be less than half of the equilibrium heights. This result was at first supposed to diminish the cogency of the tidal evidence as to the rigidity of the Earth. Lord Rayleigh, however, has pointed out that on the assumption that the ocean is not interrupted by land barriers running north and south, free steady motions, consisting of currents running along parallels of latitude, would be generated, and would cause the tides of long period, calculated on the dynamical theory, to fall decidedly short of their theoretical equilibrium values f. Since the great oceans are interrupted by such barriers, it is unlikely that the tides of long period on a rigid nucleus, and in oceans of the actual shapes and depths, would fail to follow sensibly the equilibrium law. 185 A. Applications of the theory of Elasticity to Geophysical questions. Lord Kelvin's work on the solution of the equations of elastic equilibrium for an incompressible solid sphere, subject to its own gravitation and to external disturbing forces, with the application to determine the tidal effective rigidity of the Earth, has proved to be the beginning of an extensive theory. Reference has already been made to the improvement effected by J. H. JeansJ, who led the way in the direction of including the effects of compressibility, and to the application, initiated by G. H. Darwin, to the problem of deter- mining the stresses induced in the interior of the Earth by the weight of continents and mountains. The reader, who may wish to pursue the subject, is referred to the following: — G. H. Darwin, Scientific Papers, especially vol. 1, pp. 389, 430, and vol. 2, p. 33, and " The Rigidity of the Earth," Atii del IVCongresso...Matematici, vol. 3, Roma, 1909 ; S. S. Hough, London, Phil. Trans. R. Soc. (Ser. A), vol. 187, 1896, p. 319 ; G. Herglotz, Zeitschr.f. Math. v. Phys., Bd. 52, 1905, p. 275 ; W. Schweydar, Beitrage zur Oeophysik, Bd. 9, 1907, p. 41 ; Lord Rayleigh, London, Proc. R. Soc. (Ser. A), vol. 77, 1906, p. 486, or Scientific Papers, vol. 5, p. 300 ; A. E. H. Love, London, Proc. R. Soc. (Ser. A), vol. 82, 1909, p. 73, and Some Problems of Geodynamics, Cambridge, 1911; J. Larmor, London, Proc. R. Soc. (Ser. A), vol. 82, p. 89; W. Schweydar, * G. H. Darwin, London, Proc. R. Soc, vol. 41, 1886, p. 337, reprinted in his Scientific Papers, vol. 1, Cambridge, 1907, p. 366. See also Lamb, Hydrodynamics, Chapter VIII. t Lord Rayleigh, Phil. Mag. (Ser. 6), vol. 5, 1903, p. 136, reprinted in his Scientifi,c Papers, vol. 5, Cambridge, 1912, p. 84. X Loc. cit. ante, p. 268. 185,186] PLANE STRAIN IN A CIRCULAR CYLINDER 273 Veroff. d. kgl. Preus. geoddtischen Institutes (Neue Folge), No. 54, 1912; K. Terazawa, London, Phil. Trans. R. Soc. (Ser. A.), vol. 217, 1916, p. 35, and /. Coll. Sci, Tokyo, vol. 37, 1916, Art, 7 ; H. Lamb, London, Proc. R. Soc. (Ser. A.), vol. 93, 1917, p. 293; J. H. Jeans, London, Proc. R. Soc. (Ser. A), vol. 93, 1917, p. 413. Other references will be found in these works. 186. Plane strain in a circular cylinder*. Methods entirely similar to those of Articles 173 and 175 may be applied to problems of plane strain in a circular cylinder. Taking r and 6 to be polar coordinates in the plane (oo, y) of the strain, we have, as plane harmonics of integral degrees, expressions of the type r" (a„ cos nO + /3„ sin nO), in which a„ and /3„ are constants, and as analogues of surface harmonics we have the coefficients of r" in such expressions. We may show that the analogue of the solution (13) of Article 173 is 2{\ + 'Aix) n + l\ dx ' dy in which An and 5„ are functions of the type a„ cos ?i0 + /3„ sin nO, and the functions -^jr are plane harmonic functions expressed by equations of the form (u,v)-^\An^^, ^«„»; + 2(x + 8;.)^ n + l [ dx ' dy )' -^^^^ ^»--4(^»3 +8^(^-3 (^«) dy^ The equations (55) would give the displacement in a circular cylinder due to given displacements at the curved surface, when the tractions that maintain these displacements are adjusted so that there is no longitudinal displacement. When the tractions applied to the surface are given, we may take XX„, 1_„_i by the equations All these functions are plane harmonics of the degi-ees indicated by the * Cf. Kelvin and Tait, Nat. Phil., Part ii. pp. 298—300. The problem of plane stress in a circular cylinder was solved by Clebsch, Elasticitdt, § 42. L. B. 18 .(58) 274 EXAMPLES OF SOLUTION [CH. XI suffixes. The surface tractions can be calculated from equations (55). We find two equations of the type a" 2 {n - 1) \n X + Sfi from which we get .(60) X + dfia and thus An, Bn can be expressed in terms of Xn, Yn- As examples of this method we may take the following* : — (i) X„ = acos25, F„=0. In this case we find (ii) Xn=acoa2d, Tn=asm26. In this case we find (iii) A'„=a cos 45, F„=0. In this case we find 187. Applications of curvilinear coordinates. We give here some indications concerning various researches that have been made by starting from the equations of equilibrium expressed in terms of cmwilinear co- ordinates. (a) Polar coordinates. Lame's original solution of the problem of the sphere and spherical shell by means of series was obtained by using the equations expressed in terms of polar coordinates +. The same equations were afterwards employed by C. W. Borohardtf, who obtained a solution of the problem of the sphere in terms of definite integrals, and by C. Chree§, who also extended the method to problems relating to approximately spherical boundaries ||, obtaining solutions in the form of series. The solutions in series can be built up by means of soHd spherical harmonics ( FJ expressed in terms of polar coordinates, and related functions ( U) which satisfy equations of the form V^U= V„. * The solutions in these special cases will be useful in a subsequent investigation (Chap. XVI). t J. de Math. (Liouville), t. 19 (1854). See also Lei;om fur les coordonn(es currilignes, Paris, 1859. J Loc. cit. ante, p. 106. § Cambridge Phil. Soc. Trans., vol. 14 (1889). Ii Amer. J. of Math., vol. 16 (1894). 186, 187] IN SERIES 275 (6) CylindHcal coordinates. Solutions in series have been obtained* by observing that, if J^ is the symbol of Bessel's function of order n, «**"*"'"* ^„ (/!;?•) is a solution of Laplace's equation. It is not difficult to deduce suitable forms for the displacements Ur, Ug, Ug. The case in which Ug vanishes and % and u^ are independent of 6 will occupy us presently (Article 188). In the case of plane strain, when «, vanishes and u^ and m^ are independent of z, use may be made of the str^s-function (of. Article 144 sv/pra). The general form of this function expressed as a series proceeding by sines and cosines of multiples of 6 has been given by J. H. Michellt. (e) Plane strain in non-circular cylinders. When the boundaries are curves of the family a = const., and u is the real part of a function of the complex variable x+iy, we know from Article 144 that the dilatation A and the rotation sr are such functions of x and y that (X + 2^) A + i^iizs is a function of x + iy, and therefore also of a+i(3, where /3 is the function conjugate to a. For example, let the elastic solid medium be bounded internally by an elliptic cylinder. We take a; + iy=c cosh (a + 1/3), so that the curves a = const, are confocal ellipses, and 2c is the distance between the foci. Then the appropriate forms of A and sr are given by the equation (X + 2/i) A 4- i2/xiEr = Se""""' ( J„ cos K/3 + 5„ sin m/3). If we denote by h the absolute value of the complex quantity d (a + iffjjd (x+iy), then the displacements u^ and Ua are connected with A and or by the equations In the case of elliptic cylinders ujk and ujh can be expressed as series in cosn/3 and sinw^ without much difficulty. As an example J we may take the case where an elliptic cylinder of semi-axes a and b is turned about the line of centres of its normal sections through a small angle (p. In this case it can be shown that the displacement produced outside the cylinders is expressed by the equations (d) Solids of revolution. If r, 6, z are cylindrical coordinates, and we can find a and ^ as conjugate functions of z and r in such a way that an equation of the form a = const, represents the meridian curve of the surface of a body, we transform Laplace's equation v2F=0 to the form * L. Pochhammer, J./. Math. {Crelle), Bd. 81 (1876), p. 33, and C. Chree, Cambridge Phil. Soc. Trans., vol. 14 (1889). t London Math. Soc. Proc, vol. 31 (1900), p. 100. J The problem was proposed by E. B. Webb. For a different method of obtaining the solution see D. Edwardes, Quart. J. of Math., vol. 26 (1893), p. 270. The corresponding problem .for a rigid ellipsoid, embedded in an elastic solid medium, and turned through a small angle about a principal axis, is discussed by E. Daniele, II Nuovo Cimento (Ser. 6), t. 1, 1911. 18—2 276 SYMMETRICAL STRAIN [CH. XI where J denotes the absohite vaUie of d{s + ir)ld{a + tff). If we can find solutions of this equation in the cases where V is independent of 6, or is proportional to sin nd or cos hO, we can obtain expressions for the dilatation and the components of rotation as series. Wangerin* has shown how from these solutions expressions for the displacements can be deduced. The appropriate solutions of the above equation for V are known in the case of a number of solids of revolution, including ellipsoids, cones and tores. 188. SymmetrioaJ strain in a solid of revolution. When a solid of revolution is strained symmetrically, so that the dis- placement is the same in all planes through the axis of revolution, we may express all the quantities that occur in terms of a single function, and reduce the equations of equilibrium of the bod}' strained by surface tractions only to a single partial differential equation. Taking r, 6, z to be cylindrical coordinates, we have the stress-equations of equilibrium in the forms drr drz rr—66_ drz dzz ^^_« dr dz r ' dr dz r ^ ' Writing U, w for the displacements in the directions of r and z, and sup- posing that there is no displacement at right angles to the axial plane, we have the expressions for the strain-components du u dw du dw ^ ,,„, ^rr=-^, ««» = -. «« = a^' ^'■' = ^+d^' erfl = e,e = 0. ...(62) We begin bj putting, by analogy with the corresponding theory of plane strain, oraz Then the second of equations (61) gives us ^_o^ ld<#,. no arbitrary function of r need be added, for any such function can be included in (f>. We observe that e„=^ (re^g), and write down the equivalent equation in terms of stress- components, viz. : rr-(r6d-(rzz=^{{d6 -a-rr-a-'zz)r}, and hence we obtain the equation (l-t- 0-) {ri- -ee) = r - {ee -T-(rz?) = 0, * ArcUv f. Math. {Grunert), vol. 55 (1873). The theory has been developed further by P. Jaerisoh, J. f. Math. (Crelle), Bd. 104 (1889). The solution for an ellipsoid of revolution with given surface displacements has been expressed in terms of series of spheroidal harmonica by 0. T«lone, Moma, Ace. Line. Rend. (Ser. 6), t. 14 (1905). 187, 188] IN A SOLID OF REVOLUTION 277 and we may put dd=(rV^. All the stress-components- have now been expressed in terms of two functions (p and B. The sum G of the principal stresses is expressed in terms of (j) by the equation e = ?T + ^-|- 22 = (1 -Ho-) V20, and, since e is an harmonic function, we must have V*<^ = 0. The functions (f) and R are not independent of each other. To obtain the relations between them we may proceed as follows: — The equation Ujr=eg^ can be written U=r(e6-<7n'-a7z)IE, or 0=-{l + a-)rRIE; and then the equation rz=ji£ri can be written 8w^ 2(1+0-) og 1+0- as 3r~ E crdz E ^ bz ' Also the equation e„={zz- iTrr — a-dO)IE can be written The equations giving dw/dr and dwjdz are compatible if and, if we introduce a new function Q by means of the equation dr 8?" ' we have g = (l-.)V^<^, where, as before, no arbitrary function of z need be added. The stress-components are now expressed in terms of the functions (j) and Q which are connected by the equation last written. The equations giving dw/dr and dw/dz become, when Q is introduced, ■" ¥~ E dr\dz dzj' dz~ E dz\dz dz)' We may therefore express U and w in terms of Q and by the formulas l+o-/3a 3<^\ l-f-o-rSQ d+Q. Then we have dz^ ^ r di- r dr ^ dr^ a?2 dr^ ^^ ^ dr^ ' and we have also The first of equations (61) would enable us at once to express rz in terms of a function x such that ^=dxldz. We therefore drop all the subsidiary functions and retain x only. In accordance with the above detailed work we assume -If"^-|j}. «'?=lh-;5?}. S-3^^{(2-.)V.^-g) (63, Then the first of equations (61) gives us - = |:{(l--)^=X-'5}. (64) and the second is satisfied by this value of rz if v^a; = o (65) The stress-components are now expressed in terms of a single function j(^ which satisfies equation (65)*. The corresponding displacements are easily found from the stress-strain relations in the forms E drdz' E 1^ ' '^^dr^^rdr ....(66) 189. Symmetrical strain in a cylinder. When the body is a circular cylinder with plane ends at right angles to its axis, the function x 'will have to satisfy conditions at a cylindrical surface r = a, and at two plane surfaces z = const. It must also satisfy equation (65). Solutions of this equation in terms of r and z can be found by various methods. The equation is satisfied by any solid zonal harmonic, i.e. by any function of the form (r2-|-^2)»+i (,.2^. ^s)-4 a,nd also by the product of such a function and (r^ -i- z^). All these functions are rational integral functions of r and z, which contain even powers of r only. Any sura of these functions each multiplied by a constant is a possible form for x- * A method of expressing all the quantities in terms of a single function, which satisfies a partial differential equation of the fourth order different from (65), has been given by J. H. Miohell, London Math. Soc. Proc, vol. 31 (1900), pp. 144—146. 188, 189] IN A CYLINDER 279 The equation (65) is satisfied also by any harmonic function of the form e±*^ /„ {kr), where k is any constant, real or imaginary, and J^ {x) stands for Bessel's function of zero order. It is also satisfied by any function of the form e±**r -^ Jo{kr), for we have V^ L^^'r ^ J„ {kr)\ = - 2k'e'^'"' J, (kr). When k is imaginary we may write these solutions in the form Jo (iKr) (A cos KZ + B sin kz) + r -j-Jo(iKr) (Ccos Kz + Dsia kz), . . .(67) in which « is real and A, B, C, D are real constants. Any sum of such expressions, with different values for k, and different constants A, B, G, D, is a possible form for ;>(;. The formulae for the displacements U, w that would be found by each of these methods have been obtained otherwise by C. Chree*. They have been applied to the problem of a cylinder pressed between two planes, which are in contact with its plane ends, by L. N. G. Filonf. Of the solutions which are rational and integral in r and z, he keeps those which could be obtained by the above method by taking % to contain no terms of degree higher than the seventh, and to contain uneven powers of z only. Of the soliations that could be obtained by taking j^ to be a series of terms of type (67), he keeps those which result from putting k = mr/c, where n is an integer and 2c is the length of the cylinder, and omits the cosines. He finds that these solutions are sufficiently general to admit of the satisfaction of the following conditions : (i) the cylindrical boundary r = a is free from traction ; (ii) the ends remain plane, or w = const, when z = ±c; (iii) the ends do npt expand at the perimeter, ot U=0 when r = a and z= ± c; (iv) the ends are subjected to a given resultant pressure. He shows also how a correction may be made when, instead of condition (iii), it is assumed that the ends expand by a given amount. The results are applied to the explanation of certain discrepancies in estimates of the strength of short cylinders to resist crushing loads, the discrepancies arising from the employment of different kinds of tests; and they are applied also to explain the observation that, when cylinders (or spheres) are compressed between parallel * Cambridge Phil. Soc. Trans., vol. 14 (1889), p. 250. t London, Phil. Trans. R. Soc. (Ser. A), vol. 198 (1902). Filon gives in the same paper the solutions of other problems relating to symmetrical strain in a cylinder. 280 CYLINDER UNDER TERMINAL PRESSURE [CH. XI planes, pieces of an approximately conical shape are sometimes cut out at the parts subjected to pressure. Instead of taking the second solution of equation (65) in terms of Bessel's functions to be expressed by e^'^^r-r- {J^{kr)], we may take it to have the form ze^'J^ (kr). The expressions, which would thus be obtained for the displacements U, w, have been utilized by F. Purser, loc. cit. ante, p. 145. CHAPTER XII. VIBRATIONS OF SPHERES AND CYLINDERS. 190. In this Chapter we shall illustrate the method explained in Article 126 for the solution of the problem of free vibrations of a solid body. The free vibrations of an isotropic elastic sphere have been worked out in detail by various writers*. In discussing this problem we shall use the method of Lamb and record some of his results. When the motion of every particle of a body is simple harmonic and of period 27r/p, the displacement is expressed by formulae of the type u = Au' cos (pt + e), V = Av' cos (pt + e), w = Aw' cos (pt + e), ...(1) in which u', v', w' are functions of oc, y, z, and A is an arbitrary small con- stant expressing the amplitude of the vibratory motion. When the body is vibrating freely, the equations of motion and boundary conditions can be satisfied only Up is one of the roots of the " frequency equation," and u', v', w' are " normal functions." In general we shall suppress the accents on u', v', w', and treat these quantities as components of displacement. At any stage we may restore the amplitude-factor A and the time-factor cos (pt + e) so as to obtain complete expressions for the displacements. The equations of small motion of the body are where ^J-+^ + ^^ (3) da; oy az When u, v, w are proportional to cos {pt + e) we obtain the equations (X + ^,)(^^, ^^, ^^) + ^V^{u,v,w) + pp^(u,v,w) = (4) * Reference may be made to P. Jaerisch, J./. Math. {Crelle), Bd. 88 (1880) ; H. Lamb, Lotidon Math. Soc. Proc., vol. 13 (1882); C. Chree, Gamhridge Phil. Soc. Tratts., vol. 14 (1889). 282 SOLUTION OF THE EQUATIONS OF VIBRATION [CH. XII Differentiating the left-hand members of these equations with respect to «, y, z respectively, and adding the results, we obtain an equation which may be written (V''+/iOA=0 (5) where }e=fp\{\^1ix) (6) Again, if we write /c^ = p^p/ya, (7) equations (4) take the form We may suppose that A is determined so as to satisfy equation (5), then one solution (mi, Vi, Wi) of the equations last written is 1 /9A aA 9A\ dy and a more complete solution is obtained by adding to these values for W], v-i, W], complementary solutions {u^, v^, Wj) of the system of equations (V2+«'')m2 = 0, {V^+k')v^ = 0, (V'+«^)w2 = 0, (8) and |l? + |?!.^ + ^^ = (9) ox dy az When these functions are determined the displacement can be written in the form (m, v,w) = A (mi + u^, «i + «2, Wi + Wa) cos (pi + e) (10) 191. Solution by means of spherical harmonics. A solution of the equation (V^ + A") A = can be obtained by supposing that A is of the form f(r)Sn, where r'=af+y^ + z^, and *S„ is a spherical surface harmonic of degree v. We write R^ instead of f{r). Then rRn is a solution of Riccati's equation g + »--"-^)(*)-o, of which the complete primitive is expressible in the form fl dy^An sin hr + Bn cos hr r An and 5„ being arbitrary constants. The function r'^Sn is a spherical solid harmonic of degree n. When the region of space within which A is to be determined contains the origin, so that the function A has no singularities in the neighbourhood of the origin, we take for A the formula A = l The function ■\(f„ (x) is expressible as a power series, viz. : ( — )" f .K^ a;* "I i/'n(*) = j_3_5__^2TO+l") f ~2(2ji + 3)"'"2.4.(2?i + 3)(2re + 5)~'"J ' ""^^^^ which is convergent for all finite values of x. It is an "integral function." It may be in terms of a Bessel's function by the formula ^/'«W = (-)»i^'(2T).^;-(''+«/„+J(^) (U) It satisfies the differential equation /(^2 2(»i+l) d A , , , „ nr.\ The functions i/^„ (.!!;) for consecutive values of n are connected by the equations ^^^^^^^^ = ^'^n{^)=-i^n-2(.^)-{^n-l)i,^.,{x) (16) The function *■„ (x) determined by the equation , , , /I dYfcoax\ which has a pole of order 2re+l at the origin, and is expressible by means of a Bessel's function of order —{n+^), satisfies equations (15) and (16). In like manner solutions of equations (8) and (9) which are free from sin- gularities in the neighbourhood of the origin can be expressed in the forms where Cr„, 7„, F„ are spherical solid harmonics of degree n, provided that these harmonics are so related that du,_^dv,_^dm^^ (9Us) dx dy dz One way of satisfying this equation is to take f/„, F„, Wn to have the forms where x» is a spherical solid harmonic of degree w; for with these forms we have ?f^ + ^^ + ?f5 = and xUr. + yVr, + zWn = 0. dx dy oz A second way of satisfying equation (9 his) results from the observation that 284 DISPLACEMENT IN A [CH. XII curl (?t2, V3, Wa) satisfies the same system of equations (8) and (9) as (ws, v^, w^). If we take m/, v^, w.^ to be given by the equations we find such formulae as where ■^n («»') means A-^n {Kr)ld (kt). By means of the identity and the relations between yj/ functions with consecutive sufiixes, the above formula is reduced to the following : — of which each term is of the form Un'fn(Kr). In like manner the other components of curl (V, Vs, w^) can be formed. Hence, taking %„ and (pn+i to be any two solid harmonics of degrees indicated by their suffixes, we have solutions of the equations (8) and (9) in such forms as .=s ^«(-)(^t-t-1?-^' .(20) The corresponding forms of v^ and w^ are obtained from this by cyclical interchange of the letters cc, y, z. 192. Formation of the boundary -conditions for a vibrating sphere. We have nosv to apply this analysis to the problem of the free vibrations of a solid sphere. For this purpose we must calculate the traction across a spherical surface with its centre at the origin. The components Xr, Yr, ^r of this traction are expressed, as in Article 175, by formulae of the type rXy '^ A 9 / , 3m /oi\ =- xA + ^(ux + vy +'wz) + r-^ u (.^i) In this formula A has the form given in (11), viz. : Sw^i/^n (Ar), and w, v, w have such forms as 1 3A^„ ^ ^ ■' y' dz dy CUB J dy (22) lf2^-(-)---a-.C^) 191, 192] VIBRATING SPHERE 285 We find r 9A wa; + ^2/ + m = - ^^ g^ + 2 (n + 1) {t„ («-r) + /cV^i|r„+, («r)} ^„+i, or (23) This formula gives us an expression for the radial displacement {ux + vy + wz)/r. In forming the typical terms of a;A, ^ {ux + vy + wz), r ^ m we make continual use of identities of the type (19) and of the equations satisfied by the i|r functions. We shall obtain in succession the contributions of the several harmonic functions a)„, 0„, %„ to each of the above expressions. The function &)„ contributes to a;A the terms B^l+-«{-fe--'"«a(^)} W and the functions (^„, 'x^^ contribute nothing to a;A. The function m^ contributes to 9 {ux + vy + wz)/dx the terms - 1 \{n + 1) hrf„' {hr) + /„ and ^n are connected with each other by the compatible equations (33) and (34). 195. Vibrations of the first class* . When the vibration is of the first class the displacement is of the form where K.'^=p'^pjjx ; and the possible values of ^ are determined by the equation (m-l)i/^„(Ka) + (cai|r„'(Ka) = (38) The dilatation vanishes. The radial displacement also vanishes, so that the displacement at any point is directed at right angles to the radius drawn from the centre of the sphere. It is also directed at right angles to the normal to that surface of the family ;(„ = const, which passes through the point. The spherical surfaces determined by the equation V'n ('='') ==^ ^^^ "nodal," that is to say the displacement vanishes at these surfaces. The spherical surfaces determined by the equation (« - 1) '/'« (t'') + K>-f „' (Kr) = 0, in which k is a root of (38), are "anti-nodal," that is to say there is no traction across these surfaces. If ^i, k^, ... are the values of k in ascending order which satisfy (38), the anti-nodal surfaces corresponding with the vibration of frequency (2ir)~'^J{fi.lp) k, have radii equal to n^aJK,, xja/K,, ... Kg_ia/)Cj. If ra=l we have rotatory vibrations^. Taking the axis of z to be the axis of the harmonic xi^ the displacement is (m, d, w) = A cos {pt+f) i/fi (/cr) {y, —X, 0), so that every spherical surface concentric with the boundary turns round the axis of z through a small angle proportional to ^^{Kr), or to {nr)'^ cos Kr - {Kr)~^smKr. The possible values of k are the roots of the equation i|^i'(Ka) = 0, or tan /ca = 3Ka/(3- K^a^). * The results stated in this Article and the following are due to H. Lamb, loc. cit. p. 281. t Modes of vibration analogous to the rotatory vibrations of the sphere have been found for any solid of revolution by P. Jaerisch, J.f. Math. (Crelle), Bd. 104 (1889). 194-196] OF A SPHERE 289 The lowest roots of this equation are — =1-8346, 2-8950, 3-9225, 4-9385, 5-9489, 6-9563,.... The number tt/ko is the ratio of the period of oscillation to the time taken by a wave of distortion* to travel over a distance equal to the diameter of the sphere. The nodal surfaces are given by the equation tan «»•= v=^^^{hr\ w=^-y^t; {hr), (40) and the frequency equation is 6o=0, or ^o{ha) + -^^ha^^ {ha) = Q, (41) which is tan ha 1 ha l-i(K2/A2)^2(^2- There are, of course, no radial vibrations when the material is incompressible. When K-/h'=3, the six Jowest roots of the frequency equation are given by — = -8160, 1-9285, 2-9359, 3-9658, 4-9728, 5-9774. r The number ir/ha is the ratio of the period of oscillation to the time taken by a wave of dilatation t to travel over a distance equal to the diameter of the sphere. Spheroidal vibrations. When n = 2 and a>2 and (^2 ^.re zonal harmonics we have what may be called spheroidal vibrations, in which the sphere is distorted into an ellipsoid of revolution becoming alternately prolate and oblate according to the phase of the motion. Vibrations of this type would tend to be forced by forces of appropriate period and of the same type as tidal disturbing forces. It is found that the lowest root of the frequency equation for free vibrations of this type is given by Ka/n- = -848 when the material is incompressible, and by Ka/7r = -840 when the material fulfils Poisson's condition. For a sphere of the same size and mass as the Earth, supposed to be incompressible and as rigid as steel, the period of the gravest free vibration of the type here described is about 66 minutes. * The velocity of waves of distortion is (/i/p)*- See Chapter XIll. t The velocity of waves of dilatation is {{\ + 2ii)lp\^. See Chapter XIlI. L. E. 19 290 RADIAL VIBRATIONS OF A SPHERICAL SHELL [CH. XII 197. Further investigations on the vibrations of spheres. The vibrations of a sphere that would be forced by surface tractions proportional to simple harmonic functions of the time have been investigated by Chree*. Free vibrations of a shell bounded by concentric spherical surfaces have been discussed by Lambt, with special reference to the case in which the shell is thin. The influence of gravity on the free vibrations of an incompressible sphere has been considered by BromwichJ. He found, in particular, that the period of the " spheroidal " vibrations of a sphere of the same size and mass as the Earth and as rigid as steel would be diminished from 66 to 55 minutes by the mutual gravitation of the parts of the sphere. A more general discussion of the effects of gravitation in a sphere of which the material is not incompressible has been given by Jeans §. It has been proved that, when both gravity and compressibility are taken into account, the period of spheroidal vibrations of a sphere of the same size and mass as the Earth, as rigid as steel, and having a Poisson's ratio equal to |-, would be almost exactly one hour||. 198. Radial vibrations of a hollow sphere H. The radial vibrations of a sphere or a spherical shell may be investigated very simply in terms of polar coordinates. In the notation of Article 98 we should find that the radial displacement U satisfies the equation vr^ r or r^ and that the radial traction rr across a sphere of radius r is The primitive of the differential equation for U may be written „_ d / A am hr+B cos />r\ d{hi) \ Tr ; ' and the condition that the traction rr vanishes at a spherical surface of radius r is [(X + 2/i) {(2 - A^r^) sin hr - 2hr cos Ar} + 2X {hr cos hr - sin hr)] A +l{\ + 2iJ.){{2-A''r^)coahr + 2hrsmhr}-2X{hrsmhr+coahr)]B=0. When the sphere is complete up to the centre we must put B=0, and the condition for the vanishing of the traction at r = a is the frequency equation which we found before. In the case of a spherical shell the frequency equation is found by eliminating the ratio A : B from the conditions which express the vanishing of rr at r=a and at r=b. We write so that 2X/(X + 2/i) = 2-», and then the equation is vha + {h^a^-v) tan ha _ vhb + {h^l^- v) tan hb {li'a^ — v)-vha tan ha {M' — v)- vhb tan hb ' In the particular case of a very thin spherical shell this equation may be replaced by 3 i'Aa+(A%^ — j')tan/ja_ da (Jfia^ -v)-vha tan Iw, ' * hoc. cit. p. 281. t London Math. Soc. Proc, vol. 14 (1883). t London Matli. Soc. Proc, vol. 30 (1899). § London, Phil. Tram. R. Soc. (Ser. A), vol. 201 (1903). li A. E. H. Love, Some Problems of Geodynamics. •^ The problem of the radial vibrations of a solid sphere was one of those discussed by Poisson in his memoir of 1828. See Introduction, footnote 36. 197-199] VIBRATIONS OF A CIECULAE CYLINDER which is and we have therefore 291 2 sec2 hob {A%2 _ „ (3 _ „)} = 0, Aa=V{>'(3-i/)}. In terms of Poisson's ratio o- the period is 199. Vibrations of a circular cylinder. We shall investigate certain modes of vibration of an isotropic circular cylinder, the curved surface of which is free from traction, on the assumption that, if the axis of z coincides with the axis of the cylinder, the displace- ment is a simple harmonic function of z as well as of t*. Vibrations of these types would result, in an unlimited cylinder, from the superposition of two trains of waves travelling along the cylinder in opposite directions. When the cylinder is of finite length the frequency of fi-ee vibration would be determined by the conditions that the plane ends are free from tractioij. We shall find that, in general, these conditions are not satisfied exactly by modes of vibration of the kind described, but that, when the radius of the cylinder is small compared with its length, they are satisfied approximately. We use the equations of vibration referred to cylindrical coordinates r, d, z. The equations are dr laA dd = (^ + ^>^^V-fe-'>^W^'^ 3^ dz r dr dz ' dr 2/U. d^r .(42) in which _ 1 d(rur) 1 du) du^ r dr r dd dz ' .(43) and _ 1 duz due ^"""■-rdd' dz dUr ■ dr ' ^"^'-rK dr dd)' •••^**^ so that ■sTr, ■are, ■=^z satisfy the identical relation 1 d {rm^ 1 9tn-9 Sct^ r dr r dd dz :0. .(45) The stress-components rr, rO, rz vanish at the surface of the cylinder r = a. These stress-components are expressed by the formulse -^ , A o 9«r -^ (Idur , d fue\] -^ fdur , du, )■ .(46) * The theory is effectively due to L. Poohhammer, /. /. Math. (Crelle), Bd. 81 (1876), p. 324. It has been discussed also by C. Chree, loc. cit. p. 281. 19—2 292 TORSIONAL AND LONGITUDINAL [CH. XII In accordance with what has been said above we shall take Ur, «», u^ to be of the forms M,= C7e'(»^+P« UB=Ve'^y^+P*\ w^=TFe'(T«+3'0, (47) in which U, V, W are functions of r, 0. 200. Torsional vibrations. We can obtain a solution in which U and W vanish and V is independent of 6. The first and third of equations (42) are satisfied identically, and the second of these equations becomes Vr+-V^--.V+>c'^V=0, (48) where k''-' = p'pjfj^ — 7^. Hence V is of the form BJ^ (ic'r), where 5 is a constant, and Jj denotes Bessel's function of order unity. The conditions at the surface r = a are satisfied if k' is a root of the equation d (Ji («'a)| da ( a One solution of the equation is k = 0, and the corresponding form of V given by equation (48) is F = Br, where 5 is a constant. We have therefore found a simple harmonic wave-motion of the type M^ = 0, Ue = Bre'-<-T^+P*\ u^ = 0, (49) in which y' = p^pl/J-- Such waves are waves of torsion, and they are propa- gated along the cylinder with velocity '\/{fj./p)*. The traction across a normal section 2 = const, vanishes if du^jdz vanishes ; and we can have, therefore, free torsional vibrations of a circular cylinder of length I, in which the displacement is expressed by the formula - = cos -J- 5„ cos (^— y'^ 4- 6J, (50) ?! being any integer, and the origin being at one end. 201. Longitudina Ivibrations. We can obtain a solution in which V vanishes and U and W are independent of 6. The second of equations (42) is then satisfied identically, and from the first and third of these equations we find or' r dr d'THe 1 8to-« ■nr^ , ^-^ + - 3 ^ -I- /«; Vfl = 0, or' r or r' where h'' = p'p/(X + 2iJ.)- y\ K'^=p'plfi-y'' (52) * Qi. Lord Rayleigh, Theory of Sound, Chapter VII. .(51) 199-201] VIBRATIONS OF A CIRCULAR CYLINDER 293 We must therefore take A and ot^, as functions of r, to be proportional to Jo{h'r) and J-^{K'r). Then to satisfy the equations we have to take U and W to be of the forms .(53) where A and G are constants. The traction across the cylindrical surface r = a vanishes if A and G are connected by the equations A z^ d^Jp (h'a) p'pX da' X + 2fj, Jo {h'a) da '\ 2Ay^-^^+G[2y^-P^)j,(.-a)^0. ..(54) On eliminating the ratio A :G we obtain the frequency equation. When the radius of the cylinder is small we may approximate to the frequency by expanding the Bessel's functions in series. On putting Jo (h'a) = 1 - ^h'^a' + ^h'*a\ J (ic'a) = ^(/c'a - the frequency equation becomes ■i«'W), e-v).'»(i-^') h'^l - ^a%'') + - X, p'p (l-ia%") fj, \ + 2/JL + 2y'K' (1 - faV'O ah''' (1 - ^a^h'') = 0. It is easily seen that no wave-motion of the type in question can be found by putting k' = 0. Omitting the factor «'a and the terms of order a', we find a first approximation to the value of p in terms of 7 in the form p^y^(E/p), (55) where E, = /j. (3X. + 2fi)/(\ + fi), is Young's modulus. The waves thus found are " longitudinal " and the velocity with which they are propagated along the cylinder is s/iEjp) approximately*. When we retain terms in a', we find a second approximation! to the velocity in the form p = y^{Elp){\-{, Ve = Vsme&^y''+Pt>, Wj= Fcos ^6''''^+**', ...(58) where IT, V, W are functions of r. Then we have A= cos6le"T^+^"f^+- + - + f'yl \dr r r ' 1v!r=- sin ^e^v^+P't {— + 477) , 2l!78 = cos 06* 'I'^+.P" ilr^ U - ^~\ , 1vT,= sin (9 e"v^+*" f I? + - + -^ \tir r r J .(59) 201, 202] VIBRATIONS OF A CIRCULAR CYLINDER 295 From equations (42) we may form the equation d^A 1 9A A , , , „ W+rTr-f^ + ^'^ = ^' (60) where h'^ is given by the first of equations (52); and it follows that A can be written in the form ^ = -^~^AJ,(h'r)cosee^'y-'^i'", (61) where *1 is a constant. Again, we may form the equation which, in virtue of (45), is the same as ^ + ;:-9;r-;:r + «^-. = 0, (62) where k'^ is given by the second of equations (52). It follows that 2m ^ can be written in the form 2OTz=/c'2C/i(«V)sin0e">^+^«, (63) where G is a constant. We may form also the equation _& „ =_:^_^nr 1 8 /"^^gTsN d dm, which, in virtue of (45), is the same as In this equation 2'5r2 has the value given in (63), and it follows that 2^-, can be written in the form 2^,= |,^C ^II^ + ,B^ JA^i sin ^eMr-.'> (65) where ^ is a constant. The equations connecting the quantities U, V, W with A, «r,., BTj can then be satisfied by putting dr or r r r dr } (66) W = iAyJ:,{h'r) - iBk'^ J^iK'r). When these forms for U, V, W are substituted in (58) we have a solution of equations (42). Since m, sin +ug cos vanishes when r = 0, the motion 296 FLEXURAL VIBRATIONS OF A CYLINDER [CH. XII of points on the axis of the cylinder takes place in the plane containing the unstrained position of that axis and the line from which 6 is measured ; and, since u^ vanishes when r = 0, the motion of these points is at right angles to the axis of the cylinder. Hence the vibrations are of a " transverse " or " flexural " type. We could form the conditions that the cylindrical surface is free from traction. These conditions are very complicated, but it may be shown by expanding the Bessel's functions in series that, when the radius a of the cylinder is very small, the quantities "p and 7 are connected by the approximate equation * p- = \aY{Elp), (67) where E is Young's modulus. This is the well-known equation for the frequency pl^ir of flexural waves of length 277/7 travelling along a cylindrical bar. The ratios of the constants A,B,C which correspond with any value of 7 are determined by the conditions at the cylindrical surface. When the cylinder is terminated by two normal sections z = and z = I, we write m/l for the real positive fourth root of 4ip'p/a^E. We can obtain four forms of solution by substituting for 17 in (52), (.58), (66) the four quantities ± m/l and + i,in/l successively. With the same value of p we should have four sets of constants A, B, C, but the ratios A : B : C in each set would be known. The conditions that the stress-components zz, zB vanish at the ends of the cylinder would yield sufficient equations to enable us to eliminate the constants of the types A, B, G and obtain an equation for p. The condition that the stress-component zr vanishes at the ends cannot be satisfied exactly ; but, as in the problem of longitudinal vibrations, it is satisfied approximately when the cylinder is thin. * Cf. Lord Eayleigh, Theory of Sound, Chapter VIII. CHAPTEE XIII. THE PROPAGATION OF WAVES IN ELASTIC SOLIP MEDIA. 203. The solution of the equations of free vibration of a body of given form can be adapted to satisfy any given initial conditions, when the frequency equation has been solved and the normal functions determined; but the account that would in this way be given of the motion that ensues upon some local disturbance originated within a body, all points (or some points) of the boundary being at considerable distances from the initially disturbed portion, would be difficult to interpret. In the beginning of the motion the parts of the body that are near to the boundary are not disturbed, and the motion is the same as it would be if the body were of unlimited extent. We accordingly consider such states of small motion in an elastic solid medium, extending in- delinitely in all (or in some) directions, as are at some time restricted to a limited portion of the medium, the remainder of the medium being at rest in the unstressed state. We begin with the case of an isotropic medium. 204. Waves of dilatation and waves of distortion. The equations of motion of the medium may be written If we differentiate the left-hand and right-hand members of these three equations with respect to *■, y, z respectively and add the results, we obtain the equation (^ + 2;.)V^A = p^ (2) If we eliminate A from the equations ( 1 ) by performing the operation owrl upon the left-hand and right-hand members we obtain the equations 9^ H^''{^x, ■^y, ^z) = PkT2('^x, ■^y, ■^z) (3) 298 PROPAGATION OF WAVES THROUGH [CH. XIII If A vanishes the equations of motion become fi^-{ii,v, w) = p^^{u, V, w) (4) If ■sTj;, -CTy, OT^ vanish, so that (u, v, lu) is the gradient of a potential , we may put V=^ for A, and then we have In this case the equations of motion become (X + 2/i)V\ii, V, w) = p^^{u, V, w) (5) Equations (2), (3), (4), (5) are of the form 3!^ = c2V=d,; (6) for A, c^ has the value {X + 2/j,)/p; for nr^,... it has the value p./ p. The equation (6) will be called the " characteristic equation." If (^ is a function of t and of one coordinate only, say of x, the equation (6) becomes which may be integrated in the form (j)=f(a;-ct) + F{x + ct), f and F denoting arbitrary functions, and the solution represents plane waves propagated with velocity c. If is a function of t and r only, r denoting the radius vector from a fixed point, the equation takes the form ct^ r Q-fi ^ ^" which can be integrated in the form ^_ f{r-ct) , Fir-^cC) and again the solution represents waves propagated with velocity c. A function of the form i-'^f{r~et) represents spherical waves diverging from a source at the origin of r. We learn that waves of dilatation involving no rotation travel through the medium with velocity {(X + 2/u.)/pj*, and that waves of distortion involving rotation without dilatation travel with velocity {/J-lp}^- Waves of these two types are sometimes described as " irrotational " and " equivoluminal " respectively*- * Lord Kelvin, Phil. Mag. (Ser. 5), vol. 47 (1899). The result that in an isotropic solid there are two types of waves propagated with different velocities is due to Poisson. The recognition of the irrotational and equivoluminal characters of the two types of' waves is due to Stokes. See Introduction. 204, 205] AN ISOTROPIC SOLID MEDIUM 299 If plane waves of any type are propagated through the medium with any velocity c we may take u, v, w to be functions of Ix + my + nz — ct, in which l,m,n are the direction-cosines of the normal to the plane of the waves. The equations of motion then give rise to three equations of the type pc'u" = {\ + fj.)l (lu" + viv" + nw") + ti{P + m? + 'n?) u", where the accents denote differentiation of the functions with respect to their argument. On elimination of u", v", w" we obtain an equation for c, viz. : (\ + 2/.-/3c0 (M-pcO» = (7) showing that all plane waves travel with one or other of the velocities found above. 205. Motion of a surface of discontinuity. Kinematical conditions. If an arbitrary small disturbance is originated within a restricted portion of an elastic solid medium, neighbouring portions will soon be set in motion and thrown into states of strain. The portion of the medium which is dis- turbed at a subsequent instant will not be the same as that which was disturbed initially. We may suppose that the disturbed portion at any instant is bounded by a surface /S. If the medium is isotropic, and the propagated disturbance involves dilatation without rotation, we may expect that the surface S will move normally to itself with velocity {(X -I- 2fji.)lp]^; if it involves rotation without dilatation, we may expect the velocity of the surface to be {/x/p}*. We assume that the surface moves normally to itself with velocity c, and seek the conditions that must be satisfied at the moving surface. On one side of the surface t, this element passes from a state of rest without strain to a state of motion and strain corresponding with the displacement (m, V, w). The change is effected by the resultant traction across the boundaries of the element, that is by the traction across h8, and the change of momentum is equal to the time-integral of this traction. The traction in question acts across the surface normal to v upon the matter on that side of the surface towards which v is drawn, so that its components per unit of area are —X^, — F„, — Z„. The resultants are obtained by multiplying these by hS, and their impulses- by multiplying by Si. The equation of momentum is therefore pSS.cSi^l', |, ^) = -(Z„, n,Z,)S,SSi, from which we have the equations fdu dv dW 'Ki'* i)=-<^"^"^-> <") 205-207] AN ISOTROPIC SOLID MEDIUM 301 In these equations du/dt, ... and X„, ... are to be calculated frona the values of )(,... on that side of S on which there is disturbance; and the equations hold at all points of S. In the case where there is motion and strain on both sides of the surface S, but the displacements on the two sides of S are expressed by diflFerent formulje, we may denote them by {ui, Vj, Wj) and {u^, t»2, %). At all points of S the displacement must be the same whether it is calculated from the expressions for Ui, ... or from those for %, We may prove that the values at S of the differential coefficients of u^, ... are connected by equations of the type cui dUi Smj Sag 8«i 3^2 ex cix _dy dy _ dz ^^ _ 3"! CW2_ 1 /cMi cu^ cos (a;, v) ~ cos (y, v) ~ cos {z, v)~ dv dv ~ c \dt it ) ' with similar equations in which u is replaced by v or by «'. If we denote the tractions calculated from (mj, Vi, Wj) by X^C), ... and those calculated from (mj, %, w^) by XJ^\ ... we may show that the values at S of these quantities and of onijct, ... are connected by tho equations 207. Velocity of waves in isotropic medium. If we write I, m, n for the direction-cosines of v, the equations (11) become three equations of the type of which the right-hand member may also be written in the form „ , ,idu dv dw\ , f 8« , 9^ ,^iOv\ f 9m dw ., diu\ ,T„, + '^r95 + "a^--^8-4 ^'^^ These equations hold at the surface S, at which also we have nine equations of the type ^' = -iz|', (14) dx c dt' ^ ^ so that, for example, , dv dv Im dv l — = m^= -5-. . ay ox c ot On substituting for du/dx, ... from (14) in (12), we obtain the equation 302 PROPAGATION OF WAVES THROUGH [CH. XIII and, on eliminating du/dt, dv/dt, dw/dtivom this and the two similar equations, we obtain the equation (7) of Article 204. The form (13) and the equations of type (14) show that equation (12) may also be written du , „ , , /du dv dw\ idv du\ fbu 'bu:\ .,„, -pCg- = (X + 2;.)Z(g_+g- + g-J-M,.(g--g-)+;..(--3^). (16) Hence it follows that, when the rotation vanishes, we have three equations of the type from which we should find that pc^ = X + 2;ii ; and, when the dilatation vanishes, we have three equations of the type dt '^r dt dt dt] from which we should find that pc" = /m. These results show that the surface of discontinuity advances with a velocity which is either {(\ + 2fi)/p]i or (fi/p)^, and that, if there is no rotation, the velocity is necessarily {(\ + 2fj,)/p]i, and, if there is no dilatation, the velocity is necessarily (p-lp)^. 208. Velocity of waves in seolotropic solid medium. Equations of the types (10) and (11) hold whether the solid is isotropic or not. The former give the six equations ,M V w r = - 1 - , e^ = -m-, fij^ = - «- , c ( w v\ ( u -.wx / ii u\ (17) in which the dots denote differentiation with respect to t, and I, m, n are written for cos {x, v), .... The equations (11) can be written in such forms as — pcii = l — +m- l-wr — , (18) ^^xx ^^xy ^^zx where W denotes the strain-energy-function expressed in terms of the components of strain. Now let ^, 7/, ^ stand for w/c, vjc, w/c. Equations (17) are a linear substitution expressing e^^, ... in terms of f, 77, f. When this substitution is carried out W becomes a homogeneous quadratic function of f, rj, f. Denote this function by II. We observe that, since eyy, e„, e^j are in- dependent of ^, we have the equation an ,dw dw dW of dexx oBxy ceo; = (21) 207-209] AN -EOLOTBOPIC SOLID MEDIUM 303 and we have similar equations for 9n/?7; and 911/3^. Hence the equations of type (18) can be written pc-f=^, pc^=g^. pc^r = ^ (19) Now suppose that H is given by the equation n = i [X„r + X^,,^ + Xssf + 2X^,7? + 2X^ff + 2X,,f7?], (20) then the equations (19) show that tf satisfies the equation Xu ~ pc^j Xjo, Xsi Xl2» "->I — P^' ^3 i Xsi, Xjs, Xss — pc^ Since ^, 17, 5' are connected with e^x, ... bv a real linear substitution, the homogeneous quadratic function 11 is necessai-ily positive, and therefore equation (21) yields three real positive values for c". The coefiBcient-s of this equation depend upon the direction (/, m, n). There are accoixiingly three real wave-velocities answering to any direction of propagation of waves*. The above investigation is effectively due to E. B. Ciiristoffelt, who has given the following method for the formation of the function n :^Let the six components of strain ''j^i ««(;.•• ^ii+'>^i'i+'h(, ^^^^Q^ii+^l + ^aC)-, and therefore the coefBcients Xu, ... in the function n are to be obtained by squaring the form XiJ-t-Xaij-l-Xjf, or we have \is=CisP+C3Bm-+CtAn^+(,c^+c^)mn + {cu+Cie) nl+{ci3+cee) Im, 209. Wave-surfaces. The envelope of the plane la!+7m/+n2=c (22) in which c is the velocity of propagation of waves iu the direction {I, m, n) is the "wave- surface" belonging to the medium. It is the surface boiinding the disturbed portion of the medium after the lapse of one unit of time, beginning at an instant when the dis- turbance is confined to the immediate neighbourhood of the origin. In the case of isotropy, c is independent of I, m, m, and is given by the equation (7); in the case of Eeolotropy e is a function of I, m, n given by the equation (21). In the general case the * For a general discossion of the three types of waves we may refer to Lord Kelvin, Baiiimore Lectures, London 1904. t Ann. di Mat. (Ser. 2), t. 8 (1S77), reprinted in E. B. Christoffel, Ges. math. Abhandlungen, Bd. 2, Leipzig 1910, p. 81. 304 WAVE-SURFACES [CH. XIII wave-surface is clearly a surface of three sheets, corresponding with the three values of c^ which are roots of (21). In the case of isotropy two of the sheets are coincident, and all the sheets are concentric spheres. Green* observed that, in the general case of aeolotropy, the three possible directions of displacement, answering to the three velocities of propagation of plane waves with a given wave-normal, are parallel to the principal axes of a certain ellipsoid, and are, therefore, at right angles to each other. The ellipsoid would be expressed in our notation by the equation (Xn, X22, ... X12) {x, y, 0)2 = oonst. He showed that, when If has the form iA{e„+eyy + eJ^ + iZ{eyJ'-4:eyye^) + iM{eJ-4e^,e^) + ^N{e^^-4£„eyy), (23) the wave-surface is made up of a sphere, corresponding with the propagation of waves of irrotational dilatation, and Fresnel's wave-surface, viz. : the envelope of the plane (22) subject to the condition c^-Lip "*" c^-Mjp "^ J^Njp (^^^ The two sheets of this surface correspond with the propagation of waves of equivoluminal distortion. Green arrived at the above expression for W as the most general which would allow of the propagation of purely transverse plane waves, i.e. of waves with displacement parallel to the wave-fronts. Green's formula (23) for IF is included in the formula (15) of Article 110, viz. : 'iW={A, B, C, F, a, H){e,„ Byy, e,,y+LeyJ' + MeJ+JVe^% which characterizes elastic solid media having three orthogonal planes of symmetry. To obtain Green's formula we have to put A=B=C, F=A-2L, G=A-2M, E=A-2K It is noteworthy that these relations are not satisfied in cubic crystals. Green's formula for the strain-energy -function contains the strain-components only; the notion of a medium for which M'=2(Zra-^2-|-il/sr/-|-#-=r/) (25) was introduced by MacOullagh t. The wave-surface is Fresnel's wave-surface. Lord RayleighJ, following out a suggestion of Rankine's, has discussed the propagation of waves in a medium in which the kinetic energy has the form ///I t^lD'+p^ (D'+p^ (S)]'^^'^^''^' (26) while the strain-energy-function has the form appropriate to an isotropic elastic solid. Such a medium is said to exhibit "seolotropy of inertia." When the medium is incom- pressible the wave-surface is the envelope of the plane (22) subject to the condition -5 + T, + -, = 0; (27) c^pi-fi Cp^-p. c'p^-p it is the first negative pedal of Fresnel's wave-surface with respect to its centre. * 'On the propagation of light in crystallized media,' Cambridge Phil. Soc. Trans., vol. 7 (1839), or Mathematical Papers, London 1871, p. 293. + 'An essay towards a dynamical theory of crystalline reflexion and refraction,' Dublin, Trans. R. Irish Acad., vol. 21 (1839), or Collected Works of James MacOullagh, Dublin 1880, p. 14.5. J 'On Double Eefraction,' Phil. Mag. (Ser. 4), vol. 41 (1871), or Scientific Papers, vol. 1, Cambridge 1899. 209, 210] poisson's integeal fobmula 305 The case where the energy-function of the medium is a function of the components of rotation as well as of the strain-components, so that it is a homogeneous quadratic func- tion of the nine quantities ^, s-, 5- , ■•■, has been discussed by H. M. Macdonald*. The most general form which is admissible if transverse waves are to be propagated independently of waves of dilatation is shown to lead to Fresnel's wave-surface for the transverse waves. The still more general case in which there is seolotropy of inertia as well as of elastic quality has been investigated by T. J. I' A. Bromwicht. It appears that, in this case, the requirement that two of the waves shall be purely transverse does not lead to the same result as the requirement that they shall be purely rotational, although the two require- ments do lead to the same result when the seolotropy, does not affect the inertia. The wave-surface for the rotational waves is derived from Fresnel's wave-surface by a homo- geneous strain. 210. Motion determined by the characteristic equation. It appears that, even in the case of an isotropic solid, much complexity is introduced into the question of the propagation of disturbances through the solid by the possible co-existence of two types of waves propagated with different velocities.^ It will be well in the first instance to confine our atten- tion to waves of a single type — irrotational or equivoluminal. The motion is then determined by the characteristic equation (6) of Article 204, viz. d'. This equation was solved by Poisson J in a form in which the value of ^ at any place and time is expressed in terms of the initial values of (ft and dip/dt. Poisson's result can be stated as follows : Let ^o and (p,, denote the initial values of „ on this sphere. Then the value of at the point {x, y, z) at the instant t is expressed by the equation <^=^(%) + 4o (28) If the initial disturbance is confined to the region of space within a closed surface 2„, then c^o and 4>o liave values different from zero at points within ^0, and vanish outside So- Taking any point within or on So as centre, we may describe a sphere of radius ct ; then the disturbance at time t is confined to the aggregate of points which are on the surfaces of these spheres. This aggregate is, in general, bounded by a surface of two sheets — an inner and an outer. When the outer sheet reaches any point, the portion of the medium * London Math. Soc. Proc, vol. 32 (1900), p. 311. t London Math. Soc. Proc, vol. 84 (1902), p. 307. t Paris, Mem. de Vlnstitut, t. 3 (1820). A simple proof was given by Liouville, J. de Math. {Liouville), t. 1 (1856). A symbolioal proof is given by Lord Eayleigh, Theory of Sound, Chapter XIV. 20 L. E. ■"" 306 MOTION RESULTING FROM [CH. XIII which is close to the point takes suddenly the small strain and velocity implied by the values of and d^/dt ; and after the inner sheet pa'sses the point, the same portion of the medium returns to rest without strain*- The characteristic equation was solved in a more general manner by Kirchhofff. Instead of a sphere he took any surface S, and instead of the initial values of (f> and d(j>/dt on S he took the values of and its first derivatives are estimated for the point Q at the instant t — r/c. Let [ must tend to zero at infinite distances in the order r~^ at least. These conditions may be expressed by saying that all the sources of disturbance are on the side of S remote from {x, y, z). Kirchhoff's formula (29) can be shown to include Poisson'st. The formula may also be written in the form *-h\i[m-'-mv'- '-> where -^i — ) is to be formed by first substituting t—rjc for i in (^ and then difierentiating as if r were the only variable quantity in [<^]/r. The formula (31) is an analogue of Green's formula (7) of Article 158. It can be interpreted in the statement that the value of <^ at any point outside a closed surface (which encloses all the sources of disturbance) is the same as that due to a certain distribution of fictitious sources and double sources on the surface. It is easy to prove, in the manner of Article 124, that the motion inside or outside S, that is due to given initial conditions, is uniquely determined by the values of either <^ or 3(^/ov at S. The theorem expressed by equation (31) can be deduced from the properties of superficial distributions of sources and double sources and the theorem of uniqueness of solution I . 211. Arbitrary initial conditions. When the initial conditions are not such that the disturbance is entirely irrotational or equivoluminal, the results are more complicated. Expressions for the components of the displacement which arises, at any place and time, from a given initial distribution of displacement and velocity, have been obtained§, and the result may be stated in the following form : — Let (mo, Vo, Wo) be the initial displacement, supposed to be given throughout a region of space T and to vanish on the boundary of T and outside T, and let (mo, *o> m'o) be the initial velocity supposed also to be given throughout T and to vanish outside T. Let a and b denote the velocities of irrotational and equivoluminal waves. Let S^ denote a sphere of radius at having its centre at the point {x, y, z), and S^ a sphere of radius ht having its centre at the same point. Let V denote that part of the volume contained between these spheres which is within T. Let r denote the distance of any point {x', y' z') within V, or on the parts of S^ and S^ that are within T, from the point {x, y, z), and let q^ denote the initial displacement at (»', y', z'), and q„ the initial velocity at the same point, each projected upon the radius vector r, * For the ease where there is a, moving surface of discontinuity outside S, see a paper by the Author, London Math. Soc. Proc. (Ser. 2), vol. 1 (1904), p. 37. t See my paper just cited. X Of. J. Larmor, London Math. Soc. Proc. (Ser. 2), vol. 1 (1904). § For references see Introduction, p. 18. Eeference may also be made to a paper by the Author in London Math. Soc. Proc. (Ser. 2), vol. 1 (1904), p. 291. 20—2 u = 308 MOTION DUE TO [CH. XIII supposed drawn from (x, y, z). Then the displacement u at (a;, y, z) at the instant t can be written 1 {[[{ 9^r~' ?Fr~'^ 8^r~') 1 fii ( av-' av-i 9v-'\ , Sr-v^. , , ago\ -4^.IJrO'»-a^+^»s^ + '^»a^J + ^l*5° + ^» + "e^i -^(i«„ + «„ + r^^»)|d -hcurl (L, M, N), and the displacement in the form (u, V, w) = gradient of ^ -h curl {F, G, H). * Cf. Stokes, loc. cit. p. 306. 211, 212] VARIABLE BODY FORCES 309 Then the eqiiations of motion of the type can be satisfied if , i, ... are given in terms of X, Y, Z by the equations (7) of Article 130, and the integrations expressed in (33) can be performed. Taking the case of a single force of magnitude ■)( (t), acting at the origin in the direction of the axis of x, we have, as in Article 130, where R denotes the distance of (,r', y\ z) from the origin. We may partition space around the point {x, y, £) into thin sheets by means of spherical surfaces having that point as centre, and thus we maj' express the integrations in (33) in such forms as where dS denotes an element of surface of a sphere with centre at (x, y, z) and radius equal to r. Now JJ{dR~^/dx')dS is equal to zero when the origin is inside S, and to 4Trr^ (9ro~V9») when the origin is outside S, r„ denoting the distance of {x, y, z) from the origin. In the former case r^ < r, and in the latter i\ > r. We may therefore replace the upper limit of integration with respect to r" by r-o, and find ^ 4nra-p dx J ^ ^\ a) Having found we have no further use for the ?• that appears in the process, and we may write r instead of ?■(,, so that r now denotes the distance of (x, y, z) from the origin. Then we have '^=-4i^7„"'^(^-^'>*' <^*) * Cf. L. Lorenz, J. f. Math. (Grelle), Bd. 5S (1861), or CEuvres Sdmtifiques, t. 2, CopenhsLgen 1899, p. 1. See also Lord Eayleigh, Theory of Sound, vol. 2, § 276. 310 MOTION DUE TO VARIABLE FORCES [CH. XIII In like manner we should find ^irp dz J ■T--^ t'x(t-t')dt'. The displacement due to the force % («) is given by the equations* (35) 1 / r\ + 4^7^^ V bj' 1 8V-1 /•'■/* ,, .,,,,^ 1 drdrn ( i)-l^^*- )}• 47r/) dxdy, 1 8 V-' /■'■/* , ,^ -,, 1 drdr{l (^ r\ 1 A rN] (36) 213. Additional results relating to motion due to body forces. (i) The dilatation and rotation calculated from (36) are given by the equations 213,= -- 111 U'-'ih .(37) 47r62p Zy\ (ii) The expressions (36) reduce to (U) of Article 130 when '^{t) is replaced by a constant. (iii) The tractions over a spherical cavity required to maintain the displacement expressed by (36) are statically equivalent to a single force parallel to the axis of x. When the radius of the cavity is diminished indefinitely, the magnitude of the force is X (0- (iv) As in Article 132, we may find the effects of various nuclei of strain t. In the case of a "centre of compression " we have, omitting a constant factor, (»'^''^)=(i'l'^)^0-31' '''' representing irrotational waves of a well-known type. In the case of a " centre of rotation about the axis of z " we have, omitting a factor, ^-' ^' -)=(!' -A' '){U{'-'i)}' ^''^ representing equivoluminal waves of a well-known type. (v) If we combine two centres of compression of opposite signs in the same way as two forces are combined to make a '-double force without moment" we obtain irrotational waves of the type expressed by the equation („, ,, ,).(JL, _£L, .^\\ljt-':-)] (40) * FormulsB equivalent to (36) were obtained by Stokes, loc. cit. p. 306. t For a more detailed discussion, see my paper cited on p. 307, footnote §. 212-214] HARMONIC FORCES 311 If we combine two pairs of centres of rotation about the axes of x and y and about parallel axes, in the same way as two pairs of forces are combined to make a centre of rotation, we obtain equivoluminal waves of the type , X / fc- c2 52 52\fl / rW ("' '-' '")=(.-^' a^ai' -g^=-gp)t n'-fe)}' (^^^ in which the displacement is expressed by the same formulae as the electric force in the field around Hertz's* oscillator. Lord Kelvin t has shown that by superposing solutions of the types (40) and (41) we may obtain the effect of an oscillating rigid sphere close to the origin. (vi) "When x if) is a simple harmonic function of the time, say x{t) = A cos pt, we find /;;V,(.-0<..'=^4{cos. (.-0 - cos, (.-3 -f sin, (.-0 +f sin, (.-3} , and complete expressions for the effects of the forces can be written down by (36) t In this case we may regard the whole phenomenon as consisting in the propagation of two trains of simple harmonic waves with velocities respectively equal to a and 6 ; but the formulae (36) show that, in more general cases, the effect produced at the instant < at a point distant ?• from the point of application of the forces does not depend on the magnitude of the force at the two instants t—r/a and t-r/b only, but also on the magnitude of the force at intermediate instants. It is as if certain effects were propagated with velocities intermediate between a and 6, as well as the definite effects (dilatation and rotation) that are propagated with these velocities § . (vii) Particular integrals of the equations of motion under body forces which are proportional to a simple harmonic function of the time (written e'^*) can be expressed in the forms pi-pt f r r p-'pr/a e' where *= - i- |JJ(x' ?^+ F' ?^+^' %^) d^dy'd,, 214. Waves propagated over the surface of an isotropic elastic solid body 1 1. Among periodic motions special importance attaches to those plane waves * Hertz, Electric Waves, English edition, p. 137. For the discussion in regard to the result see W. Konig, Ann. Phys. Ghem. {Wiedemann), Bd. 37 (1889), and Lord Eayleigh, Phil. Mag. (Ser. 6), vol. 6 (1903), p. 385, reprinted in his Scientific Papers, vol. 3, p. 142. t Phil. Mag. (Ser. 3), vols. 47 and 48 (1899). J For the effects of forces which are simple harmonic functions of the time, see Lord Eayleigh, Theory of Sound, vol. 2, pp. 418 et seq. The theory of waves due to forces of damped harmonic type, and the subsidence of vibrations caused by their communication to a surrounding elastic sohd medium, have been discussed by E. Laura, Torino Mem. (Ser. 2), t. 60 (1910), Torino Atti, t. 46 (1911), and Boma, Ace. Line. Rend. (Ser. 5), t. 21 (1 Sem.), 1912, and by 0. Tedone, Roma, Ace. Line. Rend. (Ser. 5), t. 22 (1 Sem.), 1913. § Cf. my paper cited on p. 307, footnote §, and Stokes's result recorded on p. 308. II Cf. Lord Eayleigh, London Math. Sac. Proc. , vol. 17 (1887), or Scientific Papers, vol. 2, p. 441. 312 WAVES PROPAGATED OVER THE [CH. XIII of simple harmonic type, propagated over the bounding surface of a solid body, which involve a disturbance that penetrates but a little distance into the interior of the body. We shall take the body to be bounded by the plane z = 0, and shall suppose that the positive sense of the axis of ^is directed towards the interior of the body. Then the waves in question are characterized by the occurrence, in the expressions for the quantities defining the motion, of factors of the form e~" and e'"", where r and s are real and positive. Let the direction of propagation of the waves be the axis of x, and let the dilatation A be expressed by the formula A = Pe-«e''^*-.^*', (42) where P is constant. Then p/f is the velocity of propagation. Denoting p'p/{\ + 2iJ.) by A^ as in Article 190, and remembering that A satisfies the equation (V^ + A^) A = 0, we see that r-^^f^-h? (43) A displacement answering to (42) is given by the equations (Mi,i'i,wO = (t/,0,r) /i-^Pe-'-^+MP*-/^), (44) and with this we may compound any displacement {ih,^^, w^ which satisfies div {U2,V2, Wa) = 0, ( V= + K-) (wg, v^, Ws) = 0, where, as in Article 190, «^ is written for p^pjfi- We write (m„ v„ w,) = (ts, /3,/) «-» Qe-"+m-M, (45) where /8 is constant, and s^=f^-K^ (46) The surface z — being free from traction, the equations du dw ^ dv dw ^ ^ . „ 9w _ .,^, 95 + 3^=*^' ai+9^ = «' ^^ + 2^9^ = 0. (47) in which (u, v, w) = {u^ + u^, v^ 4- v^, w^ + w^), must hold at that surface. These equations give 2rfPlh' + (s« +p) Q/«^ = 0, /8 = 0, («2 - 2A^) Pjh'' - 2r'Plh^ - 2sfQlK^ =0, (48) where K'/h^ — 2 has been written for X//jl. The equation /3 = shows that the motion is two-dimensional. There is no displacement in a direction parallel to the plane boundary and transverse to the direction of propagation. The elimination of P and Q from the remaining equations of (48) gives (j.-2y-4_P = 0, (49) or by (43) and (46) 214] SURFACE OF A BODY 313 If we write k'^ for i^jf^ and h"^ for h^jp, this equation becomes «'8_8«'8 + 24«'<'-16(l+A'^)/2 + 16/i'^ = (50) When the material is incompressible, so that h'^/K'^ = 0, equation (50) becomes a cubic for k'^, viz.: «'« - Sk'* + 24k'^ - 16 = 0, which has one real positive root a:'^ = 0'91262... and two complex roots 3-5436... + c (2-2301... ). Since k'^ is finite and A'V/c'^ is zero, equation (43) shows that r' is real, and equation (49) shows that, for the complex values of k"', 4rs//^ = - (2-5904...) ±t (6-8852...). Since the real part of s, given by this equation, has the opposite sign to r, there are no waves of the required type answering to the complex values of k'". The real value «:'= = 0-91262... gives r'=f-\ s2 = (0-08737... )/^ so that there is a wave-motion of the required type. The velocity of propa- gation is given by the equation p//= (0-9553... )V(Wp)- When the material fulfils Poisson's condition (\ = fi), so that K'^/h'^ = B, equation (50) becomes («:'2 - 4) {3k'' - I2ic'' -f- 8) = 0. The roots «'" = 4 and «'^ = 2 -)- 1 a/3 are irrelevant, since they make h'^ > 1 and r a pure imaginary. The remaining root /e'^ = 0-8453. . . gives r' = (0-7182. . .)/^ s"- = (0-1546. . .)/^ and the velocity of propagation is now given by the equation p//= (0-9194...) V(Wp). In both cases the waves travel over the surface with a velocity, which is independent of the wave-length 27r//, and slightly less than the velocity of equivoluminal waves propagated through the body. Waves of this kind are often called " Rayleigh-waves." Concerning the above type of waves Lord Rayleigh (loc. cit.) remarked : "It is not improbable that the surface waves here investigated play an important part in earthquakes, and in the collision of elastic solids. Diverging in two dimensions only, they must acquire at a great distance from the source a continually increasing preponderance." The subject has been investigated further by T. J. I'A. Bromwich* and H. Lambf. The former showed that, when gravity is taken into account, the results obtained by Lord Rayleigh are not essentially altered. The latter has discussed the * London Math. Soc. Proc, vol. 30 (1899). t Phil. Tram. Boy. Soc. (Ser. A), vol. 203 (1904). 314 EAYLEIGH WAVES [CH. XIII effect of a limited initial disturbance at or near the surface of a solid body. He showed that, at a distance from the source, the disturbance begins after an interval answering to the propagation of a wave of irrotational dilatation ; a second stage of the motion begins after an interval answering to the propa- gation of a wave of equivoluminal distortion, and a disturbance of much greater amplitude begins to be received after an interval answering to the propagation of waves of the type investigated by Lord Rayleigh. The importance of these waves in relation to the theory of earthquakes has perhaps not yet been fully appreciated*- The theory has been extended by H. Lambf to the case of a solid body bounded by two parallel planes. * See the Author's Essay Some Problems of Geodynamics, or G. W. Walker, Modem Seismology, London 1913. t London, Proc. E. Soc. (Ser. A), vol. 93, 1917, p. 114. CHAPTER XIV. TORSION. 215. Stress and strain in a twisted prism. In Article 86 (d) we found a stress-system which could be maintained in a cylinder, of circular section, by terminal couples about the axis of the cylinder. The cylinder is twisted by the couples, so that any cross-section is turned, relatively to any other, through an angle proportional to the distance between the planes of section. The traction on any cross-section at any point is tangential to the section, and is at right angles to the plane containing the axis of the cylinder and the point; the magnitude of this traction at any point is proportional to the distance of the point from the axis. When the section of the cylinder or prism is not circular, the above stress-system does not satisfy the condition that the cylindrical boundary is free from traction. We seek to modify it in such a way that all the conditions may be satisfied. Since the tractions applied at the ends of the prism are statically equivalent to couples in the planes of the ends, and the portion of the prism contained between any cross-section and an end is kept in equilibrium by the tractions across this section and the couple at the end, the tractions in question must be equivalent to a couple in the plane of the cross-section, and the moment of this couple must be the same for all cross- sections. A suitable distribution of tangential traction on the cross-sections must be the essential feature of the stress-system of which we are in search. Accordingly, we seek to satisfy all the conditions by means of a distribution of sheanng stress, made up of suitably directed tangential tractions on the elements of the cross-sections, combined, as they must be, with equal tangential tractions on elements of properly chosen longitudinal sections. We shall find that a system of this kind is adequate ; and we can foresee, to some extent, the character of the strain and displacement within the prism. For the strain corresponding with the shearing stress, which we have described, is shearing strain which involves, in general, two simple shears at each point. One of these simple shears consists of a relative sliding f,. = T("^-y), e,, = T(|^ + ^) (2) 316 THE TORSION PROBLEM [CH. XIV in a transverse direction of elements of different cross-sections ; this is the type of strain which occurred in the circular cylinder. The other simple shear consists of a relative sliding, parallel to the length of the prism, of different longitudinal linear elements. By this shear the cross-sections become distorted into curved surfaces. The shape into which any cross- section is distorted is determined by the displacement in the direction of the length of the prism. 216. The torsion problem*. We shall take the generators of the surface of the prism to be parallel to the axis of z, and shall suppose that the material is isotropic. The discussion in the last Article leads us to assume for the displacement the formulae U = — TyZ, V=TZX, W = T and yjr are mathematically identical with the velocity-potential and stream-function of a certain irrotational motion of incompressible frictionless fluid, contained in a vessel of the same shape as the prism f. This motion is that which would be set up bj' rotating the vessel about its axis with angular velocity equal to — 1. (b) The function ^^ — J (^ + y^) is mathematically identical with the velocity in a certain laminar motion of \-iscous fluid. The fluid flows under pressure through a pipe, and the section of the pipe is the same as that of the prism ^. (c) The function "^ — -^ (ix^ + y^) is also mathematically identical vrith the stream-function of a motion of incompressible frictionless fluid circulating with uniform spin, equal to unity, in a fixed cj-lindrical vessel of the same shape as the prism §. The moment of momentum of the liquid is equal to the quotient of the torsional rigidity of the prism by the rigidity of the material. The velocity of the fluid at any point is mathematically identical with the shearing strain of the material of the prism at the point. In the analogy (a) the vessel rotates as stated relatively to some frame regaitled as fixed, and the axes of x and y rotate with the vesseL The velocity of a particle of the fluid relati\'e to the fixed frame is resolved into components parallel to the instantaneous positions of the 3.xes of x and i/. These components are ccjilav and c(f>lcy. The velocity of the fluid relative to the vessel is utilized in the analogy (c). We may use the analogy in the form (a) to determine the effect of twisting the prism about an axis when the effect of twisting about any parallel axis is known. Let ^o be the * The functions are determined for a number of forms of boundary in Articles 221, 222 infru. For the special condition, necessary to secure uniqueness in the case of a boUow shaft, see Article 222 (iii). t Kelvin and Tait, Xat. Phil. Part n., pp. 242 et seg. The velocity -potential is here defined by the convention that the velocity of the fluid in the positive sense of the axis of a; is dldx, not -dipldx. X J. Boussinesq, J. de math. {LiouviUe), (Ser. 2), t. 16 (1871). § A. G. GreenhiU, Article 'Hydromechanics,' Ency. Brit., 9tb edition. 320 STRENGTH TO RESIST TORSION [CH. XIV torsion-function when the axis meets a cross-section at the origin of {x, y) ; and let <^' be the torsion-function when the prism is twisted about an axis parallel to the first, and meeting the section at a point {of, y'). Rotation of the vessel about the second axis is equivalent at any instant to rotation about the first axis combined with a certain motion of translation, which is the same for all points of the vessel. This instantaneous motion of translation is the motion of the first axis produced by rotation about the second ; and the component velocities in the directions of the axes are —y' and x', since the angular velocity of the vessel is —1. It follows that we must have o — xy'+yx'. The com- ponent displacements are therefore given by the equations u=-r(i/-y')2, i' = r{x-x')z, w=T(l>'; and the stress is the same as in the case where the axis of rotation passes through the origin. The torsional couple and the potential energy also are the same in the two cases. 219. Distribution of shearing stress. The stress at any point consists of two superposed stress-systems. In one system we have shearing stresses X^ and Fj of amounts — firy and fiTX respectively. In this system the tangential traction per unit of area on the plane z — const, is directed, at each point, along the tangent to a circle, having its centre at the origin and passing through the point. There must be equal tangential traction per unit of area on a plane passing through the axis of this circle, and this traction is directed parallel to the axis of z. In the second system we have shearing stresses X^ and Y^ of amounts fiTdldy. The corresponding tangential traction per unit of area on the plane z = const, is directed at each point along the normal to that curve of the family = const, which passes through the point, and its amount is proportional to the gradient of cp. There must be equal tangential traction per unit of area on a cylindrical surface standing on that curve of the family (j) = const, which passes through the point, and the direction of this traction is that of the axis of z. These statements concerning the stress are inde- pendent of the choice of axes of x and y in the plane of the cross-section, so long as the origin remains the same. The resultant of the two stress-systems consists of shearing stress with components X^ and Y^, which are given by the equations (3). If we put t-i(^ + 2/') = ^. (9) the direction of the tangential traction (X^, Y^) across the normal section at any point is the tangent to that curve of the family ^ = const, which passes through the point, and the magnitude of this traction is ixrd'^jdv, where dv is the element of the normal to the curve. The curves '^ = const, may be called " lines of shearing stress." The magnitude of the resultant tangential traction may also be expressed by the formula -{(s-')'-(i-)r <-) 218-220] STRENGTH TO RESIST TORSION 321 and this result is independent of the directions of the axes of x and y. If we choose for the axis of a; a line parallel to the direction of the tangential traction at one point P, the shearing stress at P will be equal to the value at P of the function fir {d

from the equation (4) and the condition (5) when the boundary of the section of the prism has one or other of certain special forms. The arbitrary constant which may be added to will in general be adjusted so that <^ shall vanish at the origin. (a) The circle. If the cylinder of circular section is twisted about its axis of figure, (^ vanishes, and we have the solution already given in Article 86 (d). If it is twisted about any parallel axis (/> does not vanish, but can be determined by the method explained in Article 218. In the latter case the cross -sections are not distorted, but are displaced so as to make an angle differing slightly from a right angle with the axis. (6) The ellipse. The function yjr is & plane harmonic function which satisfies the condition yfr — ^ (of + y'^) = const, at the boundary x^/a^ + y^jb^ = 1. If we assume for i/r a form A (x^ — y^), we find the equation {^-A)a' = i^ + A)b' (11) It follows that we must have ^ n^ — h^ «2 _ A2 ^=2^rrF^(-^-^^)' ^=-^4^^^ (12) It is clear that this solution is applicable to the case of a boundary consisting of two concentric similar and similarly situated ellipses. The prism is then a hollow elliptic tube. (c) The rectangle* - The boundaries are given by the equations x= ±a,y= ±b. The function yjr differs by a constant from ^ (3/^ -1- a^) when « = ± a and b>y>-h; it differs by the same constant from H^ + ^0 when y=±b and a>x>-a. We introduce a new function ->|r' by means of the equation yjr' = f - ^ (x' - y^) - :!^b\ Then -yjr' is a plane harmonic function within the rectangle ; and we may take ifr' to vanish on the sides y=±b, and to be equal to y^ - b^ on the sides * The corresponding hydrodynamical problem was solved by Stokes, Cambridge Phil. Soc. Trams., vol. 8 (1843), reprinted in his Math, and Phys. Papers, vol. 1, p. 16. 220, 221] THE TORSION PROBLEM 323 01= ± a. Since the boundary-conditions are not altered when we change x into — X ov y into — y, we seek to satisfy all the conditions by assuming for ■\jr' a formula of the type 2^^ cosh mx cos my. The conditions which hold at the boundaries y=±b require that m should be I (2w + 1) Tr/b, where n is an integer. If we assume that, when b>y >—b, the function y^ - 6= can be expanded in a series according to the form f - b^ = S^^,, cosh (:?^^±1>^ cos (?!^±J)^^ , we may determine the coefficients by multiplying both members of this equation by cos ((2n + 1) 'iry/2b}, and integrating both members with respect to y between the extreme values — b and 6. We should thus find ^».>cosh(?!^:tlhL^ = (-r^46^ 23 •2b V / " {2n + iyTr'' This process suggests that when b>y> — b the sum of the series i 46^ f 2YJ-r^ (2n-H)7ry „to*'' Uj(2n + 1)3'°' 26 ^^"^^ is y^ — b". We cannot at once conclude that this result is proved by Fourier's theorem*, because a Fourier's series of cosines of multiples of iry/2b represents ' a function in an interval given by the inequalities 2b>y> — 2b, and the value y^ — 6" of the function to be expanded is given only in the interval b>y>—b. If the Fourier's series of cosines contains uneven multiples of •try 12b only, the sign of every term of it is changed when for y we put 2b — y; it follows that, if the series (13) is a Fourier's series of which the sum is y2 — ^2 -when b>y>0, the sum of the series when 2b>y >b isb' — (26 — yy. Now we inay show that the Fourier's series for an even function of y, whicjfi has the value y^ — 6" when b>y >0, and the value 6^ — (26 — yf when 2b>y >b, is in fact the series (13). We may conclude that the form of -v|r is cosh (^'' + ^^-^'= 1, w s ., /2V2, (-Y 26 (2n + l)Try ^b^ + i(x^- u") - 46^ - 2 7„ ^— -jT, TT—rr^ cos ^^ — ./ ^ , "^ , WJ n-=o{^n + iy (2w + 1) 7ra 26 26 and hence that sinh(?ni>^ . AiJ^y ^ (-) 26 . (2n + l)7r.vt .... d) = -xy + 4i¥[-] 2 ,^ tt; TT sm^^ ,,/ ^ '. ...(14) ^ ^ VW „=o (2« + 1)' ,i2n + l)'na 26 ^o(2» + l)%p„t^ (2« + l)7ra 26 * Observe, for example, that the Fourier's series of cosines of multiples of iri//26 which has the sum y'' - 6* throughout the interval 2b>y> -2bis t The expression for must be unaltered when x and i/, a and 6, are interchanged. For an account of the identities which arise from this observation the reader is referred to a paper by F. Purser, Messenger of Math. , vol. 11 (1882). 21—2 324 SPECIAL SOLDTIONS OF [CH. XIV 222. Additional results. The torsion problem has been solved for many forms of boundary. One method is to assume a plane harmonic function as the function \j/, and determine possible boundaries from the equation i/'-^(^^+y2) = const. As an example of this method we may take yjr to be A {x^ - Sxi/) ; if we put A= — l/6a, the boundary can be the equilateral triangle*, of altitude 3a, of which the sides are given by the equation {x - a) {x -yJZ + 2a) {x+y JZ + 2(i) = 0. Other examples of this method have been discussed by Saint- Venant. Another method is to use conjugate functions |, j; such that l + ii? is a function of x + iy. If these functions can be chosen so that the boundary is made up of curves along which either ^ or i; has a constant value, then y^r is the real part of a function of l + iij, which has a given value at the boundary; and the problem is of the same kind as the torsion problem for the rectangle. We give some examples of this method : — (i) A sector of a circlet, boundaries given hj r = Q, r = a, 6= +0. — We find ^ = ^'-%-5i20 + '^'f L^--' y ^^cos{(2«+l)-|J, where ^^''^^-(-)"^' [ (2>. + li.-4/3 " (2^^ + (2^ + /) . + 4^ - If we write re'-^ = ax, then ( —-3 — +1 s ,2 ( rx „2^ Afl 1 fx ^2? I dx{ I ^ i 2 ■=^'' «M li":^^ J 4/3, _, ,„. 1 {^x where | .j; | ^ 1, and i&vT'^x"^^^ denotes that branch of the function which vanishes with x. In case 7r/2^ is an integer greater than 2 the integrations can be performed, but when 7r/2/3=2 the first two terms become infinite, and their sum has a finite limit, and we find for a quadrantal cylinder 1/^-10 = — r-a;2 1og^ + tan-i.r2 + i (^^-"i) log(l+a;i)1. . For a semicircular cylinder ^-.<^ = ^[j..^-z(.+^)+i.(.Hi-2)logl±|]. (ii) For a curvilinear rectangle bounded by two concentric circular arcs and two radii, we use conjugate functions a and (3, which are given by the equation .r + iy = ce''+'^ ; we take the outer radiu.s, u. to be ce"» and the inner, h to be ce""" (so that c is the geo- metrical mean of the radii), and we take the bounding radii to be given by the equations 3= ±00. We find <^ = - ^a6e2. ^^ + 25a6/3„2|4„*„, cos Zpj Q I ^""^ 2S~ ''"^^ 28 where *„ = J cosh 2ao ,^ , ^ -1- sinh la, ^ ^osh (2!^+i)Z-?o ,i^i, (2»+l)^a„ 2po (_).sm ^ > and A^^: ^£2 {(2»-|-l)n--4/3o}(2™-l-l),r{(2«H-l)7r-l-40o} ' * See Figures 23 and 24 in Article 223. t See A. G. GreenhUl, Messenger of Math., vol. 8 (1878), p. 89, and vol. 10 (1880), p. 83. 222, 223] THE TORSION PROBLEM 325 (iii) When the twisted prism is a hollow shaft, the inner and outer boundaries being circles which are not concentric, we may use the conjugate functions |, r) determined by the equation ^+iJ/ = ctan^ (! + "/) ; and, it rj = a represents the outer boundary, and i7 = /3 the inner, we may prove* that i//' = 2c2 2 (-)" e~ "^ coth /3 sinh m (i; - o) + c""" coth a sinh k (/3 - t)) cos n^. M=i ■ sinh M (/3 - a) In this example the differential equation and boundary-condition for yjr would still be satisfied if a term of the form Arj were added to the expression given for ijr. The conjugate function (p would then contain a term of the form A§, and the displacement w, or rtj), would then be many- valued. To secure a one-valued expression for w it would be necessary to put .4 = 0. A similar result holds for any hollow shaft. (iv) When the boundaries are confocal ellipses and hyperbolas we may use the con- jugate functions ^, rj determined by the equation x + ii/=c cosh (|4-n)). In the case of a hollow tube, of which the section is bounded by two confocal ellipses §,, and li , we may prove t that V' = J<;' sinh2(|o-|)-Hsinh2(|-|i) sinh 2 do -li) cos 2jj. 223. Graphic expression of tHe results. (a) Distortion of the cross-sections. The curves <^ = const, are the contour lines of the surface into which any cross-section of the prism is distorted. These curves were traced by Saint-Venant for a number of forms of the boundary. Two of the results are shown in Fig. 22 and Fig. 23. In both Fig. 22. cases the cross-section is divided into a number of compartments, 4 in Fig. 22, 6 in Fig. 23, and (p changes sign as we pass from any compartment to an adjacent * H. M. Maodonald, Gamhridge Phil. Soc. Proc, vol. 8 (1893). t Cf. A. G. Greenhill, Quart. J. of Math., vol. 16 (1879). Other examples of elliptic and hyperbolic boundaries are worked out by Pilon, loc. cit., p. 321. 326 DISTORTION OF CROSS-SECTIONS OF A TWISTED PRISM [CH. XIV compartment, but the forms of the cuires <^ = const, are unaltered. If we think of the axig of the prism as vertical, then the curved surface into which any cross-section is strained lies above its initial position in one compartment and below it in the adjacent compart- ments. Saint- Venant showed that the sections of a square prism are divided in this way into 8 compartments by the diagonals and the lines drawn parallel to the sides through the centroid. When the prism is a rectangle, of which one pair of opposite sides is much longer than the other pair, there are only 4 compartments separated by the lines drawn parallel to the sides through the centroid. The limiting case between rectangles which are di\'ided into 4 compartments and others which are divided into 8 compartments occurs when the ratio of adjacent sides is 1'4513. The study of the figures has promoted comprehension of the result that the cross-sections of a twisted prism, of non-circular section, do not remain plane. Fig. 23. (6) Lines of shearing stress. The distribution of tangential traction on the cross-sections of a twisted prism can be represented graphically by means of the lines of shearing stress. These lines are deter- mined by the equation Fig. 24. 223-225] DISTRIBUTION OF SHEARING STRESS 327 They have the property that the tangential traction on the cross-section is directed at any point along the tangent to that curve of the family which passes through the point. If the curves are traced for equidifierent values of o, the tangential traction at any point is measured by the closeness of consecutive curves. In the case of the prism of elliptic section V' - i (^' +y') = - (^'6' +fa^)l{a^ + 62), and the lines of shearing stress are therefore concentric similar and similarly situated ellipses. In the case of the equilateral triangle and the lines of shearing stress are of the forms shown in Fig. 24. 224. Analogy to the form of a stretched membrane loaded uniformly*. Let a homogeneous membrane be stretched with uniform tension T and fixed at its edge. Let the edge be a given curve in the plane of .•;;, y. When the membrane is subjected to pressure, of amount p per unit of area, it will undergo a small displacement s, and is a function of x and y which vanishes at the edge. The equation of equilibrium of the membrane is The function ^Tzjp is determined by the same condition as the function * of Article 219, provided that the edge of the membrane is the same as the bounding curve of the cross- section of the twisted prism. It follows that the contour lines of the loaded membrane are identical with the lines of shearing stress in the cross-section of the prism. Further the torsional rigidity of the prism can be represented by the volume contained between the surface of the loaded membrane and the plane of its edge. We have seen already in Article 216 that the torsional rigidity is given by the equation ^-//{(S-')'-(l")"}-* or, in terms of % we have = 2/i I [■^dxdy, g2^ g2^ since * vanishes at the edge and ^^^-t- ^^^-1-2=0. It follows that the volume m question is {p/4iiT) C. 225. Twisting couple. The couple can be evaluated from (6) of Article 216 when the function j> is known. We shall record the results in certain cases. * The analogy here described was pointed out by L. Prandtl, Phys. Zeitschr., Bd. 4 (1903), it affords a means of exhibiting to the eye the distribution of stress in a twisted prism. The method is further developed by A. A. GrifiBth and G. I. Taylor, Engineering, London, vol. 104 (1917), pp. 655, 699. 328 MAGNITUDE OF TWISTING COUPLE [CH. XIV (a) The circle. If a is the radius of the circle the twisting couple is ^fiTira* (15) (b) The ellipse. From the value of (j) in Article 221 (6) we find that the twisting couple is fjLT7ra'b'j{a' + ¥) (16) (c) The rectangle. From the result of Article 221 (c) we find for the twisting couple the formula /XT lab (a= + 6=) - ^T^ab {a- - b') + ^f^r^ (|)' [ [ja; ^ _ y g| dxdy, where <1> stands for the series . , (2n + 1) TTX . (2n + 1) ttw / >„ sinh '^ ^ — sin ^^ ^, ^ ^ ^ (-)" 26 26 nZoC^n + iy (2n + 1) ira cosn 2ft Taking one term* of the series, we have a term of the integral, viz. : (zl! ZL (2n + 1)'' cosh ((2/( + 1) 7ra/26} 26 j ^ ^J/{. sinh ^^^^-^ cos <^^^^ ,(2n+l)'rrx . (2?H-l)7r«l , , - y cosh 2ft sin ^ 2b^\ ^'^^y- Now /•" . , (2/! + 1) TT.r 26 r , (2n + 1) -Ka X sinh -r^ dx = t^t — — 2a cosh -^^ ^ J -a 26 (2m + 1) tt 26 26 2sinh(?^^±^' (2n + 1) TT 26 J -a 26 (2n + l)7r 26 /•* (2w + l)7r2/, 26 „, ,, J./"^^--^- ^^=(-2;m)^2(-l)», Hence the twisting couple is equal to |;.ra63 + f^)V.a63 2 ^J_— ^.ft.fiV v ^ tanh ^'"+^)"" ^'^ VW '^ „=o(2n + l)^ ^ W„ro(2n + l)' 26 Since 2 (2m + 1)-^ is 7rV96, we may write down the value of the twisting couple in the form 225, 226] TORSION of jeolotropio prism 329 The series in (17) has been evaluated by Saint- Venant for numerous values of the ratio a : b. When a > 36 it is very nearly constant, and the value of the twisting couple is neai-ly equal to lirabA -^-- (3-361) . For a square the couple is (2-2492) /ira*. The twisting couple was also calculated by Saint- Venant for a number of other forms of section. He found that the resistance of a prism to torsion is often very well expressed by replacing the section of the prism by an ellipse of the same area and the same moment of inertia*. The formula for the twisting couple in the case of an ellipse of area A and moment of inertia / is fn- A*/4fr^I. 226. Torsion of seolotropic prism. The theory which has been explained in Article 216 can be extended to a prism of seolotropic material when the normal section is a plane of symmetry of structure. Taking the axis of s to be parallel to the generators of the bounding surface, we have the strain- energy-function expressed in the form belonging to crystalline materials that correspond with the group Cg (Article 109). The displacement being expressed by the formulse (1), the stress-components that do not vanish are X^ and Y^, and these are given by the equations The equations of equilibrium are equivalent to the equation which must hold over the area of the cross-section.; and the condition that the bounding surface may be free from traction is satisfied if the equation C55 g| cos {X, v) -t- C44 g| cos (y, v) + C45 |g? cos (^, l;) -I- p| COS (y, v)^ =c^ycos{x, I/) — C44«cos(y, v) — Ci^ {x 00s {x, v)—ycos{y, v)} holds at all points of the bounding curve. Exactly in the same way as in the case of isotropy, we may prove that the differential equation and the boundary-condition are compatible, and that the tractions across a normal section are equivalent to a couple of moment T // j 644^^ -1- C65y2 _ 2645*^3^ -f C44^- ?^ - Cjsy g| -h 645 f;c g| - y ^H rfa; rfy. The analysis is simpUfled considerably in case 045=0. If we put L for <;44 and M for C55, the differential equation may be written 8:1:2 +^ 8y2 " ' and, ii f {x, y)=0 is the equation of the bounding curve, the boundary-condition may be written M^^+Lg--^=My^-Lxf-. ox ex dy ay '' ox oy We change the variables by putting /Z+M , /L+M ., , L+M Then <(>' satisfies the equation 2^{LM)' 3^' 8V_o * Saint- Venant, Paris, G. R., t. 88 (1879). 330 TORSION OF ^OLOTROPIC PRISM [CH. XIV The equation /(a;, y) = becomes F(.i/, y') = 0, where so that ^^-^f I ^^ ^-^^ / ^L _ and the boundary-condition is transformed into ex doc' Zy' dy' dx dy' ' which is ^ = w' cos {a/, v) - x! cos (V, c '), m if dv is the element of the normal to the transformed boundary. Thus ^ can be found for any boundary if 0' can be found for an orthographic projection of that boundary ; and the problem of finding <^' is the simple torsion problem which we considered before. As an example we may take a rectangular prism with boundaries given by x=±_a, y=±b. We should find that the formula for ^ is 6--XV+ /^^'^'l (-)" ^^^^ 3in ^^^ + ^^"^ ''"'^ 2bJM and that the twisting couple is expressed by the formula ifr.6s j^_w(iy I --J— ^tanhi ^"+'>;;-^^ l. This formula has been used by W. Voigt in his researches on the elastic constants of crystals. [See Article 113.] 226 A. Bar of varying circular section. When the twisted bar is isotropic and of circular section, but the radius of the circle is a function of the position of its centre on the axis of the bar, the. displacement of any point is directed at right angles to the axial plane passing through the point, just as in the case of a bar of uniform circular section. Let v denote this displa^cement. Then, using cylindrical polar coordinates r, 6, z, with an axis of z coinciding with the axis of the bar, we have the components of strain and stress expressed by the equations A A A - 9^ A dv V e„=0. eee=0, 6^^ = 13, 60^ = ^^, ejr=0, e^8 = g , for V is a function of z and r, but is independent of 0. The equations of equilibrium [Article 59 (i)] reduce to the single equation d^v Idv V d'v - — I -l- — = 0. or^ rdr r^ dz^ This equation may be written dr dr\rj\ dz I dz\r. 226, 226 AJ BAR OF VARYING CIRCULAR SECTION 331 showing that there exists a function y}r which has the properties expressed by the equations dr\rj dz ' dz \rj dr The function -^jr satisfies the differential equation obtained by eliminating v/r from these equations, viz. : Sr" r dr dz^ The stress-components, which do not vanish, are expressed by the equations "^ ~ r' dr ' "^^ " T^dz' and the condition that the bounding surface is free from traction takes the form i|r = const, at the boundary. The above theory is due to J. H. Michell*. It was re-discovered by A. Fbpplt, and further developed by F. A. WillersJ, who investigated, in particular, an approximate solution for a bar consisting of two portions, each portion being a circular cylinder, and the two portions having the same axis but' different radii. An obvious particular solution of the equation for ■v/f is ■^ = Ar*, where A is constant. By this solution the displacement and stress in a bar of uniform circular section are expressed in terms of ■'), by putting z=r'cosff, r=r'am6', 6='. The equation becomes If we assume for yjr an expression of the form where ■!/(•„ is a function of 6', we And that •>//■„ must satisfy the equation ^-3cotfl'^r+«(«-3)>^.=0; and if we then put yjrn^-am^d'.Xn, and' write fi' for cos 6', this equation becomes and a solution is X»=(l-/''')^2{^n-2(F')}, * London, Proc. Math. Soc, vol. 31, 1900, pp. 140, 141. t Milnchen, Akad. d. Wiss. Sitzungsber., Bd. 35, 1905, pp. 249, 504. J Zeitschr.f. Math. u. Phys., Bd. 55, 1907, p. 225. 332 LOCAL PERTURBATIONS DUE TO [CH. XIV where P„_2 denotes the zonal surface harmonic (Legendre's coefl&cient) of degree « — 2. Thus yp- can he of the form where 4 is a constant. The case where t/^ is of the form Ar* is included in this formula by putting 71 = 4. In all the solutions of this type i/^ is a rational function of z and r. Another method of obtaining particular solutions of the equation for •»//■ is to assume that TJr is of the form e*'".R, where iJ is a function of r. It then appears that R satisfies the equation so that we may write where .4 is a constant, and J^ {kr) denotes Bessel's function of order 2. 22^ B. Distribution of traction over terminal section. In the theory of torsion, developed in this Chapter, the twisting couple is supposed to be applied by means of tangential tractions exerted upoa the terminal sections, and these tractions are supposed to be distributed over the sections according to determinate laws. When the external forces, whose resultant is the twisting couple, are distributed in some other way over the terminal sections, or the neighbouring portions of the cylindrical boundary, the theory avails for the determination of the stress in all parts of the twisted bar except those near to the ends ; but near the ends there are " local perturbations." (Of Articles 89 and 133.) The nature of the local perturbations may be illustrated by means of the analysis in Article 226 A. It will be sufficient to examine the case of a circular cylinder of radius a, twisted by tractions of the type 6z distributed over the terminal section z = Q. We shall suppose that z is positive within the cylinder. Then a solution of the equation for i/r can be written 00 m = l where t, A^, A^,... are constants, and h^, k^,... are the roots, in order of increasing magnitude of the equation J^ (ka) = 0. The corresponding value of 6z at the section z = is given by the equation (^^)z=o = fJ.\rr+ 2 Ankn Ji (knr) [ »=i for ^. |; {r'J2 (kr)} = k {/; (kr) + ^ J, (Ar)| = kJ,(kr), where the accent denotes differentiation of the function J^ [kr) with respect to its argument kr, and J^ (kr) denotes Bessel's function of order 1. 226 A, 226 b] distribution of terminal forces 333 The equation J, (kr) = - J/ (kr) + ^^Ji (kr) shows that k^, k^, ... are the roots of the equation and the equation shows that Jyika) = ^JAka), Hence the twisting couple, which is I r {6z)z=o ^Trrdr, Jo is ^/j,'n-a*T, and the terms in /g i^nr) contribute nothing to this couple for any of the values of k which can occur. It is known* that an arbitrary function of r can be expanded, within the interval a > r > 0, in a series of the form where the k's are roots of the equation J-;{ka)IJ^{ka) = llka. Thus we see that the assumed formula for y^r can represent the effect of any forces of the type dz which are statically equivalent to the couple ^fiirO^r. The occurrence of the factors e"*»^ shows that the effects due to the distribu- tion of the forces constituting the couple, as distinguished from their resultant moment, diminish exponentially as the distance from the terminal section increases. The analysis of this Article was given eflfeotively by F. Purser, Dublin, Proc. R. Irish Acad., vol. 26, Sect. A, 1906, p. 54, afterwards in a more general form by O. Tedone, Roma, Ace. Line. Rend. (Ser. 5), t. 20, (Sem. 2), 1911, p. 617. The corresponding theory for twisting couple applied by means of tractions, exerted upon a portion of the cylindrical boundary, can be worked out by means of solutions of the equation for \//- of the form {A cos kz + B sin kz) r^ J^ {ikr). This theory was obtained effectively by another method by L. N. G. Filon, London, Phil. Trans. R. Soc. (Ser. A), vol. 198, 1902, p. 147, afterwards more completely by A. Timpe, Math. Ann., Bd. 71, 1912, p. 480. * See Lord Bayleigh, Theory of Sound, vol. 1, § 203. CHAPTER XV. THE BENDING OF A BEAM BY TERMINAL TRANSVERSE LOAD. 227. Stress in a bent beam. In Article 87 we described the state of stress in a cylinder or prism of any form of section held bent by terminal couples. The stress at a point consisted of longitudinal tension, or pressure, expressed by the formula tension = — Mxjl, where M is the bending moment, the plane of {y, z) contains the central-line, the axis of x is directed towards the centre of curvature, and I is the moment of inertia of the cross-section about an axis through its centroid at right angles to the plane of bending. In Article 95 we showed how an extension of this theory could be made to the problem of the bending of a rectangular beam, of small breadth, by terminal transverse load. We found that the requisite stress-system involved tangential traction on the cross-sections as well as longitudinal tensions and pressures, but that the requisite tension, or pressure, was determined in terms of the bending moment by the same formula as in the case of bending by terminal couples. This theory will now be generalized for a beam of any form of section*. Tangential tractions on the elements of the cross-sections imply equal tangential tractions, acting in the direction of the central-line, on elements of properly chosen longitudinal sections, the two tangential tractions at each point constituting a shearing stress. It is natural to expect that the stress-system which we seek to determine consists of longitudinal tensions, and pressures, determined as above, together with shearing stress, involving suitably directed tangential tractions on the elements of the cross-sections. We shall verify this antici- pation, and shall show that there is one, and only one, distribution of shearing stress by means of which the problem can be solved. 228. Statement of the problem. To fix ideas we take the central-line of the beam to be horizontal, and one end of it to be fixed, and we suppose that forces are applied to the cross- * The theory is due to Saint- Venant. See Introduction, footnote 50, and p. 20. 227, 228] SHEARING STRESS IN A BENT BEAM 335 section through this end so as to keep the beam in a nearly horizontal position, and that forces are applied to the cross-section containing the other end in such a way as to be statically equivalent to a vertical load W acting in a line through the centroid of the section. We take the origin at the fixed end, and the axis of z along the central-line, and we draw the axis of x vertically downwards. Further we suppose that the axes of x and y are parallel to the principal axes of inertia of the cross-sections at their centroids. We denote the length of the beam by I, and suppose the material to be isotropic. We consider the case in which there are no body forces and no tractions on the cylindrical bounding surface. t^ % w Pig. 25. The bending moment at the cross-section distant z from the fixed end is W{1 — z). We assume that the tension on any element of this section is given by the equation Z^ = -W(l-z)xlI, (1) where I stands for the integral I x^dxdy taken over the area of the cross- section. We assume that the stress consists of this tension Z^ and shearing stress having components Xz and F^, so that the stress-components X^, Yy, Xy vanish ; and we seek to determine the components of shearing stress X^, and Fz. Two of the equations of equilibrium become dX^/dz = 0, dYJdz = 0, and it follows that Xz and Fj must be independent of z. The third of the equations of equilibrium becomes ^ —^ f-f^?- (^) The condition that the cylindrical bounding surface is free from traction is ^^cos(«, I') -f^cos (y, i/) = (3) The problem before us is to determine Xz and Yz as functions of x and y in accordance with the following conditions :— (i) The differential equation (2) is satisfied at all points of the cross- section of the beam. (ii) The condition (3) is satisfied at all points of the bounding curve of this section. , 336 DETERMINATION OF THE SHEARING STRESS [CH. XV (iii) The tractions on the elements of area of the terminal cross-section {z = I) are statically equivalent to a force W, directed parallel to the axis of X, and acting at the centroid of the section. (iv) The stress-system in which Xx=Yy= Xy=: 0, Z^ is given by (1), and Xz, Yz satisfy the conditions already stated, is such that the conditions of compatibility of strain-components (Article 17) are satisfied. 229. Necessary type of shearing stress. The assumed stress-system satisfies the equations X,= Yy=Xy = 0, Z,= -W{l-z)xlI, ^^ = ^ = 0, and consequently the strain-components satisfy the equations W{l-z)x _ de^_dey,_ fizz— pr ) "xx — ^yy— "t!zz> ''xy—'J, "^ — "^ — — "j where E and o- denote Young's modulus and Poisson's ratio for the material. The equations of compatibility of the type O ^yy O ^zz _ 6j/2 dz^ dy^ dydz are satisfied identically, as also is the equation dxdy dz \ dx dy dz . The remaining equations of compatibility of this type become l_ /djyz _ dezx \ ^ Q A (^ _ ^^\ ^ _ 2(7 ly dx V dx dy.) ' dy\dx dy J ~ El ' From these equations we deduce the equation dx dy " EI y' where 2t is a constant of integration ; and from this equation it follows that Byz and Bzx can be expressed in the forms eyz-rx + ^, e,x^-ry+g + -^f (4) where 0o is a function of x and y. On substituting from these equations in the formulse X^^fiez^ and Yz = /J^yz, and using the relation /j, = ^E/{1 + a), we see that equation (2) takes the form 8^0 a^ 2(l+a)Jf _„ dxl^'^df^ EI ^~"' 228, 229] IN A BENT BEAM 337 and condition (3) takes the form -^ = T{y cos {x, v)-x cos {y, v)] - ^ 2/' cos {x, v). These relations are simplified by putting W <\>, = T--^^[x + l is the torsion function for the section (Article 216), and % is a function which satisfies the equation S+p=» <") at all points of a cross-section, and the condition ^ = - {^o-a;^ + (1 - \a) y^} cos {x, v)-{2 + and x must be one-valued. Cf. Article 222 (iii) supra. We have therefore shown that the problem stated in Article 228 admits of one, and only one, solution. 230. Formulae for the displacement. The displacement can be deduced from the strain without determihing the forms of (p and x- The details of the work are as follows : — We have the equation dw _ W{l-z)x &■" m ' from which we deduce the equation Wl 1 W '^^ " El''''^2 ^/'^^' + '^'' (^^) where ^' is a function of x and y. Again, we have the equations du Wl W \ dz EI^ %Ef ^y^^y^cx ox', of which the second is obtained from (11) and the second of (4). These two equations are compatible if ox' El 229, 230] IN A BENT BEAM Again, we have the equations 339 dv Wl W and these are compatible if dz Zy hy ' j Oil O'O Further, by differentiating the left-hand member of the equation r— + ^=0 with respect to z, we obtain the equation -^-^— + ^ — y = 0. The three equations for 0o - 0' show that we must have W 4)' = ' in the equations for dujdz and dv/dz, we obtain the equations ^f^=-ry + f^{lz-iz^-i 'El w = T<^ - -^ {X {IZ- i Z') + X +, "'f'] -_^^i^+ V^ * I ■ \ ...(12) 22—2 340 SOLUTION OF THE PROBLEM OF FLEXURE [CH. XV in which a, ^, 7, a', /3', 7' are constants of integration. These equations give the most general possible form for the displacement (u, v, w) when the stress is determined by the conditions stated in Article 228. The terms of (12) that contain a, /3, 7, a', /3', 7' represent a displacement which would be possible in a rigid body, and these constants are to be determined by imposing some conditions of fixity at the origin. (Cf. Article 18.) We have supposed that the origin is fixed, and we must therefore have a' = 0, /3' = 0. We shall, in general, suppose that the additive constants in the expressions for a. In case b = a this result reduces to that already found for the circle. (d) Confocal ellipses. By an analysis similar to the above the problem might be solved for a section bounded by two confocal ellipses. The result could not be expressed rationally in terms of x and y. Taking |o and ^i to be the values of | which correspond with the outer and inner boundaries, and writing c for {ofi — b")^, we may show that X = <^ cos ?; [(J - J a-) cosh ^ - (i + i tr) {cosh |o cosh ^i cosh (|o 4- ^i) cosh ^ - sinh ^0 sinh ^^ sinh (^o + li) sinh ^}] + o3cos3,[Acosh3|-(^ + 3^.) -'"'^g°'=°^^^^/TJ^)-^^^^^^^^ (16) (e) The rectangle. The equations of the boundaries are x= ±a, y=±b. The boundary- condition at a; = + a is g = -{|^a^ + (l_^o.)2/=}, (b>y>-b). The boundary-condition at y = + 6 is 9y ^ = + (2 + 0-) 6a;, {a>x>- a). We introduce a new function %' by the equation x' = X-i(2 + ^)(a=^-3a;2/0 (17) Then x is a plane harmonic function within the rectangle, dy^jdy vanishes at y=±b, and the condition at « = + a becomes ^^ = -(l + a)a^ + af. Now when b>y> — b the function y'' can be expanded in a Fourier's series as follows : — ,.^V^^v(zI%os!7. d it' n=i n" Hence ^ can be expressed in the form . , mrx TT^ n=i n , rnra b cosh -T- and, by means of this and (17), t^; can be written down. (/) Additional results. The results for the circle and ellipse are included in the formula 281, 232] FOE CERTAIN FORMS OF SECTION 843 the solution for the ellipse was first found by adjusting the constants A and B of this formula, and several other examples of the same method were discussed by Saint- Venant. Among sections for which the problem is solved by this formula we may note the curve of which the ordinate is given by the equation y=±h\{\-x^la^)''\, {x>-a). The corresponding function x is X=-a2^+i(l-itr)(^3_3^^2). When and % have been found. In each of the particular cases that we have solved T vanishes. This is due to the symmetry of the sections. An example of an unsymmetrical form of section for which the analysis could be worked out is shown in Fig. 28, which represents the cross-section of a hollow tube with a cavity placed excentrically. (Cf. Article 222, Result iii.) (f) Anticlastic curvature. The terms of u, v, as given by (12), which depend on x, y, but not on t, represent changes of shape of the 346 DISPLACEMENT AND SHEARING STRESS [CH. XV cross-sections in their own planes. These changes are of the same kind as those described in Article 88. It follows that the neutral plane is deformed into an anticlastic surface. The strained central-line is one of the lines of curvature of this surface ; the correspondiug centres of curvature are below the neutral plane, and the corresponding radii of curvature are expressed by the formula EIj W {I — z). The other centre of curvature of the surface, at any point of the central-hne, is above the neutral plane; and the corre- sponding radii of curvature are expressed by the formula EljaWiJ—z). (g) Distortion of the cross-sections into curved surfaces. The expression for w may be written W w=T — jrf * {iz — \z^) — ^a; + SoX — EI X- das + xy^ ....(26) The term t(j> corresponds with the twisting of the beam by the load, and we know that it represents a distortion of the cross-sections into curved surfaces. The terms — x{W {Iz — ^z'')/EI + /3] represent a displacement by which the cross-sections become at right angles to the strained central-line. The term s„x represents a displacement by which each cross-section is turned back, towards the central-line, through an angle So. as explained in (c) above. The remaining terms in W/EI represent a distortion of the cross-sections into curved surfaces, independent of that which depends upon t^. If we construct the surface which is given by the equation ' = -m{^-'(Mh'^]^''t'' (2^) and suppose it to be placed so that its tangent plane at the origin coincides with the tangent plane of a strained cross- section at its centroid, the strained cross-section will coincide with this surface. 232, 233] IN A BENT BEAM 347 In the case of a circular boundary the value of the right-hand member of (27) is and the contour lines of the strained cross-section are found by equating this expression to a constant. Some of these lines are traced in Fig. 29. 233. Distribution of shearing stress. The importance of the transverse component Y^ of the tangential traction on the cross-sections may be seen in the case of the elliptic boundary. When a is large compared with h, the maximum value of Y^ is small compared with that of Xg ; as the^ratio of 6 to a increases, the ratio of the maximum of Y^ to that of Xj increases ; and, when h is large compared with a, the maximum of Fj is large compared with that of X;,. Thus the importance of Fj increases as the shape of the beam approaches to that of a plank. We may illustrate graphically the distribution of tangential traction on the cross-sections by tracing curves, which are such that the tangent to any one of them at any point is in the direction of the line of action of the tangential traction at the point. As in Article 219, these curves may be called " lines of shearing stress." The differential equation of the family of curves is da,/X,= dy/Y„ (28) or ^^+{2+a)xy^dx-^^ + ^aay> + {l-^a)f^dy = 0. Since dXJdx + dYijdy is not equal to zero, the magnitude of the shearing stress is not measured by the closeness of neighbouring curves of the family. As an example we may consider the case of the elliptic boundary. The differential equation is -[3^,{2»'(i+-')+»n-g^,{(i'+j»)""+a-i')»i-i''(»'-rt-j']*, and this may be expressed in the form 2^.^/(1 .,.cr)aHcr&2}--{2(l-fo-)a2-H 62} -H^' {2 (l-|---^,{i/{h-iz^)+x'+yx^}. .(30) 233, 234] IN A BENT AND TWISTED BEAM 349 When the direction of the load is not that of one of the principal axes of the cross- sections at their centroids, we may resolve the load, P say, into components W and W parallel to the axes of x and y. The solution is to be obtained by combining the solutions given in Articles 229, 230 with that given here. Omitting displacements which would be possible in a rigid body we deduce from the expressions (12) and (30) the equations of the strained central-line in the form W W and this line is therefore a plane curve in the plane W'xir= Wyll. The neutral plane is determined by the equation e^=0, and, since W W «^'=-^j(J'-^)x-^,{l-z)y, this is the plane Wxjl+ W'y/r = 0. The neutral plane is therefore at right angles to the plane of bending. The load plane is given by the equa,tion ylx= W'l W. Since / and /' are respectively the moments of inertia of the cross-section about the axes of y and x, the result may be expressed in the form : — The traces of the load plane and the neutral plane on the cross-section are conjugate diameters of the ellipse of inertia of the cross-section at its ceutroid*. (6) Combined strain. We may write down the solution of the problem of a beam held bent by terminal couple about any axis in the plane of its cross-section, by means of the results given in Article 87 ; we have merely to combine the results for two component couples about the principal axes of the cross-section at its centroid. By combining the solution of the problems of extension by terminal tractive load [Articles 69 and 70 (A)], of torsion (Chapter xiv.), of bending by couples, and of bending by terminal transverse load, we may obtain the state of stress or strain in a beam deformed by forces applied at its ends alone in such a way as to be statically equivalent to any given resultant and resultant moment. In all these solutions the stress-components denoted by J^^, Yy, Xy vanish. As regards the strength of a beam to resist bending we may remark that, when the linear dimensions of the cross-section are small compared with the length, the most important of the stress-components is the longitudinal tension, and the most important of the strain-components is the longitudinal extension, and the greatest values are found in each case in the sections at which the bending moment is greatest, and at the points of these sections which are furthest from the neutral plane. The condition of safety for a bent beam can be expressed in the form : — The maximum bending moment must not exceed a certain limiting value. The condition of safety of a twisted prism was considered in Article 220. The quantity which must not, in this case, exceed a certain limiting value is the shear ; and this is generally greatest at those points of the boundary which are nearest to the central-line. When the beam is at the same time bent and twisted, the components of stress which are different from zero are the longitudinal tension Z^ due to bending and the shearing stresses X^ and T^. If the length of the beam is great compared with the linear dimensions of the cross-section the values of Z^ near the section = and the terms of X^ and T^ that depend upon twisting can be comparable with each other, and they are large compared * The result was given by Saint- Venant in the memoir on torsion of 1855. 350 THEORY OF COMBINED STRAIN [CH. XV with the terms of X, and I\ that are due to bending. For the purpose of an estimate of strength we might omit the shearing stresses and shearing strains that are due to bending, and take account of those only which are due to twisting. In any case in which the stress-components X^, V^, Z^ are different from zero and Xx, y^, Xy vanish, the principal stress-components can be found by observing that the stress-quadrio is of the form 2 (2Z,a; -+-2 F^y+Zj0) = const., and therefore one principal plane of stress at any point is the plane drawn parallel to the central-line to contain the direction of the resultant, at the point, of the tangential tractions on the cross-section. The normal traction on this plane vanishes, and the values of the two principal stresses which do not vanish are i^,±M-2'.H4(J'/+J7)]* (31) In any such case the strain-quadric is of the form ^\_-aZ,{x'+y''+z^) + {\ + a-)z{'iX,x+iY,y + Z,z)\==const., and the principal extensions are equal to -^. ^-^$^±w[^^^+4W+r,=^)]*, (32) ^E %E the first of these being the extension of a line at right angles to that principal plane of stress on which the normal traction vanishes. (c) ^olotropio material. The complexity of the problem of Article 228 is not essentially increased if the material of the beam is taken to be seolotropic, provided that the planes through any point, which are parallel to the principal planes, are planes of symmetry of structure. We suppose the axes of 30, y, z to be chosen in the .same way as in Article 228, and assume that the strain- energy-function has the form \{A, B, C, F, 0, H) {e,„ e„„, ej' + i {Le^^^^MeJ+Nexy^). We denote the Young's modulus of the material for tension in the direction of the axis of z by E, and we denote the Poisson's ratios which correspond respectively with con- tractions parallel to the axes of x and y and tension in the direction of the axis of z, by ay and 0-2. We assume a stress- system restricted by the equations IF, Xt —Ty — Xy — Q, Z^ = -{l-z)x. .(33) Then we may show that X, and Y^ necessarily have the forms : ,i>*(|-,)-^^f[|+,,^.?z^-.-^.. 2i ■]■ where and x are solutions of the same partial differential equation which respectively satisfy the following boundary conditions : — cos(^, v) M^S + cos (j/, v) L i = CCS {x,v) My- cos (i/,v)Lx, cos(^, .) J/g + cos(y, v)l'^ = -cos {x, .) J/^K^H-^"^"'"^^'"^ .(34) 2i - cos {y, v) {E- Ma-j) xy. ,.(35) 234, 235] CEiTiciSMS or certain methods 351 Further we may show that the displacement corresponding with the stress-system expressed by (33) and (34) necessarily has the form ; W ' V= TZ!l) + -^j.{l-z) ^^ ^y\ bx by J by\ bx by J bx by '. by^ '\ bx' bt bxby J ba? a=e„„<»' . 9%^(«) 9%V^ ^ ^ (15) .(16) by^ bxby J ba? by^ bxby In all these equations the terms containing z and the terms independent of z must vanish separately. The relations between components of stress and components of strain take such forms as E (e,/)0 + e»,<»') = X„'"0 + Z^'») - <7 (F^<"^ + F,<») + Z.^^z + ^,(»)), in which the terms that contain z, and those which are independent of z, on the two sides of the equations must be equated severally. Selecting first the terms in z, we observe that all the letters with index (1) satisfy the same equations as are satisfied by the same letters in. Article 237, and it follows that we may put e«™ = ei - KyX - K^y, p (\) — p (1) — _ „-« (1) p (1) = in which ei, «i, k-1, Ti are constants, and ^ is the torsion function for the cross-section. Again,, selecting the terms independent of z, we find from the first two of equations (12) .(17) /f,.r,,-,x,»,«,=//{.f-g'.'f!)- ,[iXJt^ZY^^^^^ \ bx ' by j] = ly {cos {x, v) XJ'i + cos {y, v) Xj,""} - x {cos {x, v) Xy^"^ + cos {y, v) Fj,'»'} ds, which vanishes by the first two of equations (13). Also we have by (17) JJ{a;F,« -yZ/)} dxdy = iMT,^\\^^ + f+x^^-y^^dxdy, 358 BEAM UNDER STRESS WHICH VARIES [CH. XVI where the integral on the right is the coefficient of /* in the expression for the torsional rigidity of the beam. It follows that Tj must vanish, and hence that Z^w and F^w vanish. This conclusion is otherwise evident ; for if t^ did not vanish we should have twist of variable amount tiZ maintained by tractions at the ends. The torsional couples at different sections could not then balance. By selecting the terms independent of z in the third of equations (12) and conditions (13) we find the differential equation dx dy and the boundary-condition Z2(°' cos {x, v) + Fj"" cos {y, v) = 0, which are inconsistent unless \Z^'-'Ulxdy=0. Since ^j"' = E{ei — k^x — K^y), this equation requires Ci to vanish. We may now rewrite equations (17) in the form e„(» =-K,x- K^y, e,,w = e,,,") = - o-e,,'", e^,,"' = e,^w = e^^™ = 0. . . .(18) Since X^"' and Fj'^' vanish, we find, by selecting the terms independent of z in the first two of equations (12) and conditions (13), that XJ>\ F^»",Xy<»> vanish and that e^^^"^ is a linear function of x and y. We may therefore put e^z'"' = 6„ - K,x - K^y, e^«" = ey,}"^ = - <7e^^<»', e»;„'»i = 0, (19) where eo, ko, «o' are constants. Equation (16) is satisfied identically. Further, by selecting the terms independent of z in the third of equations (12), and the third of conditions (13), and in equations (16), we find, as in Articles 229 and 234 (a), that ezj;'"' and ej/j'"' must have the forms o m (20) where t^ and x are the flexure functions for the cross-section, corresponding with bending in the planes of {x, z) and (y, z), and Tq is a constant. We have shown that, in the body with a cylindrical boundary, the most general state of stress consistent with the conditions that no forces are applied except at the ends, and that the stress-components are linear functions of 2, has the properties (i) that X^ and Y^ are independent of z, (ii) that X^, Vy, Xy vanish. Thus the only stress-component that 238, 239] UNIFORMLY ALONG ITS LENGTH 359 depends upon z is Z^ which is a linear function of z. Conversely, if there are no body forces and Xj, }"„, Z„ all vanish, the equations of equilibrium become Sz ' 3a ' 'fix dy dz ' and it follows from these that JC^ and Y^ are independent of z and that Z^ is a linear func- tion of z. Thus the condition that the stress varies uniformly along the beam is the same as the conditions that X,., Yy, Xy vanish*. The most general state of strain which is consistent with the conditions (i) that the stress varies uniformly along the beam, (ii) that no forces are applied to the beam except at the ends, consists of extension due to terminal tractive load, bending by transverse forces, and by couples, applied at the terminal sections, and torsion produced by couples applied to the same sections about axes coinciding with the central-line. The resultant force at any section has components parallel to the axes of x, y, z which are equal to — EIk^, —ETk-1, Eeo, where I = ji x^dxdy and F = i\ y^dxdy; and the resultant couple at any section has components about axes parallel to the axes of x, y which are equal to - EI' («„' + K^z), EI {Ko + K,Z), and a component about the axis of z which is equal to ,.njf[x^ + f + x^^-y^£)dxdy + /.«, jj I ^ ?^ - 2/ ll + (2 + 1<7) ^^y - (1 - ic,'=W" (36) Thus the constants k^, k^' are determined in terms of the load per unit of length. If the body forces and the surface tractions on the cylindrical bounding surface give rise to a couple about the axis of z, the moment of this couple is M/3 (a; F - yX) dxdy + \ (xY^ - yX^) ds, and from equations (32) and (34) we find that this expression is equal to \xY,<-''>-yX,<'y\dxdy. -//«■ * J. H. Michel], loc. cit., p. 359. t E. Almansi, Roma, Ace. Line. Rend. (Ser. 6), t. 10 (1901). 239-241] OF THE UNIFORMLY LOADED BEAM 363 On substituting iiej^'> for Z/^' and fiSyJ-''^ for Yp, and using the expressions given in (23) for eJ-'^'> and ej,j('>, we have an equation to determine tj. When no twisting couple is applied along the length of the beam, and the section is symmetrical with respect to the axes of x and y, Tj vanishes. The constants k^, k^, Ti depend, therefore, on the force- and couple- resultants of the load per unit of length. The terms of the solution which contain the remaining constants 6o, Kq. «o', «i, «/; to are the same as the terms of the complete solution of the problem of Article 238. These constants depend therefore on the force- and couple-resultants of the tractions applied to the terminal sections of the beam. Since the terms containing k^, «/, ti alone would involve the existence of tractions on the normal sections, the force- and couple-resultants on a terminal section must be expressed by adding the contributions due to the terms in k^, k^', Tj to the contributions evaluated at the end of Article 238. The remaining constants 6o, ... are then expressed in terms of the load per unit of length and the terminal forces and couples. When the functions r'2,K^z). The distribution over the cross-section of the tangential tractions X^ and Y^ which are statically equivalent to these resultants is the same as in Saint-Venant's solutions (Chapter xv.). When there is twist t„ -1- t-^z, the tractions X^ and Fj which accompany the twist are distributed over the cross-sections in the same way as in the torsion problem (Chapter xiv.). The stress-component Z^ is not equal to Eczz because the stress-components Xx, Yy are not zero, but the force- and couple-resultants of the tractions Z^ on the elements of a cross-section can be expressed in terms of the constants of the solution without solving the problem of plane strain. The resultant of the tractions Z^ is the resultant longitudinal tension. The moments of the tractions Z^ about axes drawn through the centroid of a cross-section parallel to the axes of y and x are the components about these axes of the bending moment at the section. To express the resultant longitudinal tension we observe that jjZzdxdy = \\z("Hxdy =^^ [£'e„(») -f a (X^<«) -f- F/«))] dxdy. Now we may write down the equations = I X [X^'»i COS («, v) + Zj,(»' cos {y, v)] ds + [[« (Z^'" + pX) dxdy. 241, 242] OF THE UNIFORMLY LOADED BEAM 365 The integral Fj,"" dxdy may be transformesl in the same way, and hence we find the formula [j^, dxdy=\\^ [£'e^(«) + + pF)] dxdy + a{{xX, + yY,)ds (39) Since the resultant longitudinal tension is the same at all sections, and is equal to the prescribed terminal tension, this equation determines the constant to- To express the bending moments, let M be the bending moment in the plane of {x, z). Then M=-jjxZ,dxdy, (40) and therefore we have dM dz = - J I a; (Z/) + 2zZ^^'>) dxdy = EI{k, + 2zk,). This equation shows that M is expressible in the form i/ = ^/(«;„ + «x^ + A:2^^) + const (41) In like manner we may show that the bending moment in the plane of (y, z) is expressible in the form EF (/to' + K-^z + /CaV) + const. We shall show immediately how the constants may be determined. 242. Relation between the curvature and the bending moment. We shall consider the case in which one end ^r = is held fixed, the other end z = l is free from traction, and the load is statically equivalent to a force W per unit of length acting at the centroid of the cross-section in the direction of the axis of «* The bending moment M is given by the equation M={W{l-z)\ (42) and the comparison of this equation with (41) gives the equations K, = -Wl/EI, K^ = \WIEI. (43) We observe that, if the constant added to the right-hand member of (41) were zero, the relation between the bending moment and the curvature would be the same as in uniform bending by terminal couples and in bending * The important case of a beam supported at the ends, and carrying a load W per unit of length, can be treated by compounding the solution for a beam with one end free, bent by the nniform load, with that for a beam bent by a terminal transverse load equal to - \Wl. 366 NATURE or THE SOLUTION OF THE PBOBLEM [CH. XVI by terminal load. The constant in question does not in general vanish. To determine it we observe that the value of M at ^ = is and therefore (44) Now we may write down the equations x{X^^o) + Yy^o^)dxdy /I i [^ {x^ - f) Z,'"' + xyXy^"^] + 1 {i {x^ - f) Xym + xyYy^^^\ (0). dxdy = \[\{x'-f)X^ + xyYJ\ds+^^\_\{x-'-f){pX+XP) + xy{pY+Y,''-^^)\dxdy. Hence we have the result M-EI{Kf,-\-K-i,Z + Ki z^) = - \\Ex {ej<''> + Kox) dxdy -a-\[^{x^- y'') X. + xyYJ] ds - /^ + iTZ^}, y=a(l-(r)xy, w=-2a(TXZ, and the curvature of the central-Hne is 2<7a. If we consider a sHce of the beam between two normal sections as made up of filaments having a direction transverse to that of the beam, and regard these filaments as bent by forces applied at their ends, it is clear that the central-line ol the beam must receive a curvature, arising from the contractions and extensions of the longitudinal filaments, in exactly the same way as transverse filaments of a beam bent by terminal load receive a curvature. The tendency to auticlastic curvature which we remarked in the case of a beam bent by terminal loads affords an explanation of the production, by distributed loads, of some curvature over and above that which is related in the ordinary way to the bending moment. This explanation suggests that the effect here discussed is likely to be most important in such structures as suspension bridges, where a load carried along the middle of the roadway is supported by tensions in rods attached at the sides. 243. Extension of the central-line. The fact that the central-line of a beam bent by transverse load is, in general, extended or contracted was noted long ago as a result of experiment*, and it is not difficult to see beforehand that such a result must be true. Consider, for example, the case of a beam of rectangular section loaded along the top. There must be pressure on any horizontal section increasing from zero at the lower surface to a finite value at the top. With this pressure there must be associated a contraction of the vertical filaments and an extension of the horizontal filaments. The value of the extension of the horizontal central-line is determined by means of the formula (39). Since the stress is not expressed completely by the vertical pressure, this extension is not expressed so simply as the above argument might lead us to infer. The result that fo^O may be otherwise expressed by saying that the neutral plane, if there is one, does not contain the central-line. In general the locus of the points at which e„ vanishes, or there is no longitudinal extension, might be called the " neutral-surface.'' If it is plane it is the neutral plane. 244. Illustrations of the theory. (a) Fonn of the solution of the problem, of plane strain. When the body force is the weight of the beam, and there are no surface tractions, we may make some progress with the solution of the problem of plane strain (Article 239) without finding x- In this case, putting X—g, F=0, we see that the solution of the stress-equations (32) can be expressed in the form^ J'y = g - \eJo) _ 2<,^ [x + (1 +ix^+Z.vy^)-\eJ'>)+y.K^{\+la-)a\v+i,i.) = -^ fj^KiO^ (51) The surface tractions arising from the terms in iK^a^x can be annulled by superposing the stress-system ^x' = 0, F;=-,.<2(l-(-JT)a%, Ay») = (52) The surface tractions arising from the terms in ix.a-Ki{a^ — Zxy'^) can be annulled by super- posing the stress-system XJ = iicTK^x (Ja;2 -%y'' + -i^a^), F; = ;ao-K2 « ( - A^^ + fy^ + 1«^). Ay<') = ,i,(rK2.y{-f,r2+j_(y2_a2)} (53) The stress-components Xj^', ly, Ay) are therefore determined, and thus the problem of plane strain is solved for a circular boundary. I fmd the following expressions for the stress-components in a circular cylinder bent by its own weight : — -V. = ^ [(5 -1- 2 40/7r, or (span/depth) > 4-25 nearly. When this condition is satisfied there are two such points. The positions of these points can be determined experimentally, since they are characterized by the absence of any doubly refractive property of the glass, and the actual and calculated positions were found to agree very closely. A general theory of two-dimensional problems of this character has been given by L. N. G. I^lon*. Among the problems solved by him is included that of a beam of infinite length to one side of which pressure is applied at one point. The components of displacement and of stress were expressed by means of definite integral, and the results are rather difficult to interpret. It is clear that, if the solution of this special problem could be obtained in a manageable form, the solution of such questions as that discussed by Stokes could be obtained by synthesis. Filon concluded from his work that Stokes's value for the horizontal tension requires correction, more especially in the lower half of the beam, but that his value for the vertical pressure is a good approxi- mation. As regards the question of the relation between the curvature and the bending moment, Filon concluded that the Bernoulli-Eulerian theorem is approximately verified, but that, in applying it to determine the deflexion due to a concentrated load, account ought to be taken of a term of the same kind as the so-called "additional deflexion due to shearing" [Article 235 (e)]. Consider for example a beam BG supported at both ends and carrying a concentrated load W at the middle point A (Fig. 33). Either part, AC or 44w JW. Y w Fig. 33. * London, Phil. Trans. B. Soc. (Ser. A), vol. 201 (1903). Eeference may also be made to a thesis by C. Eibi^re, Sur divers cas de la flexion des prismes rectangles, Bordeaux, 1888. 374 EFFECT OF SUEFACE LOADING OF BEAMS [CH. XVII AB, of the beam might be treated as a cantilever, fixed at A and bent by terminal load ^F acting upwards at the other end; but Saint-Venant's solution would not be strictly applicable to the parts AB or A C, for the cross-sections are distorted into curved surfaces which would not fit together at ^. In Saint-Venant's solution of the cantilever problem the central part of the cross- section at A is vertical, and the tangent to the central-line at A makes with the horizontal a certain small angle s^. [Article 232 (c).] Filon concluded from his solution that the deflexion of the centrally loaded beam may be determined approximately by the double cantilever method, provided that the central-line at the point of loading A is taken to be bent through a small angle, so that AB and AC are inclined upwards at the same small angle to the horizontal. He estimated this small angle as about fs^. The correction of the central deflexion which would be obtained in this way would be equivalent, in the case of a narrow rectangular beam, to increasing it by the fraction 45c?2/16?2 of itself, where I is the length of the span, and d is the depth of the beam. The correction is therefore not very important in a long beam. It must be understood that the theory here cited does not state that the central-line is bent through a small angle at the point immediately under the concentrated load. The exact expression for the displacement shows in fact that the direction is continuous at this point. What the theory states is that we may make a good approximation to the deflexion •by assuming the Bemoulli-Eulerian curvature-theorem — which is not exactly true — and at the same time assuming a discontinuity of direction of the central-line — which does not really occur. 245 a. Further investigations. Filon* has verified his theory experimentally by means of polarized light. The subject has been investigated in a simpler way by H. Lambt. He treats the problem as one of generalized plane stress (Article 94), and considers the case of a series of equal loads applied at a series of points, situated at regular intervals along the length of an infinite beam. He finds an expression for the deflexion consisting of three terms. The first term is identical with the deflexion given by the BernouUi-Eulerian theory. The additional deflexion expressed by the second term is of the order d^ja^ as compared with that expressed by the first term, d denoting the depth of the beam, and a the distance between consecutive load-points ; and this additional deflexion is represented by a zig-zag line whose successive straight portions make very obtuse angles with one another at the load-points. The third term is very small except in the immediate neighbourhood of the load-points, where it has the effect of rounding off' the angles of the zig-zag. Lamb concludes that the Bemoulli- Eulerian theory is " entitled to considerable respect." The matter has been discussed from a different point of view by J. DougallJ. He considers an infinite circular cylinder to which external forces are applied in any manner, and finds the solution for concentrated force at any point, either within the cylinder or on the surface. He shows that the particular solutions of which this general solution is composed fall into two distinct classes. The first class consists of Saint-Venant's six solutions answering to simple extension, bending by terminal couples, torsion, and bending * Phil. Man. (Ser. 6), vol. 23, 1912, p. 63. t Atti d. IV congr. internazionale d. matematici, t. 3, Kome 1909, p. 12. X Edinburgh, R. Soc. Trans., vol. 49, 1914, p. 895. 245, 246] CONTINUOUS BEAMS 375 by terminal transverse load, along with displacements possible in a rigid body. The solutions of the second class are defined in terms of harmonic fmictions of the type e^'l" Jn ifirla) cos n{6- 6^), where a is the radius of the cylinder, r, 6, z are cylindrical coordinates referred to the axis of the cylinder as axis of 2, /3 is a root of a certain transcendental equation independent of a, n is an integer, and J'„ the symbol of a Bessel's function of order n. The modes of equilibrium expressed by the solutions of the first class are described as " permanent free modes," those expressed by solutions of the second class as " transitory free modes." The distinction between permanent and transitory modes had been arrived at by Dougall in an investigation concerning the theory of elastic plates, cited in Article 313 infra, and a general account of them was given by him in Proc. of the fifth international Congress of Mathema- ticians, Cambridge 1913, p. 328. The occurrence of the factor sSV" in the solutions expressing transitory modes indicates the nature of these modes as local perturbations. (Cf. Article 226b supra.) The permanent modes answering to displacements possible in a rigid body are required for fitting together the solutions on the two sides of a section to which forces are applied. The general conclusion to be drawn from Dougall's work is favourable to the Bernoulli -Eulerian theory. Dougall indicates the extension of his methods to cylinders of sections other than circular. The analysis for a circular cylinder has also been discussed by O. Tedone, Roma, Aoc. Line. Rend. (Ser. 5), t. 13 (Sem. 1), 1904, p. 232, and t. 21 (Sem. 1), 1912, p. 384. 246. The problem of continuous beams *- In what follows we shall develope the consequences of assuming the Bernoulli-Eulerian curvature-theorem to hold in the case of a long beam, of small depth and breadth, resting on two or more supports at the same level, and bent by transverse loads distributed in various ways. We shall take the beam to be slightly bent in a principal plane. We take an origin anywhere in the line of the supports, and draw the axis of x horizontally to the right through the supports, and the axis of y vertically downwards. The curvature is expressed with sufficient approximation by (Py/dx\ The tractions exerted across a normal section of the beam, by the parts for which x is greater than it is at the section upon the parts for which x is less, are statically equivalent to a shearing force N, directed parallel to the axis of y, and a couple G in the plane of (x, y). The conditions of rigid-body equilibrium of a short length Aa; of the beam between two normal sections yield the equation ^+i\^=0 (1) ax The couple G is taken to be expressed by the equation "-^S <^> where B is the product of Young's modulus for the material and the moment of inertia of a normal section about an axis through its centroid at right * The theory was initiated by Navier. See Introduction, p. 22. Special cases have been dis- cussed by many writers, among whom we may mention Weyraueh, Aufgaben zur Theorie elastischer Korper, Leipzig, 1885. 376 CONTINUOUS BEAMS [CH. XVII angles to the plane of {x, y)*. The senses of the force and couple, estimated as above, are indicated in Fig. 34. Except in estimating B no account is taken of the breadth or depth of the beam. G N Fig. 34. In the problems that we shall consider the points of support will be taken to be at the same level. At these points the condition 2/ = must be satisfied. At a free end of the beam the conditions iV = 0, G = must be satisfied. At an end which rests freely on a support (or a "supported" end) the conditions are y = 0, G = 0. At an end which is " built-in " (encastre) the direction of the central-line may be taken to be prescribed f. In the problems that we shall solve it will be taken to be horizontal. The displace- ment y is to be determined by equating the flexural couple G at any section, of which the centroid is P, to the sum of the moments about P of all the forces which act upon any portion of the beam, terminated towards the left at the sectionj. This method yields a differential equation for y, and the constants of integration are to be determined by the above special conditions. The expressions for y as a function of x are not the same in the two portions of the beam separated by a point at which there is a concentrated load, or by a point of support, but these expressions must have the same value at the point ; in other words, the displacement y is continuous in passing through the point. We shall assume also that the direction of the central-line, or dyjdx, is continuous in passing through such a point. Equations (1) and (2) show that the curvature, estimated as (Pyjdx^, is continuous in passing through the point. The difference of the shearing forces N calculated from the displacements on the two sides of the point must balance the concen- trated load, or the pressure of the support; and thus the shearing force, and therefore also (Py/day', is discontinuous at such a point. 247. Single span. We consider first a number of cases in which there are two points of support situated at the ends of the beam. In all these cases we denote the length of the span between the supports by I. * B is often called the " flexural rigidity." t Such an end is often described as ' ' clamped. ' ' J This is, of course, the same as the sum of the moments, with reversed signs, of all the forces which act upon any portion of the beam terminated towards the right at the section. 246, 247] EFFECT OF LOAD ON ONE SPAN (a) Terminal forces and couples. Z77 AY M„ M: Fig. 35. Let the beam be subjected to forces Fand couples M^ and M^ at the ends A and B. The forces F must be equal and opposite, and, when the senses are those indicated in Fig. 35, they must be expressible in terms of l/i and M^ by the equation The bending moment at any section a; is {I — x)Y + M^, ov Mo(l-x)/l + M,x/l. The equation of equilibrium is accordingly Integrating this equation, and determining the constants of integration so that y may vanish at « = and a,t x = l, we find that the deflexion is given by the following equation : > By = -^l-'a)(l-x){M„{2l-x) + M^(l + x)} (3) The deflexion given by this equation may be described as "due to the couples at the ends of the span.'' (b) Uniform load. Supported ends. z\ "Z\ Fig. 36. Taking w to be the weight per unit of length of the beam, we observe that the pressures on the supports are each of them equal to ^wl. The moment about any point P of the weight of the part BP of the beam is ^w(l — w^), and therefore the bending moment at P, estimated in the sense already explained, is the sum of this moment and —^wl{l — x), or it is — ^wx (I — x). 378 EFFECT OF LOAD [CH. XVII The equation of equilibrium is accordingly Integrating this equation, and determining the constants of integration so that y may vanish at a; = and at a; = Z, we find the equation By = ^wx{l - x){P + x{l - x)] (4) If we refer to the middle point of the span as origin, by putting x=\l + x', we find By = ^w{\P-x''){^^P-x''). (c) Uniform load. Built-in ends. The solution is to be obtained by adding to the solution in case (6) a solution of case (a) adjusted so that dyjdx may vanish at :r=0 and x = l. It is clear from sj'mmetry that J/j = .1/i) and T= 0. We have therefore 'By=^.v{l-x){l^-\-lx-x^)-lM^i.l-x\ where M is written for J/g or M-^. The terminal conditions give Jf = ^wP, and the equation for the deflexion becomes B2/ = :^wx^{l-xf, or, referred to the middle point of the span as origin of x', it becomes By=iiw(iP-x'^y. (d) Concentrated load. Supported ends. W/l W0 Z.\ "ZS w Fig. 37. Let a load W be concentrated at a point Q in AB, at which x = ^. We shall write |' for l — ^, so that AQ = ^ and BQ = ^'. The pressures on the supports A and B are equal to W^'/l and W^/l respectively. The bending moment at any point in AQ, where f >«> 0, is - W^'x/l; and the bending moment at any point in BQ, where I > x > ^, is — W^ (I - x)/l. The equations of equilibrium are accordingly d'y inAQ B^ = -^ F|' dx' I oe, -^Q ^di = d'y _ W^ I (l-x). 247] ON ONE SPAN 379 We integrate these in the forms B (?/ - a; tan a) = - ^ Z-' Ff V, B{y-(l- a;)tan/3] = - i l-'W^{l - xf, where tana and tan/3 are the downward slopes of the central-line at the points A and B. The conditions of continuity of y and dyjdx at Q are B |; tan a - ^ Z-i Tf f f = Bf ' tan /3 - ^ Z"' F^p, B tan a - ^ ?-i TTf f = - B tan jS + i Z"' TFf |'=- These equations give B tan a = i l-^ W^^ (^ + 2r), B tan /3 = J Z-^ If ^f (2? + f )■ Hence in J.Q, where f > a; > 0, we have B2/ = iZ-Fr|?(? + 2r)^f-*i, (5) and in J5Q, where l>x>^, we have By=ii-F^{r(2r + n(^-^)-(Z-^)n (6) We observe that the deflexion at any point P when the load is at Q is equal to the deflexion at Q when the same load is at P. The central deflexion due to the weight of the beam, as determined by the solution of case (6), is the same as that due to | of the weight concentrated at the middle of the span. (e) Concentrated load. Built-in ends. Fig. 38. To the values of By given in (5) and (6) we have to add the value of "By given in (3), and determine the constants M(, and J/j by the conditions that dyjdx vanishes at i!;=0 and at ^•=i!. We find from which M^=W^^'^ll\ M^=Wi^'jl\ Hence in AQ, where |>a;>0, we have and in BQ, where l>x>^, we have ■By^\l-^Wp{l-xY{Z^'.v-^{l-x)}. 380 THE THEOREM OF [CH. XVII We notice that the deflexion at P when the load is at Q is the same as the deflexion at Q when the same load is at P. The points of inflexion are given by cPyldx'^=Q, and we find that there is an inflexion at P^ in AQ where APi = AQ . ABI{ZAQ + Bq). In like manner there is an inflexion at P^ in BQ where BP^ = BQ.ABI{ZBQ+AQ). The point where the central-line is horizontal is given by dyldx=0. If such a point is in AQ it must be at a distance from A equal to twice APi, and for this to happen AQ must be >BQ. Conversely, if AQrAa'' + B^, where A and B are constants, and a and (3 are the roots of the quadratic x^ + ix+\=0, or we have a=-2+V3, )3 = -2-V3. The constants A and B are to be determined from the values of M &t the first and last supports. (c) Uniform load on each span. Let Wj^g denote the load per unit of length on the span AB, and Wbo that on BC. Then we find, in the same way as in case (a), the equation of three moments in the form 382 GRAPHIC METHOD OF SOLUTION OF THE [CH. XVII (d) Concentrated load on one span. Let a load W be concentrated at a point Q in BC given by a;=f The deflexion in AB is given, in accordance vi^ith the results of Article 247 (a), by the equation By = -i(x-a){b-a:)[M^{2h-x-a) + MB(b + x-2a)]/{b-a), and that in BQ is given by By = iW[i^-b)(c-^)(2c-b-^)(x-b)-(c-^)ix-by]/ic-b) -i{x-b){c-x){Ms(2c-x-b) + Mo(c + x-2b)}/{c-b). The condition of continuity of dyjdx at a; = 6 is ^{M^+mB){b-a) = ^W{^-b){c-^){2c-b-m<^-^)-^(:'^^B + Mc){c-h), and the equation of three moments for A, B, C is therefore Iab{Ma + -^Ms) + ha {'2Mb + Mc) = WIbqIqc{1 + IqcIIbc), (8) where Ibq and Iqc are the distances of Q from B and C. In like manner if D is the next support beyond G, the equation of three moments for B, C, D is he {Mb + 2Mc) + lci> {2Mc + Mj))= WIjbqIqc (1 + IsQlhc) (9) 249. Graphic method of solution of the problem of continuous beams*. The equation of equilibrium (2), viz. B j^ = G, is of the same form as the equation determining the curve assumed by a loaded string or chain, when the load per imit length of the horizontal projection is proportional to - G. For, if T denotes the tension of the string, m the load per unit length of the horizontal projection, and ds the element of arc of the catenary curve, the equations of equilibrium, referred to axes drawn in the same way as in Article 246, are ,dx , d fmdy\ . dx T^ = const. = Tsay, -^r^j + m- = 0, and these lead, by elimination of T, to the equation It follows that the form of the curve assumed by the central-line of the beam in any span is the same as that of a catenary or funicular curve determined by forces proportional to GBx on any length Bx of the span, provided that the funicular is made to pass through the ends of the span. The forces GSx are to be directed upwards or downwards according as G is positive or negative. * The method is due to Mohr. See Introduction, footnote 99. 248, 249] PROBLEM OF CONTINUOUS BEAMS 383 The tangents of such a funicular at the ends of a span can be determined without finding the funicular, for they depend only on the statical resultant and moment of the fictitious forces GSx. To see this we take the ends of the span to be a; = and w=l, and integrate the equation (2) in the forms dx -y- X =rdx, (^-^)| + 2/ I {I - on) :s '^^' G and hence we obtain the eqxiation ax ^ G , x~ ax- B ■(\l-x) J X from which it follows that dxJo -f J (l-x)G dx, dx, ' scG IB dx. VZ \axj I These values depend only on the resultant and resultant moment of the forces Ghx, and therefore the direction of the central-line of the beam at the ends of the span would be determined by drawing the funicular, not for the forces Ghx, but for a statically equivalent system of forces. The flexural couple G at any point of a span AB may be found by adding the couple calculated from the bending moments at the ends, when there is no load on the span, to the couple calculated from the load on the span, when the ends are "supported." The bending moment due to the couples at the ends of the span is represented graphically by the ordinates of the line A'B' in Fig. 40, where AA' and BB' represent on any suitable scale the bending moments at A and B. The bending moment due to AG A' Fig. 40. Fig. 41. uniform load on the span is equal to - ^wx {I - x), as in Article 247 (b), and it may be represented by the ordinates of a parabola as in Fig. 41. The bending moment due to a concentrated load is equal to — Wx (l - ^)/l, when ^>x>0, and to — W{l-x) ^/l, when l>x>^,asm Article 247 (d); and it may be represented by the ordinate of a broken line as in Fig 42. The 384 GRAPHIC METHOD OF SOLUTION OF THE [CH. XVll bending moment due to the load on the span may be represented in a general way by the ordinate of the thick line in Fig. 43. A',.:, B f c i . represented by the area of the triangle AA'B, acting upwards through that point of trisection g of AB which is nearer to A, (ii) a force c/)', represented by the area of the triangle A' BE, acting upwards through the other point of trisection g of AB, (iii) a force F^ represented by the area contained between AB and the thick line in Fig. 43, acting downwards through the centroid of this area. We take the line of action of F to meet AB in the point G. When the load on the span is uniform, F = ^wP, and G is at the middle point of AB. When there is an isolated load, F= | W^ (l - ^), and (r is at a distance from A equal to ^{1 + f). The forces F and the points G are known for each span, and the points g, g are known also. The forces ^, (/>' are unknown, since they are propor- tional to the bending moments at the supports, but these forces are connected by certain relations. Let A^, A^,... denote the supports in order, let ^i, ^Z, F-^ denote the equivalent system of forces for the first span A^A-^^, and so on. Let Ifo, il/i, M^,... denote the bending moments at the supports. Then we observe, for example, that 0/ : ^2 = i/i.J.o4i : iHfj. ^iJ.2, and therefore the ratio (^1' : ^2 is known. Similarly the ratio ^2' : <^3 is known, and so on. If the forces <^, <^', as well as F, were known for any span, we could construct a funicular polygon for them of which the extreme sides could be made to pass through the ends of the span. Since the direction of the central-line of the beam is continuous at the points of support, the extreme sides of the funiculars which pass through the common extremity of two consecutive spans are in the same straight line. The various funicular polygons belonging to the different spans form therefore a single funicular polygon for the system of forces consisting of all the forces ^, ^', F. 250. Development of the graphic method. The above results enable us to construct the funicular just described, and to determine the forces ^, or the bending moments at the supports, when the 24i), 250] PROBLEM OF CONTINUOUS BEAMS 385 bending moments at the first and last supports are given. We consider the case where these two bending moments are zero*, or the ends of the beam are "supported." We denote the sides of the funicular by 1, 2, 3, ... so that the sides 1, 3, 6,... pass through the supports A^, A^, A^,... . Fig. 44. We consider the triangle formed by the sides 2, 3, 4. Two of its vertices lie on fixed lines, viz.: the verticals through ^r/ and g^. The third vertex V^ also lies on a fixed line. For the side 3 could be kept in equilibrium by the forces (/)i' and 02 and the tensions in the sides 2, 4, and therefore Fj is on the line of action of the resultant of 0/ and ^a; but this line is the vertical through the point Oi, where ai5ra=^i5'i' and a^gi = A^g^, for (pi : 02=-4i5'i' : -Ai5'2- Again, the point Ca where the side 2 meets the vertical through A,, is determined by the condition that the triangle formed by the sides 1 and 2 and the line A^C^ is a triangle of forces for the point of intersection of the sides 1 and 2, and A0C.2 represents the known force i\ on the scale on which we represent forces by lines. Since the vertices of the triangle formed by the sides 2, 3, 4 lie on three fixed parallel lines, and the sides 2 and 3 pass through the fixed points G2 and A^, the side 4 passes through a fixed point Ci, which can be constructed by drawing any two triangles to satisfy the stated conditions. In the above the point Cj may be taken arbitrarily, but, when it is chosen, A^Gi represents the constant horizontal component of the tension in the sides of the funicular on the same scale as that on which AoC^ represents the force Fj. We may show in the same way that the vertices of the triangle formed by the sides n, 6, 7 lie on three fixed vertical lines, and that its sides pass through three fixed points. The vertical on which the intersection Fg of • The sketch of the graphic method given in the text is not intended to be complete. For further details the reader is referred to M. L6vy, loc. cit. p. B80. A paper by Perry and Ayrton in London, Proc. R. Soc, vol. 29 (1879), may also be consulted. The memoir by Canevazzi cited in the Introduction, footnote 99, contains a very luminous account of the theory. L. E. 25 386 GRAPHIC METHOD FOR CONTINUOUS BEAMS [CH. XVII the lines 5 and 7 lies passes through the point a^, where a^ffs = A^gi and a^gl = A.^g^. The fixed point G^, through which the side 5 passes, is on the vertical through C,, and at such a distance from Cj that this vertical and the sides 4 and 5 make up a triangle of forces for the point of intersection of the sides 4 and 5. The line CjCj then represents the force ^2 on a certain scale, which is not the same as the scale on which A^G^ represents ^1, for the horizontal projection of GJ^Jt, represents the constant horizontal component tension in the funicular on the scale on which GJu^ represents F-:,. Since C4 is known, the ratio of scales in question is determined, and Cj is therefore determined. The side 6 passes through the fixed point A^, and the fixed piiint C^ through which the side 7 passes can be constructed in the same way as C'i was constructed. In this way we construct two series of points Gj, C5, ... Cji-,, ... and Gi, G,, ... Csjfc+i, .... We construct also the series of points «], Oj, ... aj, ... , where a^g^ = Akgic+i and aj^gjc+i = Aiegk- -By aid of these series of points we may construct the required funicular. Consider the case of n spans, the end An, as well as A^, being simply supported. The line joining G^n-i to An is the last side (3?i— 1) of the funicular, since the force (^„', like (^,, is zero. The side (3n — 2) meets the side (3w- 1) on the line of action of F^, and passes through the point Gj„_2. Let this side (3w — 2) meet the vertical through «„_i in F,i_i. Then the line Vn-\ G-in-i is the side (3?i— 4). The side (3« — 3) is determined by joining the point where the side (3n. — 2) meets the vertical through gn to the point where the side (3n — 4) meets the vertical through g'n-i. This side (3n. — 3) necessarily passes through .4„_, in consequence of the mode of construction of the points G. Proceeding in this way we can construct the funicular. When the funicular is constructed we may determine the bending moments at the supports by measurement upon the figure. For example, let the side 4 meet the vertical through A^ in S^. Then A^S^ and the sides 3 and 4 make up a triangle of forces for the point of intersection of 3 and 4. The horizontal projection of either of the sides of this triangle which are not vertical is \A-^A„_. Hence 4,(Si represents the force (^2 on the same scale as ^^1.42 represents the horizontal tension in the sides of the funicular. Thus i4iS,/-4i.42 represents the force ^2 on a constant scale. But (^2 represents the product of Ml and A^A^ also on a constant scale. Hence A^SJAiA^ repre- sents the bending moment at A^ on a constant scale. In like manner, if the side ^k + \ meets the vertical through ^^ in the point S^, then At:SklAkAu.^.i' represents the bending moment at .4^. CHAPTEE XVIII. GENERAL THEORY OF THE BENDING AND TWISTING OF THIN RODS. 251. Besides the problem of continuous beams there are many physical and technical problems which can be treated as problems concerning long thin rods, and, on this understanding, are capable of approximate solution. In this Chapter we shall consider the general theory of the behaviour of such bodies, reserving the applications of the theory for subsequent Chapters. The special circumstance of which the theory must take account is the possibility that the relative displacements of the parts of a long thin rod may be by no means small, and yet the strains which occur in any part of the rod may be small enough to satisfy the requirements of the mathematical theory. This possibility renders necessary some special kinematical investigations, subsidiary to the general analysis of strain considered in Chapter i. 252. Kinematics of thin rods*. In the unstressed state the rod is taken to be cylindrical or prismatic, so that homologous lines in different cross-sections are parallel to each other. If the rod is simply twisted, without being bent, linear elements of different cross-sections which are parallel in the unstressed state become inclined to each other. We select one set of linear elements, which in the unstressed state are parallel to each other and lie along principal axes of the cross-sections at their centroids. Let 8/ be the angle in the strained state between the directions of two such elements which lie in cross-sections at a distance Ss apart. Then Um Sf/Bs measures the twist. 8«=0 When the rod is bent, the twist cannot be estimated quite so simply. We shall suppose that the central-line becomes a tortuous curve of curvature 1/p and measure of tortuosity 1/2. We take a system of fixed axes of x, y, z of which the axis of z is parallel to the central-line in the unstressed state, and the axes of x, y are parallel in the same state to principal axes of the * Cf. Kelvin and Tait, Nat. Phil., Part I, pp. 94 et seq., and Kirohhoff, J.f. Math. {Grelle), Bd. 56 (1839), or Ges. AbhamUungen (Leipzig 1882), p. 285, or Vorlesungen iiber math. Physik, Mechanik, Vorlesnng 28. 25—2 388 EXTENSION CURVATURE AND TWIST [CH. XVIII cross-sections at their centroids. Let P be any point of the central-line, and, in the unstressed state, let three linear elements of the rod issue from P in the directions of the axes of x, y, z. When the rod is deformed these linear elements do not in general continue to be at right angles to each other, but by means of them we can construct a system of orthogonal axes of x, y, z. The origin of this system is the displaced position P^ of P, the axis of z is the tangent at P^ to the strained central-line, and the plane {x, z) contains the linear element which, in the unstressed state, issues from P in the direction of the axis of x. The plane of {x, z) is a " principal plane " of the rod. The sense of the axis of x is chosen arbitrarily. The sense of the axis of z is chosen to be that in which the arc s of the central-line, measured from some assigned point of it, increases ; and then the sense of the axis of y is determined by the condition that the axes of x, y, z in this order are a right-handed system. The system of axes constructed as above for any point on the strained central-line will be called the " principal torsion-flexure axes " of the rod at the point. Let P' be a point of the central-line near to P, and let P/ be the displaced position of P.. The length Ssj of the arc Pj P/ of the strained central-line may differ slightly from the length Bs of PP'. If e is the extension of the central-line at Pj we have licQ (SsJBs) = {l + e) (1) a«=o The extension e may be zero. For any application of the mathematical theory of Elasticity to be possible, it must be a small quantity of the order of , the strains contemplated in the theory. Suppose the origin of a frame of three orthogonal axes of x, y, z to move along the strained central-line of the rod with unit velocity, and the three axes to be directed always along the principal torsion-flexure axes of the rod at the origin of the frame. We may resolve the angular velocity with which the frame rotates into components directed along the instantaneous positions of the axes. We shall denote these components by k, k', t. Then k and k' are the components of curvature of the strained central-line at Pj, and t is the twist of the rod at Pj. These statements may be regarded as definitions of the twist and com- ponents of curvature. It is clear that the new definition of the twist coincides with that which was given above in the case of a rod which is not bent, and that K, K are the curvatures, as defined geometrically, of the projections of the strained central-line on the planes of (y, z) and {x, z), and therefore the resultant of k and k is a vector directed along the binorraal of the ."ftrained central-line and equal to the curvature 1/p of this curve. 253. Kinematical formulae. We investigate in the first place the relation between the twist of the rod and the measure of tortuosity of its strained central-line. • Let I, m, n denote 252, 253] OF A THIN EOD 389 the direction-cosines of the binormal of this curve at Pj referred to the prin- cipal torsion-flexure axes at Pj, and let V, m', n' denote the direction-cosines of the binormal at P/ referred to the principal torsion-flexure axes at P/. Then the limits such as Km {l' — l)/Ssi are denoted by dl/dsi, .... Again let 1+ Bl, ... denote the direction cosines of the binormal at Pi' referred to the principal torsion-flexure axes at Pj. We have the formulae* lim Sl/Ssi = dl/dsi — mr + uk', lim Sm/Ssi = dm/dsi — nK+ It, «s,=0 lim Sn/Ssi = dn/dsi — Ik' + vik. 8si=0 The measure of tortuosity 1/S of the strained central-line is given by the formula 1/t' = lim [{Biy + (8my + (8n)=']/(Ss,)S 6*1=0 and the sign of 2 is determined by choosing the senses in which the principal normal, binormal and tangent of the curve are drawn. We suppose the prin- Fig. 45. cipal normal (marked n in Fig. 45) to be drawn towards the centre of curvature, and the tangent to be drawn in the sense in which Sj increases, and we choose the sense in which the binormal (marked b in the figure) is drawn in such a way that the principal normal, the binormal and the tangent, taken in this order, are parallel to the axes of a right-handed system. Now we may put 1 = fcp = — cos /, m = k'p — siny, h = 0, where p is the radius of curvature ; and then ^tt -/ is the angle between the * Cf . E. J. Eouth, Dynamics of a system of rigid bodies (London 1884), Part 11, Chapter I. 390 KINEMATICAL FORMULA [CH. XVIII principal plane (x, z) of the rod and the principal normal of the strained central-line. On substituting in the expression for 1/2 ^ and making use of the above convention, we find the equation •(2) _dl 1 ^~ds/S' in which tan /= — («'/«) (3) The necessity of introducing such an angle as /into the theory was noted by Saint- Venant*. The case in which / vanishes or is constant was the only one considered by the earlier writers on the subject. The linear elements of the deformed rod which issue from the strained central-line in the direction of the principal normals of this curve are, in the unstressed state, very nearly coincident with a family of lines at right angles to the central-line. If / vanishes or is constant these lines are parallel in the unstressed state. We may describe a state of the bent and twisted rod in which / vanishes or is constant as such that the rod, if simply unbent, would be prismatic. When/ is variable the rod, if simply unbent, would be a twisted prism, and the twist would be df/dsi. With a view to the calculation of k, k! , t we take the axes of x, y, z at Pi to be connected with any system of fixed axes of x, y, z by the orthogonal scheme ■(4) X y z x k OTl «i y h m2 »2 z h ms Ms in which, for example, Z„ rwi, «, are the direction-cosines of the axis of x at Pj referred to the fixed axes. We have the nine equations dljds^ = l^T-l3 k', dyds, = l,K-l^T, dls/dsi = Iik' -I^k, ' dm^lds^ = m,iT-jn^K', drrh/ds^ = msK-tnjT, dms/dsi = 7rhK' — m^K, -...(5) dnJdsi = n^T-nsK', dn^/dsi = VsK -n^r, di%/dsi = niK' -n^K, which express the conditions that the axes of x, y, z are fixed, while those of X, y, z are moving with the angular velocity («, k , t)\. From these we obtain such equations as K=^l. dU 'ds.-^'^'Ts;^''' dn. The differentiations with respect to s, may, since e is small, be replaced by difierentiations with respect to s, provided that the left-hand members of * Paris, C.J?., t. 17 (1843). t Cf. B. J. Bouth, loc. cit. p. 389. 253] EELATING TO A THIN EOD 391 the equations are multiplied by 1 + e. If k, k, t are themselves small, and quantities of the order ex are neglected, the factor l + e may be replaced by unity. If K, K, T are not regarded as small quantities, a first approximation to their values can be obtained by replacing 1 + e by unity. For the esti- mation of K, K, T we may therefore ignore the distinction between ds^ and ds and write our formulae , dL dvu diu , , dig dn^ dus , dL diih dn, T = Lj -J- + V%> -J— + ?l3 J- . as OS as .(6) The direction-cosines L, ... can be expressed in terms of three angles 6, yjr, (f), as is usual in the theory of the motion of a rigid body. Let 6 be the angle which the axis of z at Pj makes with the fixed axis of z, yjr the angle which a plane parallel to these axes makes with the fixed plane of (x, z), (j) the angle which the principal plane {x, z) of the rod at Pi makes with the plane zPi^. Then the direction-cosines in question are expressed by the equations ?i= -sini/r sin ^-l-cos (^cosi^cosS, nii = cosi/rsia(^ + sinA/rcos0cosd, »i=— sinflcos^, "j ij= - sin >/f cos <^ — cos i/f sin 1^ cos d, OT2=cosi/rcos(^-sin\/^sin0cos5, ^2= sindsini^, >- ?3= sinflcosi/f, 7713 = sin dain^, %= cos 5. j (7) Fig. 46. 392 EQUATIONS OF EQUILIBRIUM [CH. XVIII The relations connecting dff/ds, d\}rjds, d4>/ds with k, k\ t are obtained at once from Fig. 46 by observing that k, k , r are the projections on the principal torsion-flexure axes at Pj of a vector which is equivalent to vectors dOjds, d-\jr/ds, d(l>/ds localized in certain lines. The line P^^ in which d6/ds is localized is at right angles to the plane zP^z, and d'^/ds and d(f)/ds are localized in the lines PjZ and P^z. We have therefore the equations K = -=- sin — -J- sin tf cos c')n{x + ^) + Z.||«+|^'«+^^+(l+e)(^:«'-U)< 256] BENT AND TWISTED ROD 397 For the dififerences of the y- and z-coordinates we have similar expressions with mi, ma, m^ and n^, n^, n^ respectively in place of ?,, l^, I,. Since the scheme (4) is orthogonal, the result of squaring and adding these ex- pressions is dx) + r^ + r^ ds' dr) dx dr) Z + ( 1 + ^ ) m + ^ n + (1 + e) n {t (a; + f ) - «f } dx ^+^'^+^s''+('^ +''>'' {^ + "(y + v)- x' {«!+ 0} ...(14) and this is r-j^. We have therefore expressed r^^ in the form of a homogeneous quadratic function of I, m, n. Now, the strains being small, r^ is nearly equal to r, and we can write rj2 = r^ (1 + 2e), where e is the extension in the direction I, m, n. Further we shall have « = ^xxi'^ + SyyTn? + CzzU^ + Byzinn + ezxnl + e^ylm, where the quantities ea;x, ■■■ are the six components of strain. The coefficient of I in the first line of the expression (14) must be nearly equal to unity, and the coefficients of m and n in this line must be nearly zero. Similar statements mutatis mmtandis hold with regard to the coefficients of I, m, n in the remaining lines. We therefore obtain the following expressions for the components of strain : d^ ^xx — o ^. ' ^yy — and dx dr) dy' dy dx' .(15) dx + {\ + e)[K'^-r{y + r,)], ^«z — d^ dr) + '^ + (1+6){t(^+^)-<}, dy ds 3? ezz = e + ^+{l + e)\K{y + r))-K'{x+^)}. .(16) In obtaining the formulae (15) and (16) we have not introduced any approximations except such as arise from the consideration that the strains are " small," and, in particular, that e, being the extension of the central-line, must be small. But we can see, without introducing any other considerations, that the terms of (16), as they stand, are not all of the same order of magni- tude. In the first place it is clear that the terms — ry, tx, Ky, — k'x must be small ; in other words, the linear dimensions of the cross-section must be small compared with the radius of curvature of the central-line, or with the reciprocal of the twist. Such terms as «'^, rrj, ... are small also. We may 398 NATURE OF THE STRAIN IN A [CH. XVIII therefore omit the products of e and these small quantities, and rewrite equations (16) in the forms .(17) ezi = e — KX + Ky + ^ — k ^ + Krj. f Now the position of the origin of x, y, and that of the principal plane of {x, z), are unaffected by the displacement (f, ??, f), and therefore this displace- ment is subject to the restrictions : (i) ^> V< ? vanish with x and y for all values of s, (ii) dr)jdx vanishes with x and y for all values of s. We conclude that, provided that the strain in the rod is everywhere small, the necessary forms of the strain-components are given by equations (15) and (17), where the functions ^, r), f are subject to the restrictions (i) and (ii). 257. Approximate formulae for the strain. We have now to introduce the simplifications which arise from the con- sideration that the rod is " thin." The quantities f, tj, f may be expanded as power series in x and y, the coefficients in the expansions being functions of s; and the expansions must be valid for sufficiently small values of x and y, that is to say in a portion of the rod near to the central-line*. There are no constant terms in these expansions because ^, r), f vanish with x and y. Further d^jdx and 9^/9?/ must be small quantities of the order of admissible strains, and therefore the coefficients of those terms of f which are linear in X and y must be small of this order. It follows that f itself must be small of a higher order, viz., that of the product of the small quantity d^jdx and the small coordinate x. Similar considerations apply to 17 and f. As a first step in the simplification of (17) we may therefore omit such terms as —rr), k'^. When this is done we have the formulaef 9? 9| dl; dv , 9? ,,,, '- = dx-^ + ds' ^^^ = a2/ + ™ + a^- e^^e-.x + Ky + ^,...(18) and these with (1.5) are approximate expressions for the strain-components. * The expansions may not be valid over the whole of a cross-section. The failure of Cauohy's theory of the torsion of a prism of rectangular cross-section (Introduction, footnote 85) sufficiently Illustrates this point. But the argument in the text as to the relative order of magnitude of such terms as ry and such terms as tt; could hardly be affected by the restricted range of validity of the expansions. + It may be observed that Saint-Venant's formulse for the torsion of a prism are included in 256-258] BENT AND TWISTED ROD 399 Again we may observe that similar considerations to those just adduced in the case of ^ apply also in the case of 9f/9s ; this quantity must be of the order of the product of the small quantity d^^/dxds and the small coordinate a-, which is the same as the order of the product of the small quantity d^/dx and the small fraction xjl, where Hs a length comparable with (or equal to) the length of the rod. Thus, in general, d^/ds is small comjiared with d^/dx. Similar considerations apply to dri/ds and d^lds*. As a second step in the simplificationof (17) we may omit 9f/'"'s, drj/ds, d^lds and obtain the formulajf e«=^-Ty, ey^ = ^+Tx, e^, = e - k'x + ki/ (19) Again we may observe that in Saint-Venant's solutions already cited e vamshes, and in some solutions obtained in Chapter xvi. e is small compared with k'x. In many important problems e is small compared with such quan- tities as Tx or k'x. ^Vhenever this is the case we may make a third step in the simplification of the formulae (17) by omitting e. They would then read ^2^ = 9^-"^. ^^ = 0^ + '^'^' ezz = -K:x + Ky (20) With these we must associate the formulae (15), and in the set of formulae we may suppose, as has been explained, that f, ij, fare approximately independent of s. It appears therefore that the most important strains in a bent and twisted rod are (i) extension of the longitudinal filaments related to the curvature of the central-line in the manner noted in Article 232 (b), (ii) shearing strains of the same kind as those which occur in the torsion problem discussed in Chapter xiv., (iii) relative displacement of elements of any cross-section parallel to the plane of the section. The last of these strains is approxi- mately the same for dififerent cross-sections provided that they are near together. 258. Discussion of the ordinary approximate theory. To determine the stress-resultants and stress-couples we require the values of the stress-components X^, T^, Z^. Since Zz = (1 -I- 0-) (1 - la-) ^°' ^^^ + «w) + (1 - "■) ^^}' the fonnalee (15) and (18) by putting |=i|=0 ; and his formulse for bending by terminal load are included by putting 1= -ffcxy + Jo-jc'li^-j^), ri=irK'xy + i *o'> To. Then k^, ko' are the components of the initial curvature, and t,, is the initial twist. If I/Sq is the measure of tortuosity of the central-line at any point, and ^tt — /» is the angle which the principal plane of {xg, z^ at the point makes with the principal normal of the central-line, we have the formulae ta.nfo = -Ko'/K„, To = l/'S.„ + dfJds, (25) which are analogous to (2) and (3) in Article 253. When the rod is further bent and twisted, we may construct at each point on the strained central-line a system of " principal torsion-flexure axes," in the same way as in Article 252, so that the axis of z is the tangent of the strained central-line at the point, and the plane of {sc, z) contains the linear element which, in the unstressed state, issues from the point and lies along the axis of ajo. By means of this system of axes we determine, in the same way as before, the components of curvature of the strained central-line and the twist of the rod. We shall denote the components of curvature by «!, «/, and the twist by Ti. The equations of equilibrium can be written down, by the method of Article 254, in the forms dT ds - Nk,' + N'k, + Z = 0, .(26) and dG ds dG; ds -G't, + Hk,'-N' + K = 0, Hk, + Gt, + N + K' = 0, .(27) JTT ^-Gk,'-\-G'k, + & = 0. ds The rod could be held straight and prismatic by suitable forces, and, according to the ordinary approximation (Article 255), the stress-couples at any cross-section would be —Ak^, — Bkq, — Ctq. The straight prismatic rod could be bent and twisted to the state expressed^ by Kj, k/, tj and then, according to the same approximation, there would be additional couples Ak^, Bki, Gti. The stress-couples in the rod when bent and twisted from the state expressed by «o. «o'; Tq to that expressed by «;,, k/, Ti would then be given by the formulae* G = A(k,-,c,), G' = B{k^-k-), H=G{t,-t„) (28) * These formulsB, due to Clebsch, were obtained also, by a totally different process, by A. B. Basset, Amer. J. of Math., vol. 17 (1895). 26—2 404 RODS NATURALLY CURVED [CH. XVIII It is clear from the discussion in Article 258 that these formulce can be used with greater certainty if the rod is subjected to terminal forcea and couples only than if forces are applied to it along its length. It may be noted that, even when the cross-section of the rod has kinetic symmetry so that A=B, the flexural couples are not equivalent to a single couple about the binormal of the strained central-line unless ki'/ko'=kiIkq. When this condition is satisfied the flexural couple is of amount B (l/pj- l/po); where p, and p„ are the radii of curvature of the central-line in the stressed and unstressed states. The above method of calculating the stress-couples requires the ratios of the thickness of the rod to the radius of curvature and to the reciprocal of the twist to be small of the order of small strains contemplated in the mathematical theory of Elasticity. Unless this condition is satisfied the rod cannot be held straight and untwisted without producing in it strains which exceed this order. It is, however, not necessary to assume that this condition is satisfied in order to obtain the formulfe (28) as approximately correct formulae for the stress-couples. "We may apply to the question the method of Article 256, and take account of the initial curvature and twist by means of the equations lr=bx-yTf)bs, mr = hy+XTf)bs, nr=hs{\- K^x-^-Kf^y), where y stands for Kf,y - k^x. We should then find instead of (14) In deducing approximate expressions for the strain-components we denote by [y] any quantity of the order of the ratio (thick ness)/(radius of curvature) or (thickness)/(reciprocal of twist), whether initial or final, and by [e] any quantity of the order of the strain. Thus T^y and r^y are of the order [y] ; Z^jdx and {Ki-Kf,)y are of the order [e]. If, in the above expression for r^^, we reject all terms of the order of the product [y] [e] as well as all terms of the order [e]^, we find instead of (19) the formulae ^^x = ^,-(.Ti-To)y, ey^=^+{ri-Ta)x, e^ = e + {Ki-Ko)y-{Ki-KQ')x. From these we could deduce the formulae (28) in the same -way as (12) are deduced from (19), and they would be subject to the same limitations. CHAPTER XIX. PROBLEMS CONCERNING THE EQUILIBRIUM OF THIN RODS. 260. Kirchhoff's kinetic analogue. We shall begin our study of the applications of the theory of the last Chapter with a proof of Kirchhoff's theorem*, according to which the equations of equilibrium of a thin rod, straight and prismatic when un- stressed, and held bent and twisted by forces and couples applied at its ends alone, can be identified with the equations of motion of a heavy rigid body turning about a fixed point. No forces or couples being applied to the rod except at the ends, the quantities X, Y, Z and K, K', @ in equations (10) and (11) of Article 254 vanish. Equations (10) of that Article become ^-N'r+TK'=0, ^-Tk + Nt = 0, ^-Nk' + N'k = 0,...{1) as as as which express the constancy, as regards magnitude and direction, of the resultant of N, N', T; and, in fact, this resultant has the same magnitude, direction and sense as the force applied to that end of the rod towards which s is measured. We denote this force by R. Equations (11) of Article 254 become, on substitution from (12) of Article 255, and omission of K, K\ @, a'^^-{B-G)k't=N', B^-{G-A)tk = -N, cJ-(A-S)««' = 0. (2) The terms on the right-hand side are equal to the moments about the axes of X, y, z oi a force equal and opposite to R applied at the point (0, 0, 1). We may therefore interpret equations (2) as the equations of motion of a top, that is to say of a heavy rigid body turning about a fixed point. In this analogy the line of action of the force R (applied at that end of the rod towards which s is measured) represents the vertical drawn upwards, s repre- sents the time, the magnitude of R represents the weight of the body, A,B,G represent the moments of inertia of the body about principal axes at the fixed point, (k, k, t) represents the angular velocity of the body referred to the * G. Kirchhoff, loc. cit., p. 387. 406 THEOREM OF THE KINETIC ANALOGUE [CH. XIX instantaneous position of this triad of axes. The centre of gravity of the body is on the <7-axis at unit distance from the fixed point; and this axis, drawn from the fixed point to the centre of gravity at the instant s, is identical, in direction and sense, with the tangent of the central- line of the rod, drawn in the sense in which s increases, at that point Pj of this line which is at an arc-distance s from one end. The body moves so that its principal axes at the fixed point are parallel at the instant s to the principal torsion-flexure axes of the rod at Pj. On eliminating N and N' from the third of equations (1) by the aid of equations (2), we find the equation ds as as or, by the third of (2), ^^{T+^ (Ak' + Bk'^ + Gt')} = 0, giving the equation T-\-\{Ak'+Bk'^+ Gr^) = Q.ons\, (3) This equation is equivalent to the energy-integral of the equations of motion of the kinetic analogue. 261. Extension of the theorem of the kinetic analogue to rods naturally curved*. The theorem may be extended to rods which in the unstressed state have curvature and twist, provided that the components of initial curvature kq, kq' and the initial twist tj, defined as in Article 259, are constants. This is the case if, in the unstressed state, the rod is straight but not prismatic, in such a way that homologous transverse lines in different cross-sections lie on a right helicoid ; or if the central-line is an arc of a circle, and the rod free from twist ; or if the central-line is a portion of a helix, and the rod has such an initial twist that, if simply unbent, it would be prismatic. When the rod is bent and twisted by forces and couples applied at its ends only, so that the components of curvature and the twist, as defined in Article 259, become kj, ki , ri, the stress-resultants N, N', T satisfy the equations '^^-N'r, + TK^=0, '^-T., + Nr,=0, ^^-N.^+N'K,=0 (4) These equations express the result that N, N', T are the components, parallel to the principal torsion-flexure axes at any section, of a force which is constant in magnitude and direction. We denote this force, as before, by R. Since the stress-couples at any section are A {k^ — kq), B{k{ -k^), C{ti — t^ we have the equations C-~-A{ki- Ko) Ki+B{ki'- Ka) Ki = 0. * J. Larmor, London Math, Soc. Proc, vol. 15 (1884). .(5) 260-262] THE PROBLEM OF THE ELASTICA 407 The kinetic analogue is a rigid body turning about a fixed point and carrying a flywheel or gyrostat rotating about an axis fixed in the body. The centre of gravity of the flywheel is at the fixed point. The direction-cosines I, m, n of the axis of the flywheel, referred to the principal axes of the body at the point, and the moment of momentum h of the fly- wheel about this axis, are given by the equations — AKa=hl, —BKQ—hm, —CTQ = hn (6) The angular velocity of the rigid body referred to principal axes at the fixed point is (kj, ki', ti) and the interpretation of the remaining symbols is the same as before. 262. The problem of the elastica*. As a first application of the theorem of Article 260 we take the problem of determining the forms in which a thin rod, straight and prismatic in the unstressed state, can be held by forces and couples applied at its ends only, when the rod is bent in a principal plane, so that the central-line becomes a plane curve, and there is no twist. The kinetic analogue is then a rigid pendulum of weight R, turning about a fixed horizontal axis. The motion of the pendulum is determined completely by the energy-equation and the initial conditions. In like manner the figure of the central-line of the rod is determined completely by the appropriate form of equation (3) and the terminal conditions. We take the plane of bending to be that for which the flexural rigidity is B. Then k and t vanish, and the stress-couple is a flexural couple 0', = Bk', in the plane of bending. The stress-resultants are a tension T and a shearing force N, the latter directed towards the centre of curvature. Let d be the angle which the tangent of the central-line at any point, drawn in the sense in which s increases, makes with the line of action of the force R applied at the end from which s is measured (see Fig. 47). Then we have T=—Rcosd, and k'=- dOjds, and the equation (3) becomes -Rcoad + \B {d0/dsf = const (7) In the kinetic analogue B is the moment of inertia of the pendulum about the axis of sus- pension, and the centre of gravity is at unit distance from the axis. The line drawn from the centre of suspension to the centre of gravity at the instant s makes an angle with the vertical drawn downwards. * The problem of the elastica was first solved by Euler. See Introduction, p. 3. The systematic application of the theorem of the kinetic analogue to the problem was worked out by W. Hess, Math. Ann., Bd. 25 (1885). Numerous special oases were discussed by L. Saalsehiitz, Der belastete Stab, Leipzig, 1880. 408 THE PROBLEM OF THE ELASTICA [CH. XIX Equation (7) can be obtained very simply by means of the equations of equilibrium. These equations can be expressed in the forms T = -Rcos0, N = -Rsm0, ^ + N = 0, as from which, by putting G' = — B(dd/ds), we obtain the equation B(d'd/d^) + R sin = 0, (8) and equation (7) is the first integral of this equation. The shape of the curve, called the elastica, into which the central-line is bent, is to be determined by means of equation (7). The results take different forms according as there are, or are not, inflexions. At an inflexion d0/ds vanishes, and the flexural couple vanishes, so that the rod can be held in the form of an inflexional elastica by terminal force alone, without couple. The end points are then inflexions, and it is clear that all the inflexions lie on the line of action of the terminal force R — the line of thrust. The kinetic analogue of an inflexional elastica is an oscillating pendulum. Since the interval of time between two instants when the pendulum is momentarily at rest is a constant, equal to half the period of oscillation, the inflexions are spaced equally along the central-line of the rod. To hold the rod with its central- line in the form of a non-inflexional elastica terminal couples are required as well as terminal forces. The kinetic analogue is a revolving pendulum. In the particular case where there are no terminal forces the rod is bent into an arc of a circle. The kinetic analogue in this case is a rigid body revolving about a horizontal axis which passes through its centre of gravity. If the central-line of the rod, in the unstressed state, is a circle, and there is no initial twist, the kinetic analogue (Article 261) is a pendulum on the axis of which a flywheel is symmetrically mounted. The motion of the pendulum is independent of that of the fly- wheel, and in like manner the possible figures of the central-line of the rod when further bent by terminal forces and couples are the same as for a naturally straight rod. The magnitude of the terminal couple alone is altered owing to the initial curvature. 263. Classification of the forms of the elastica. (a) Inflexional elastica. Let s be measured from an inflexion, and let a be the value of 6 at the inflexion «=0. We write equation (7) in the form ^^VIs) +'S(cos"-cos5) = (9) To integrate it we introduce Jacobian elliptic functions of an argument u with a modulus /■ which are given by the equations u = 8s'{RIB), k=smha (10) Then we have ^ = 2kGn{u + K), B.m\e=ksa{u + K), (11) where K is the real quarter period of the elliptic functions. To determine the shape of the 262, 263] FORMS OF THE ELASTICA 409 curve, let x, y be the coordinates of a point referred to fixed axes, of which the axis of x coincides with the line of thnist. Then we have the equations - ds.lds = cos fl, dj/ds = sin fl, and these equations give x = .(12) y=-2*y(|)c°(«+^. where B am ti denotes the elliptic integral of the second kind expressed by the formula fu Ea,mu= I dn^udu, Jo and the constants of integration have been determined so that x and y may vanish with s. The inflexions are given by cos 5 = cos a, or sn'(u + K) = l, and therefore the arc between two consecutive inflexions is 2 ^{BjR) . K, and the inflexions are spaced equally along the axis of X at intervals 2 ^(BjR) {2Ea.mK- E). The points at which the tangents are parallel to the line of thrust are given by sin 6=0, or sn (^ + K) An{u + E) = 0, so that u is an uneven multiple of K. It follows that the curve forms a series of bays, separated by points of inflexion and divided into equal half-bays by the points at which the tangents are parallel to the line of thrust. Thechangeof the form of the curve as the angle a increases is shown by Figs. 48 — 55 over- leaf. When a > Jtt, X is negative for small values of m, and has its numerically gi-eatest negative value when u has the smallest positive value which satisfies the equation dvL^{u + E)=-^. Let Ml denote this value. The value of u for which x vanishes is given by the equation ■M = 2 {^am {u + E) — Eaxa E). When u exceeds this value, x is positive, and x has a maximum value when u = 'iE — ui . Figs. 50 — 52 illustrate cases in which xjj is respectively greater than, equal to, and less than |xii, |. Fig. 53 shows the case in which xx=0 or 2^amA'=A'. This happens when a = 130° approximately. In this case all the double points and inflexions coincide at the origin, and the curve may consist of several exactly equal and similar pieces lying one over another. Fig. 54 shows a case in which 'iEa.w.E Fig. 56. in which the constants of integration are chosen so that x vanishes with s, and the axis of X is parallel to the line of action of R, and at such a distance from it that the force R and the couple — B (ddjds) which must be applied at the ends of the rod are statically equivalent to a force R acting along the axis of x. The curve consists of a series of loops lying altogether on one side of this axis. The form of the curve is shown in Fig. 56. 264. Buckling of long thin strut under thrust* The limiting form of the elastica when a is very small is obtained by writing 6 for sin in equation (8). We have then, as first approximations, e = acos{s^{R/B)], x = s, J=a^/{B/R) sin {x^/{R/B)}, ...(17) so that the curve is approximately a curve of sines of small amplitude. The distance between two consecutive inflexions is Tr\/{B/R). It appears therefore that a long straight rod can be bent by forces applied at its ends in a direction parallel to that of the rod when unstressed, provided that the length I and the force R are connected by the inequality PR>'n^B (18) If the direction of the rod at one end is constrained to be the same as that of the force, the length is half that between consecutive inflexions, and the inequality (18) becomes l^R>\ir^B (19) r^ The theory was initiated by Euler. See Introduotion, p. 3. 412 BUCKLING OF STRUT UNDER THRUST [CH. XIX If the ends of the rod are constrained to remain in the same straight line, the length is twice that between consecutive inflexions, and the inequality (18) becomes l'R>4<7r"-B (20) These three cases are illustrated in Fig. 57. Any of these reaults can be obtained very easily without having recourse to the general theory of the elastica. We take the second case, and suppose that a long thin rod is set up vertically and loaded at the top with a weight R, while the lower end is constrained to remain vertical*. Let the axes of x and y be the vertical line drawn upwards through the lowest point and a horizontal line drawn through the same point in the plane of bending, as shown in Fig. 57 6. If the rod is very slightly bent, the equation of equilibrium of the portion between any section and the loaded end is, with sufficient approximation, -.Bg + iJ(y,-y) = 0, where yi is the displacement of the loaded end. The solution of this equation which satisfies the conditions that y vanishes with x, and that y=yi when x = i, is r _ An{{l-i)J{RIB) }-\ ^-^'L mn{l^{RIB)] J' and this solution makes dy/dx vanish with x if cos {I ^{R/B)}=0. Hence the least value of I by which the conditions can be satisfied is ^n- ^{B/R). It should be noted that, in the approximate theory here explained there occurs an indeterminate quantity, a or yi, and the approximate theory cannot be made to yield the value of this quantity. For example it cannot be made to determine the terminal deflexion in the problem illustrated in Fig. 57 bf. The theory of the elastioa leads easily to the result that, when this deflexion is small compared with I, its amount is This remark does not invalidate the statement that, when a is small, the elastica is nearly identical with a curve of sines. As a matter of fact, if the distance between consecutive inflexions and the maximum ordinates of a curve of sines and an elastica are the same, and the maximum ordinate is small compared with the distance between consecutive inflexions, the curves coincide to a high order of approximation. From the above we conclude that, in the case represented by Fig. 57 6, if the length is slightly greater than ^Tr'J{B/R), or the load is slightly greater than \7r^B/V, the rod bends under the load, so that the central-line assumes the form of one half-bay of a curve of sines of small amplitude. If the length of the rod is less than the critical length it simply contracts under the load. If the length is greater than the critical length, and the load is truly central while the rod is truly cylindrical, the rod may simply contract; but the equi- librium of the rod thus contracted is unstable. To verify this it is merely necessary to show that the potential energy of the system in a bent state is less than that in the contracted state. * We neglect the weight of the rod. The problem of the bending of a vertical rod under its own weight will be considered in Article 276. t Cf. K. W. Burgess, Phys. Rev., March, 1917. 264, 265] ENERGY METHOD 413 265. Computation of the strain-energy of the strut. Let the length I be slightly greater than^7r^(5/iJ). Let &> denote the area of the cross-seotion of the rod, and E the Young's modulus of the material. If the rod simply contracts, the amount of the contraction is RjEa, the loaded end descends through a distance RljEa, and the loss of potential energy on this account is JPljEa. The potential energy of contraction is ^IPljEa). The potential energy lost in the passage from the unstressed state to the simply contracted state is, therefore, ^RH/Ea. If the rod is of sufficient length to be deformed into one half-bay of a curve of the elastica family we may show that the loss of energy incurred in passing from the unstressed state to the bent state exceeds that incurred in passing from the unstressed state to the simply contracted state. The deformed rod is, of course, contracted as well as bent, and the amount of the contraction at any point is {R cos 6)1 Ea. The length of the deformed rod being denoted by I', we have .=^7(1), ,=,[,I^,{M|^-,}] («, Here K is the real quarter-period of elliptic functions of modulus sin ^a, ( = ^), and a is the small angle which the central-line of the rod at the loaded end makes with the line of action of the load. The potential energy lost through the descent of the load ia R(1- I cose dsj. The potential energy of bending is J5 I {d6jdsfds,orR\ {coa6-cos,a)ds. The potential energy of contraction is ^Ea \ (^ cos BjEaf ds or ^ (B^jEa) U' - I sin^ 6ds) . Hence the loss of potential energy incurred in passing from the unstressed state to the bent state is RU + rcoaa-^ I coad dsi -^^ U' - ain^edsi, and the excess of the potential energy in the simply contracted state above that in the bent state is Rh + rooaa-2j cos ds\ - ^ -g^U + l' - I ain^ dsX . In the terms of this expression which contain the factor R^jEm, we may identify I with I', because these terms are already small of the second order in the strain-components. Hence the expression may be written R(l+l'coaa-'2.\ coa ds\ - -^ ( I' - i aia'0ds\ (22) To evaluate this expression we have the results expressed by (21) and the result fl' „ , „ /2Ea.mK _\ I ooa 0ds=l I -^ — ~ )' and we require the value of I sin^ ds. Now sin2 = 4/:2 sn^ (m + K) Aa^{u + K) = 4k^' ^ (cn2 a/dn* u), 414 FAILURE BY BUCKLING [CH. XIX /" d?« _ k^ anucnu 2(l+/;'2) f du u f da P sn M en M 1 /"j , , Hence T sin^ eds = ^l'{l-i:^-(l- U^) (A'am A')/A'}. The expression (22) can now be evaluated in the form -f[.-l(.-..-a-.*,?^}]. To determine the sign of this expression we expand the functions of h which occur as far as terms in k*. We have /i = J,r (l+iF + ^F), Ea.mK=in (1 - |P - jft/t*), (^amZ)/r=l - JP-^V^S and the expression becomes iRl'k*{l-3RIEa,), which is certainly positive for any feasible value of RIEa>*. It will be observed that, if the rod is but slightly bent, V is nearly equal to I (1 —R/Eo)),^ and the condition for the existence of an elastica satisfying the terminal conditions is l'>]iT^(BIR), so that, in strictness, the condition (19) of Article 264 should be replaced by l^R>{n'^B{\+^RIEu>), I being the unstrained length of the strut. Conditions (18) and (20) should be modified in the same way. The correction is of no practical importance. 266. Resistance to buckling. The strains developed in the rod, whether it is short and simply contracts or is long and bends, are supposed to be elastic strains, that is to say such as disappear on the removal of the load. For Euler's theory of the buckling of a long thin strut, explained in Article 264, to have any practical bearing, it is of course necessary that the load required, in accordance with inequalities such as (19), to produce bending should be less than that which would produce set by crushing. This condition is not satisfied unless the length of the strut is great compared with the linear dimensions of the cross-section. In view of the lack of precise information as to the conditions of safety in general (Chapter iv.) and of failure by crushing (Article 189), a precise estimate of the smallest ratio of length to diameter for which this condition would be satisfied is not to be expected. The practical question of the conditions of failure by buckling of a rod or strut under thrust involves some other considerations. When the thrust is not truly central, or its direction not precisely that of the rod, the longitu- dinal thrust is accompanied by a bending couple or a transverse load. The * This Article has been revised. The necessity for revision was pointed out by S. Timosohenko, ' Sur la stability des syst^mes ^lastiques,' Paris, Ann. des ponts et cfumssdes, 1913. 265-267] ELASTIC STABILITY 415 contraction produced by the thrust R is RjEto. When the thrust is not truly- central, the bending moment is of the order Re, where c is some linear dimension of the cross-section, and the extension of a longitudinal filament due to the bending moment is of the order Rc^jB^ which may easily be two or three times as great, numerically, as the contraction RjEm. The bending moment may, therefore, produce failure by buckling under a load less than the crushing load. Again, when the line of thrust makes a small angle /8 with the central-line, the transverse load R sin ^ yields, at a distance compar- able with the length I of the rod, a bending moment comparable with IR sin ^ ; and the extension of a longitudinal filament due to this bending moment is comparable with IRc sin ^jB. Thus even a slight deviation of the direction of the load from the central-line may produce failure by buckling in a fairly long strut. Such causes of failure as are here considered can best be discussed by means of Saint- Venant's theory of bending (Chapter xv.): but, for a reason already mentioned, a precise account of the conditions of failure owing to such causes is hardly to be expected. It is clear that such considerations as are here advanced will be applicable to other cases of buckling besides that of the buckling of a rod under thrust. The necessity for them was emphasized by E. Lamarle*. His work has been discussed critically and appreciatively by K. Pearson f. In recent years the conditions of buckling have been the subject of considerable discussion|. 267. Elastic stability. The possibility of a straight form and a bent form with the same terminal load is not in conflict with the theorem of Article 118, because the thin rod can, without undergoing strains greater than are contemplated in the mathe- matical theory of Elasticity, be deformed in such a way that the relative displacements of its parts are not small§. The theory of the stability of elastic systems, exemplified in the discussion in Articles 264, 265, may be brought into connexion with Poincar6's theory of " equilibrium of bifurcation ||." The form of the rod is determined by the extension e at the loaded end and the total curvature a; and these quantities depend upon the load R, the length I and flexural rigidity B being regarded as constants. We might represent the state of the rod by a point, determined by the coordinates e and a, and, as R varies, the point would describe a curve. * ' Mem. sur la flexion du bois,' Ann. des travaux publics de Belgique, t. 4 (1846). + Todhnnter and Pearson's History, vol. 1, pp. 678 et seq. t Eeferenoe may be made to the writings of J. Kiibler, C. J. Kriemler, L. Prandtl in Zeitschr. d. Deutschen Ingenieure, Bd. 44 (1900), of Kubler and Kriemler in Zeitschr. f. Math. u. Phys., Bde. 45-47 (1900-1902), and the dissertation by Kriemler, ' Labile u. stabile Gleichgewichts- figuren...auf Biegung beanspruchter Stabe... ' (Karlsruhe, 1902), also to the paper by Timosohenko cited in Article 265 and to that by Southwell to be cited in Article 267 a. § Cf. G. H. Bryan, Cambridge Phil. Soc. Proc, vol. 6 (1888). II Acta Mathematica, t. 7 (1885). 416 ELASTIC STABILITY [CH. XIX When R is smaller than the critical load, a vanishes, and the equilibrium state, defined by e as a function of R, is stable. When R exceeds the critical value, a possible state of equilibrium would still be given by a = 0; but there is another possible state of equilibrium in which a does not vanish, and in this state a and e are determinate functions of R, so that the equilibrium states for varying values of R are represented by points of a certain curve. This curve issues from that point of the line o = which represents the extension, or rather contraction, under the critical load. Poincarc^ describes such a point as a " point of bifurcation," and he shows that, in general, there is an " exchange of stabilities " at such a point, that is to say, in the present example, the states represented by points on the line a = 0, at which e numeri- cally exceeds the extension under the critical load, are unstable, and the sta- bility is transferred to states represented by points on the curve in which a=^0. 267 A. Southwell's method. Another method of investigating problems of elastic stability has been proposed by R. V. Southwell*. We may most conveniently explain this method in its application to the problem of Article 264, and especially the case illustrated in Fig. 57 b. Similar principles are involved in any other application of the method, and other examples will be found in Chapters xxiv. and XXIV. a. We consider the series of configurations in which the rod can be held by gradually increasing R. For very small values of R the rod is simply con- tracted. We suppose the contracted rod with given R to suffer a very small displacement, by which it becomes slightly bent. If the value of R is such that the slightly bent state can be maintained without altering R, the equilibrium in the contracted state is critical, and any further increase in R results in buckling. The point emphasized by Southwell is that the superior limit to the values of R consistent with stability, and the accompanying con- traction, are relatively considerable. It follows that the simplified methods of Article 258 A do not avail without modification to determine the small trans- verse displacement which can occur if R slightly exceeds this limit. The necessary modification will appear in what follows. Let Si denote the contracted length of a portion of the rod measured from the lower end, and, as in Article 258 A, let u, v, w, /3 specify the displacement of the rod from the simply contracted state. Let T,, denote the tension in the simply contracted state, and let the value of T in the bent state be expressed as To + T'. Then the component curvatures and the twist in this state are given by the equations "" ds,'' "'ds,'' ^~rf^ <^^^^ * ' On the general theory of elastic stability,' London, Phil. Trans. R. Sac. (Ser. A), vol. 213 (1913), p. 187. 267, 268] STABILITY OF THE ELASTICA 417 We omit terms of the second order in u, v, w, /3. Then equations (11) of Article 254 become and equations (10) of the same Article become asi* dSi^ dsi* dsj!' ds The terminal conditions at the lower end (s^ = 0) are M = 0, v = 0, /S = 0, ^ = 0, T- = 0. asi dsi The terminal conditions at the upper end (s^ = l^) are 6^ = 0, O' = 0, H=0, N-T,'^ = 0, N'-T„^=0, To + T' = -R. asi asi Now To= — R, and we have therefore T'= 0. Also we see that we must have /8= 0. The most general possible form for u is A^ cos WiR/B) Si) + 5i sin W(B/B) s,j + G^s^ + I)„ where A^, B^, G-i, B^ are constants. The terminal conditions require £j = (7, = 0, A = -^i, cos{V(-R/^)^i}=0. Similar considerations apply to v, and the condition that the equilibrium may be critical is the same as that previously obtained. 268. Stability of inflexional elastica. When the lower end of the loaded rod is coustrained to remain vertical, and the length I slightly exceeds ^n^^B/R), a possible form of the central-line is a curve of sines of small amplitude having two inflexions, as in Fig. 58 (6) overleaf. Another possible form is an elastica illustrated in Fig. 58 (c). In general, if n is an integer such that l(in + l) rr>l y/{RIB)>i {2n-l) TT (24) n forms besides the unstable straight form are possible, and they consist respectively of • 1, 3, ... 2n- 1 half-bays of different curves of the elastica family. The forms of these curves are given respectively by the equations A'=Z^(i2/5)x[l, J, ...,1/(271-1)] (25) We shall show that all these forms except that with the greatest K, that is the smallest number of inflexions, are unstable*. Omitting the practically unimportant potential energy due to extension or contraction of the central-line, we may estimate the loss of potential energy in passing from the unstressed state to the bent state in which there are r-fl inflexions, in the same way as in Article 265, as R[l {I +COS a)-2 j cosdds], (26) * The result is opposed to that of L. Saalschiitz, Der belastete Stab (Leipzig, 1880), but I do not think "that his argument is quite convincing. The result stated in the text agrees with that obtained by a different method by J. Larmor, loc. cit., p. 406. It is supported also by the investi- gation of M. Born, ' XJntersuchungen u. d. Stabilitat d. elastischen Linie in Ebene u. Eaum, unter verschiedenen Grenzbedingungen,' (Diss.) Gottingeu, 1906. 27 418 STABILITY OF THE ELASTICA [CH. XIX and this is (2r4-l)>/(5fl) (4ff^,-4£'^-2/i'A'), (27) where E^ is written for E&mK,., and the suffix r indicates the number {r + l) of inflexions. We compare the potential energies of the forms with r + \ and s + 1 inflexions, s being greater than r. Since , , (2>- + l)^, = (2s + l)/r„ (28) the potential energy in the form with s + 1 inflexions is the greater if (2s + l) (2£', + ffA')>(2'- + l) (2£; + ^A'). Since this condition is But, since dk\ 1 + 2k d. dK\-\ _ _ 2k{\-k^) /dKy . [(] - -; V' "^ ^ dkJJ = Z~ \dkj it follows that (1-A^) ( 1 + 7? tt ) diminishes as k increases. Now when «>r, K^KKr, and /t, < ir ; and therefore the inequality (29) is satisfied. D Fig. 58. In the case illustrated in Fig. 58 the three possible forms are (a) the unstable straight form, (6) the slightly bent form with two inflexions, (c) the bent form with one inflexion. The angle a for the form (c) is given by ^=f tt, and it lies between 175° and 176°. It may be observed that the conclusion that the stable form is that with a single inflexion is not in conflict with Poincare's theory of the exchange of stabilities at a point of bifurcation, because the loci in the domain of e and a which represent forms with two or more inflexions do not issue from the loeus which represents forms with one inflexion but from the locus a = which represents straight forms. The instability of forms of the elastica with more than the smallest possible number of inflexions between the ends is well known as an experimental fact. Any particular case 268, 269] ROD SUBJECT TO TERMINAL FORCES 419 can be investigated in the same way as the special case discussed above, in which the tangent at one end is, owing to constraint, parallel to the line of thrust. An investigation of this kind cannot, however, decide the question whether any particular form is stable or unstable for displacements in which the central-line is moved out of its plane. This question has not been solved completely. One special case of it will be considered in Article 272 (e). Other cases are considered by M. Born, loo. cit., p. 417. 269. Rod bent and twisted by terminal forces and couples. We resume now the general probleiU of Article 260, and express the directions of the principal torsion-flexure axes at any point Pj on the strained central-line by means of the angles 6, y^,

, sin (9 sin <^, cos 6') (30) Equation (3) of Article 260 becomes ^ (Ak^ + Bk"- +G'T^} + R cos e = const (31) Since the forces applied at the ends of the rod have no moment about the line PjZ, the sum of the components of the stress-couples about a line drawn through the centroid of any section parallel to this line is equal to the corre- sponding sum for that terminal section towards which s is measured. We have therefore the equation — Ak sin 9 cos + Bk sin sin (/> + Gt cos 6 = const (32) The analogue of this equation in the problem of the top expresses the constancy of the moment of momentum of the top about a vertical axis drawn through the fixed point. The equations (31) and (32) are two integrals of the equations (2) of Article 260, and, if a third integral could be obtained, dOjds, d-<}r/ds, d(l}/ds would be expressible in terms of 6, yjr, cos^ ajr, d-^jds = cos a/r, d(j)/ds = t — sin a cos ajr. (35) P From equations (2) of Article 260 we find (N, N') = (— cos (f>, sin ) [Gt cos^ ajr — £ sin a cos' ajr'], WSMH. and then from equations (30) we find B, = Gt cos ajr — B sin a cos' ajr'^^ (36) The terminal force is of the nature of tension or pressure according as the right-hand member of (36) is positive or negative. (See Fig. 59.) For the force to be of the nature of tension, t must exceed B sin a cos a/Gr. The axis of the terminal couple lies in the tangent plane of the cylinder at the end of the central-line, and the components of this couple about the binormal and tangent of the helix at thi.s point are B cos^ ajr and Gt. The components of the same couple about the tangent of the circular section and the generator of * Cf. Kirchhoff, loc. cit., p. 387. 269-271] SPIRAL SPRINGS 421 the cylinder at the same point are, therefore, Rr and K, where K is given by the equation ^ = C't sin a + 5cos'a/r (37) It follows that the rod can be held so that it has a given twist, and its central-line forms a given helix, by a wrench of which the force R and the couple K are given by equations (36) and (37), and the axis of the wrench is the axis of the helix. The force and couple of the wrench are applied to rigid pieces to which the ends of the rod are attached. The helical form can be maintained by terminal force alone, without any couple ; and then the force is of magnitude B cos^ afr^ sin a, and acts as thrust along the axis of the helix. In this case there must be twist of amount — B cos' alCr sin a. The form can be maintained also by terminal couple alone, without any force ; and then the couple is of magnitude 5 cos a/r, and its axis is parallel to the axis of the helix. In this case there must be twist of amount B sin a cos a/Cr. When the state of the rod is such that, if simply unbent, it would be prismatic, d(f>/ds vanishes, and the twist of the rod is equal to the measure of tortuosity of the central-line (of. Article 253). To hold the rod so that it has this twist, and the central-line is a given helix, a wrench about the axis of the helix is required ; and the force R and couple K of the wrench are given by the equations R= - {B - O) ain a cos^ alr% K={B cos^ a+C sin^ a) cos ajr. 271. Theory of spiral springs*. When the sections of the rod have kinetic symmetry, so that A = B, and the unstressed rod is helical with such initial twist that, if simply unbent, it would be prismatic, we may express the initial state by the formulae Ko = 0, «„' = cos'-' a/r, tj = sin a cos ajr (38) By suitable terminal forces and couples the rod can be held in the state expressed by the formulae «! = 0, Kj' — cos' ajr^, tj = sinati cosai/7-i, (39) where n, «i are the radius and angle of a new helix. The stress-couples at any section are then given by the equations , _ / cos^ai cos'«\ tT_n { ^^^ "' ^°^ "i ^^^ " °°^ " A ~' ~ \ Vi r J ' Vn r J ' and the stress-resultants are given by the equations iV=0, T=N'tana^, _ COS' Oj / sin «i cos «! _ sin a cos a\ _ ^ sin «! cos a^ / cos' a, _ cos' a \ All the equations of Article 259 are satisfied. The new configuration can be maintained by a wrench of which the axis is the axis of the helix, and the force R and couple K are given by the equations ^ cos «] /sin «! cos a, sin a cos a^ „ sin o-i / cos' a, cos' a \ -^ _, . /sinaiCOSKi sin a cos a\ , „ /cos'ai cos'a \ if=Csma,( 'j^ --) + Bcosa.[— —).] * Cf. Kelvin and Tait, Nat. Phil., Part ii., pp. 139 et seq. (40) 422 SPIRAL SPRINGS [CH. XIX The theory of spiral springs is founded on this result. We take the spring in the unstressed state to be determined by the equations (38), so that the central-line is a helix of angle a traced on a cylinder of radius r, and the principal normals and binormals in the various cross-sections are homologous lines of these sections. We take I to be the length of the spring, and h to be the length of its projection on the axis of the helix, then the cylindrical coordinates r, 6, z of one end being r, 0, 0, those of the other end are r, •)(_, h, where j^ = (Z cos a)/r, A = Zsina (41) We suppose the spring to be deformed by a wrench about the axis of the helix, and take the force R and couple K of the wrench to be given. We shall suppose that the central-line of the strained spring becomes a helix of angle oti on a cylinder of radius i\, and that the principal normals and binormals continue to be homologous lines in the cross-sections. Then R and K are expressed in terms of «i and r-^ by the equations (40). When the deformation is small we may write r + hr and a 4- 8« for r^, a,, and suppose that small changes Spj; and hh are made in ^^ and h. We have hh = (i cos a) 8a, ^X~ ~ [(' ^^^ '*)/''] ^* ~ [(^ cos a)/?'^] 8r, from which Sa = (8h)l{l cos o), (Sr)/!^ = — (sin a. Bh + r cos a . Sx)/ii~ cos" a. TT t. sin a cos a . Sr cos 2a » Hence 6 = — sm a cos a -- -I Sa Bh . By = cos o -^ — h sm a -p , , 5. cos= a . , S^ ., • S« and b = — cos" a -- — 2 sin a cos a — cos a -, sin a It follows that the force R and the couple K are expressed in terms of I, r, a, Bh, Bx by the equations 1 ^^1? ^^^ ^°^' " "•" -^ ^^"^^ "^ Bh + (G- B) sin a cos a . rBx\ K =^[{0- B) sin a cos a .Bh + (C sin" a + B cos" a) rBx]. ...(42) If the spring is deformed by axial force alone* without couple, the axial displacement 8/i and the angular displacement S^ are given by the equations ., ,„/sin2a , cos'''a\ _ ^ , . /I IN „ * The results for this case were found by Saint- Venant, Paris, C. R ,t. 17 (1843). A number of special cases are worked out by Kelvin and Tait, loc. cit., and also by J. Perry, Applied MecMnics (London, 1899). The theory has been verified experimentally by J. W. Miller, Phys. Rev., vol. 14 (1902) . The vibrations of a spiral spring supporting a weight so great that the inertia of the spring may be neglected have been worked out in accordance with the above theory by L. E. Wilberforce, Phil. Mag. (Ser. 5), vol. 38 (1894). 271,272] VARIOUS PROBLEMS OF EQUILIBRIUM 423 If the cross-section of the spring is a circle of radius a, 1/C- IjB is AalEna^ where cr is Poisson's ratio and E is Young's modulus for the material. Hence both bh and 8^ are positive. In the same case fir is negative, so that the spring is coiled more closely as it stretches. 272. Additional results. (a) Rod subjected to termincd couples. "When a rod which is straight and prismatic in the unstressed state is held bent and twisted by terminal couples, the kinetic analogue is a rigid body moving under no forces. The analogue has been worked out in detail by W. Hess*. When the cross-section has kinetic symmetry so that A = B, the equations of equilibrium show that the twist t and the curvature (K^-t-K'^)^ are constants, and that, if we put as in Article 253 then B{dflds) = {B-C)T. It follows that the measure of tortuosity of the central-line is CtIB, and, therefore, that this line is a helix traced on a circular cylinder. If we use Euler's angles 6,^,<^3a in Article 253, aad take the axis of the helix to be parallel to the axis of z in Fig. 46 of that Article, 6 is constant, and \ir-e is the angle a of the helix. The axis of the terminal couple is the axis of the helix, and the magnitude of the couple is B cos air, as we found before, r being the radius of the cylinder on which the helix lies. (6) Straight rod with initial twist. When the rod in the unstressed state has twist tq and no curvature, and the cross- section has kinetic symmetry so that A = B, the rod can be held bent so that its central-line has the form of a helix (a, r), and twisted so that the twist is ri,by a wrench about the axis of the helix ; and the force R and couple K of the wrench are found by writing r, - tq for t in equations (36) and (37) of Article 270. (c) Rod bent into circular hoop and twisted uniformly. When the rod in the unstressed state is straight and prismatic, and the cross-section has kinetic symmetry, one of the forms in which it can be held by terminal forces and couples is that in which the central-line is a circle, and the twist is uniform along the length. The tension vanishes, and the shearing force at any section is directed towards the centre of the circle, and its amount is CV/r, where r is the radius of the circle. {d) Stability of rod subjected to twisting couple and thrust. When the rod, supposed to be straight and prismatic in the unstressed state, is held twisted, but without curvature, by terminal couples, these couples may be of such an amount as could hold the rod bent and twisted, li A=B the central-line, if it is bent, must be a helix. When the couple K is just great enough to hold the rod bent without dis- placement of the ends, the central-lino just forms one complete turn of the helix, the radius r of the helix is very small, and the angle a of the helix is very nearly equal to \n. We have the equations K=CT=:Br~^ cos a, lcoaa = 27rr, where t is the twist, and I the length of the rod. Hence this configuration can be maintained if 2ir/l=KIB. We infer that, under a twisting couple which exceeds 2wBll, the straight twisted rod is unstable. * Math. Ann., Bd. 23 (1884). 424 STABILITY UNDER TERMINAL THRUST AND COUPLE [CH. XIX This question of stability may be investigated in a more general manner by supposing that the rod is held by tei-minal thrust R and twisting couple ^ in a form in which the central-line is very nearly straight. The kinetic analogue is a symmetrical top which moves so that its axis remains nearly upright. The problem admits of a simple solution by the use of fixed axes of x, y, z, the axis of z coinciding with the axes of the applied couples and with the line of thrust. The central-line is near to this axis, and meets it at the ends. The twist r is constant,' and the torsional couple CV can be equated to K with sufScient approximation. The flexural couple is of amount Bjiy, where p is the radius of curvature of the central-line, and its axis is the binormal of this curve. The direction-cosines of this binormal can be expressed in such forms as ^dj (Pv. dz d^y\ ''\ds d^~ds'ds^)' and therefore the components of the flexural couple at any section about axes parallel to the axes of x and y can be expressed with sufficient approximation in the forms -B- b"-^,. Fig. 60. For the equilibrium of the part of the rod contained between this section and one end we take moments about axes drawn through the centroid of the section parallel to the axes of X and y, and we thus obtain the equations -+/i:^-y/e=o, 5^!|+z^+x/j= ds' ds .(43) 0. The complete primitives are X = £i sin (?i«-l- ci) + Li sin (jz* + f 2)1 y = Zi cos (ji s -I- e 1) -I- Z2 cos (.j'j s -f c 2), where ij, L^, ei, e^ are arbitrary constants, and q^, q^ are the roots of the equation Bq^ + Kq-R = Q. The terminal conditions are (i) that the coordinates x and y vanish at the ends s = and s = l, (ii) that the axis of the terminal couple coincides with the axis of z. The equations (4.3) show that the second set of conditions are satisfied if the first set are satisfied. We have therefore the equations Xj sin €1 4- Z2 fiin f 2 = 0, Zj cos f 1 + L^ cos eg = 0, and ti sin (ji ?-!-€,) -I- Z2 sin (?'2^ + f2) = 0, Zioos {qil-\-ei) + L2, cos {q2l + (^ = 0. On substituting for Z2 cos cj and Z2 sin e^ from the first pair in the second pair, we find the equations Zi{sin (^J-|-ei)-sin(g'2?+ei)} = 0, Ly{GQa{qil + ei) - cos.{qil + (i)} = 0, from which it follows that q^l and q.^l differ by a multiple of 27r. The least length I by which the conditions can be satisfied is given by the equation 27r/^=|gi-2'2|, ^2_ K^ R °^ P iB^'^B' 272] STABILITY OF FLAT BLADE BENT IN ITS PLANE 425 The rod subjected to thrust R aud twisting couple K is therefore unstable if p <4fi2"'"£ .(44) This condition* includes that obtained above for the case where there is no thrust, and also that obtained in (18) of Article 264 for the case where there is no couple. If the rod as subjected to tension instead of thrust, R is negative, and thus a sufiicient tension will render the straight form stable in spite of a large twisting couple. («) Stability of flat blade bent in its plane f. Let the section of the rod, be such that the flexural rigidity B, for bending in one principal plane, is large compared with either the flexural rigidity A, for bending in the perpendicular plane, or with the torsional rigidity C. This would be the case if, for example, the cross-section were a rectangle of which one pair of sides is much longer than Fig. 61. the other pair. Let the rod, built in at one end so as to be horizontal, be bent by a vertical transverse load R applied at the other end in the plane of greatest flexural rigidity. We shall use the notation of Article 253, and suppose, as in Article 270, that the line of action of the load R has the direction and sense of the line Pjz, and we shall take the plane of (z, x) to be parallel to the vertical plane containing the central-line in the unstressed state. If the length I, or the load R, is not too great, while the flexural rigidity B is large, the rod will be slightly bent in this plane, in the manner discussed in Chapter xv. But, when the length, or load, exceed certain limits, the rod can be held by the terminal force, directed as above stated, in a form in which the central- line is bent out of the plane (x, z), and then the rod will also be twisted. It will appear that the defect of torsional rigidity is quite as influential as that of flexural rigidity in rendering possible this kind of buckling. Let 3 be measured from the fixed end of the central-line, and let Xj, yi, zj be the coordinates of the loaded end of this line. Let x, y, z be the coordinates of any point Pi on the strained central-line. For the equilibrium of the part of the rod contained between the section drawn through Pi and the loaded end we take moments about axes drawn * The result is due to A. G. Greenhill, Proc. Inst. Mech. Engineers, 1883. t Cf. A. G. M. Michell, Phil. Mag. (Ser. 5), vol. 48 (1899), and L. Prandtl, ' Kippersoheinungen ' (Diss.), Niirnberg, 1899. The problem here solved and other problems, oonoerning the stability of a flat blade under various conditions of loading and support, are considered by J. Prescott, Phil. Mag. (Ser. 6), vol. 36, 1918, p. 297. 426 STABILITY OF FLAT BLADE BENT IN ITS PLANE [CH. XIX through P, parallel to the fixed axes. Using the direction-cosines defined by the scheme (4) of Article 253, we have the equations -{AKh + B<'h + CTk) + {ji-j) R = 0, \ -{AKmi + BK'm2+Cnn3)~{xi-x)R=0, V (45) AK-iix + BKrii + CTn^ =0. J When we substitute for k, k, t from equations (8) of Article 253, and for ly, ... from equations (7) of the same Article, we have AkIi + Bk'12+Ct13 = [ - ( J sin" + 5 oos^ (p) sin \lf + (A-B) sin cos sm-^sm6 + Ccos'^smdcosd]-Y, A Kmi + Bk' in2 + Crm^ = [{A sin" + 5 003^0) oosi/' + (4 -B) sin cos sin •\/^cos5]-i- + Csin 8am-\jf-^ -[{Acos'')smyj/am6cosd + (A -5)sin0cos0cos\|fsinfl- Csin\/fsin6cosfl]--^, A Kill + B k' 712 + Crrig = -{A-B) sin cos sin fl ^ + (7 cos 6^ + {A sin25cos20 + Ssin2 5 sin20 + Ccos"^)^ . In equations (45) we now approximate by taking A and G to be small compared with 5, and 6 to be nearly equal to Jtt, while and ^ are small, and also by taking Xj to be equal to I and x to be equal to s. We reject all the obviously unimportant terms in the expressions for (^1 kZi + ...), .... We thus find the equations Since dy j ds = mi=ain 6 am ■^ = ^ nearly, we deduce from the first and second equations of this set the equation and from the second and third equations of the same set we deduce the equation A'^-=R(l-s); and, on eliminating dy\rjds between the two equations last written, we find the equation ^S + ?(^-^)''^=^ (46) This equation can be transformed into Bessel's equation by the substitutions k = i{l-sfRN{AO, = ^ (?-«)* (47) It becomes and the primitive is of the form = [4'Jj(|) + 5'/_j(a](?-.)4, (48) where A' and ZC are constants. Now when s=l, d\jrjds vanishes, and the twisting couple Ct vanishes ; hence d(f>lds vanishes. This condition requires that A' should vanish. Further, vanishes when »=0, and thus the critical length is given by the equation J __ i (f)=0 at i=^PR/J{AC), or 2.6 AC^'"^^ ' 2.4... (2n). 6.14:.. .(8n-2) A^O'^'" 272, 273] ROD BENT BY LOCAL FORCES 427 The lowest root of this equation for R^l^/ACia 16 nearly, and we infer that the rod bent by terminal transverse load in the plane of greatest flexural rigidity is unstable if l>y{AC)*IR^, where y is a number very nearly equal to 2. The result has been verified experimentally by A. G. M. Michell and L. Prandtl. It should be observed that the rod, if of such a length as that found, will be bent a good deal by the load R, unless B is large compared with A and C, and thus the above method is not applicable to the general problem of the stability of the elastica for displacements out of its plane. 273. Rod bent by forces applied along its length. When forces and couples are applied to the rod at other points, as well as at the ends, and the stress-couples are assumed to be given by the ordinary- approximations (Article 255), forms are possible in which the rod could not be held by terminal forces and couples only. When there are no couples except at the ends, the third of equations (11) of Article 254 becomes C^^-{A-B)kk' = 0, and this equation shows that to hold the rod bent to a given curvature without applying couples along its length, a certain rate of variation of the twist along the length is requisite. In other words a certain twist, indeterminate to a constant pres, is requisite. When there are no applied couples except at the ends, and the curvature is given, while the twist has the required rate of variation, N and N' are given by the first two of equations (2) of Article 260. The requisite forces X, Y, Z of Article 254 and the tension T are then connected by the three equations (10) of that Article. We may therefore impose one additional con- dition upon these quantities. For example, we may take Z to be zero, and then we learn that a given rod can be held with its central-line in the form of a given curve by forces which at each point are directed along a normal to the curve, provided that the rod has a suitable twist. Similar statements are applicable to the case in which the rod, in the unstressed state, has a given curvature and twist. As an example* of the application of these remarks we may take the case of a rod which in the unstressed state forms a circular hoop of radius vg, with one principal axis of each cross-section inclined to the plane of the hoop at an angle /„, the same for all cross- sections. We denote by B the flexural rigidity corresponding with this axis. The initial state is expressed by the equations K„= -j-j-icos/o, Ko'=>-o"'sin/o, t-o=0. Let the rod be bent into a circular hoop of radius n, with one principal axis of each cross-section inclined to the plane of the hoop at an angle /i, the same for all cross-sections. The state of the rod is then expressed by the equations Ki=-ri-icos/i, Ki' = ri-isin/i, ri = 0. * Cf. Kelvin and Tait, Nat. Phil., Part ii., pp. 166 et seq. 428 FORM OF SUSPENDED WIRE [CH. XIX To hold the rod in this state forces must be applied to each section so as to be equivalent to a couple about the central-line ; the amount of this couple per unit of length is -— (4 sin /i cos/o - B cos/i sin/o) -—^{A-B) sin/j cos/i . 273 A. Influence of stiffness on the form of a suspended wire. As another example of the equilibrium of a thin rod under forces applied along its length, we consider the problem of a wire suspended from two fixed points at the same level*- We shall suppose that the wire is stretched taut under a high terminal tension, so that its central-line at any point is but slightly inclined to the horizontal, and denote the inclination of the tangent to the horizontal by 6. Then in the equations (10) and (11) of Article 254 and (12) of Article 255 we have to put K = T = V, K = ^r > as JQ as ' K=K'^@ = 0, X^-wcos0, Z = -wsmd, Y=0, where w is the weight of the wire per unit of length. The equations become -p- + r T w cos ^ = 0, as as dT .^d0 . . r. -J-- N -, w sin ^ = 0. ds ds Elimination of T and N yields the equation j,{d fd^e /de\ ded'O] r . . /d^e A//dey) ,, ^ ids W^j + d^M-'"r'''^^{d? °°^ V/[W \ = ^' which can be integrated in the form „ (fd^e „\ /de , d'9 . J , , .^ idd where a is a constant of integration. On putting ^ = 0, we see that a is the value of the tension T at a point where ^ = 0, so we shall write To for a. The equation can then be integrated again in the form d^d B -r^ sec 6 + ws = To tan 6, where no constant need be added if s is measured from a point where ^ = 0. If, as was supposed, 6 is everjrwhere small, this equation may be replaced by the simpler equation B'l^-To.e^-.s, * A. E. Young, Phil. Mag. (Ser. 6), vol. 29, 1915, p. 96. 273, 274] ROD BENT BY NORMAL PEESSDBE 429 and integrated in the form ^ = a cosh Xs + /3 sinh Xs + yj^ s, where X is written for V(2V-B), and a and /3 are arbitrary constants, and then a must be put equal to zero because vanishes with s. If the ends of the wire at the supports are constrained to be horizontal, and the length of the wire between the supports is I, we have ^ = when s = + i ^) and then '^ T.smhixr The length of the wire between the supports exceeds the distance between the supports by cos 6) ds, or approximately and this is J \2V \ ^ sinh IXl) w' f^ l^ U-" P ^24 X^ SXtsmh^Xl 16sinh^|x;;l The first term in this expression gives the excess length calculated by neglect- ing the stiffness, and the remaining terms give the correction for stiffness. 274. Rod bent in one plane by uniform normal pressure. We consider next the problem of a rod held bent in a principal plane by normal pressure which is uniform along its length. The quantity X of Article 254 expresses the magnitude of this pressure per unit of length. Let F denote the resultant of the shearing force JV and the tension T at any Cross-section, F^, Fy its components parallel to fixed axes of x and y in the plane of the bent central-line. We may obtain two equations of equilibrium by resolving all the forces which act upon any portion of the rod parallel to the fixed axes. These equations are ^F^ + xf- = 0, .}f,-xP = 0. ds ds ds ' ds It follows that the origin can be chosen so that we have F, = -yX, F, = xX; and therefore the magnitude of F at any point P of. the strained central-line is rX, where r is the distance OP, and the direction of F is at right angles to OP. This result can be expressed in the following form : — Let Pi and P^ be any two points of the strained central-line, and let F^ and F2 be the resultants of the shearing force and tension on the cross-sections through Pi and Pj, the senses of Pj and F^ being such that these forces arise from the action of 430 ROD BENT BY NORMAL PRESSURE [CH. XIX the rest of the rod on the portion between Pi and P^. From Pj, Pa draw lines PiO, P^O at right angles to the directions of Pj, P^. We may regard the arc P^P^ as the limit of a polygon of a large number of sides, and this polygon as in equilibrium under the flexural couples at its ends, the forces P,, Pj, and a force Xhs directed at right angles to any side of the polygon of which the length is hs. The forces are at right angles to the sides of the figure formed by OP^, OP^ and this polygon, and are proportional to them; and the lengths of OP-, and OP^ are F-^jX and Pa/X. The senses in which the lines must be drawn are indicated in Fig. 62 * Fig. 62. Let r denote the distance OP. Then ,dr N=- ds ds ' The stress-couple G' satisfies the equation ^H = -N=rX- ds ds ' Hence we have G' = \Xr'Jr const. In the particular case where the central-line in the unstressed state is a straight line or a circle, the curvature l/p of the curve into which it is bent is given by the equation £/p = JZr^-fconst (49) The possible forms of the central-line can be determined from this equationf. 275. Stability of circular ring under normal pressure. When the central-line in the unstressed state is a circle of radius a, and the rod is very slightly bent, equation (49) can be written in the approximate form ZBIa^ (50)* 276. Height consistent with stabilityf. As a further example of the equilibrium of a rod under forces applied along its length, we consider the problem of a vertical column, of uniform material and cross-section, bent by its own weight. Let a long thin rod be set up in a vertical- plane so that the lower end is constrained to remain vertical, and suppose the length to be so great that the rod bends. Take the origin of fixed axes of x and y at the lower end, draw the axis of x vertically upwards and the axis of y horizontally in the plane of bending. (See Fig. 63.) For the equilibrium of the portion of the rod contained between any section and the free end, we resolve along the normal to the central-line, and then, since the central-line is nearly coincident with the axis of x, we find the equation ^ I dx' where W is the weight of the rod. The equation of equilibrium dG/ds + N — can, therefore, be replaced by the approximate equation dx' Fig. 63. B^ + W'-^P = 0, I •(■51) where p is written for d,y/dx. The terminal conditions are that dp/dx vanishes at x = l, and y and p vanish at x = 0. Equation (51) can be transformed into Bessel's equation by the substitu- tions ^=i\/(S)^^-^)*' ^='?(^-^)* (52) * The result is due to M. Levy, loc. cit. t The theory is due to A. G. Greenhill, Cambridge Phil. Soc. Proc , vol. 4 (1881). It has been discussed critically by C. Chree, Cambridge Phil. Soc. Proc, vol. 7 (1892). 432 HEIGHT CONSISTENT WITH STABILITY [CH. XIX It becomes and the primitive is of the form p = iA'J^{^) + B'J_^m{l-x)\ (53) where A' and B' are constants. To make dp/dx vanish at x = l we must have A' = 0, and to make p vanish at x = we must have J_i (f ) = at f = m WjB)^. Hence the critical length is given by the equation 1 U 4- <^— V' : 1- =0 3.2 B ^■••^^ > 3.6...(3?i).2.5...(3n-l) 5" The lowest root of this equation for Z' TF/5 is (7 . 84 . . . ), and we infer that the rod will be bent by its own weight if the length exceeds (2.80 ...)n/{B/W). The numerical value agrees with that obtained by a different method by S. Timoschenko, loc. cit., p. 414. Greenhill {loc. cit, p. 405) has worked out a number of cases in which the rod is of varying section, and has applied his results to the explanation of the forms and growth of trees. CHAPTER XX. VIBRATIONS OF RODS. PROBLEMS OF DYNAMICAL RESISTANCE. 277. The vibrations of thin rods or bars, straight and prismatic when unstressed, fall naturally into three classes : longitudinal, torsional, lateral. The "longitudinal" vibrations are characterized by the periodic extension and contraction of elements of the central-line, and, for this reason, they will sometimes be described as "extensional." The "lateral" vibrations are characterized by the periodic bending and straightening of portions of the central-line, as points of this line move to and fro at right angles to its unstrained direction ; for this reason they will sometimes be described as "flexural." In Chapter xii. we investigated certain modes of vibration of a circular cylinder. Of these modes one class are of strictly torsional type, and other classes are effectively of extensional and flexural types when the length of the cylinder is large compared with the radius of its cross-section. We have now to explain how the theory of such vibrations for a thin rod of any form of cross-section can be deduced from the theory of Chapter xviii. In order to apply this theory it is necessary to assume that the ordinary approximations described in Articles 255 and 258 hold when the rod is vibrating. This assumption may be partially justified by the observation that the equations of motion are the same as equations of equilibrium under certain body forces — the reversed kinetic reactions. It then amounts to assuming that the mode of distribution of these forces is not such as to invalidate seriously the approximate equations (21), (22), (23) of Article 258. The assumption may be put in another form in the statement that, when the rod vibrates, the internal strain in the portion between two neighbouring cross-sections is the same as it would be if that portion were in equilibrium under tractions on its ends, which produce in it the instan- taneous extension, twist and curvature. No complete justification of this assumption has been given, but it is supported by the results, already cited, which are obtained in the case of a circular cylinder. It seems to be legitimate to state that the assumption gives a better approximation in the case of the graver modes of vibration, which are the most important, than in L. E. 28 434 EXTENSIONAL, TORSIONAL AND FLEXURAL MODES [CH. XX the case of the modes of greater frequency, and that, for the former, the approximation is quite sufficient. The various modes of vibration have been investigated so fully by Lord Rayleigh* that it will be unnecessary here to do more than obtain the equations of vibration. After forming these equations we shall apply them to the discussion of some problems of dynamical resistance. 278. Extensional vibrations. Let w be the displacement, parallel to the central-line, of the centroid of that cross-section which, in the equilibrium state, is at a distance s from some chosen point of the line. Then the extension is dw/ds, and the tension is Eo) (dw/ds), where E is Young's modulus, and m the area of a cross- section. The kinetic reaction, estimated per unit of length of the rod, is pa (dHu/df), where p is the density of the material. The equation of motion, formed in the same way as the equations of equilibrium in Article 254, is ^3^ = ^97 (1) The condition to be satisfied at a free end is dw/ds = ; at a fixed end w vanishes. If we form the equation of motion by the energy-method (Article 115) we may take account of the inertia of the lateral motion t by which the cross-sections are extended or contracted in their own planes. If x and y are the coordinates of any point in a cross- section, referred to axes drawn through its centroid, the lateral displacements are - a-x (dwjds), — ay (^wjds), where cr is Poisson's ratio. Hence the kinetic energy per unit of length is »'-{©'--©}. where K is the radius of gyration of a cross-section about the central-line. The potential energy per unit of length is and, therefore, the variational equation of motion is where the integration with respect to s is taken along the rod. In forming the variations we use the identities ow dSw dho . /dio . \ dw d&w d'w , d /dw ^ \ Tt-Tt+w^'"=wt u n' Ts^+w *"" =3; [di n ' fd^d^__d*w_ \_dfdhe_d8w_dHv_ \ d_(dhodhw 'd^w \ \dsdt dsdt dsm-' J ~ ds [Mi 'W ~ 9^2 ) "^ di {Mt "37 ~ 3^23^ / ' * Theory of Sound, Chapters vii. and viii. t The lateral strain is already taken into account when the tension is expressed as the product of E and m (dwjds). If the longitudinal strain alone were considered the constant that enters into the expression for the tension would not be E but \ + 2(U. 277-280] OF VIBRATION OP THIN RODS 435 and, on integrating by parts, and equating to zero the coefficient of 8w under the sign of double integration, we obtain the equation PyW.-"^ mfi) = ^ds^ (2) By retaining the term pcrK^d*w/ds^df' we should obtain the correction of the velocity of wave-propagation which was found by Pochhammer and Chree (Article 201), or the correction of the frequency of free vibration which was calculated by Lord Rayleigh*. 279. Torsional vibrations. Let ^p• denote the relative angular displacement of two cross-sections, so that d-\jr/ds is the twist of the rod. The centroids of the sections are not displaced, but the component displacements of a point in a cross-section parallel to axes of *' and y, chosen as before, are — ffry and ^j/x. The torsional couple is C(d-\jrjds), where C is the torsional rigidity. The moment of the kinetic reactions about the central-line, estimated per unit of length of the rod, is pooK^d^'^ldt'^). The equation of motion, formed in the same way as the third of the equations of equilibrium (11) of Article 254, is ,.^.S=c|i (3) The condition to be satisfied at a free end is dyjr/ds = ; at a fixed end yjr vanishes. When we apply the energy-method, we may take account of the inertia of the motion by which the cross- sections are deformed into curved surfaces. Let be the torsion- function for the section (Article 216). Then the longitudinal displacement is (j) (dyjr/ds), and the kinetic energy of the rod per unit of length is *'[-CI)"HSJ)7*'*]- The potential energy is J C (d^/dsY, and the equation of vibration, formed as before, is By inserting in this equation the values of C and jcjj^da that belong to the section, we could obtain an equation of motion of the same form as (2) and could work out a correction for the velocity of wave-propagation and the frequency of any mode of vibration. In the case of a circular cylinder there is no correction and the velocity of propagation is that found in Article 200. 280. Flexural vibrations. Let the rod vibrate in a principal plane, which we take to be that of (x, z) as defined in Article 252. Let u denote the displacement of the centroid of any section at right angles to the unstrained central-line. We may take the angle between this line and the tangent of the strained central- line to be dujds, and the curvature to be S^w/a*^. The flexural couple G' is Bd^u/ds\ where B = Eak'^, k' being the radius of gyration of the cross-section * Theory of Sound, § 157. 28—2 436 FLEXURAL VIBRATIONS OF RODS [CH. XX about an axis through its centroid at right angles to the plane of bending. The magnitude of the kinetic reaction, estimated per unit of length, is, for a first approximation, pw {o^u/dt-), and its direction is that of the displacement u. The longitudinal displacement of any point is -x{dit/ds); and therefore the moment of the kinetic reactions, estimated per unit of length, about an axis perpendicular to the plane of bending is p'»k'^ (dhi/dsdt^). The equations of vibration formed in the same way as the first of the set of equations (10) and the second of the set of equations (11) of Article 254 are and, on eliminating iV, we have the equation of vibration „ (p^k-^,)=-Ek-p^ (5) If "rotatory inertia" is neglected we have the approximate equation Pd¥ = -^'' d?' (^^ and the shearing force N at any section is — Ea)k''d^u/ds\ At a free end d^ujds^ and d'ujd^ vanish, at a clamped end u and du/ds vanish, at a "supported" end u and d^ulds^ vanish. By retaining the term representing the effect of rotatory inertia we could obtain a correction of the velocity of wave-propagation, or of the frequency of vibration, of the same kind as those previously mentioned*. Another correction, which may be of the same degree of importance as this when the section of the rod does not possess kinetic symmetry, may be obtained by the energy-method, by taking account of the inertia of the motion by which the cross-sections are distorted in their own planes t. The components of displacement parallel to axes of ,r and y in the plane of the cross-section, the axis of x being in the plane of bending, are o'^ti on and the kinetic energy per unit of length is expressed correctly to terms of the fourth order in the linear dimensions of the cross-section by the formula i"{®'-'^'--'^iSi.*^{^))' where k is the radius of gyration of the cross-section about an axis through its centroid drawn in the plane of bending. The term in cT{k'^-k^) depends on the inertia of the motion by which the cross-sections are distorted in their planes, and the term in if^ depends on the rotatory inertia. The potential energy is expressed by the formula * Cf. Lord Eayleigh, Theory of Sound, § 186. + The cross-sections are distorted into curved surfaces and inclined obliquely to the strained central-line, but the inertia of these motions would give a much smaller correction. 280, 281] LONGITUDINAL IMPACT 437 The variational equation of motion is = 0. In forming the variations we use the identities 9i! os'^iB!;^ e< Zs^Zr ds^ofi it \ ds^bt^ ot as well as identities of the types used in Article 278. The resulting equation of motion is B-^' a-)+*M J^,V -Ef^'t^ (7) Corrections of the energy such as that considered here will, of course, affect the terminal conditions at a free, or supported end, as well as the differential equation of vibration. Since they rest on the assumption that the internal strain in any small portion of the vibrating rod contained between neighbouring cross-sections is the same as in a prism in which the right extension, or twist, or curvature is produced by forces applied at the ends and holding the prism in equilibrium, they cannot be regarded as very rigorously established. Lord Eayleigh (loc. cit.) calls attention to the increase of im- portance of such corrections with the frequency of the vibration. We have already remarked that the validity of the fundamental assumption diminishes as the frequency rises. 281. Rod fixed at one end and struck longitudinally at the other*. We shall illustrate the application of the theory of vibrations to problems of dynamical resistance by solving some problems in which a long thin rod is thrown into extensional vibration by shocks or moving loads We take first the problem of a rod fixed at one end and struck at the other by a massive body moving in the direction of the length of the rod. We measure t from the instant of impact and s from the fixed end, and we denote by I the length of the rod, by m the ratio of the mass of the striking body to that of the rod, by V the velocity of the body at the instant of impact, by w the longitudinal displacement, and by a the velocity of propagation of extensional waves in the rod. The differential equation of extensional vibration is a^ = ^ 9i^ ^^^ The terminal condition at s = is w = 0. The terminal condition at s=lis the equation of motion of the striking body, or it is ^^9^ = -^^^ (^^ * Cf. J. Boussinesq, Applications des potmtiels . . . , pp. 508 et leq., or Saint-Venant In the 'Annotated Clebsch,' Note finale du § 60 and Ghangements et additions. A new and powerful method of solving problems of the kind here discussed has been devised by T. J. I'A. Bromwieh, London, Proc. Math. Soc. (Ser. 2), vol. 15 (1916), p. 401, and further developed by him in Phil. Mag. (Ser. 6), vol. 37 (1919), p. 407. 438 LONGITUDINAL IMPACT [CH. XX since the pressure at the end is, in the notation of Article 278, - Ew (dw/ds), and Ewja' is equal to the mass of the rod per unit of length. The initial condition is that, when t = 0, w; = for all values of s between and I, but at s = Z lim (dwldt)=-V, (10) i= +0 since the velocity of the struck end becomes, at the instant of impact, the same as that of the striking body. We have to determine w for positive values of t, and for all values of s between and I, by means of these equations and conditions. The first step is to express the solution of the differential equation (8) in the form w ==f{at- s) + F{at + s), (11) where /and F denote arbitrary functions. The second step is to use the terminal condition at s = to eliminate one of the arbitrary functions. This condition gives in fact f(at) + F(at) = 0, and we may, therefore, write the solution of equation (8) in the form w=fiat-s)-f{at + s) (12) The third step is to use the initial conditions to determine the function / in a certain interval. We think of / as a function of an argument ^, which may be put equal to at — s or at + s when required. Since dw/ds and dw/dt vanish with t for all values of s between and I we have, when l>^>0, -/'(-0-/'(f) = 0, /'(-f)-/'(f) = 0. Hence it follows that, when 1>^>-1, f'(^) vanishes and /(f) is a constant which can be taken to be zero ; or we have the result when 1>^>-1, /(0 = (13) The fourth step is to use the terminal condition (9) at s = ^ to form an equation by means of which the value of /(^ as a function of ^ can be determined outside the interval l> ^>-l. The required equation, called the "continuing equation*," is ml [/" (at - 1) -/" {at + 0] =/' {at -l)+f' {at + 1), or, as it may be written, /"(r) + (iM)/'(o=/"(r-2o-(i/mo/'(i:-2Z) (m) We regard this equation in the first instance as an equation to determine /'(D- The right-hand member is known, it has in fact been shown to be zero, in the interval U>^>1. We may therefore determine the form of /'(t) in this interval by integrating the equation (14). The constant of integration is to be determined by means of the condition (10). The function * Equation promotrice of Saint- Vcnant. 281] ROD STRUCK AT ONE END 439 /' (f) will then be known in the interval 31>^>1, and therefore the right- hand member of (14) is known in the interval 51 > ^> Bl. We determine the form of/' (f) in this interval by integrating the equation (14), and we deter- mine the constant of integration by the condition that there is no discon- tinuity in the velocity at s = Z after the initial instant. The function /' (f) will then be known in the interval 5l> ^> 31. By proceeding in this way we can determine/' (?) for all values of f which exceed — I. The integral of (14) is always of the form where is a constant of integration. When 3l> ^> I the expression under the sign of integration vanishes, and f (f) is of the form Ce~^/'"'- Now the condition (10) gives a[f'{-l + 0)-f'{l + 0)] = -V,oTf'il + 0)=V/a. Hence Ge~'^l^ = Vja, and we have the result when3«>f>Z, /'(?) = -«- «-«/«' (16) We observe that /' (?) is discontinuous at ? = Z. When 5Z > ? > 3Z we have f" (^_ 2^) - (l/mO/'(?- 2i) = - 2 {Vjmla) e-(^-3i)Mi, and equation (15) can be written /'(?) = Ge-f/™^ - 2 ( YImla) {^-3l)e- if-s')/™'. The condition of continuity of velocity at s = Z at the instant t = 2lja gives f'(l-0)-f'{3l-0)=f'(l + 0)-f'(3l + 0), V V a a giving C=(F/a)(eV».+ e3/m). Hence, when 5l>^> 31, no = ^e-'f-''''»' + ^{i - |^(r- 3o} e-'f-«^'/'»^ (17) When 7Z > ? > 5Z we have /" (? - 20 - ^/' (r- 20 = - ^ [e- 'f-""^'"' + 2e- ^ > 51, V /'(f) = _. e- ^>l, 51 > ^ > 31, ... and determine the constants of integration so that f(l) = and /(f) is continuous. We find the following results : when Sl>^> I, /(f) = (wiZF/a)(l - e-'f-"/""} ; when 51>^>21, when 11 >^> 51, /(f) = ~{^- e-'^-^'""^} +~b- + ^i (?- 30^ e-«-MM^ \ /(f) = y -(19) ilV + J^.(f-50Je-'f-w/».'; The solution expresses the result that, at the instant of impact, a wave of compression sets out from the struck end, and travels towards the fixed end, where it is reflected. The motion of the striking body generates a continuous series of such waves, which advance towards the fixed end, and are reflected there. 281, 282] ROD STRUCK AT ONE END 441 In the above solution we have proceeded as if the striking body became attached to the rod, so that the condition (9) holds for all positive values of t ; but, if the bodies remain detached, the solution continues to hold so long only as there is positive pressure between the rod and the striking body. When, in the above solution, the pressure a,t s = l becomes negative, the impact ceases. This happens when f {at -1)+/' {at + 1) becomes negative. So long as 2l>at >0 this expression is equal to {Vja) e^ <"/">! which is positive. When il>at >2l, it is Ze-a,«r . --A at -21 .[,,,,.(,_ ti?)], which vanishes when 2aat>2l if 2 + e""2/'"< 4/m. Now the equation 2 + e"2/"'=4/»?i has a root lying between m=l and m=2, viz.: to = 1-73.... Hence, if m< 1-73, the impact ceases at an instant in the interval 4Z/a>i>2f/a, and this instant is given by the equation i! = -(2 + rB+^me-2/'»). If m>l'73 we may in like manner determine whether or no the impact ceases at an instant in the interval 6Z/a><>4Z/a, and so on. It may be shown also that the greatest compression of the rod occurs at the fixed end, and that, if m< 5, its value is 2 (I + e~2/™) V/a, but, if m>5, its value is approximately equal to (1 +^ot) Via. If the problem were treated as a statical problem by neglecting the inertia of the rod, the greatest compression would be Jm{Vla). For further details in regard to this problem reference may be made to the authorities cited on p. 437. 282. Rod free at one end and struck longitudinally at the other*. Wheu the end « = is free, dwjds vanishes at this end for all values of t, or we have -/' {at)+F' {at) = 0. Hence we may put F{Q=f{() and write instead of (12), w=f{at — s)+f{at + s), and, as before, we find that /(f) vanishes in the interval 1>^>—1. The continuing equation is now /" (f) + (iM)/(C= -f'(c-m+aM)f{C-2l) and the discontinuity of/' (f) at f =? is determined by the equation a{f {-l+0)+f' {l+0)]= - V, or f{l + 0)= - V/a. Hence we find the results : when 31>C>1, f {0 = - - e-'f" '""", Vm! when 51>C>31, /'(f)= - J«-'f"'""''+I{l -J-^(f-30}e-(f- =')""'. Now the extension at s = l is f{at+l)-f' {at -I), and, until t=2lla, this is -(F/a)e-«"'"', which is negative, so that the pressure remains positive until the instant t = 2l/a ; but, immediately after this instant, the extension becomes ( V/a) (2 - e-^""), which is positive, so "* Cf . J. Boussinesq, loc. cit. , p. 437. 442 DYNAMICAL RESISTANCE [CH. XX that the pi-essure vanishes, and the impact ceases at the instant t=2l/a, that is to say after the time taken by a wave of extension to travel over twice the length of the rod. The wave generated at the struck end at the instant of impact is a wave of compression ; it is reflected at the free end as a wave of extension. The impact ceases when this reflected wave reaches the end in contact with the striking body. The state of the rod and the velocity of the striking body at this instant are determined by the above formulae. The body moves with velocity Fe-^/™ in the same direction as before the impact ; and the rod moves in the same direction, the velocity of its centre of mass being mV (l-e-"/^). The velocity at any point of the rod is 2 Fe-'/"» cosh {s/ml), and the extension at any point of it is 2 ( V/a) e "* sinh {s/ml), so that the rod rebounds vibrating. 283. Rod loaded suddenly. Let a massive body be suddenly attached without velocity to the lower end of a rod, which is hanging vertically with its upper end fixed. With a notation similar to that in Article 281, we can write down the equation of vibration in the form w^"-^^"-^' ^^^^ and the value of w in the equilibrium state is ^gs{2,l — s)/a\ Hence we write w = lgs{2l-s)/a^ + w, (21) and then w' must be of the form w' = 4,{at-s)-(j>{at + s), (22) and, as before, we find that, in the interval 1>^>—1, 4>(X) vanishes. The equation of motion of the attached mass is ^-^ -^-m.. <-> which gives the continuing equation f (r)+^jf (n=<^"(?-20- ^^>l, \ ,^'(f) = _-^TO?{l-e-'{at-s)-<\)' (at + s), and, at the fixed end, this is equal to lgla^-2' (at), or lglai + 2 (g/a V)f{at), where / is the function so denoted in Article 281. The maximum value occurs when f{at) = 0. Taking m, = l, so that the attached mass is equal to the mass of the rod, we find from (16) that/'(a<) does not vanish before t = 3l/a, but from (17) that it vanishes between t—Zlja and t=bl/a if the equation l+e^l-2{C-Zl)/l} = has a root in the interval 5Z>f>3Z. The root is f=;{3 + |(l + l/e^)}, or f = Z (3-568), which is in this interval. The greatest extension at the fixed end is ^ {1 + 2« - 2-M8 [ _ 1 + e2 |i + 2 (0-568)}] }, or (Ig/a'^) (l+4e-»-668), or (3-27) Igla^. The statical strain at the fixed end, when the rod supports the attached mass in equilibrium, is 2lgla^, and the ratio of the maximum dynamical strain to this is 1-63 ; 1. This strain occurs at the instant i = (3-568) l/a. Taking to=2, so that the attached mass is twice the mass of the rod, we find from (16) that/' (at) does not vanish before t = 3lla, but from (17) that it vanishes between t=Zlla and t=blla if the equation l + e{l-(f-30/?} = has a root in the interval 51>(>ZI. The root is f=Z(4+l/e), or f=i (4-368), which is in this interval. The greatest extension at the fixed end is ^ {1 + 4e-i (3-388) [ - I + (1 + 1 -368) «] }, or Igja^ {l + 8e~''-^ or {b-04) Igja^. The statical strain in this case is Slgja^, and the ratio of the maximum dynamical strain to the statical strain is 1-68 : 1. This strain occurs at the instant <= (4-368) l/a. Taking »i = 4, so that the attached mass is four times the mass of the rod, we find from (17) that/'(a<) does not vanish before t=5lja, but from (18) that it vanishes between t=5lla and t—llja if the equation ^-hU-^im e*+[l -(f- 50/Z+i (f- blW]e = () has a root in the interval 7Z>f>5Z. The smaller root is f =2(6183), which is in this interval. The greatest extension at the fixed end is where f is given by the above equation. The extension in question is therefore % [9+8«-i('-i83) {2e-i- (1-183)}], which is found to be {9'18) (Ig/a^). The statical strain in this case is b{lgja^), and the ratio of the maximum dynamical strain to the statical strain is 1-84 nearly. This strain occurs at the instant t= (6-183) Ija. The noteworthy result is that, even when the attached mass is not a large multiple of the mass of the rod, the greatest strain due to sudden loading does not fall far short of the theoretical limit, viz. twice the statical strain. (Of. Article 84.) The principles to be applied to problems involving sudden changes of longitudinal motion have been perhaps 444 DYNAMICAL RESISTANCE [CH. XX sufiaciently exemplified in this Article and the two preceding. An example of practical imporUnce is solved in a paper by J. Perry " Winding ropes in mines," Phil. Mag. (Ser. 6), vol. 11, 1906, p. 107. 284. Longitudinal impact of rods. The problem of the longitudinal impact of two rods or bars has been solved by means of analysis of the same kind as that in Article 281*. It is slightly more complicated, because different undetermined functions are required to express the states of the two bars; but it is simpler because these functions are themselves simple. The problem can be solved also by considering the propagation of waves along the two rodsf . The extension e and velocity v at the front of an extensional wave travelling along a rod are connected by the equation e = - vja. (Cf. Article 205.) The same relation holds at any point of a wave of compression travelling entirely in one direction, as is obvious from the formula w =f{at - s) which characterizes such a wave. When a wave of compression travelling along the rod reaches a free end, it is reflected ; and the nature of the motion and strain in the reflected wave is most simply investigated by regarding the rod as produced indefinitely, and supposing a wave to travel in the opposite direction along the continuation of the rod in such a way that, when the two waves are superposed, there is no compression at the end section. It is clear that the velocity propagated with the " image " wave in the continuation of the rod must be the same as that propagated with the original wave, and that the extension propagated with the "image' wave must be equal numerically to the compression in the original wave J:. Now let I, V be the lengths of the rods, supposed to be of the same material and cross-section §, and let V, V be their velocities, supposed to be in the same sense. We shall take I > I'. When the rods come into contact the ends at the junction take a common velocity, which is determined by the condition that the system consisting of two very small contiguous portions of the rods, which have their motions changed in the same very short time, does not, in that time, lose or gain momentum. The common velocity must therefore be ^{V-\- V). Waves set out from the junction and travel along both rods, and the velocity of each element of either rod, relative to the rod as a whole, when the wave reaches it, is \(V~V'), so that the waves are waves of compression, and the compression is ^(F~F')/a. To trace the subsequent state of the shorter rod Z', we think of this rod as continued indefinitely beyond the free end, and we reduce it to rest by impressing on the whole system a velocity equal and opposite to V. At the ♦ Saint- Venant, J. de Math. (Liouville), [Sir. 2), t. 12 (1867). t Cf. Kelvin and Tait, Nat. Phil., Part i., pp. 280, 281. t Cf. Lord Eayleigh, Theory of Sound, vol. 2, § 257. § Saint-Venant, loc. cit. , discusses the case of different materials or sections as well. 283, 284] LONGITUDINAL IMPACT OF RODS 445 instant of impact a positive wave* starts from the junction and travels along the rod; the velocity and compression in this wave are i(F~F') and ^{V»^V')/a. At the same instant a negative "image" wave starts from the section distant 2i'from the junction in the fictitious continuation of the rod; the velocity and extension in this " image " wave are |( y~ V) and ^( F~ V')/a. After a time I'ja from the instant of impact both these waves reach the free end, and they are then superposed. Any part of the actual rod in which they are superposed becomes unstrained and takes the velocity V'^V. When the reflected wave reaches the junction, that is to say after a time 21' /a from the instant of impact, the whole of the rod I' is moving with the velocity V^^ V, and is unstrained. Hence, superposing the original velocity V, we have the result that, after the time taken by an extensional wave to travel over twice the length of the shorter rod, this rod is unstrained and is moving with the velocity V originally possessed by the longer rod. To trace the state of the longer rod I from the beginning of the impact, we think of this rod as continued indefinitely beyond its free end, and we reduce it to rest by impressing on the whole system a velocity equal and opposite to V. At the instant of impact a positive wave starts from the junction and travels along the rod; the velocity and compression in this wave are ^ ( F~ V) and J ( F^ V')/a. At the same instant a negative "image" wave starts from the section distant 21 from the junction in the fictitious continua- tion of the rod; the velocity and extension in this "image" wave are iCF^F') and |(F~F')/a. After a time 21' ja from the instant of impact the junction end becomes free from pressure, and a rear surface of the actual wave is formed. Hence, the rod being regarded as continued indefinitely, the wave of compression and the "image" wave of extension are both of length 21'. Immediately after the instant 21' /a the junction end becomes unstrained and takes zero velocity. Hence, superposing the original velocity F, we see that this end takes actually the velocity F, so that the junction ends of the two rods remain in contact but without pressure. The state of the longer rod I between the instants 2l'/a and 2l/a is determined by superposing the waves of length 21', which started out at the instant of impact from the junction end and the section distant 21 from it in the fictitious continuation of the rod. After a time greater than l/a these waves are superposed over a finite length of the rod, terminated at the free end, and this part becomes unstrained and takes a velocity F~ V, the velocity — F being supposed, as before, to be impressed on the system. The state of the rod at the instant 2l/a in the case where I > 21' is different from the state at the same instant in the case where I < 21'. Iil> 21' the wave of compression has passed out of the rod, and the wave of extension occupies a length 21' * An extensional wave is "positive" or "negative" according as the velocity of the material is in the same sense as the velocity of propagation or in the opposite sense. 446 DYXAMICAL RESISTANCE [CH. XX terminated at the junction. The strain in this portion is extension equal to -|(F^F')/a and the velocity in the portion is i(F~F'), the velocity - Fbeing impressed as before. The remainder of the rod is unstrained and has the velocity zero. Hence, superposing the original velocity F, we see that a length I - -21' terminated at the free end has at this instant the velocity V and no strain, and the remainder has the velocity |( F+ F') and extension ^( F~ V')la. The wave in the rod is now reflected at the junction, so that it becomes a wave of compression travelling away from the junction, the compression is ^( F'x- V')la and the velocity of the junction end becomes V. The ends that came into contact have now exchanged velocities, and the rods separate. If I < 21' the waves of compression and extension are, at the instant 2l/a, superposed over a length equal to 21' — I terminated at the free end, and the rest of the rod is occupied by the wave of extension. The velocity — V being impressed as before, the portion of length 21' — I terminated at the free end is unstrained and has the \elocity V~V', and the remaining portion has extension ^{V'^V')/a and velocity |(F~^F'). Hence, superposing the original velocity F, we see that a length 21' — I terminated at the free end has at the instant the velocity V and no strain, and the remainder has the velocity i( F + F') and the extension l(V'>'V')/a. The wave is reflected at the junction, as in the other case, and the junction end takes the velocity V. In both cases the roHs separate after an interval equal to the time taken by a wave of extension to travel over twice the length of the longer rod. The shorter rod takes the original velocity of the longer, and rebounds without strain; while the longer rebounds in a state of vibration. The centres of mass of the two rods move after impact in the same way as if there were a " coefficient of restitution " equal to the ratio l' : I. 284 A. Impact and vibrations. Reference has already been made in the Introduction (pp. 25, 26) to the suggestion that the phenomena of impact, and, in particular, the existence of the Newtonian " coefficient of restitution " might be traced to the presence after impact of some energy existing in the form (jf vibrations of the bodies which have come into collision* The result which has just been obtained appeared, at first sight, to corroborate this suggestion; but the difficulty arose that the result is not verified by experiment. This difficulty led Voigtf to imagine that some special conditions must hold near the ends of two rods which impinge longitudinally, or, in other words, that the rods should be thought of as separated by a layer of transition, in which the determining circumstance is the geometrical character of the terminal surfaces. This matter has been further investigated by J. E. Sears+. He made an elaborate * See Kelvin and Tait, Xat. Phil, Part i , §§ 302—304. t See Introduction, footnote 118. + Cambridge Proc. Phil. Soc, vol. 14, 1908, p. 257, and Cambridge Trans. Phil. Soc, vol. 21, 1912, p. 49. 284, 285] DYNAMICAL RESISTANCE 447 series of experiments on the longitudinal impact of metal rods with rounded ends, and constructed a theory, according to which the state of a small portion of either rod near the ends that come into contact is determined by Hertz's theory of impact (Chapter Vill. supra), while the state of the remaining portions is determined by Saint-Venant's theory, described in Article 2S4. Sears' theory was confirmed by experiment. In regard to the general question of vibrations set up in bodies by impact reference may be made to Lord Rayleigh, Phil. Mag. (Ser. 6), vol. 11, 1916, p. 283, or Scientific Papers, vol. 4, p. 292. Further experiments on impact are described by B. Hopkinson, loc. cit. ante, p. 115. 285. Problems of dynamicaJ resistance involving transverse vibration. The results obtained in Articles 281 — 284 illustrate the general character of dynamical resistances. Similar methods to those used in these Articles cannot be employed in problems that involve transverse vibration for lack of a general functional solution of the equation (6) of Article 280*. In such problems the best pi-ooedure seems to be to express the displacement as the sum of a series of normal functions, and to adjust the constant coefficients of the terms of the series so as to satisfy the initial conditions. For examples of the application of this method reference may be made to Lord Rayleigh + and Saint- VenantJ. A simplified method of obtaining an approximate solution can sometimes be employed. For example, suppose that the problem is that of a, rod "supported" at both ends and struck by a massive body moving with a given velocity. After the impact let the striking body become attached to the rod. At any instant after the instant of impact we may, for an approximation, regard the rod as at rest and bent by a certain transverse load applied at the point of impact. It wiU have, at the point, a certain deflexion, which is determined in terms of the load by the result of Article 247 {d). The load is equal to the pressure between the rod and the striking body, and the deflexion of the rod at the point of impact is equal to the displacement of the striking body from its position at the instant of impact. The equation of motion of the striking body, supposed subjected to a force equal and opposite to this transverse load, combined with the conditions that, at the instant of impact, the body has the prescribed velocity, and is instantaneously at the point of impact, are sufficient conditions to determine the displacement of the striking body and the pressure between it and the rod at any subsequent instant. In this method, sometimes described as Cox's method §, the deflexion of the rod by the striking body is regarded as a statical effect, and thus this method is in a sense an anticipation of Hertz's theory of impact (Article 139). It has already been pointed out that a similai- method was used also by Willis and Stokes in their treatment of the problem of the travelling load 1 1. * Fourier's solution by means of definite integrals, given in the Bulletin des Sciences a la Societe phihtiiatique, 1818, (cf. Lord Eayleigh, Theory of Sound, vol. 1, § 192), is applied to problems of dynamical resistance by J. Boussinesq, AppHcation.< dfs Potenticls, pp. 456 et seq. t Theory of Sound, vol. 1, § 168. t See the ' Annotated Clebseh,' Xote du § 61. § H. Cox, Cambridge Phil. Soc. Trans., vol. 9 (1850). Cf. Todhunter and Peareon's History. vol. 1, Article 1435. II See Introduction, p. 26. 448 WHIRLING OF SHAFTS [CH. XX A somewhat similar method has been employed by Lord Rayleigh* for an approximate determination of the frequency of the gravest mode of transverse vibration of a rod. He set out from a general theorem to the effect that the frequency of any dynamical system, that would be found by assuming the displacement to be of a specified type, cannot be less than the frequency of the gravest mode of vibration of the system. For a rod clamped at one end and free at the other, he showed that a good approximation to the frequency may be made by assuming the displacement of the rod to be of the same type as if it were deflected statically by a transverse load, concentrated at a distance from the free end equal to one quarter of the length. This method has been the subject of some dis- cussion t. It has been shown to be apphcable to the determination, of the frequency of the gravest mode of transverse vibration of a rod of variable cross-section |. It has been shown also that a method of successive approximation to the various normal functions for such a rod, and their frequencies, can be founded upon such solutions as Lord Rayleigh's when these solutions are regarded as first approximation s§. 286. The whirling of shafts||. A long shaft rotating between bearings remains straight at low speeds, but when the speed is high enough the shaft can rotate steadily in a form in which the central-line is bent. The shaft is then said to " whirl." Let u be the transverse displacement of a point on the central-line, CI the angular velocity with which the shaft rotates. When the motion is steady the equation of motion, formed in the same way as equation (6) in Article 280, is -pn^a = -Ek'^^^, (26) and the solution of this equation must be adjusted to satisfy appropriate conditions at the ends of the shaft. We shall consider the case in which the ends s = and s = I are " supported." The equation is the same as that for a rod executing simple harmonic vibrations of speed 27r/fi. In order that the equation ^^■''S = ^"^" (27) may have a solution which makes u and 2/0, ^0. with origin at the unstrained position P of the particle, will be denoted by u, v, w. The rod will receive a new curvature and twist, defined, as in Articles 252 and 259, by means of a moving system of "principal torsion-flexure axes." We recall the conventions that the axis of 2 in this system is directed along the tangent of the strained central-line at the point Pi to which P is displaced, and that the plane of («, z) is the tangent plane at Pj of the surface made up of the aggregate of particles which, in the unstressed state, lie in the plane of {sc^, z^ at P. We have denoted the components of curvature and the twist of the strained central-line at Pj by 1^1, «/, Tj. When the displacement (w, v, w) of any point of the central-line is known, the tangent of the strained central-line at any point is known, and it is clear that one additional quantity will suffice to determine the orientation of the axes of {x, y, z) at Pj relative to the axes of (xo, 1/0, ^0) at P. We shall take this quantity to be the cosine of the angle between the axis of x at Pj and the axis of y^ at P, and shall denote it by /3. The relative orientation of the two sets of axes may be determined by the orthogonal scheme of transformation H yo h a; Li M, A\ y X3 M.J, N, z Xs M, N, •(1) in which, for example, L^ is the cosine of the angle between the axis of x at Pj and the axis of x^ at P. We shall express the cosines L^, ... , the com- ponents of curvature «i, ki and the twist tj in terms of u, v, w, /3. 289. Orientation of the principal torsion-flexure axes. The direction-cosines L^, M3, N^, are those of the tangent at Pi to the strained central-line referred to the axes of x^, y^, z^ at P Now the co- ordinates of Pi referred to these axes are identical with the components of displacement u, v, w. Let P' be a point of the unstrained central-line near to P, let' hs be the arc PF, and let hx^, hj^, 8^0 be the coordinates of P' referred to the axes of «„, ?/„, «» at P, also let ^, t), ^ be the coordinates of P/, the displaced position of P, referred to the same axes. The limits such as 29—2 452 KINEMATICAL FORMULA FOR [CH. XXI iim {^-u)IBs are the direction-cosines L,,.... Let («', v', w') be the dis- placement of P' referred to the axes of .r„, y„, Zo at P', and {U', V, W) the same displacement referred to the axes of x^, y^, «o at P. Then it V, r) = (S^o + U', Sy, + r, Bzo + W). The limits of Sx^/Ss, By^/Bs, Bzo/Bs are 0, 0, 1. The limits of (u - u)/Bs,... are dulds, ... and we have the usual formulae connected with moving axes in such forms as ,. U' — u du , lim — 5: = -, VTo + WK„ . 8s=o OS as Hence we obtain the equations is = -^ - OTo + W«:„', M3=-£-WK„ + ttr„ F3=l+-£-UKo' + VK„. ...{2) The equation L^' + M^ -\- N^^=\ leads, when we neglect squares and products of u, V, w, to the equation -1 UKo +VK„=0, (3) which expresses the condition that the central-line is unextended. In conse- quence of this equation we have i^3 = 1. The direction-cosines of the axes of x, y at P,, referred to the axes of Xo, yo, Zo at P, are determined by the conditions that M^ is /S and that the scheme of transformation (1) is orthogonal and its determinant is 1. These conditions give us A = l, Jlf, = /3, N, = -L„ Z.,= -/3, M,= \, N, = -M,.\ ^^^ These equations might be found otherwise from the formulae (7) of Article 253 by writing Xi, ... instead of /j, ... , taking 6 to be small, and putting /8 for + yfr. They are, of course, correct to the first order in the small quantities u, v, w, /8. 290. Curvature and twist. For the calculation of the components of curvature and the twist we have the formulae (6) of Article 253, in which Ki, ... are written for k In those formulae Zj, ... denoted direction-cosines of the axes of a;, y, z referred to fixed axes. Here we have taken L-^, ... to denote the direction-cosines of the axes of X, y, z at Pj referred to the axes of x^, y„, z^ at P If P' is a point near to P, so that the arc PP' = Bs, and P/ is the displaced position of P', we may denote by Z/, . . . the direction-cosines of the axes of x, y, z at P/ referred to the axes of x^, y^, z„ at P', and then the limits such as lim (A' — Li)/Bs are s«=o 289-291] CURVED BOD SLIGHTLY DEFORMED 453 the differential coefficients such as dLijds. Let the fixed axes of reference for ^1, ... be the axes oi x^, y^, Za at P, and let l^ + hl^, ... denote the direction- cosines of the axes of x, y, z at P/ referred to these fixed axes. Then the limits such as lim SZ]/8s are the differential coefficients such as dli/ds. It is clear that, at P, l^ = L-^, ... but that dli/ds=^dLJds, .... We have in fact the usual formulse connected with moving axes, viz. : — dlj/ds = dLJds — M-iTo + N^ko, dvii/ds = dMJds — NiK,, + L^t^, dnjds = dNJds — L^Ko' + M^k^, with similar formulae for dl^/ds, . . . and dlg/ds, .... In the formulae (6) of Article 253 we write k^, ... ior k,... , put Wj = iVs = 1, replace k,... by the values found for ii, ... in (2) and (4), and substitute the values just found for dl-^jds, Rejecting terms of the second order in the small quantities u, v, w, l3, we obtain the equations «i = «o + P«o - -^- Tolls, Tl = To + -J- + KaLs + Ko'Mg, j in which L3 and Ms are given by the first two of equations (2). 291. Simplified formulae. The formulae are simplified in the case where /, = ^tt. In this case the axis of Xo, which is a principal axis of a cross-section at a point of the unstrained central-line, coincides with the principal normal of this curve at the point. When this is the case we have ICo=0, Ki,' = l/po, T(,= l/So, du V w ^ _dv u ^ I .(5) _/3 d (^,^\_]u(^_Ld>0 iu= -Xi{a?IB)[eln~\{l-co!ie-\esi.n6)'\, u=dw/d6. The displacements are clearly the same at any two points symmetrically situated on opposite sides of this diameter. We may deduce the value of u at any point, and we may prove that the diameter which coincides with the line of thrust is shortened by {(7r2-8)/4ff} (XjaS/S), while the perpendicular diameter is lengthened by {(4-,r)/2,r}(Zia3/5)t. (v) When the rod forms a complete circular ring of weight W, which is suspended from a point in its cir- cumference, we measure 6 from the highest point, and * Cf. H. Lamb, London Math. Sac. Proc, vol. 19 (1888), p. 365. text under the numbers (i) — (v) are taken from this paper. t These results are due to Saint- Venant, Paris, G. R., t. 17 (1843). Fig. 66. The results given in the 456 BENDING OF INCOMPLETE CIRCULAR RING [CH. XXI find for the displacement «; at a point for which iT>d>0 the value w= - W(a^/B) (Stt)-! {(5- 7r)2 sin5 -4 (5-5r) (1 -cos 6)-n^amd} ; the displacement is the same at the corresponding point in the other half of the ring. In this case we may prove that the amounts by which the vertical diameter is lengthened and the horizontal diameter shortened are the halves of what they would be if the weight W were concentrated at the lowest point. (vi) When the rod forms a complete circular ring which rotates with angular velocity a about one diameter*, taken as axis of y, its central-line describes a surface of revolution of which the meridian curve is given by the equations x=asmd + ^{m afla^jB) sva? 6, y=acose + ^{inufla^lB) (l-cosS^^i where m denotes the mass of the ring per unit of length, and 6 is measured from the diameter about which the ring rotates. This diameter is shortened and the perpendicular diameter lengthened by the same amount J {may^a^jB). (h) Incomplete circular ring bent out of its plane. As before we take a for the radius of the circle, and specify a point on it by an angle 6; and we take the plane 'o^^y e /' A\ of the circle to be that principal plane of the rod for which the flexural rigidity is B. We consider the case where the rod is bent by a load W, applied at the end 6 = 0. in a direction at right angles to this plane, and is fixed at the end ^ = 0, so that the tangent at this point is fixed in direction, and the transverse ^ linear element which, in the unstressed state, is directed towards the centre of the circle is also fixed in directionf. Then m, v, w, /3, du/dd, dvjdd vanish with 6. The stress-resultants N, N', T at any section are statically equivalent to the force W, of which the direction is parallel to that of the axis of y„ at any section, and we have, therefore, N = ^W, N' = W, T={W la), {dvjdd) (13) The equations of moments are, therefore, %^H = aW, f.^-a^W, f-(?^0 (14) From the first and third of these, combined with the conditions that and H vanish when 6 = a, we find G = -aW sm(a-d), H = aW {1 - cos (a - 0)} (15) * G. A. V. Peschka, Zeitschr.f. Math. u. Phys. {SchlSmilch) , Bd. 13 (1868). t The problem has been discussed by Saint- Venant, Paris, C. JR., t. 17 (1843), and by H. Eesal, J. de Math. (Liouville), (S^r. 3), t. 3 (1877). The treatment of the incomplete circular ring as a thin rod is a first approximation to a theory of the bending of curved beams. For a discussion of curved beams reference may be made to J. J. Guest, London, Proc. B. Soc. (Ser. A), vol. 95, 1918, p. 1. 292, 293] Now we have VIBRATING CIRCULAR RING 457 a^ Kde^ .(16) and from these equations and the terminal conditions at = we can obtain the equations V + a^ = -^r {& - sin a + sin (a - 6)], v = -jr- 1(0 - sin 6) — sin a (1 - cos 6)} } -(17) + ^ Wa^ (j, + -J j [6 cos {a -6)- sin d cos a}. We may prove also that u and w are small of the order v^. 293. Vibrations of a circular ring. We shall illustrate the application of the theory to vibrations by con- sidering the free vibrations of a rod which, in the unstressed state, forms a circular ring or a portion of such a ring, and we shall restrict our work to the case where the cross-section of the ring also is circular. We denote the radius of the cross-section by c, and that of the circle formed by the central-line by a, and we take the displacement u to be directed along the radius drawn towards the centre of the latter circle. The equations of motion, formed as in Articles 278—280, are and dN dd + T = ma dN'- de ~ d^ dG dd + H- N'a = -Ic^'m dG' de + Na ^^^'^dtAd'e^ dH de -G = ^c^ma d'0 df ' ST 961" 3% d¥de' w iV^ = ma^, (18) .(19) in which m is the mass of the ring per unit of length, and (20) E being the Young's modulus and yu, the rigidity of the material of the ring. The above equations with the condition dw de = u. .(8 his) yield the equations of motion. 458 VIBRATING CIRCULAR RING [CH. XXI It is clear that the above system of equations falls into two sets. In the first set v and ^8 vanish, and the motion is specified by the displacement u or w, these variables being connected by equation (8); in this case we have flexural vibrations of the ring in its plane. In the second set u and w vanish, and the motion is specified by v or ^, so that we have flexural vibrations in- volving both displacement at right angles to the plane of the ring and twist. It may be shown in the same way that the vibrations of a curved rod fall into two such classes whenever the central-line of the unstressed rod is a plane curve, and its plane is a principal plane of the rod at each point. In case the central-line is a curve of double curvature there is no such separation of the modes of vibration into two classes, and the problem becomes extremely complicated*. (a) Fleaniral vibrations in the plane of the ring. We shall simplify the question by neglecting the "rotatory inertia." This amounts to omitting the right-hand member of the second of equations (19). We have then „ _ Eire^ (d*w d-w\ d^w _dN' The normal functions for free vibration are determined by taking w to be of the form W cos {pt -\- e), where IF is a function of d. We then have the equation a^TF B^ir a=F/ 4«2ay\ 4may "a^ +^"3^ ^WA ~:Ei^)'"~E^^^^■ T\lQ. complete primitive is of the form K = 3 1F= S (j4, cos njd -I- 5, sin njd), K=l where n^, Us, n^ are the roots of the equation 71'' {n'' -lf = {n^+\) (imaylETTC^). If the ring is complete n must be an integer, and there are vibrations with n wave-lengths to the circumference, n being any integer greater than unity. The frequency is then given by the equation f E;^ n^in^-iy ^ 4ma* n^ + l ^ ' * The vibrations of a rod of which the natural form is helical have been investigated by J. H. Michell, loc. cit., p. 450, and also by the present writer, Cambridge Phil. Soc. Tram., vol. 18 (1899). t The result is due to E. Hoppe, J.f. Math. (Crelle), Bd. 73 (1871). 293] VIBEATING CIRCULAR RING 459 When the ring is incomplete the frequency equation is to be obtained by forming the conditions that N, T, G' vanish at the ends. The result is difficult to interpret except in the case where the initial curvature is very slight, or the radius of the central-line is large compared mth its length. The pitch is then slightly lower than for a straight bar of the same length, material and cross-section*. (6) Fleooural vibrations at right angles to the plane of the ring. We shall simplify the problem by neglecting the " rotatory inertia," that is to say we shall omit the right-hand members of the first and third of equations (19); we shall also suppose that the ring is complete. We may then write v=Vcos{n6+a) cos {pt + e), ^ = B' cos (nO + a) cos {pt + e), where V, B', a, e are constants, and n is an integer. From the first and third of equations (19) and the second of equations (18) we find the equations n' {aB' + n'V) + ^ «» {aB' + F) = ^^ F, ^n'{aF+V) + {aB'-\-n-^V) = 0, ' from which we obtain the frequency equation f Ej^n^Sr^ (22) where a is Poisson's ratio for the material, and we have used the relation JS" = 2/u. (1 + a). It is noteworthy that, even in the gravest mode {n — 2), the frequency differs extremely little from that given by equation (21) for the corresponding mode involving flexure in the plane of the ring. (c) Torsional and emtensional vibrations. A curved rod possesses also modes of free vibration analogous to the torsional and extensional vibrations of a straight rod. For the torsional vibrations of a circular ring we take u and to to vanish, and suppose that v is small in comparison with a/S, then the second of equations (18) and the first of equations (19) are satisfied approximately, and the third of equations (19) becomes approximately For a complete circular ring there are vibrations of this type with n wave-lengths to the circumference, and the frequency pjin is given by the equation p2=^'(l-ho- + »'^) (23) When n=0, the equations of motion can be satisfied exactly by putting v=0 and taking j8 to be independent of 6. The characteristic feature of this mode of vibration is that each * The question has been discussed very fully by H. Lamb, loe. cit., p. 455. + The result is due to J. H. Miohell, loc. cit., p. 450. 460 VIBRATING CIRCULAR RING [CH. XXI circular cross-section of the circular ring is turned in its own plane through the same small angle about the central-line, while this line is not displaced*. For the extensional modes of vibration of a circular ring we take v and /3 to vanish, and suppose that equation (8) does not hold. Then the extension of the central-line is a~^ (dii>ldd — u), and the tension T is Evc'^a~^ {dwjdO - u). The couples G, fi" and the shearing force N' vanish. The expressions 'for the couple 0' and the shearing force N contain c* as a factor, while the expression for T contains c^ as a factor. We may, there- fore, for an approximation, omit O' and N, and neglect the rotatory inertia which gives rise to the right-hand member of the second of equations (19). The equations to be satisfied by u and w are then the first and third of equations (18), viz. : ma TT^ ■■ The displacement in free vibrations of frequency pftn is given by equations of the form u={A sin nd+B cos n6) cos (pt + e), w=n{A cosnB — B sin n6) cos {pt + fj, where p^ = s (\ + rfi) (24) When »i=0, w vanishes and u is independent of 6, and the equations of motion are satisfied exactly. The ring vibrates radially, so that the central-line forms a circle of periodically variable radius, and the cross-sections move without rotation. The modes of vibration considered in (c) of this Article are of much higher pitch than those considered in (a) and (6), and they would probably be difficult to excite. * The result that the modes of vibration involving displacements v and /S are of two types was recognized by A. B. Basset, London Math. Soc. Proc, vol. 23 (1892), and the frequency of the torsional vibrations wasfound by him. CHAPTER XXII. THE STRETCHING AND BENDING OF PLATES. 294. ' Specification of stress in a plate. The internal actions between the parts of a thin plate are most appro- priately expressed in terms of stress-resultants and stress- couples reckoned across the whole thickness. We take the plate to be of thickness 2h, and on the plane midway between the faces, called the " middle plane," we choose an origin and rectangular axes of x and y, and we draw the axis of z at right angles to this plane so that the axes of x, y, z are a right-handed system. We draw any cylindrical surface C to cut the middle plane in a curve s. The edge of the plate is such a surface as G, and the corresponding curve is the " edge-line." We draw the normal v to s in a chosen sense, and choose the sense of s so that v, s, z are parallel to the directions of a right-handed system of axes. We consider the action exerted by the part of the plate lying on that side of G towards which v is drawn upon the part lying on the other ' side. Let hs be a short length of the curve s, and let two generating lines of G be drawn through the extremities of hs to mark out on G an area A. The tractions on the area A are statically equivalent to a force at the centroid of A and a couple. We resolve this force and couple into components directed along V, s, z. Let [T], [S], [N] denote the components of the force, [H], [G], [K] those of the couple. When Ss is diminished indefinitely these quantities have zero limits, and the limit of [K]/Ss also is zero, but [T]/8s, ... [G]/Ss, may be finite. We denote the limits of [T]/Ss, ... by T, .... Then T, S, N are the components of the stress-resultant belonging to the line s, and H, G are the components of the stress-couple belonging to the same line. T is a tension, S and N are shearing forces tangential and normal to the middle plane, G' is a flexural couple, and II a torsional couple. When the normal v to s is parallel to the axis of x, s is parallel to the axis of y. In this case we give a suffix 1 to T, .... When the normal v is parallel to the axis of y, s is parallel to the negative direction of the axis of x. In this case we give a suffix 2 to T, .... The conventions in regard to the senses of these forces and couples are illustrated in Fig. 68. 462 STRESS-RESULTANTS AND STRESS-COUPLES [CH. XXII For the expression -of T, ... we take temporary axes of x, y', z which are parallel to the directions of v, s, z, and denote by X'^:, ... the stress-com- ponents referred to these axes. Then we have the formulae* 7'-*S, Fig. 68. ^^O, T=f X'^dz, S=r X'ydz, N^(- X'.dz, J -h J -h J -ft = f -zX'y.dz, G=r zX'„,dz; J -h J -ft H and, in the two particular cases in which v is parallel respectively to the axes of X and y, these formulae become /•ft and Ti = f X^dz, S, = r Xydz, N, = r X,dz^ J -h J -h J -h H, = j'' -zXydz, G,= f zX^dz, rh rh rh S,= . -Xydz, T,= Yydz, N,= F, J -h , J -h J -h , rh rh Oi=\ zYydz, H^=\ zX'ydz. J ~k J —ft We observe that in accordance with these formulse 82 = — Si, H2= — Hi ■(1) .(2) .(3) 295. Transformation of stress-resultants and stress-couples. When the normal v to the curve s makes angles 6 and ^-rr — 6 with the axes of X and y,T, S, ... are to be calculated from such formute as rh '= X'^dz, J -h in which the stress-components X'^, ... are to' be found from the formulas (9) of Article 49 by putting li = cos 9, nil = sin 0, l^ = - sin 6, m^ = cos 0, n^ = n^ = Z, = mj = 0, Wj = 1. * It is assumed tliat the plate is but slightly bent. Cf. Article 328 in Chapter XXIV. 294-296] EQUATIONS OF EQUILIBRIUM OF A PLATE 463 We find T=T,cos'e + T^sm'e + S^sm20, \ S = i(T-^- T,) sin 2d +■ S, cos 261, N = N^ COB d + N^ sine, [ (4) G = G, cos^ d + G^ sin^i d - E, sin 26, 1 H=^(G^-G^) sin 26 + H^cos2$. I Instead of resolving the stress-resultants and stress-couples belonging to the line s in the directions v, s, z we 'might resolve them in the directions X, y, z. The components of the stress-resultant would be : parallel to x, T cos 6 — Ssm 6, or T^ ess ^ -1- >Si sin d,'\ ' parallel to y, Tsin6 +S cos 6, or T^ sin ^ -|- /Sfj cos ^, > (5) parallel to z, Ni cos 6 + 'N^ sm6; ' ] and those of the couple "would be : about an axis parallel to x, H cos 6 — G sin 0, or Hi cos 6 ^G^ sin 6,\ about an axis parallel to y, H siD.6 + G cos 6, or G^ cos 6 — Hi sin 6.) 296. Equations of equilibrium. Let C denote, as before, a cylindrical surface cutting the middle plane at right angles in a curve s, which we take to be a simple closed contour. The external forces applied to the portion of the plate within G may consist of body forces and of surface tractions on the faces {z = h and z = — h)oi the plate. These external forces are statically equivalent to a single force, acting at the centroid P of the volume within G, and a couple. Let [X'], [ F'], [Z'] denote the components of the force parallel to the axes of x, y, z, and [_L'\ [ilf' ], [,N''\ the components of the couple about the same axes. When the area m within the .curve s is diminished indefinitely by contracting s towards P, the limits of [Z '], ... [i'], . . . are zero and the limit of [iV']/m also is zA'o, but the lipiits of [X']/&), ... may be finite. We denote the limits of [X']/+^> + Z' = 0, ^> + ^^^-hF = 0, ^' + ^^-^^' = 0. ...(11) ox oy dx ay ox ay We transform the equations (10) in the same way and simplify the results by using equations (11). The third equation is identically satisfied. We thus find two equations which hold at every point of the middle plane, viz. 9^'_^^ + i^, + Z' = 0, f}-^S-N, + M' = (12) ox oy ox Oy Equations (11) and (12) are the equations of equilibrium of the plate. 297. Boundary-conditions. In a thick plate subjected to given forces the tractions specified by A'„, Yy,Zy, where v denotes the normal to the edge, have prescribed values at every point of the edge. When the plate is thin, the actual distribution of 296, 297] BOUNDARY-CONDITIONS AT THE EDGE OF A PLATE 465 the tractions applied to the edge, regarded as a cylindrical surface, is of no practical importance. We represent therefore the tractions applied to the edge by their force- and couple-resultants, estimated per unit of length of the edge-line, i.e. the curve in which the edge cuts the middle surface. It follows from Saint- Venant's principle (Article 89) that the effects produced at a distance from the edge by two systems of tractions which give rise to the same force- and couple-resultants, estimated as above, are practically the same. Let these resultants be specified by components T, S, N and H, G in the senses previously assigned for T, S, N and H, G the normal to the edge- line being drawn outwards Let the stress-resultants and stress-couples belonging to a curve parallel to the edge-line, and not very near to it, be calculated in accordance with the previously stated conventions, the normal to this curve being drawn towards the edge-line ; and let limiting values of these quantities be found by bringing the parallel curve to coincidence with the edge-line. Let these limiting values be denoted by T, S, N and H, G. It is most necessary to observe that the statical equivalence of the applied tractions and the stress-resultants and stress-couples at the edge does not require the satisfaction of all the equations ?=T, ^=S, i7=N, B=H, 0==G. These five equations are equivalent to the boundary-conditions adopted by Poisson*. A system of four boundary -conditions was afterwards obtained by Kirchhofff, who set out from a special assumption as to the nature of the strain within the plate, and proceeded by the method of variation of the energy-function. The meaning of the reduction of the number of conditions from five to four was fii-st pointed out by Kelvin and TaitJ. It lies in the circumstance that the actual distribution of tractions on the edge which give rise to the torsional couple is immaterial. The couple on any finite length might be applied by means of tractions directed at right angles to the middle plane, and these, when reduced to force- and couple-resultants, estimated per unit of length of the edge-line, would be equivalent to a distribution of shearing force of the type N instead of torsional couple of the type H. The required shearing force is easily found to be — 9H/9s. This result is obtained by means of the following theorem of Statics: A line-distribution of couple of amount H per unit of length of a plane closed curve s, the axis of the couple at any point being normal to the curve, is statically equivalent to a line- distribution of force of amount — dHjds, the direction of the force at any point being at right angles to the plane of the curve. * See Introduction, footnote 36. Poisson's solutions of special problems are not invalidated, because in all of them H vanishes. + See Introduction, footnote 125. J Nat. Phil, first edition, 1867. The same explanation was given by J. Boussinesq in 1871. See Introduction, footnote 128. L. B. 30 466 BO DND AEY-CONDITION S [CH. XXII The theorem is proved at once by forming the force- and couple-resultants of the line- distribution of force —dBjds. The axis of s being at right angles to the plane of the curve, the force at any point is directed parallel to the axis of z, and the force-resultant is r 7\TT expressed by the integral I — -^ds taken round the closed curve. This integral vanishes. The components of the couple-resultant about the axes of x and y are expressed by the integrals I -y^ ds and \x^ ds taken round the curve. If v denotes the direction of the normal to the curve, we have dff , and I -y^ ds= i H^ ds= I Scos{x, v) ds, lx-^ds=l—B-^ds=jII cos (y, v) ds, the integrations being taken round the curve. The expressions I JI cos (x, v) ds and l^cos(y, v)ds are the values of the components of the couple-resultant of the line- distribution of couple ff. The theorem may be illustrated by a figure. We may think of the curve s as a polygon H-5H H A H4-6H V H-5H H H + 3H Fig. 69. length 8s, or, in the limit, with a line-distribution of force of a large number of sides. The couple H8s, belonging to any side of length 8s, is statically equivalent to two forces each of magnitude ff, directed at right angles to the plane of the curve in opposite senses, and acting at the ends of the side. The couples belonging to the adjacent sides may similarly be replaced by pairs of forces of magnitude S+8H or IT- Bff as shown in Fig. 69, where 8ff means (dH/ds) Ss. In the end we are left with a force - dS at one end of any side of dff/ds. From this theorem it follows that, for the purpose of forming the equations of equilibrium of any portion of the plate contained within a cylindrical surface C, which cuts the middle surface at right angles in a curve s, the torsional couple H may be omitted, provided that the shearing stress-resultant N is replaced by N -dH/ds*- Now the boundary-conditions are limiting forms of the equations of equilibrium for certain short narrow strips of the plate ; the contour in which the boundary of any one of these strips cuts the middle plane consists of a short arc of the edge-line, the two normals to this curve at the ends of the arc, and the arc of a curve parallel to the edge-line intercepted between these normals. The limit is taken by first bringing the parallel curve to coincidence with the edge-line, and then diminishing the length of the arc of the edge-line indefinitely. In accordance with the above * This result might be used in forming the equations of equilibrium (11) and (12). The line- integrals in the third of equations (9) and the first two of equations (10) would be written and these can be transformed easily into the forms given in (9) and (10). /("-'#)"■■ /|-o.i..+,(»-?51tfc ("-'#)! * 297] AT THE EDGE OF A PLATE 467 theorem we are_ to form these equations by omitting H and H, and replacing N and N by N-dH/ds and N-BH/as. The boundary-conditions are thus found to be T = T, S = S, M-dH/ds = -N-dR/ds, G = G. These four equations are equivalent to the boundary-conditions adopted by KjrchhofiF. In investigating the boundary-conditions by the process just sketched we observe that the terms contributed to the equations of equilibrium by the body forces and the tractions on the faces of the plate do not merely vanish in the limit, but the quotients of them by the length of the short arc of the edge-line which is part of the contour of the strip also vanish in the limit when this length is diminished indefinitely. If this arc is denoted by 8s we have such equations as lim {hs)--^ilx'dxdy = 0, lim {Ss)--^ ( ( {L' + vZ')da;dv = 0, Ss=0 J J Ss=0 J J the integration being taken over the area within the contour of the strip. The equations of equilibrium of the strip lead therefore to the equations lim (Ss)-'^\{Tooad-Saiae)ds=0, lim {&s)-i [ {Tame + Scose)ds==0, 8»=o J Ss=n J lim (Ss)-^llocose-ix;(jV-~\\ds = 0, in which the integrations are taken all round the contour of the strip, and T, ... denote the force- and couple-resultants of the tractions on the edges of the strip, estimated in accordance with the conventions laid down in Article 294. We evaluate the contributions made to the various line-integrals by the four lines in which the edges of the strip cut the middle plane. Since the parallel curve is brought to coincidence with the edge-line, the contributions of the short lengths of the two normals to this curve have zero limits ; and we have to evaluate the contributions of the arcs of the edge-line and of the parallel curve. Let v„ denote the direction of the normal to the edge-line drawn outwards. The contributions of this arc may be estimated as {T cos {x, vo) - S cos (y, vo)} Ss, {T cos (y, vq) + S cos {x, !/„)} Ss, ] N - "g- [ S«i and J -Gcos(y, j'Q)-l-y (N--^U 8s, -^ G cos (;», i'o)-A'(N—g-j|- 8s. In evaluating the contributions of the arc of the parallel curve, we observe that the con- ventions, in accordance with which the T, ... belonging to this curve are estimated, require the normal to the curve to be drawn in the opposite sense to i/„, and the curve to be described in the opposite sense to the edge-line, but the arc of the curve over which we integrate has the same length 8s as the arc of the edge-line. In the limit when the parallel curve is brought to coincidence with the edge-line we have, in accordance with these conventions, T=f, S=S, N=-N, a=0, H=H, dHlds=-dE/ds, and cos6=— cos {x, vo), sin 5 = - cos (y, vo). 30—2 468 BOUNDARY-CONDITIONS [CH. XXII Hence the contributions of the arc of the parallel curve may be estimated as I - fcos {x, i/„) + S cos (y, vq)} 8s, {-fcoa (y, vo) - S cos {x, vq)} Ss, | - #+ -g^ I Ss, and \Ocos(t/,vo)+^(-^+-^jiSs, | -tfcos(a.', vo) -«(- iV'+— j[ 8s. On adding the contributions of the two arcs, dividing by &s, and equating the resulting expressions to zero, we have the boundary-conditions in the forms previously stated. In general we shall omit the bars over the letters T,..., and write the boundary-conditions at an edge to which given forces are applied in the form y=T, S=S, i\^_^ = N-^, G = G (14) At a free edge T, S, N — dHjds, G vanish. At a "supported" edge the displacement w of a point on the middle plane at right angles to this plane vanishes, and T, S, G also vanish. At a clamped edge, where the inclination of the middle plane is not permitted to vary, the displacement (u, v, w) of a point on the middle plane vanishes, and 9w/9i' also vanishes, v denoting the direction of the normal to the edge-line. The effect of the mode of application of the torsional couple may be illustrated further by an exact solution of the equations of equilibrium of isotropic solids*. Let the edge- line be the rectangle given hy x= ± a, i/= ±b. The plate is then an extreme example of a flat rectangular bar. When such a bar is twisted by opposing couples about the axis of X, so that the twist produced is t, we know from Article 221 (c) that the displacement is given by 26^2 - (_)n "n*! 2/i ™ U provided that the tractions by which the torsional couple is produced are expressed by the formulae cosh (^'' + ^)^y ,in {^n + l)^z . , (2«-|-l)wy (2w+])n-2 2^A» (-)» -"'^-^A^-^^-lA^ X^— flT — 2" 2 T^ n=o(2n-|-l)2 ,(2M + l)_7r6 There are no tractions on the faces z= ±h or on the edges ?/= +6. The total torsional couple on the edge x=a is Vm^A36-^.A4(1)' I _2^tanh(?^ VV n=o(2M-l-l)* 2A 1)776 and of this one-half is contributed by the tractions A'y directed parallel to the middle plane, and the other half by the tractions 1\ directed at right angles to the middle plane. * Kelvin and Tait, Nat. Phil., Part 11., pp. 267 et scq. 297; 298] AT THE EDGE OF A PLATE 469 When the plate is very thin the total torsional couple is approximately equal to i^fiTh^b, so that the average torsional couple per unit of length of the edge-lines a;= +a is approximately equal to §;utA'. At any point which is not near an edge y=+h, the state of the plate is expressed approximately by the equations ► u^—ryzi v=—tzx, w=Txy. The traction Xy is nearly equal to — 2^t2 at all points which are not very near to the edges y= +6, and the traction X^ is very small at all such points. The distribution of traction on the edge .r=a is very nearly equivalent to a constant torsional couple such as would be denoted by H^, of amount ^firh?, combined with shearing stress-resultants such as would be denoted by N^, having values which differ appreciably from zero only near the corners {a;=a, y= +6), and equivalent to forces at the corners of amount ^iirh^. At a distance from the free edges y= + b which exceeds three or four times the thickness, the stress is practically expressed by giving the value - 2;xt2 to the stress-component Xy and zero values to the remaining stress-components. The greater part of the plate is in practically the same state as it would be if there were torsional couples, specified by Hi = ^li.Th? at all points of the edges x= +a, and 11^= -|/xrA^ at all points of the edges y=+b. Thus the forces at the corners may be replaced by a statically equivalent distri- bution of torsional couple on the free edges, without sensibly altering the' state of the plate, except in a narrow region near these edges. Within this region the value of the torsional couple ff2, belonging to any line y= const., which would be calculated from the exact solution, diminishes rapidly, from -^/irh^ to zero, as the edge is approached. The rapid diminution of ^2 is accompanied, as we should expect from the second of equations (12), by large values of i\^i. If we integrate iVj across the region, that is to say, if we form the integral I J^f^dt/, taken over a length, equal to three or four times the thickness, along any line drawn at right angles to an edge y=b or y=-b and terminated at that edge, we find the value of the integral to be very nearly equal to + ^firh?. This remark enables us to understand why, in the investigation of equations (14), the third of equations (13), viz. lim (8s)~M(iV — ^)ds=0, where the integration is taken round the contour of a " strip," as was explained, should not be replaced by the equation lim {is)~^\Nds=0, and also why the latter equation does not lead to the _ «s=o _l _ result N= N. When iV, H, are calculated from the state of strain which holds at a distance from the edge, and equations (14) are established by the method employed above, it is implied that no substantial difference will be made in the results if the linear dimensions of the strip, instead of being diminished indefinitely, are not reduced below lengths equal to three or four times the thickness. When the dimensions of the strip are of this order, the contributions made to the integral I Nds by those parts of the contour which are normal to the edge-line may not always be negligible ; but, if not, they will be practically balanced by the contributions made to I - {dff/ds) ds by the same parts of the contour*. 298. Relation between the flexural couples and the curvature. In Article 90 we found a particular solution of the equations of equili- brium of an isotropic elastic solid body, which represents the deformation of * Of. H. Lamb, London Math. Soc. Proc, vol. 21 (1891), p. 70. 470 CURVATUHE PBODUCED BY COUPLES [CH. XXII; a plate slightly bent by couples applied at its edges. To express the result which we then found in the notation of Article 294 we proceed as follows : — On the surface into which the middle plane is bent we draw the principal tangents at any point. We denote by s^, Sj the directions of these lines on the unstrained middle plane, by i?,, R^ the radii of curvature of the normal sections of the surface drawn through them respectively, by G/, ff/ the flexural couples belonging to plane sections of the plate which are normal to the middle surface and to the lines Sj, S2 respectively. We determine the senses of these couples by the conventions stated in Article 294 in the same way as ii s^, S2, z were parallel to the axes of a right-handed system. Then, according to Article 90, when the plate is bent so that Ej , B^ are constants, and the directions Si, $2 are fixed, the stress-resultants and the torsional couples belonging to the principal planes of section vanish, and the flexural couples Gi, G2 belonging to these planes are given by the equations G/ = -Z)(l/Ei + a//?,), G2'=-I){l/R2 + <7/R,), (15) where, with the usual notation for elastic constants, D = ^Eh'l{l-+G2 sin^ . E, i?, '^'"VW^^RT. sm^d> cos^(^ /cos" A sin"d) G, = -D ir, = iZ)(l-a)sin20(l-l). Again, let w be the displacement of a point on the middle plane in the direction of the normal to this plane, and write _dHf d'w d'w "'-da^' "'-df' ^ = a^ (1^) Then the indicatrix of the surface into which the middle plane is bent is given, with sufficient approximation, by the equation «!«" + K^y^ + 2Txy = const. ; and, when the form on the left is transformed to coordinates f, 17, of which the axes coincide in direction with the lines s,, s^, it becomes eiR. + v'IR2. «1 = 298, 299] CURVATURE PRODUCED BY COUPLES 471 Hence we have the equations cos^ A sin^ (b sin^ d> cos'' 6 . / 1 1 \ and the formulse for G^, G^, H^ become G, = - Z) («! + (r«,,), (?, = -Z)(«;2+[cos^.f5..P).sin^.f^..g).(l-.)sin..||], ^=i)(l-.)[sin^cos^(^-g) + (cos^^-sin^^)|j]. We may transform these equations, so as to avoid the reference to fixed axes of x and y.*, by means of the formulae 5-=cosfl5--smfl =-, 5-=cos55 — hsm^s-, 5- = 0, -5- = -, ...(19) OS oy ox ov ox oy ov os p ^ where p' is the radius of curvature of the curve in question. We find _ _ rav /3V , lawNi „ _,, , 3 /3w\ ,_, ^=-^W^''[w^-p'¥.)]^ ^=^^^-''>Tv\js) ^^°) These equations hold whenever the stress-couples are expressed by the formulae (18). In the problem of Article 90 we found for the potential energy of the plate, estimated per unit of area of the middle plane, the formula hD Ih^J-'^'-^wi RIR2 or, in our present notation, ii) [(«i + «,)^- 2(1'-^) (/.,«, -tO] (21) We shall find that this formula also is correct, or approximately correct, in a wide class of problems. 299. Method of determining the stress in a plate f- We proceed to consider some particular solutions of the equations of equilibrium of an isotropic elastic solid body, subjected to surface tractions only, which are applicable to the problem of a plate deformed by given forces. * Cf. Lord Eayleigh, Theory of Sound, vol. 1, § 216. t The method was worked out briefly, and in a much more general fashion, by J. H. Mlohell, London Math. Soe. Proc, vol. 31 (1900), p. 100. 472 METHOD OF DETERMINING THE STRESS [CH. XXII These solutions will be obtained by means of the system of equations for the determination of the stress-components which were given in (iv) of Article 92. It was there shown that, besides the equations we have the two sets of equations and _^^ 1 a^© ^_ 1 a^@ „,y _ 1 a^® .„.. where © = Z^+Fj, + Z, (25) It was shown also that the function @ is harmonic, so that V^© = 0, and that each of the stress-components satisfies the equation Vy = 0. We shall suppose in the first place that the plate is held by forces applied at its edge only. Then the faces z= ±h are free from traction, or we have ^2= rj = ^j = when z=±h. It follows from the third of equations (22) that dZ^jdz vanishes at z = h and at z = — h. Hence Z^ satisfies the equation V*.Zz = and the conditions Z^ = 0, ZZ^jdz =0 at z= ±h. If the plate had no boundaries besides the planes z = ±h, the only possible value for Z^^ would be zero. We shall take Z^ to vanish*. It then follows from the equations Ve = 0, V% = - (1 -I- o-)-'9'©/^2;^ that is of the form @„ -i- z®^, where @„ and ©1 are plane harmonic functions of oc and y which are independent of z. For the determination oi X^, Y^ we have the equations dx dy ' ' \ + adx- ' 1+0-92/' and the conditions that Xj = F^ = at « = + A. A particular solution is given by the equations ^'-21+a^^ ''dx' ^'-^l + a^^ '' dy ^^^' We shall take X^ and F^ to have these forms. When X^, Y^, Z^ are known general formulae can be obtained for X^, Yy, Xy. If ©1 is a constant, Xz and F^ vanish as well as Z^, and the plate is then in a state of " plane stress." If ©i depends upon x and y the plate is in a state of "generalized plane stress" (Article 94). We shall examine separately these two cases. In like maimer, when the plate is bent by pressure applied to its faces, we find a particular solution of the equation V^Zg = which yields the prescribed values oi Z^ at z = h and z = — h, and we deduce the most general form of © which is consistent with this solution. We proceed to * J. H. Michell, loc. cit., calls attention to the analogy of this procedure to the customary treatment of the condenser problem in Electrostatics. 299-301] PLATE IN A STATE OF PLANE STRESS 473 find particular solutions of the equations satisfied by Z^ and 7^ and to deduce general formulae for X^, Yy, Xy. 300. Plane stress. When Z^, Y^, Z^ vanish throughout the plate there is a state of plane stress. We have already determined in Article 145 the most general forms for the remaining stress-components and the corresponding displacements. We found for © the expression © = @o + /3^, (27) where ©„ is a plane harmonic function of x and y, and ;S is a constant. The stress-components X^, Yy, Xy are derived from a stress-function x by the formulae y _9'X and x has the form where Z J^ Z=--^ " dxdy' .(28) .(29) .(30) 21-1-0- If we introduce a pair of conjugate functions |^, tj of a; and y which are such that dx~dy~ " dy~ dx' ^'*^'' the most general forms for %„ and Xi can be written Xo = M+f, X^ = i^i^ + f) + F, (32) where / and F are plane harmonic functions. The displacement (m, v, w) is then expressed by the formulae 1 /'fc , /3 ,1 2S®«^ 1 + 0- 9 / , x\ u = ^i^^ + ^x. + l.z^-^)-^--(Xo + ^X^ 1 + a d dy I " E dy (%o .+ ^xO, .(33) «' = -;! (i/8 (a;= + f + <7Z') + ^^d the other on /8, %i. These two systems are independent of each other. 301. Plate stretched by forces in its plane. Taking the (©o> Xo) system, we have the displacement given by the equations ''-_EV^*'^^ dx) E dx' 1 / _Li .,9@oN l+o-9Xo w- E' (34) 474 PLATE STRETCHED IN ITS PLANE [CH. XXII where ^o is of the form ^x^ +f, ©o and / are plane harmonic functions, and ^, r) are determined by (31). The normal displacement of the middle plane vanishes, or the plate is not bent. The stress is expressed by the formulae X.= df Xo- V - ^' Z„ = - 21 +a 1 a- 21 + / 1 .(35) dxdyK'^'' 21 + 0-' The stress-resultants- T^, T2, Si are expressed by the equations Ti df 2%o- 31 + ^ ■ hm, )' 1 + 0- 1 o- h'@, h' @o). .(36) 31 + 0- The stress-resultants JSf^, N^, and the stress-couples Gj, 0^, H^ vanish. The equations (11) and (12), in which X' , Y', Z', L', M' vanish, are obviously satisfied by these forms. When we transform the expressions for T^, T^, S^ by means of the equa- tions (4), we find that, at a point of the edge-line where the normal makes an angle with the axis of x, the tension and shearing-force T, 8 are given by the equations S=Lmecosd(~- ^' -^°^^^4}(''^° .'©0) . dfj dxdy] V"'^" 3 1 -f + T^' (*^) so that the curvature is expressed by the equations E^ E dx^' "' E^ E df ^~ E d^- From these equations and the equation ^-^x^ = P> ^^ find so that the formulse (18) hold. The stress-couples at the edge are expressible in the forms 301-303] PLATE BENT TO A STATE OF PLANE STRESS 477 and, if the edge is subjected to given forces, G and dHjds must have pre- scribed values at the edge. Since ^^i satisfies the equation '^ix^~^' *^® formulae (41') for G and dHjds are not sufficiently general to permit of the satisfaction of such conditions. It follows that a plate free from any forces, except such as are applied at the edge and are statically equivalent to couples, will not be in a state of plane stress unless the couples can be expressed by the formulae (41). Some particular results are appended. (i) When the plate is bent to a state of plane stress the sum of the principal curvatures of the surface into which the middle plane is bent is constant. (ii) In the same case the potential energy per unit of area of the middle plane is given exactly by the formula (21). (iii) A particular case will be found by taking the function F introduced in equations (32) to be of the second degree in x and y. Then xi also is of the second degree in x and y, . and we may take it to be homogeneous of this degree without altering the expressions for the stress-components. In this case w also is homogeneous of the second degree in x and y, and Ki, Kj, r are constants. The value of ^i is W Xi=-k J3^2 [("2 + '^''i) ^^ + (-^i + ffK2) y - 2 (1 - 0-) TXy\ and the stress-components which do not vanish are given by the equations WEE X^=-Y:--^z{K^ + a<^, P"j, = - J— ^z(K2 + o-f]), ^^^~\^^^'^- (iv) This case includes that discussed in Article 90, and becomes, in fact, identical with it when the axes of x aud y are chosen so that t vanishes, that is to say so as to be parallel to th^ lines which become lines of curvature of the surface into which the middle plane is bent. Another special sub-case would be found by taking the plate to be rectangular, and the axes of x and y parallel to its edges, and supposing that ki and kj vanish, while t is constant. We should then find u= —ryz, v= — TZX, w=Txy. The stress-resultants and the flexural couples (?i, O2 vanish, and the torsional couples Bi and B2 are equal to ± Z) (1 - cr) t. The result is that a rectangular plate can be held iu the form of an anticlastic surface v=Txy by torsional couples of amount D{\ — ^''=~2-7r;^)^f'+^- ^*^) where i^i is a plane harmonic function. Thus the form of x\ and therewith also that of Xx, Yy, Xy, is completely determined. The displacement is determined by the equations of the types ^±-^ (X rrV rr7\ ^^ dv 2(l + ^V,'w, ^ ov I ..,(54) where p denotes the radius of curvature of the curve. At a boundary to which given forces and couples are applied and N — dH/os have given values. The solution is sufficiently general to admit of the satisfaction of such boundary-conditions. The solution expressed by (48) is exact if the applied tractions at the edges are distributed in accordance with (62), in which w satisfies (50) ; but, if they are distributed otherwise, without ceasing to be equivalent to resultants of the types N, G, H, the solution represents the state of the plate with sufficient approximation at all points which are not close to the edge. The potential energy per unit of area can be shown to be i.[(v...,-,a-,){S^-(|^)}] 10 \jdx^ dx^ 3y2 dy'^ dxdy dxdy J ,„ m^ r/3Vw\2 /3vj%Y- + *l-aLV dx ) +\dy J_ 420(1-0-) ^ \_ dx^ 92^2 [dxdyjj ^^^> The results here obtained include those found in Article 302 by putting ©1 = 13. Equations (53) show that the stress-couples are not expressed by the formulas (18) unless the sum of the principal curvatures is a constant or a linear function of a; and y. In like manner the formula (21) is not verified unless the sum of the principal curvatures is constant ; but these formulae yield approximate expressions for the stress-couples and the potential energy when h is small. The theory which has been given in Art. 301 and in this Article consists rather in the specification of forms of exact solutions of the equations of equilibrium than in the determination of complete solutions of these equations. The forms contain a number of unknown functions, and the complete solutions are to be obtained by adjusting these functions so as to satisfy certain differential equations such as (50) and certain boundary-conditions. These forms can represent the state of strain that would be produced in a plate of any shape by any forces applied to the edge, in so far as these forces are expressed adequately by a line-distribution of force, specified by components, T, S, N — 9H/9s, and a line-distribution of flexural couple G. 304-306] CIECULAR PLATE CENTRALLY LOADED 481 305. Circular plate loaded at its centre*- The problem of the circular plate supported or clamped at the edge and loaded at the centre may serve as an example of the theory just given. If a is the radius of the plate, and r denotes the distance of any point from the centre, we may take w to be a function of r only, and to be given by the equation = 2^1°S-r + ^' ('«) where W, A are constants, and then we have on any circle of radius r W and the resultant shearing force on the part of the plate within the circle is W. Hence W is the load at the centre of the plate. The complete primitive of (56) is /_92_ 1 ^\ \3r''' r drj ^(^rnog-^+r^'^+iAr^ + B + C\ogr, where B and C are constants of integration. If the plate is complete up to the centre, G must vanish, and we take therefore the solution yr=^-^{rnog-^+rA+iAr^ + B. The flexural couple G at any circle r=a is given by the equation We may now determine the constants A and B. If the plate is supported at the edge, so that w and O vanish at /•=«, we find -=^[^'-^l°sf-*^:(«-^)-H^Sf5(a-.^)], (57) and the central deflexion, which is the value of — w at r=0, is If the plate is clamped at the edge, so that w and 3w/3r vani.sh at r=a, we have '=8-^[''^°s;-*('''-'')]' ^'') and the central deflexion is Wa^/lGirB. If the plate is very thin the central deflexion is greater when it is supported at the edge than when it is clamped at the edge in the ratio (3 + 0-) : (1 + 0-), which is 13 : 5 when a-=J. 306. Plate in a state of stress which is Tiniform, or varies uniformly, over its plane. When the stress in a plate is the same at all points of any plane parallel to the faces of the plate the stress-components are independent of x and y, and the stress-equations of equilibrium become 3^-0 2-Zf-O ^-^^=0 If the faces of the plate are free from traction it follows that X,, Y^, Z, vanish, or the plate is in a state of plane stress. The most general state of stress, independent of x and y, which can be maintained in a cylindrical or prismatic body by tractions over its curved surface can be obtained by adding the solutions given in (iv) of Article 301 and (iii) of * Besults equivalent to those obtained here were given by Saint- Venant in the ' Annotated Clebsch,' ^oU du § 45. L. E. 31 482 UNIFOKMLY VARYING STRESS [CH. XXII Article 302. In these oases the stress is uniform over the cross-sections of the cylinder or prism. When the stress-components are linear functions of x and y the stress varies uniformly over the cross-sections of the cylinder or prism. We may determine the most general possible states of stress in a prism when the ends are free from traction, there are no body forces, and the stress-components are linear functions of x and y. For this purpose we should express all the stress-components in such forms as X, = XJx + X^'y + XJS>\ where X^, A'/, XJf^ are functions of z. When we introduce these forms into the various equations which the stress-components have to satisfy, the terms of these equations which contain x, or y, and the terms which are independent of x and y must separately satisfy the equations. We take first the stress-equations of equilibrium. The equation dx dy dz ' combined with the conditions that X^ vanishes at 0= ±A, gives us the equations x;=o, x;'=o, and in like manner we have the equations x;=o, x,"=o, ''-^+x^'+x;'=o, V^X,= - :r-^ ^-^ takes the form d'^X^fdz^^cowataxit, so that d^XJdz^=0. Since X,, satisfies )V=o, f;'=o, ^+T;-Hr;'=o. It follows that A'j and Y^ are independent of x and y. The third of the stress-equations becomes therefore dZJdz = 0, and, since Z^ vanishes at the faces of the plate (z=±h), it ^■anishes everywhere. Again e is of the form xe'+ ye" + &'■'>), where e', 0", e(") are functions of z, and, since 6 is an harmonic function, they must be linear functions of z. The equation 1 d^e J+a-c this equation and vanishes a.t z= ±h, it must contain z^-h^ as a factor, and since it is independent of x aud y it must be of the form A (z^-h^), where A is constant. Like statements hold concerning F^. It follows that, if a cylindrical body with its generators parallel to the axis of z is free from body forces and from traction on the plane ends, the most general type of stress which satisfies the condition that the stress-components are linear functions of x and y is included under the generalized plane stress discussed in Article 303 by taking Qq and 61 to be linear functions of a- and y and restricting the auxiliary plane harmonic functions / and Fi introduced in equations (32) and (47) to be of degree not higher than the third. It may be shown that, in all the states of stress in a plate which are included in this category, the stress-components are expressible in terms of the quantities €,, f2, nr, which define the stretching of the middle plane, and kj, kj, t, which define the curvature of the surface into which this plane is bent, by the formulae E \ X^ = ^ _ ^ {fi + erf 2 - ("i -I- ~^' ^' 44" ^ ^' 4>h'' and the conditions that X^ and Y^ vanish at 2 = ^ and at ^ = — ^. A particular solution is X, = ^h-'p(h''-z')x, Y, = ^h-'p(h-'-z^)y, (64) and, as in Article 299, we take X^ and Y^ to have these values. To determine X^, Yy, Xy we have the equations dXx dXy _ 'Spxz dXy dYy _ Spyz 'dx^'dy" "ihF ' 'd^ '^ ~dy ~ U^ ' Xc,+ Yy= {h-'p i-{2 + a)z' + 3z{ia + <^) (*■' + f) + f>''] + 2A']- ] (65) 31—2 484 PLATE BENT BY PRESSURE [CH. XXII To satisfy the first two of these equations we take X^, Yy, Xy to have the forms ^~8F^ ^-^^ Bi/^' ''~ %h'^ ^y' dx" " dxdy' where x ncmst satisfy the equation and then the remaining equations of (65) show that must be a linear function of x and y. As in previous Articles, this function may be taken to be zero without altering X^, Yy or Xy, and therefore % must have the form where Xi" ^^^ Xo" ^i"® functions of x and y which satisfy the equations v.V = -i(i-'^)g(*^+2/0 + f|. v,V'=ii'; (66) and we may take for %/', xo" the particular solutions 3 X." = - 1 (1 - <^) I (*^ + yj + T^f (*'^ + f),\ .(67) xo" = hp (^' + y')- J More general integrals of the equations (66) need not be taken because the arbitrary plane harmonic functions that might be added to the solutions (67) give rise to stress-systems of the types already discussed. The expressions which we have now found for X^, Yy, Xy are Yy = ip + ip^A'«'- + f + h^)-M^-'^)Plsi^^' + f)-^-~^p'^,, k..(68) ^v^Tsi^-<^)P}~z«y- « The stress-components being given by (62), (64) and (68), the corresponding displacement is given by the formulae « = - ^- g [(2 -a)z'-2h^z-2h'-Ul-a)z {x^ + f)l ^ = - ■^ t-- M-^ [('^ - <^) ^' - 3^'^ - 2/1' - f (1 - cr) ^ (x» + y% w (69) E 16 A' 307; 308] APPLIED TO ONE FACE 485 It is noteworthy that when the displacement is expressed by these formulae the middle plane is slightly stretched. We have, in fact, when z =0 du dv , .^ . p dx dy The stress-resultants and stress-couples are given by the formulae T, = \ph, T, = \ph, S, = 0, ^ N'i = ^poo, N^ = \py, G^ = ^ {(3 + where y is a constant. These values are l-o-^a^ 1 + 0- „ , ,„„ ^= — r^ + ^E-y' ^=i^'/3- If we put 3p /3 + a- 2 3-0- ,^ 3{l-a-)pa^ /5-)-o- a^ , 8-|-tr-|-o-2 A'\ ^-¥\'W''-^^0^)' y 2P 1,1+^64+ l-a^ ioj' the values of w and O at r = a, as given by the solutions in Article 302 and in Article 307, become identical. We may now combine the three solutions so as to satisfy the conditions (77) at r=a. We find the following expressions for the components of displacement »=f [4<^-^^W^pi^3 + <^)«'-(l + '^)'-'}+A|(2 + 9 dy ' dx dy ' It follows that V and U would be conjugate functions of x and y which vanish at the edge-line, they would therefore vanish everywhere. The form of w^ is given by the equation w = 4 [(1 + (85) where w is given by (83). In this case the middle plane is bent without extension. Linear elements of the plate which, in the unstressed state, are normal to the middle plane do not remain straight, nor do they cut at right angles the surface into which the middle plane is bent. 311. Plate bent by uniformly varying pressure and clamped at the edge. We seek to satisfy the conditions (80) at the edge -line by a synthesis of the solutions in Articles 301, 303 and 308. For u and v we have the forms 1 1 in which the unknown functions must be chosen so that u and v vanish at the edge-line. We may show in the same way as in Article 310 that these conditions cannot be satisfied in more than one way. The unknown functions depend upon the shape of the edge-line. * The result was commimioated to the Author by Prof. G. H. Bryan. 310-312] PLATE BENT BY TRANSVERSE FORCES 491 When this line is a circle or an ellipse the conditions may be satisiied by assuming for I, V, ©0. xo tte forms where aj, 0i, y^ are constants. For a circle of radius a we should find 3po(l + o-) _ y(,(3 + 5o-) , fflV(l+o-)jPo '~ 6-2o- ' '^i~4(6-2o-) ' ^i~ 6-2o- ' and thence u= — -^ — s— ct('^^~''^)) v=0. o — zir ^ For an ellipse given by the equation iiflla!' + i/^/b^=l we should find (l+o-)(a^ + 26^)yo {a'(l +3o-) + 26^ (H-(r)}po o- (l + cr) ppg^S ^ "' 2a2(l-_o.) + 462 ' '^^ 4{2a2(l-(k^ + o-ATi) + gQ ,^ _ . 9ph\ H, = D(1- + 9^ + ^' = o ?^-^ + i\r=o ^-?^-i\r, = o. dx dy ' dx dy ^ ' dx dy By eliminating N^ and i^a from these we obtain the equation dx^ dy^ dxdy ' and by substituting from (17) and (18) in this equation we find the equation i)V>=^' (92) The stress-couples G, H at the edge are given in accordance with (17) and (18) by the formulae * For authorities in regard to the approximate theory, see Introduction, pp. 27 — 29. A general justification on the same lines as that of the corresponding theory for rods (Article 258) will be found in Article 329 of Chapter XXIV. A very elaborate investigation of exact solutions for various distributions of load has been given by J. Dougall, Edinburgh, R. Soc. Trans., vol. 41 (1904). In this investigation the correctness of the approximate theory is verified for all cases of practical importance. 312-314] OF THE BENDING OF PLATES 493 To find an expression for the shearing force N in the direction of the normal to the plane of the plate we observe that dHA 9y ^ ' " " " \^y dx )' and then on substituting from (17) and (18) we find the formula iV" = -Z)|-v/w (93) To determine the normal displacement w of the middle plane we have the differential equation (92) and the boundary-conditions which hold at the edge of the plate. At a clamped edge w and d^jdv vanish, at a supported edge w and G vanish, at an edge to which given forces are applied N — oHjds and G have given values. The same differential equation and the same boundary-conditions would be obtaiaed by the energy method by assuming the formula (21) for the potential energy estimated per unit of area of the middle plane *. In all the solutions which we have found the differential equation (92) is correct whether the formulse (18) and (21) are exactly or only approximately correct f. The solutions that would be obtained by the approximate method described in this Article differ from the exact solutions that would be obtained by the methods described in previous Articles only by very small amounts depending on the small corrections that ought to be made in the formulae (18) for the stress-couples. In general the form of the bent plate is determined with sufficient approximation by the method of this Article. There is a theory of the existence of solutions of equation (92), in which Z' is a given function of x and y, subject to special conditions at the edge of the plate. The case where the edge is clamped has been chiefly considered. For this theory reference may be made to J. Hadamard, Paris, Mem. ...par divers savants, t. 23, 1908, A. Korn, Paris, Ann. ic. tiorm. (S^r. 3), t. 25, 1908, and G. Lauricella, Acta math., t. 32, 1909. 314. Illustrations of the approximate theory. (a) Circular plate loaded symmetrically \. When a circular plate of radius a supports a load Z' per unit of area which is a function of the distance r from the centre of the circle, equation (92) becomes Ml^l^rli'T^W^^'l^' the direction of the displacement w being the same as that of the load Z'. We shall record the results in a series of cases. * The process of variation is worked out by Lord Bayleigh, Theory of Sound, § 215. t A more general form which includes (92) in the special oases previously discussed is given by J. H. Michell, loc. cit., p. 471. + The general form of the solution and the special solutions (i)— (iv) were given by Poisson in his memoir of 1828. See Introduction, footnote 36. Solutions equivalent to those in (v) and (vi) were given by Saint- Venant in the 'Annotated Clebsch,' Note du § 45. 494 APPROXIMATE THEORY [CH. XXII (i) When the total load ir is distributed uniformly and the plate is supported at the edge (ii) When the total load R^is distributed uniformly and the plate is clamped at the edge W (iii) When the load W is concentrated at the centre and the plate is supported at the edge (iv) When the load 11' is concentrated at the centre and the plate is clamped at the edge ''' = 8i["'''^°S7 + *^"'"'''^]- (v) When the total load W is distributed uniformly round a circle of radius 6 and the plate is supported at the edge, w takes different forms according as »• > or < 6. We find (vi) When the total load IF is distributed uniformly round a circle of radius 6 and the plate is clamped at the edge, we find (6) Application of the method of inversion*. The solutions given in (iii) and (iv) of (a), or in Article 305, show that, in the neighbourhood of a point where pressure P is applied, the displacement w in the direction of the pressure is of the form (P/87rZ))«-2 1ogr + f, where f is an analytic function of X and y which has no singularities at or near the point, and r denotes distance from the point. Since w satisfies the equation Vj''w = at all points at which there is no load we may apply the method of inversion explained in Article 154. Let 0' be any point in the plane of the plate, P any point of the plate, P' the point inverse to P when 0' is the centre of inversion, of, y' the coordinates oi P', R' the distance of P' from 0', w' the function of icf, -if into which w is transformed by the inversion. Then ii'^w' satisfies the equation 34 04 04 Vi'*(JS'%')=0, where V,'-* denotes the operator .—4 + g-M+ 2 , ay ^3y*^ 9.2/29^2 ■ It is clear that, if w and 3w/9i/ vanish at any bounding curve, /J'^w' and d{R'^w')ldv' vanish at the transformed boundary, 1/ denoting the direction of the normal to this boundary. * J. H. Michell, London Math. Soc. Proc, vol. 34 (1902), p. 223. 314] OF THE BENDING OF PLATES 495 We apply this method to the problem of a circular plate clamped at the edge and loaded at one point 0. Let 0' be the inverse point of with respect to the circle, C the centre of the circle, and a its radius, also let c be the distance of from C. The solution for the plate clamped at the edge and supporting a load WaXCis where r denotes the distance of any point P from C. Now invert from 0' with constant of inversion equal to a^jc^-a'. The circle inverts into itself, C inverts into 0, P inverts into P so that, if OP'=R and aP' = R\ we have Fig. 70. IJence R'^w' is R r R^ BirD R^ [- a4iJ2 , cR' ^\f~^wy\' ^;[-^'»^S**e-— )]■ It follows that the displacement w of a circular plate of radius a clamped at the edge and supporting a load W at a point distant c from the centre is given by the equation -8S[-^^l°sS+KS^'^-^^)]' (^^) where R denotes the distance of any point of the plate from 0, and R' denotes the distance of the same point from the point C, inverse to with respect to the circle. We may pass to a limit by increasing a indefinitely. Then the plate is clamped along a straight edge and is loaded at a point 0. If 0' is the optical image of in the straight edge, the displacement in the direction of the load is given by the equation w = g-J[-iJMog J'+i(fl'2-iJ2)], (95) where R, R' denote the distances of any point of the plate from the points and G. The contour lines in these two cases are drawn by Michell (Jioc. cit.). (c) Rectangular plate. Let the origin be taken at one corner and the axes of x and y along two edges, let the other edges bo given by x=%a, y = 2b. We expand Z' in the form Z' = 22Z„ . mnx . nirv where m and n are integers. Wi = Then a particular solution of equation (92) is 16Z' „ sin {mirxlia) sin (mry/ib) -i^ 22Z,„„ ; ' mya* + n^lb* + 2m'n^la^b^ ' If the edges x=0 and x=2a are supported this solution satisfies the boundary-con- ditions at these edges. If all four edges are supported the solution satisfies all the conditions, but if the remaining edges are not simply supported we have to find a solution 496 APPROXIMATE THEORY [OH. XXII W2 of the equation Vi*W2 = so that the sum Wi + W2 may satisfy the conditions aty=0 and y= 26. We assume for Wj the form * miTX 2a ' where I',, is a function oiy but not of x. Then Y^ satisfies the equation and the complete primitive is of the form r„=4„oosh^ + 5„sinh^+y^4„/cosh^ + £„.'sinh^j, where A^, B^, A^, B^ are undetermined constants. Tliese constants can be adjusted so as to satisfy the boundary-conditions at .y=0 and y = 26t. Among other examples of the application of the theory to rectangular plates we may note the problem presented 'by the flexure of a broad very thin band (such as a watch spring) into a circle of radius comparable with a third proportioual to its thickness and its breadth. Attention was drawn to this problem in Kelvin and Tait's Nat. Phil., Part II, § 717, and a solution was given by H. Lamb, Phil. Mag. (ser. 5), vol. 31, 1891, p. 182. The complete analytical solution of the problem of a rectangular plate with clamped edges, bent by pressure applied to one face, has not been made out, but an approximate method of solution has been devised by W. Ritz, J. f. Math. {Crelle), Bd. 135, 1909, p. 1, reprinted in Oes. Werke Walther Ritz, Paris, 1911. The case of a square plate subjected to uniform pressure on one face is worked out by Ritz's method by C. G. Knott in Edin- burgh, Proc. R. Soc, vol. 32, 1912, p. 390. (d) Transverse vibrations of plates. The equation of vibration is obtained at once from (92) by substituting for Z' the 9^w expression - 2pA j-^ . We have 8% 8''w aV _ 2ph 8V 3ari"^o/'*' a^%2~ D 3^2 '■^^^i When the plate vibrates in a normal mode w is of the form W cos {pt + e), where W is a function of x and y which satisfies the equation yw S4W 3W 3p(l-a2)p2 3.J;* ^ 3/ "^ 3.j;%2 ~ Eh^ ' and the possible values of p are to be determined by adapting the solution of this equation to satisfy the boundary-conditions. From the form of the coefficient of W in the right-hand member of this equation it appears that the frequencies are proportional to the thickness, and inversely proportioual to the square of the linear dimension of the area within the edge-line. * This step was suggested by M. L^vy, Paris, C. R., t. 129 (1899). + The case of four supported edges is discussed by Saint- Venant in the 'Annotated Clebseh,' Note du § 73. A number of cases are worked out by E. Estanave, 'Contribution k I'^tude de I'equOibre felastique d'une plaque...' (Tliise), Paris, 1900. Elastic constants are sometimes measured by observing the central deflexion of a rectangular plate supported at two opposite edges and loaded at the centre; see A. E. H. Tutton, Phil. Tram. E. Soc. (Ser. A), vol. 202 (1903). 314] VIBRATIONS OF PLATES 497 The theory of those modes of transverse vibratiou of a circular plate in which the displacement is a function of distance from the centre was made out by Poiason*, and the numerical determination of the frequencies of the graver modes of vibration was effected by him. In this case the boundary-conditions which he adopted become identical with Kirchhoffs boundary-conditions because the torsional couple H belonging to any circle concentric with the edge- line vanishes. The general theory of the transverse vibrations of a circular plate was obtained subsequently by Kirchhofft, who gave a full numerical discussion of the results. The problem" has also been discussed very fully by Lord Hayleigh|. The complete analytical solution of the problem of free vibrations of a square or rectangular plate has not yet been made out, but an approximate method of solution has been devised by W. Ritzg. The case of elliptic plates has been considered by E. Mathieull and A. Barth^l^myl. (e) Extendonal vibrations of plates. We may in like manner investigate those vibrations of a plate which involve no transverse displacement of points of the middle plane, by taking the stress-resultants Ti , T2, Si to be given by the approximate formulae, [cf. (iv) of Article 301], „ 2Ek /3u 3v\ „ 2^A /dv 3u\ „ Eh /Bu 9vN or the potential energy per unit of area of the middle plane to be given by the formula Eh The equations of motion are /9u Svy jBuBv ,/3u,3vyn .(97) 8^1 ^9^1 „ ,3% 951^9^2 „ ,92v d^+d^ = ^P^W^ 9^+9^=^''^9P' 3^u,,n ^^'^^ln^^9'^ p(l-.r'') 9% \ 9^2+^(^-'^^9p+i(^+'')8:^ = ^^" P' , ,, ,92v 9V , ,, , , 92d p(1-(t2)92v At a free edge the stress -resultants denoted by T, S vanish. The form of the equations shows that there is a complete separation of modes of .vibration involving transverse displacement, or flexure, from those involving displacement in the plane of the plate, or extension, and that the frequencies of the latter modes are independent of the thickness, while those of the former are proportional to the thickness**. • * In the memoir of 1828 cited in the Introduction, footnote 36. t J.f. Math. (Crelle), Bd. 40 (1850), or Ges. Abhandlimgen, p. 237, or Vorlesungen iiber math. Physik, Mechanik, Vorlesung 80. t Theory of Sound, vol. I, Chapter x. § Ann. Fhys. (4te Folge), Bd. 28, 1909, p. 737, or Ges. Werke Walther Ritz, p. 265. See also Lord Eayleigh, Phil. Mag. (Ser. 6), vol. 22, 1911, p. 225. II J. de Math. (Lioumlle), (S^r. 2), t. 14 (1869). IT Mem. de VAcad. de Toulouse, t. 9 (1877). ** Equations equivalent to (97) were obtained by Poisson and Cauchy, see Introduction, foot- notes 36 and 124. Poisson investigated also the symmetrical radial vibrations of a circular plate, obtaining a frequency equation equivalent- to (107), and evaluating the frequencies of the graver modes of this type. L. p.. 32 498 EXTENSION AL VIBRATIONS OF PLATES [CH. XXII The equations of vibration (97) may be expressed very simply in terms of the areal dilatation A' and the rotation nj, these quantities being defined analytically by the equations ,_3u 3v Sv 3ii dx ■ 3y' dx 3y ^ ' The equations take the forms 3a' ,, , Bar l-o-^gsu 9a' „ , 3ar 1 - o-^ S^v a^-(i-<^) 3^=/'^^a<^. a^+(i-<^)a^=p-^r 3? ^^^^ These forms can be transformed readily to any suitable curvilinear coordinates. Consider more particularly the case of a plate with a circular edge-line. It is appropriate to use plane polar coordinates ?■, 6 with origin at the centre of the circle. Let (J, V be the projections of the displacement of a point on the middle plane upon the radius vector and a line at right angles to the radius vector. Then we have u=f'oose- Fsin^, \=Usm d+Vcoae, (100) and ^' = _^+_+__, 2..= g^ + ---^, (101) and the stress-resultants belonging to any circle »■ = const, are T, S, where The equations of vibration give ^''^-^^-dF' ^'^—E-~ "dfi (^°^) We put U^DnCosnecospt, F= F„sin9i5cos^<, (104) where U„ and F„ are functions of r, and we write <2=p (1-0-2)^7^, K'-'=2p{l+a-)p^jE. (105) Then A' is of the form A' J„{Kr) cos neoospt, and nr is of the form -S'./„ (kV) sin ?i5 cos^i, where A' and J? are constants, and J„ denotes Bessel's function of order n. The forms of U and V are given by the equations U=[A''-^+nB'^'joo,necospt, F= -[»^"^ + fi^]sin.^cosp^, and with these forms we have ^'= - -^kV, (itr) cos nd COS pt, 2nr = .BK'V„(/cV)sin n6 cos pt. We can have free vibrations in which F vanishes and U is independent of 6 ; the frequency equation is dJi {ko) da + -J^{Ka)=0 (107) a a being the radius of the edge-line. We can also have free vibrations in which V vanishes and V is independent of 6 ; the frequency equation is dJ-i {K'a) _ Jj (k'os) da a .(108) These two modes of symmetrical vibration appear to be the homologues of certain modes of vibration of a complete thin spherical shell (cf. Article 335 infra). The mode m 314] EXTENSION AL VIBRATIONS OF PLATES 499 which U vanishes and V is independent of 6 is the homologue of the modes in which there is no displacement parallel to the radius of the sphere. The mode in which V vanishes and V is independent of 6 seems to be the homologue of the quicker modes of symmetrical vibration of a sphere in which there is no rotation about the radius of the sphere. In the remaining modes of extensional vibration of the plate the motion is compounded of two : one characterized by the absence of areal dilatation, and the other by the absence of rotation about the normal to the plane of the plate. The frequency equation is to be formed by eliminating the ratio A : B between the equations (109) These modes of vibration seem not to be of sufficient physical importance to make it worth while to attempt to calculate the roots numerically. 32—2 CHAPTER XXIII. INEXTENSIONAL DEFORMATION OF CURVED PLATES OR SHELLS. 315. A CURVED plate or shell may be described geometrically by means of its middle surface, its edge-line, and its thickness. We shall take the thick- ness to be constant and denote it by 2/i, so that any normal to the middle surface is cut by the faces in two points distant h from the middle surface on opposite sides of it. We shall suppose that the edge of the plate cuts the middle surface at right angles ; the curve of intersection is the edge-line. The case in which the plate or shell is open, so that there is an edge, is much more important than the case of a closed shell, because an open shell, or a plane plate with an edge, can be bent into an appreciably different shape without producing in it strains which are too large to be dealt with by the mathe- matical theory of Elasticity. The like possibility of large changes of shape accompanied by very small strains was recognized in Chapter xvill. as an essential feature of the behaviour of a thin rod ; but there is an important difference between the theory of rods and that of plates arising from a certain geometrical restriction. The exten- sion of any linear element of the middle surface of a strained plate or shell, like the extension of the central-line of a strained rod, must be small. In the case of a rod this condition does not restrict in any way th^ shape of the strained central-line; and this shape may be determined, as in Chapters Xix. and XXI., by taking the central-line to be unextended. But, in the case of the shell, the condition that no line on the middle surface is altered in length restricts the strained middle surface to a certain family of surfaces, viz. those which are applicable upon the unstrained middle surface*- In the particular case of a plane plate, the strained middle surface must, if the displacement is inextensional, be a developable surface. Since the middle surface can undergo but a slight extension, the strained middle surface can differ but slightly from one of the surfaces applicable upon the unstrained middle surface ; in other words, it must be derivable from such a surface by a displacement which is everywhere small. * For the literature of the theory of surfaces applicable one on another we may refer to the Article by A. Voss, 'Abbildung und Abwickelung zwei^r Flaohen auf einander ' in Ency. d, math. Wiss., ni. D 6a. 315, 316] CURVATURE OF SHELL BENT WITHOUT EXTENSION 501 316. Change of curvature in inextensional deformation. We begin with the case in which the middle surface is deformed without extension by a displacement which is everywhere small. Let the equations of the lines of curvature of the unstrained surface be expressed in the forms a = const, and /3 = const., where a. and /3 are functions of position on the sur- face, and let R^, R^ denote the principal radii of curvature of the surface at a point, jRi being the radius of curvature of that section drawn through the normal at the point which contains the tangent at the point to a curve of the family ^ (along which a is variable). When the shell is strained without ex- tension of the middle surface, the curves a = const, and /3 = const, become two families of curves drawn on the strained middle surface, which cut at right angles, but are not in general lines of curvature of the deformed surface. The curvature of this surface can be determined by its principal radii of curvature', and by the angles at which its lines of curvature cut the curves a and ^. Let "5" + S "p" s-nd p- -|- S ^ be the new principal curvatures at any point. Since the surface is bent without stretching, the measure of curvature is un- altered*, or we have \Ri Rj XR^ RJ R1R2 ' or,- correctly to the first order in 8 -5" ^^^ ^ ^ > 4«4+4-s4=o (1) M^ JXi Ml XI2 Again let yfr be the angle at which the line of curvature associated with the principal curvature -=- + 8 p- cuts the curve /8 = const, on the deformed surface, and let i?/, R2' be the radii of curvature of normal sections of this surface drawn through the tangents to the curves /3 = const, and a = const. In general y{r must be small, and R/, R2 can differ but little from R^, R^. The indicatrix of the suface, referred to axes of x and y which coincide with these, tangents, is given by the equation |, + 1, + .y tan 2t(^,-^,) = const. Referred to axes of ^ and 77 which coincide with the tangents to the lines of curvature, the equation of the indicatrix is * The theorem is due to Gauss, 'Disquisitiones generales circa superficies curvas,' GSttingm Comm. Rec, t. 6 (1823), or Werke, Bd. 4, p. 217. Cf. Salmon, Geometry of three dimemions, 4th edition, p. 355. 502 CURVATURE OF A THIN SHELL [CH. XXIII and therefore we have — — = — + — + si + s — i?i' jRj ill -^2 J^i -^2 ...(2) The bending of the surface is determined by the three quantities Kj, k^, t defined by the equations .(3) The curvature l/R of the normal section drawn through that tangent line of the strained middle surface which makes an angle w with the curve /3 = const, is given by the equation 1 cos^o) sin" ft) - d7 = p/ H — WT- + ^'^ sin ft) cos ft), K Jfi JI2 and the curvature \\TL of the corresponding normal section of the unstrained middle surface is given by the equation 1 cos" ft) sin" ft) -R xi] -R2 so that the change of curvature in this normal section is given by the equation ■p7 — p = '^1 cos" ft) + /C2 sin" ft) + 2t sin o) cos ft) (4) We shall refer to Kj, k^, t as the changes of curvature. In general, if R^ ^ i2j, equations (2) give, correctly to the first order, Sp-=«i, 8p- = /<:o, W+^ = 0. Jll JXj JI2 iti For example, in the case of a cylinder, or any developable surface, if the lines /8 = const, are the generators, tc^ vanishes, and tan 2i|r = — 2Ti?2- The case of a sphere is somewhat exceptional because of the indeterminate- ness of the lines of curvature. In this case, putting R^ = R^, we find from (1) ^Rr~^Rr^R'^^' and then we have, correctly to the first order, «i + «2 = 0, tan 2ifr = 2t/(a;i — k^ = t/«i , and, correctly to the second order, f S -^j = - f 1^2 + T' = «," + t". 316, 317] SLIGHTLY BENT WITHOUT EXTENSION 503 but Kj and k^ are not equal to S ^fj- and S -=- unless t = 0, and -ir is not small unless T is small compared with «,. The result that, in the case of a cylinder slightly deformed without extension, k^=Q, or there is no change of curvature in normal sections containing the generators, has been noted by Lord Rayleigh as "the principle upon which metal is corrugated." He has also applied the result expressed here as ki/R2 + k2/Ri=0 to the explanation of the behaviour of Bourdon's gauge*. 317. Typical flexural strain. We imagine a state of strain in the shell which is such that, while no line on the middle surface is altered in length, the linear elements initially normal to the unstrained middle surface remain straight, become normal to the strained middle surface, and suffer no extension or contraction. We express the components of strain in this state with reference to axes of x, y, z, which are directed along the tangents to the curves /8 and a at a point Pj on the strained middle surface and the normal to this surface at Pj. Let P be the point of the unstrained middle surface of which Pj is the displaced position, and let is be an element of arc of a curve s, drawn on the unstrained surface, and issuing from P ; also let R be the radius of curvature of the normal sec- tion of this surface drawn through the tangent to s at P. The normals to the middle surface at points of s meet a surface parallel to the middle surface, and at a small distance z from it, in a corresponding curve, and the length of the corresponding element of arc of this curve is approximately equal to [{R — z)IR] Bsf. When the surface is bent so that R is changed into R', and z and Bs are unaltered, this length becomes {{R' — z)jR'] 8s approximately. Hence the extension:!: of the element in question is /R'-z _ R-z\ /R ■ \ R' R Let the tangent to s at P cut the curve ;8 at P at an angle eo. The direction of the corresponding curve on the parallel surface is nearly the same ; and the extension of the element of arc of this curve can be expressed as Bxx cos^ , and neglecting z^/iJiS and z^/jj^a, we obtain the result stated in the text. J Cf. Lord Bayleigh, Theory of Sound, 2nd edition, p. 411. n/— ^, or, approximately, -^{^'-^)- 504 STRAIN IN THIN SHELL BENT WITHOUT EXTENSION [CH. XXIII In the imagined state of strain e^x, e,yz, e^^ vanish. With this strain we may compound any strain by which the linear elements initially normal to the unstrained middle surface become extended, or curved, or inclined to the strained middle surface. The most important case is that in which there is no traction on any surface parallel to the middle surface. In this case the stress- components denoted by X^, Y^, Z^ vanish, and the strain-components 62^,, Byz, e„ are given by the equations eZX = 0, Byl = 0, Bz! = - {Cr/(1 - O-)} (e^X + Syy), where a is Poisson's ratio for the material, supposed isotropic. In this state of strain the linear elements initially normal to the unstrained middle surface remain straight, become normal to the strained middle surface, and suffer a certain extension specified by the value of Bzz written above. It is clear that this extension can have very little effect* in modifying the expressions for Bxx, Byy, Bxy, and we may therefore take as approximate expressions for the strain-components Bxx = - ZK^, eyy = - ZK^, e^z = ^j-— Z (Ki -I- «2), Bxy = - ^TZ, B^x = By^ = 0. . . .(5) This state of strain may be described as the typical flsxural strain. The corresponding stress-components are E W -X'a; = - j3^2'2(«i + 0-'c, + K,y-2{l-a-){K,>.,-T% (6) where B is the "flexural rigidity" |^^V(1 - c^')- In the case of a cylinder, or any developable surface, this expression becomes |i) {k^^ + 2 (1 - a) t^\. In the case of a sphere it becomes i^^h'(>c,^ + T% or i/^h'fs^', where /.t is the rigidity of the material f. * It will be seen in the more complete investigation of Article 327 below that such effects are not entirely negligible. t These are the expressions used by Lord Bayleigh, Theory of Sound, 2nd edition, Chapter X i. 317, 318] CALCULATION OF THE CHANGE OF CURVATURE 505 318. Method of calculating the changes of curvature. The conditions which must be satisfied by the displacement in order that the middle surface may suffer no extension may be found by a straightforward method. Let ASa be the element of arc of a curve /3 = const, between two curves a and a + Sa, BS^ the element of arc of a curve a = const, between two curves /3 and /3 + 8/S ; also let a;', y' , / be the coordinates of a point on the strained middle surface referred to any suitable axes. We form expressions for X, y', z in terms of the coordinates of the point before strain and of any* suitable components of displacement. Since curves on the middle surface retain their lengths, and cut at the same angles after strain as before strain, we must have a^'8*' d£dy^ ^i.'?f! = n These equations give us three partial differential equations connecting the components of displacement. The changes of curvature also may be calculated by a fairly straightforward method. The direction-cosines I, m, n of the normal drawn in a specified sense to the strained middle surface can be expressed in such forms as 1 /dy'd/ dz'dy'\ - ABKdad^^ dad$l' and the ambiguous sign can always be determined. The equations of the normal are x — x'_y — y'_z — z' I " m n ' and, if (x, y, z) is a centre of principal curvature, we have a; = «' + Ip, y = y' + mp', z = z' + np, where p' is the corresponding principal radius of curvature ; p' is estimated as positive when the normal (l, m, ■(i) is drawn from («', y', z') towards {x, y, z). If (a + Sa, /8 + S/3) is a point on the surface near to {x, y, z') on that line of curvature through {x , y\ z) for which the radius of curvature is p, the quanti- ties X, y, z, p are unaltered, to the first order in Sa, S/3, by changing a into a -f- 8a and /3 into /3-I- S;8. The quantity we have already called tan i/r is one of the two values of the ratio Bl^\Aloi. Hence tan -f and p are determined by the equations 8/8 da "^ "" 3/8 '"«-|^^+.'(Sa..|8,)=o. 506 INEXTBNSIONAL DEFORMA.TION OF A [CH. XXIII These three equations are really equivalent to only two, for it follows from the mode of formation of the expressions for I, m, n, and from the equation i^ + m" + w^ = 1, that, when we multiply the left-hand members by I, m, n and add the results, the sum vanishes identically. By eliminating the ratio Sa/Bff from two of these equations we form an equation for p', and the values of 1/p' are »■ + ^ w and "p" + ^ p" j t'y eliminating p' from two of the equations we form an equation for S/S/Sa, which determines tan yjr. We shall exemplify these methods in the cases of cylindrical and spherical shells. In more difficult cases, or when there is extension as well as change of curvature, it is advisable to use a more powerful method. One such method will be given later ; others have been given by H. Lamb* and Lord Rayleight. The results for cylindrical and spherical shells may, of course, be obtained by the general methods ; but these cases are so important that it seems to be worth while to show how they may be investigated by an analysis which presents no difficulties beyond the manipulation of some rather long expressions. The results in these oases were obtained by Lord RayleighJ. 319. Inextensional deformation of a cylindrical shell. (a) Formuke for the displacement. When the middle surface is a circular cylinder of radius a, we take the quantities a and /3 at any point to be respectively the distance along the generator dawn through the point, measured from a fixed circular section, and the angle between the axial plane containing the point and a fixed axial plane ; and we write x and <^ in place of a and /3. We resolve the displacement of the point into components : u along the generator, v along the tangent to the cir- cular section, w along the normal to the surface drawn inwards. The coordinates x', y', / of the corresponding point on the strained middle surface are given by the equations x' = x + u, y' = (a — w) cos — vsm j>, z' ={a — w) sin (^ + M cos ^. x,x Fig. 71. 1, The conditions that the displacement may be inextensional are dx d dx ad(l) These equations show that u is independent of x, and v and w are linear functions of x. If the edge-line consists of two circles x = const., u, v, w must be periodic in ^ with period 27r, and the most general possible forms are « = — 2 - £„ sin (w^ + y8„), « = S [J.„ cos (« + a„) + B^x cos (n- dxdcji) ' " ' We write down the values of dx'/dx,... , simplified by using (7), in the forms dx' ^ dv' Idu . , dw , dz' Idu , 9w . , 5- = l, ^ = -^Tsmrf)-^-cos(i, 5- = --r-7COS(^--5-sin0, dx dx ad(f> ^ dx ^ dx ad

~ dxd^l'^ p \dx d^ "^ dx d^ dx 90 dx d^J dl dm dmdl _ . • dx d(f) dx d^ ' 508 INEXTENSIONAL DEFORMATION OF A [CH. XXIII , /t \.ftx'dm dy' dl\ , /t .s, /9a;'9m dy' dl\ + ^^'Pla^a^ + a^a^~9S3 9 (• dx a 9a; ' dl d^w dm . . (, 1 /9% 9-^ = -9^- 9^='^°'^l^ + al9^^ + ^ cos d) / 9w\ We know beforehand that, when terms of the second order in u, v, w are neglected, one value of Ijp is zero and the other is Ija 4- «2 j also the value of a?i4>lhx is tan ■y^, and tan 2i|r = — 2aT. We can now write down the above equations for p and Bx/B in the forms (correct to the first order in u, v, w) 1 + ''^) — a sin + a„) + 5„« sin (« + /3„)], 1 if — \ T = - S 5„ cos («<^ + /S„). .(11) 320. Inextensional deformation of a spherical shell. (a) FormulcB for tlie displacement. When the middle . surface is a sphere of radius a we take the coordinates 319, 320] THIN SPHERICAL SHELL 509 a and /S to be ordinary spherical polar coordinates, and write d, (jj for a, ^. The displacement is specified by components u along the tangent to the meridian in the direction of increase of 0, v along the tangent to the parallel in the direction of increase of 0, w along the normal to the surface drawn inwards. The Cartesian coordinates of a point on the strained middle surface are given by the equations x' = {a — w) sin d cos ^ + u cos 6 cos ^ — v sin 0, y' = {a — w) sin 6 sin <^ + u cos sin ^ + w cos 0, '^' ' z ={a — w) cos ^ — M sin 6. The conditions that the displacement may be inextensional are WJ V+(' "■ lag J ' a sin l\d. djl r ^ = - (a - w) sin ^ + M cos + dv d4> dv (a — w)sin0 + ucos6 + ~ sin0 + cos^ + du n dw . n ;5-r cos P — « — ;r-7 Sin ^ 0(p 0(f) du n Sw . /, ,^-r cos P — B — :5-7 Sin t* 0^ Oif COS( sin^, dz' du . „ dw a ^--^>''''^-^''''^■ The conditions that the displacement may be inextensional are, to the first order in u, v, w, du -a a , ^'" w = K7,, wsnia = ucost/+ :r-; . da o

510 INEXTENSIONAL DEFORMATION OF A or, as they may be written, [CH. XXIII du dti sin 6 9<^ sin 9<^ sin 6 dd sin 6 The last two of these equations show that w/sin 6 and w/sin ^ are conjugate functions of log (tan ^6) and 0. If the edge-line consists of two circles of latitude, u,v, w must be periodic in j> with period 2ir, and the most general possible forms for them are M = sin ^ S V = sin d S w= 2 -4„ tan" H cos (n<^ + a„) + £„ cot" ^ cos (nt/) + y8„) a g An tan" ^ sin (n<^ + a„) - 5„ cot" ^ sin (n^ + j8„) (n + cos 9) An tan" ^ cos (n^ + a'„) ,6' .(13) — (w — cos 0) 5„ cot" q cos (w + ^). The former is equivalent to a translation (-^jcosa, .^isina, 0) and a rotation A^a'^ (sin a, cos a, 0) ; and the latter is equivalent to a translation (Bicoafi, -5isin/3, 0) and a rotation Bia~^ (sin fi, cos 0, 0). It appears from what has just been said that all the displacements obtained from (13) by putting ?i = or 1 are possible in a rigid body, and the terms for which n has these values may be omitted from the summations. Similar results can be proved in the case of cylindrical shells. If the edge-line consists of one circle of latitude, and the pole ^ = is in- cluded, we must omit from (13) the terms in cot" ^ 6, {n> 1), for these terms become infinite at the pole. If the sphere is complete the terms in tan" J 6, (n > 1), must be omitted also ; that is to say no inextensional displacements 320] THIN SPHERICAL SHELL 511 are possible in a complete spherical shell except such as are possible in a rigid body*. (b) Changes of curvature. "We form next expressions for the direction-cosines I, m, n of the normal to the deformed surface, by means of such formulae as a' sin [dcf) dd d4> 30 ) ' and for this purpose we first write down the expressions for dx'/dO, . . . simplified by means of equations (12). We have ^*' a ± /Sw \ • /I , ^v , . . ^ = a cos cos

— ■^'sm , ^ = — asin^— (^ +m1cos0, and dx' • a ■ J. , /Su n dw . .\ J ^ = - a sm t' sm , ^ a \d0 / ^ a\ sm0 dj>J ^ m = — sm.0svad> ( ttt? + m ) cos 0sm {v+ -. — ^ ;:— ) cos 6, a \d0 / ^ a \ sin d(j)J ^ n 1 fdw \ . n n = — cos + - h^ + M ) sin 6'. a \d0 1 Exactly as in the case of the cylinder, the principal curvatures and the direc- tions of the lines of curvature are determined by the compatible equations * The result is in aeoordance with the theorem that a closed surface cannot be bent without stretching. This theorem is due to J. H. Jellett, I>ublm Trans. E. Irish Acad., vol. 22 (1855). 512 INEXTENSIONAL DEFORMATION OF A [CH. XXIII and we therefore write down the following equations, in which we put for „ 1 /aw \ ,. 1 / 1 dw\ shortness X =-[i^+u ], i =-{v+ -^—^ r-r , a\dd J a\ sinod4>/ dl -^ = - f 1 + -^= j COS ^ sin (^ + Z sin sin (^ - ^ cos <^, j = — [ 1 + -^ ) cos ^ cos ^ + X sin ^ cos ^ + ^ sin (/>, dn and ^ = (sin ^ + X cos + ^ j sin (/> - f — cos - Y\ cos <^, = (l + ^)sin^+Xcos0, ^= - (sin ^ + X cos + -^ j cos ^ - f -x-r cos d—Yj sin , dn . n 9X ^^''''^^^■ Our procedure in this case must be a little different from that adopted in the case of the cylinder because, to the first order, the sum and product of the principal curvatures are unaltered by the strain. We therefore begin by finding the equation for tani/r, or sindBcfj/Sd. This equation may be written fdx' tani/r9a;'^ fdm tan ->/r 9to\ _ /dy' tan i/r9i/'\ fdl tan ^frdl'x ■^ sin 61 dAj^XW^ sin dMJKde'^ sin 6 dd)) ' ■ sin 61 = 0. \de smedcf>/\dB smed4>J \d6 smed(j)/\de sine d(l)J and, by direct substitution of the values written above for dso'/de,..., it is found to be /dv dY\ , tan^ /. ,aX „ a ^Y\ a [de-''de)''"^^^''\''''^de-^'°'^-^)'''^ tan^-vV f /9X - „\ /9m - dw . Now we have 961 °'de~ de[smed^J' (dX a ^t\ fdu „ dw . .\ . „ 9 / 1 9w a [ ,^-r cos e — 1 - — cos y - ■!) - _- sm ^ = sin ^ cos ^ ;575 ^ — „ ^^ \d4> I \d(p a^ j de \sin e d(p /. .9X „ . 9F\ . ./a^w dv\ .(dw \ dv I d'w a[sm0^-Xcose-^')=^sine^-^^-+^^)-cose[^^+uj-^^-^^--^^ . a r/3'«^ ^ / \ d^w ^ .dw = sme^{^^^+wj-[^-^^^-,coie^^+w where, in the last line, use has been made of the equations (12). But, since 320] THIN SPHERICAL SHELL 513 w = du/dO, and u satisfies the equation obtained by eliminating v from the second and third of (12), viz. it follows that 1 9'w , , /j9w , IS/. „.d^u , . ^ .9m \ ^.dw + cot6 ^ + w = 3-5-^ ^ - sin''^ ^^ + sm^ cos0 ;r^-u] + cotO ^r^+'w sin^Odcf}^ dd sin'' 6' ay 9^w 3612 36' 36'» — w. Hence the equation for tan i/r becomes ^ „ , 3 / 1 3w\ f/cPw \ One of the equations for determining p is 3/ d6 80- , fdn -%'*-^'{fe^-p*h'- or 30 ax 3m 30* T oJl\ . /, tt- /I 3.A , , 1 + -^ 1 sm t' + A cos 0+ KT ^^^ T 30 ^ sin y + X cos ^ + - ( ;r^ + cot ^ ~] tan -vt- a \.30 30/ '^ , 3X tan ^b• (dX \ (du , . aw\) 30 sm 6 (30 a \30 30/j ~ ^ aKW^'^'^j '^ ~^ d0\fim0d<^) But, using the notation of Article 316, we have - — = Ki cos'' i|^ + K2 sin^ ^Ir + T sin 2i/r p a It follows that = Ki (cos 2i|r + sin 2i|r tan 2T/r) = «! sec 2a^. _ 1 /a^w N 13/1 dw\ "'"aAW^"")' '^ "a^ 30 1 sin 0307- With the values of u, v, w given in (13) we now find Kl = — Ka = 2 ■ = -2 L. E. a? sin^ w' — « a?sin^0 30 .(14) An tan" 2 cos (w0 + «„) - -B„ cot" ^ cos (w0 + (8„) r ^„ tan" 2 sin (w0 + a„) + £„ cot" ^ sin (n0 + ^n) .(15) 33 514 INEXTENSIONAL VIBRATIONS [CH. XXIII 321. Inextensional vibrations. If we assume that the state of strain in a vibrating shell is that which has been described in Article 317 as the typical flexural strain, we may calculate the frequency of vibration by forming expressions for the kinetic and potential energies*. We illustrate this method in the cases of cylindrical and spherical shells. (i) Cylindrical shell. The kinetic energy, estimated per unit of area of the middle surface, is ph fduV idv\^ fdW where p is the density of the material, and u, v, w are given by (8), in which the coefficients A^, B^ are to be regarded as functions of t. The kinetic energy T of the vibrating shell is obtained by integrating this expression over the area of the middle surface. If the ends of the shell are given by x= + l, we find T=2Trpalhl ('-n'li'M.-^*'^--*'-!©' .(16) The potential energy of bending, estimated per unit of area of the middle surface, is iD [k,' + 2 (1 - 0-) T'l where rc^ and r are given by (11). The potential energy V of the vibrating shell is obtained by integrating this expression over the area of the middle surface. We find V = D7rll''-^^[n'A„''+[^nH'+2{l-a)a'}Bn''] (17) The coefficients A^, B^ in the expressions (8) for the displacement may be regarded as generalized coordinates, and the expressions for T and V show that they are " principal coordinates," so that the various modes of vibration specified by different A'b or B's, are executed independently of each other. The vibrations in which all the B's and all but one of the A's vanish are two- dimensional and take place in planes at right angles to the axis of the cylinder. The type is expressed by the equations u = 0, v = AnCosn, th and the frequency pjlir is given by the equation Eh^ w^n' - 1 )^ 1 + 6 (1 - 0-) O'l'rv'P ^ ~ 3/3 (1 - (T=') a^ n-'+l l + ZcC'ln'{n? + \)P ^ ' If either n or Ija is at all large the two values of p belonging to the same value of n are nearly equal. (ii) Spherical shell. We shall suppose the middle surface to be bounded by a circle of latitude 6 = 0., and that the pole ^ = is included. Then in (13) and (15) the co- efficients Bn vanish. . The kinetic energy T is given by the equation T = 'irpa?h'% ^^y r sin e {2 sin^ 6 + (cos 6 + rif] tan« - dO ....(20) The potential energy of bending, estimated per unit of area of the middle surface, is ^ijJi^{k^-\- t% where k-, and t are given by (15) with the B's omitted. Hence the potential energy V of the vibrating shell is given by the equation ^=f-M^.2 '■{n'-lfAn'r J e do tan»^- 2 sin' e .(21) The coefficients ^„ in the expressions for the components of displacement can be regarded as "principal coordinates "* and the frequency can be written down. In a principal mode the type of vibration is expressed by the equations d 6 u = An sin tan" „ cos n, in which An is proportional to a simple harmonic function of the time. The frequency p„/27r is given by the equation p,^=f ^^]n^(n^-l>^ (" rtan-| ^-^y (J%in^ l2sin^^+(cos^+n)^j tan»| d^ In this expression n may be any integer greater than unity. 'When the edge-line consists of two circles of latitude, so that the coefficients B occur as well as the coefficients A, the ^'s and B's are not principal coordinates, for terms containing such products as (dAJdt) . (dBJdt) occur in the expression for T. See Lord Kayleigh, Theory of Sound, second edition, vol. 1, Chapter Xa. 33—2 516 INEXTENSIONAL VIBRATIONS OF A TBIN SHELL [CH. XXIII The integrations can- always be performed. We have /: tan2"-2^ tan2"5 tan^-'+a^ 7— +2 + -T— - n-X n n+\ . 1+COS o **■ f " sin 5 {2 sin2 5 + (cos B + nf) tan^- ide=\ Jo 2 j and the second of these can be evaluated for any integral value of n. In the case of a hemisphere {a = W) Lord Eayleigh {loc. cit.) finds the frequencies Pi, Pi, Pi for n=% 3, 4 to be given by In the case of a saucer of 120° (a = \n) he finds In the case of a very small aperture in a nearly complete sphere (a = 7r nearly) the frequency calculated from the above formula* is given approximately by ^'' a'^Zp {n-af ' • Cf. H. Lamb, loc. cit., p. 506. CHAPTER XXIV. GENERAL THEORY OF THIN PLATES AND SHELLS. 322. Pormulse relating to the curvature of surfaces. For the investigations in' the last Chapter the elements of the theory of the curvature of surfaces are adequate. For the purpose of developing a more general method of treatment of the problem of curved plates or shells we shall require some further results of this theory. It seems best to begin by obtaining these results. Let a, /3 denote any two parameters by means of which the position of a point on a surface can be expressed, so that the equations a = const., /3 = const, represent families of curves traced on the surface. Let ;y; be the angle between the tangents of these curves at any point; p^ is in general a function of a and ^. The linear element ds of any curve traced on the surface is given by the formula {dsf = A^ (day + B" (d^f + 2AB cosxdad/S, (1) where A and B are, in general, functions of a, l3. Let a right-handed system of moving axes of x, y, z be constructed so that the origin is at a point (a, jS) of the surface, the axis of z is the normal to the surface at the origin, drawn in a chosen sense, the axis of x is the tangent to the curve y8 = const, which passes through the origin, drawn in the sense of increase of a, and the axis of y is tangential to the surface, and at right angles to the axis of x*. When the origin of this triad of axes moves over the surface the directions of the axes change. If t represents the time, the components of velocity of the origin are . da , r,d^ r)dl3 . . '^Tt-^^di''''^' ^dt''''^' ^' parallel to the instantaneous positions of the axes of x, y, z. The components * When the curves a = const, and (3= const, cut at right angles we suppose that the parameters o and §, and the positive sense of the normal to the surface, are so chosen that the directions in which a and ^ increase and this normal are the directions of a right-handed system of axes. 518 CURVATURE OF SURFACES [CH. XXIV of the angular velocity of the system of axes, referred to these same directions, can be expressed in the forms da d0 P^Tt^P'dt da d^ da dS "■' di ^'''df in which the quantities ^Ji, ... are functions of a and y8. The quantities jpi, ... are connected with each other and with A, B, p^ by the systems of equations (2) and (3) below. These results may be obtained as follows : Let X, y, z denote the coordinates of a fixed point referred to the moving axes. Then X, y, z are functions of a and ,3, and the conditions that the point remains fixed while the axes move are the three equations dxda dxd^ (da, c?/3\ , (da, d^\ .da, _c?/3 radt-^r^dt-^V^di+''^dt)+'[^^di-^^'dt)-''^di+^dt''°' dy da , dydfi I da d^\ / da , d^\ „ d^ . idt+idt-'(p'dt+p^dtr''['''dt+'-^i)^^-£''''>^-''' x=o. dadt dzda dzdfi ( da dfi\ , / da d8\ „ Since these hold for all values of dajdt and d^/dt, we have the six equations dx dx = -A+riy-qiZ, 8/3" £^p,z-r,x, dy_ dz dz ^=-Bcosx + r^-q2Z, ^ = - 5 sin x +i>22 - J-2^, ^„ = q2^ 33 3^- The conditions of compatibility of these equations are three equations of the form gg(g-j = g-fggj ; attd, lu forffilng the differential coefiSoients, we may use the above expressions for dxjda, .... The results must hold for all values of ^, y, z. The process just sketched leads to the equations* and r, = ■ n = 8/3 da ~ ^''^' ~ ^'^" dqi_dq2 _ a/3 da~^'P'~'''P" dx 1 fdA dB 8a •(2) fdA da Bsmx\d0 "°^^' 1 fdB dA AsmxKda ''°^ ^ 8^ )' |+fsin^ = |cos;,. .(3) • The sets of equations (2) and (3) were obtained by D. Codazzi, Paris, Mem.. ..par divert savants, t. 27 (1882). 322, 323] CURVATURE OF SURFACES 519 To express the curvature of the surface we form the equations of the normal at (a + 8a, /S + S/3) referred to the axes of x, y, z at (a, /3). The direction-cosines of the normal are, with sufficient approximation, {q^a. + gaS/S), — (^iSa + ^aS/S), 1, and the equations are a;-(J.g« + ^g;8cosx) _ y-Bl^^va.^ It follows that the lines of curvature are given by the differential equation Api (day + B (p.2 cos x + Qi sin %) (d^y + {Ap^ + B (p^ cos x + li sin %)} dad(i= 0, (4) and that the principal radii of curvature are the roots of the equation ■fi' (Piq^ - P^qi) - R {^P2 - B (pi cos x + qi sin x)) +ABsmx = 0. .. .(5) From these results the equation of the indicatrix of the surface is easily found to be -S''''^(£^"5''°*^)2''+25^2'=°°°'* (^) The measure of curvature is given by (5) and the third of (2) in the form 1 /dr\ _ diy\ AB sin xW 9a/ 323. Simplified formulse relating to the curvature of surfaces. When the curves a = const, and /3 = const, are lines of curvature on the surface the forntalse are simplified very much. In this case the axes of x and y are the principal tangents at a point, the axis of z being the normal at the point. We have X = i-^' Pi = 0. ?2 = 0. (7) and the roots of equation (5) are — A/q^ and B/p^. We shall write R,- A' R^B' ^""^ so that J?i, R^ are the radii of curvlti((Ure of normal sections of the surface drawn through those tangent lines which are axes of x, y at any point. We have also \dA ^19^ \ "■?" £3/3' "■' Ada' I (9) _AB __ a_ A aB\__a_/ia4\ j R,R~ 9aUaa/ d^KBd^j'l / ^""^ da\Rj~R,dcL' d^[Rj~R,d0 ^ ' 520 DEFORMATION OF A THIN SHELL [CH. XXIV 324. Extension and curvature of the middle surface of a plate or shell. Id general we shall regard the middle surface in the unstressed state as a curved surface, and take the curves a = const, and /3 = const, to be the lines of curvature. In the case of a plane plate a and yS may be ordinary Cartesian coordinates, or they may be curvilinear orthogonal coordinates. In the case of a sphere a and /8 could be taken to be ordinary spherical polar coordinates. Equations (7) — (10) hold in the unstressed state. When the plate is deformed the curves that were lines of curvature become two families of curves traced on the strained middle surface, which cut each other at an angle that may differ slightly from a right angle. We denote the angle by ;;^ and its cosine by ■in-, and we denote by 6i and eg the extensions of linear elements which, in the unstressed state, lie along the curves yg = const, and a = const. The quantities a and /3 may be regarded as parameters which determine a point of the strained middle surface, and the formula for the linear element is (dsy = ^'^ (1 + e,y (day + B'(l+ e,y (d^y + 2AB{1 + eO (1 + e^) vrdad^. As in Article 322, we may construct a system of moving orthogonal axes of so, y, z with the origin on the strained middle surface, the axis of z along the normal at the origin to this surface, and the axis of x along the tangent at the origin to a curve ^ = const. The components of velocity of the origin parallel to the instantaneous positions of the axes of x and y are ^(l + eO| + 5(l+e.)^f. 5(l + e.)sm^f. The components of angular velocity of the triad of axes referred to these same directions will be denoted by ,da, ,dB ,da ,dB ,da , dB P^dt^P^W ^^dt + ^^di- "-^dt+'^Tf Then in equations (2) and (3) we must replace .4 by J. (1 + e^), Bhy B(l+ e^), Pi, Pi, ■■■n by pi', Pa', ... r/. The directions of the lines of curvature of the strained middle surface, the values of the sum and product of the principal curvatures, and the equation of the indicatrix are found by making similar changes in the formulae (4) — (6). If we retain first powers only of e^, e^, ■or, equations (3) give (11) r' — IdA dnr ^BB e, dA A de^ Bd^'^da'^ Bda'^Bd^ BdB' IdB -^dA 6i dB B de^ Z 9a Ad^ Ada^Ada' . '2 324, 325] EXTENSION AND BENDING OF A THIN SHELL 521 The indicatrix of. the strained middle surface is given, to the same order of approximation, by the formula - I' (1 - eO a;» + 1^ (1 - e,) -^ ^1 2/= + 2 ^ (1 - 6,) «;2/ = const. If Ri', Ri denote the radii of curvature of normal sections of the strained middle surface drawn through the axes of x and y at any point, and t\r. the angle which one of the lines of curvature of this surface drawn through the point makes with the axis of x at the point, we have, to the same order, CT, tan 2i^ = - ^ (1 - 6,) |a-..)+|a-..)-^i» ...(12) It is clear from these formulae that, when the extension is known, the state of the strained middle surface as regards curvature is defined by the quantities -q,'/A, p,'/B, p,'/A. \We shall write -^-^ = «„ ^-^^ = «., ^' = t (13) and shall refer to «i, Kg, t as the "changes of curvature." In the particular cases of a plane plate which becomes slightly bent, and a shell which under- goes a small inextensional displacement, these quantities become identical with those which were denoted by the same letters in Chapters xxii. and xxiii. The measure of curvature is given by the formula ' \"' • ' \ 1 - 6i - e^ / 9r/ _ dr^\ \l\l.L ^^ V9/S 9a/' where r/, r/ are given by the first two of (11). When there is no extension the values of r^, r^ for the deformed surface are identical with those of r-j, r^ for the unstrained surface, and the measure of curvature is unaltered by the strain (Gauss's theorem). The sum of the principal curvatures, being equal to 1/-Ri' + ^jRi, can be found from the formulae (12). 325. Method of calculating the extension and the changes of curvature. To calculate 6i, ...pi, ... in terms of the coordinates of a point on the strained middle surface, or of the displacement of a point on the unstrained middle surface, we introduce a scheme of nine direction-cosines expressing the directions of the moving axes of x, y, z at any point relative to fixed axes of X, y, z. Let the scheme be 521 EXTENSION AND BENDING [CH. XXIV X y z X h mj Jii y h m2 «2 2 h TO3 »3 .(14) If now X, y, z denote the coordinates of a point on the strained middle surface, the direction-cosines k, m^, ih of the tangent to the curve ^= const, which passes through the point are given by the equations Ail + e,)h = £, 9v fir 4(1 + 60 «i = ^.... (15) The direction-cosines of the tangent to the curve a = const, which passes through the point are Z^ sin % -)- ^i cos % and therefore, when ts'' and isres are neglected, k, m^, "2 are given by the equations B {(1 + 6,) I, + ^«i} = g| , -B {(1 + €2) m, + mm,] = g| , £{(l + e2)n2 + ^«i) = ^ (16) The direction-cosines Z3, mg, n^ of the normal to the strained middle surface are given by the equations ?3=mi?72 — m„Ki, ms^nj^ — n^li, n, = l^m^ — l^in, (17) From equations (15) and (16) we find, correctly to the first order in i + ; 1+262 = B' dl3. 'dz\ _ 1 J3x8x 8y9y 9z9z) '^~AB\dad^'^dad^^did^]' , .(18) Again, since the line whose direction-cosines referred to the moving axes are l,, k, h, that is the axis of x, is fixed relatively to the fixed axes, the ordinary formulae connected with moving axes give us three equations of the type dli da dl, cZ/3 . ( ,da ,d0\ , f ,da ,d^\ „ and, by expressing the fixity of the axes of y and z, we obtain two other such sets of equations. From these we find the formulse , , 9^2 , dm, ,9^2 , , dk , dm, , 9^2 ^'=^'9^ + "^^l^+^'9a' P^^^^d^ + '^^W^'^'W , J dk dm, 9»i3 , , 9^3 , 9m3 9n3 , , „, q,=k^ + m,^+n,^, q, =k^ + m,j^ +n,^, } ...{19} 'da da da , , 9^1 9mi , 9»ii , 9^1 "dm, 9^1 ^^9^ + ™^"9;8+''=9;8- 325, 326] OF A THIN SHELL 523 The formulse (IS) enable us to calculate e^, e^, ■as, and the formulae (19) give us the means of calculating p/,- 326. Formulae relating to small displacements. Let u, V, w denote the components of displacement of any point on the unstrained middle surface referred to the tangents at the point to the curves /8 = const, and a = const, and the normal at the point to the surface. We wish to calculate the extension and the changes of curvature in terms of u, V, w and their differential coefficients with respect to a and /3. (a) The extension. According to the formulse (18) we require expressions for 9x/9a, ... where X, y, z are the coordinates of a point on the strained middle surface referred to fixed axes. We shall choose as these fixed axes the lines of reference for u, t), w at a particular point on the unstrained middle surface, and obtain the required expressions by an application of the method of moving axes. Let P{a,0) be the chosen point on the unstrained middle surface, P' {a + ha, ^ + &ff) a neighbouring point on this surface. The lines of reference for u, v, w are a triad of moving axes, and the position of these axes when the origin is at P' is to be obtained from the position when the origin is at P by a small translation and a small rotation. The components of the translation, referred to the axes at P, are .48a, Bh^, 0. The components of the rotation, referred to the same axes, are given by the results in Article 323 in the forms R2 Aha 94 8a as 80 'd^B'^ da A' When P is displaced to Pi and P' to Pi, the x, y, z of Pj are the same as the u, v, w of P ; the X, y, z of Pj' are X + (ax/9a) 8a + (3x/30) 8/3, ... , and the u, v, w oi P' are u + (,du/da) Sa + (du/d^) 8/3, . . .. These quantities are connected by the ordinary formulse relating to moving axes, viz. : 9z X , 9z , __ ,sBm\ dliB^ da a)' dw ^ dw .\ ( Ahd\ , 58/3 and in these formulse we may equate coefficients of 6a and 8/3. The above process leads to the following expressions for 9x/3a, Aw dy _dv u dA 9x _ . du V dA Ri ' da da dz _ dw Au dx _du vdB 9y_D ^ ud^_Bw dz ^dw_ Bv d^~d^~Ad^' W~ d/3'^Ada B,' 9/3 ~ 9/3 "^ E^ * / .(20) 524 EXTENSION AND BENDING OF A [CH. XXIV When products of u, v, w and their differential coefficients are neglected the formulas (18) and (20) give _ 1 9m V dA w ^'~Adi'^ABd^~R,' 1 9t) M 95 Bd^'^ABd^' w \ .(21) -i^ 1 9m _ _«_ cLA _ _w_ 95 '^~^9a'^59y3 AB d^ ABda' These formulas determine the extension. When the displacement is inextensional u, v, w satisfy the system of partial differential equations obtained from (21) by equating the right-hand members to zero. As we saw in particular cases, in Articles 319 and 320, the assumption that the displacement is inextensional is almost enough to determine the forms of u, v, w as functions of a and /3. (b) The changes of curvature. According to the formulae (19) we require expressions for the direction- cosines ll, ... of the moving axes referred to the fixed axes; we require also expressions for 9/i/9a, .... We shall choose our fixed axes as before to be the lines of reference for u, v, w at one point P of the unstrained middle surface. By (15), (16), (17), (20), (21) we can write down expressions for the values of Zj,... at the corresponding point P^ of the strained middle surface in the forms .(22) k= 1, ??ll = Idv u dA Ada ABd^' " 1 dw '~Ada^ M R.' k= 1 dv Ada' u dA ^ ABd&' m2=l. 1 9m; V k= 1 dw Ad^ u 1 dw ~~BdB' V = 1. These are not the general expressions for l^, ... at any point. They are expressions for the direction-cosines of the moving axes at a point on the strained middle surface, referred to the lines of reference for u, v, w at the co7'responding point of the unstrained middle surface. For these latter direction-cosines we may introduce the orthogonal scheme u V w X A My ^1 y A M^ ^2 z ^3 M, ^S 326] THIN SHELL SLIGHTLY DEFORMED Then we have the values ^^=^' ^^ = Aia-ABW' ^'^AYa^R.- T - _ i_ ?!' J. Jf_ M M- - 1 AT _ -'■ ^^ ^ 525 T — — — ^ M „_ 19w R' ^^' ^' ' .(23) and these hold for all points. We apply the method of moving axes to deduce expressions for dl^jda, ... ; and then we form the expressions for ^,', . . . in accordance with (19). The direction-cosines of the axes of x, y, 2 at a neighbouring point Pi, referred to the lines of reference for u, v, w at P', would be denoted by Li + {dLi/da) 8a + {dZi/d^) S/3, ... ; the directiou-cosines of the axes of sn, y, z at P{, referred to the fixed axes, which are the lines of reference for u, v, w at P, would be denoted by l^ + (dli/jda) 8a + (alijd^) S/3, .... Since the components of the rotation of the lines of reference for u, v, w are B80 ASa _9^^ 3^5^ R2 ' iJi ' 3/3 5 "^ 0a ^ ' we have the ordinary formulae connected with moving axes in the forms dli ^-l«^=C^»-t-*')-*(-|M!?)--.(-t). 3»ii "3^ with similar formulae in which the suffix 1 attached to I, m, n and L, M, N is replaced successively by 2 and 3. On substituting for Zj, ... the values given in (23), we find 3a " AB 30 (3w m3.4\ 1 /3m) Av\ 3/3" 22 3a\3a Bo^j' 1 34 \S3/3' 3«ii_ 3 /l 3» u 3.4 N _ 3a ~ 3^ \3 3^ "" ZS 30.y '. dmi_d (\ dv u dA\ B /Idw u\ 1 dB d^~d^\Ada AB 30/ iJa U 3« Mj'^Ada' 3% _ 3^ A 3t» u\ A^ dni_d_ /l?^ , M , j5 n dv u dA\ d^~d^\Ada'^Rj R2\Ada ABd^J' 526 and EXTENSION AND BENDING OF A THIN SHELL [CH. XXIV cl. + ---!(-- + B dp AB dp f / Idv u_ c_A\ _\_dB ip\ JTa'^ AB ds) 4 5a ' dm, \_(dw ^\ _ J_ 3^ /3j^ _ « M\ 1 8w B dH' bJ' t //to A da ' 3»2 5 3/3 I 3w I) 5 o^ "*" A + In calculating p,', ... from the formulse (19), we write for li, ... the values given in (22), and for dlifia, ... the values just found, and we observe that, since the scheme (14) is orthogonal, two of the formulae (19) can be written U, ^'' " ~ V'' Sa "•" '"=' "^ "^ "^ '^ 3)ij 3^ The process just described leads to the formulae P A d ,\ diu , dly , drtii dill dnA dp)- , _d (\ dw V ^daVBd^'^R _ldA ndw M ~Bd^\Ad^^R,, R, \da u dA^ Bd^J' + ■ R, da \A da R, IdA/ldw V ' Bd^KBd'^^R, ^' ~ Bd^^da U da ' ji_dA\ A^ ABd^J^R . fl dw v\ 'AbW^rJ' ..(24) and , _ fldw V P'^R'^dBVEdB'^R ?2 B . ^ 'd^\Bdl3 d fl dw 1 dB /I dw u '^AdaKAdiL^R ■y d/3 KA Sa "'' R, 1 dB /■! dw V Ada [sd^'^R, 1^-^+1 ('19'' u dA\ _B fl dw Ada d^KAda ABd^J R^Ada B fdv ARXda' udA ~Bd^ ^i)- ..(25) ; We can now write down the formulae for the changes of curvature in the forms _ 1^ 9^ /]_ 9w i^\ l^gj. /I aw y\ "'' Ada\A da^ Rj'^ABd^XBdB^-R^ 1 3 ^1 3w ^ ^ >^ ^ I dBfldw "'" Bd^\Bd^^ R. _ 1 9 /I 3w V ''~Ad'a\Bd'^^R, RJ "^ AB da [a da "^ u \ Rj' ^^ dA dw A^d^da' 1 dv ' AR.da .(26) 326, 327] STRAIN IN A BENT PLATE OR SHELL 527 The above formulae admit of various verifications : (i) In the case of a plane plate, when a and ^ are Cartesian coordinates, we have _3% _3% _ d^w Ki_^, K2=g^, '"-g^- These results agree with the formulse in Article 298. (ii) In the cases of cylindrical and spherical shells, the conditions that the displace- ment may be inexteusioual can be found as particular cases of the formulae (21), and the expressions for the changes of curvature, found by simplifying (26) in accordance with these conditions, agree with those obtained in Articles 319 and 320. (iii) Let a sphere be slightly deformed by purely normal displacement, in such a way that the radius becomes ce-l-6P„(cos 5), where 6 is small, P^ denotes Legendre's nth coefficient, and 6 is the co-latitude. The sum and product of the principal curvatiires of the deformed surface can be shown, by means of the formulae of this Article and those of Article 324, to be ^ + J, (« - 1) (» + 2) P„ (cos 6) and 1^ -^ |^ (» _ 1) (» + 2) P„ (cos 6), correctly to the first order in b. These are known results. (iv) For any surface, when fi, e^, rar are given by (21), and p/, ... are given by (24) and (25), equations (11) are satisfied identically, squares and products of m, v, w and their differential coefficients being, of course, omitted. 327. Nature of the strain in a bent plate or shell. To investigate the state of strain in a bent plate or shell we suppose that the middle surface is actually deformed, with but slight extension of any linear element, so that it becomes a surface differing but slightly from some one or other of the surfaces which are applicable upon the unstrained middle surface. We regard the strained middle surface as given ; and we imagine a state of the plate in which the linear elements that are initially normal to the unstrained middle surface remain straight, become normal to the strained middle surface, and suffer no extension. Let P be any point on the unstrained middle surface, and let P be displaced to Pj on the strained middle surface. Let X, y, z be the coordinates of P^ referred to the fixed axes. The points P and Pi have the same a and ^. Let Q be any point on the normal at P to the unstrained middle surface, and let z be the distance of Q from P, reckoned as positive in the sense already chosen for the normal to the surface. When the plate is displaced as described above, Q comes to the point Qi of which the coordinates are ■a-^-liZ, j + rrisZ, z + n^z, where, as in Article eS25, ^s, m^, n^ are the direction -cosines of the normal to the strained middle surface. The actual state of the plate, when it is deformed so that the middle surface has the assigned form, can be obtained from this imagined state by imposing an additional displacement upon the points Qi. Let ^, t), f denote the components of this additional displacement, referred to axes of x, y, z 528 STRAIN IN A BENT [CH. XXIV with origin at Pi which are drawn as specified in Article 324. Then the coordinates of the final position of Q are x + k^+ l.,v + h(z + 0, y + will + nhn + ms (2 + f ), z + n,^ + nr,r, + n,{z+^). ...(27) In these expressions ^i, ... are the direction-cosines so denoted in Article 325, X, y, z, ^, ... ^3 are functions of a and /3, and |, tj, ^ are functions of a, ^, z. We consider the changes which must be made in these expressions when, instead of the points P, Q, we take neighbouring points P', Q', so that Q' is on the normal to the unstrained middle surface at P', and the distance P'Q' is 2; + Iz, where hz is small. Let P be («, /3) and P' (a ^-la, ^+ 8^), where 8a and S/3 are small ; and let r denote the distance QQ', and I, m, n the direction-cosines of the line QQ', referred to the tangents at P to the curves /3 = const, and a = const, which pass through P and the normal to the un- strained middle surface at P. The quantities a, /3, z may be regarded as the parameters of a triply orthogonal family of surfaces. The surfaces z = const- are parallel to the middle surface ; and the surfaces a = const, and yS = const, are developable surfaces, the generators of which are the normals to the unstrained middle surface drawn at points on its several lines of curvature. The linear element QQ' or r is expressed in terms of these parameters by the formula and the projections of this element on the tangents to the curves (8 = const, a = const., drawn on the middle surface, and on the normal to this surface are It, mr, m. Hence we have the formulae ^« = 4(i-./ii;.)' ^^= B{i-ziR,y ^' = ''' (^^) In calculating the coordinates of the final position of Q we have in (27) to replace xbyx + g^Sa-^gs^,..., I, by k -r k in'8tt + r,'8/3) - k (?i'Sa + g/S/S), k by h + k {p,'8a + p,'S/3) - 1, (r/gft + r,'80), k hyk + k (?.'Sa + q,'8^) - I, {p/8a + p,'S^), * ' ' f ^ by ^ + Sz. We use also the formulae (15) and (16) for dx/da,... and the formulae (28) for Sa, 8y8, Sz. 327] PLATE on SHELL 529 Let Ti denote the distance between the final positions of Q and Q'. We express r^ as a homogeneous quadratic function of I, m, n, and deduce ex- pressions for the components of strain by means of the formula Now the difference of the x-coordinates of the final positions of Q and Q' is h + »? i(^3K-^in')- h , + {hp2'-lir^) A{l-z/R,) + k + h + L- d^ h + 9? + ;r^ «r da. A{\- z/R,) ^ S/3 5 (1 - ^/i?^) ^ 3^ c*?? Zr + dr) dn + ~ nr daA(l-z/E,) d^B(l-z/B^) dz 9? h - + 9? mr : + (-a1) nr [9a A (1 - ^/i?i) ' d^£(l- zjE;) The differences of the y- and z-coordinates can be written down by sub- stituting m^, m^, ni^ and n^, n^, n, successively for l^, l^, l^. Since the scheme (14) is orthogonal, we find the value of r^^ in the form I r,' = !■' 1 + 1 - z\R^ --%'.+%(^+r)+iiUng' B''^ B Bd^ 'dz + r^ 4.j,2 i'l /. , >-\ , ^i' fc , 1 ^'^ i-./Ea-i^^+^^-^z^+iir- -f- i--.wi«;---f'<"»-Ff^i|}-s ll- zlR,\ A^^ A"^^ Adv(i-|,)*. ^=/>'.(i-i).. When we refer to the axes of x, y, z specified in Article 324, and denote the stress-resultants and stress-couples belonging to curves which are normal to the axes of x and y respectively by attaching a suffix 1 or 2 to T, . . . , we obtain the formulae '^-\.M'-i^''' ^-/X^-i')^^' ^-/X^-i)'^^' H, = 1*^ - zXy (l - J^) dz, Q, = p^zX, {l - -|-,) dz, -(31) and ^^-f-M'-i)'-' '^-L-M'-i)'-' ""H'-M'-iH rh (32) in which P/ and P/ denote, as in Article 324, the radii of curvature of normal sections of the strained middle surface drawn through the axes of x and y. We observe that the relations /Sj + ^2 = and II1 + H2 = 0, which hold in the case of a plane plate slightly deformed, do not hold when the strained middle surface is appreciably curved. The relations between the T, S, N, G, H for an assigned direction of v and those for the two special directions x and y, which we found in Article 295 for a plane plate slightly deformed, are also disturbed by the presence of an appreciable curvature. 329. Approximate formulse for the strain, the stress-resultants and the stress-couples. We can deduce from (30) of Article 327 approximate expressions for the components of strain by arguments precisely similar to those employed in Articles 257 and 259. Since |, 17, ^ vanish with z for all values of a and /3, and d^jdz, ... must be small quantities of the order of admissible strains, 34—2 532 STKESS AND STRAIN IN A [CH. XXIV f, 7), f and their differential coefficients with respect to a and /S may, for a first approximation, be omitted. Further, for a first approximation, we may omit the products of z/R^ or z/Rz and any component of strain. In particular, since q^'/B + p^'/A is of the order eJRi, we omit the product of this quantity and z] and, for the same reason, we replace such terms as :j ^-7^ and ^Pj5- by 6i and zk^. By these processes we obtain the approximate 1 — z/Ri formulae* O ^^ ^V ^? /0Q\ ea:x = fi — ^«i> eyy = e^ — ZK^, exy = Ts- — ZZT, 6^=^, '^s/z^oT. ^zs^aX- ■■■\^^) In these f, tj, f may, for a first approximation, be regarded as independent of a and /3. In case the middle surface is unextended, or the extensional strains 6i, cj, ot are small compared with the flexural strains zk^, zk^, zt, these expressions may be simplified further by the omission of e^, eg, -nr. The approximate formulae (33) for the strain-components, as well as the more exact formulas (30), contain the unknown displacements f, rj, f, and it is necessary to obtain values for these quantities, or at any rate for their differential coefficients with respect to z, which shall 'be at least approxi- mately correct. We begin with the case of a plane plate, and take a, /3 to be Cartesian rectangular coordinates, so that A and B are equal to unity, and IjRi and I/E2 vanish. In the formulee (33) f, ■»?, f are approximately independent of a, /3. We consider a slender cylindrical or prismatic portion of the plate such as would fit into a fine hole drilled transversely through it. We may take the cross-section of this prism to be so small that within it e^, e^, ts and «], K.,, T may be treated as constants. Then the strain-components, as ex- pressed by (33), are the same at all points in a cross-section of the slender prism. If there are no body forces and no tractions on the faces of the plate, we know from Article 306 that the stress in the slender prism, in which the strains are uniform over any cross-section, is plane stress. Hence, to this order of approximation X^, Y^, Z^ vanish, and we have i = ^' L' = ^' a^^=-r^(^. + e.-^ («. + «.)! (34) The remaining stress-components are then given by the equations ^'^ = iZTfy-^ 1*1 + 0-62 - ^ («i + S, = S,= -^^, ...(36) and 0, = -D{K,+ a-K^), G2 = -D{K^ + a-K,), -H^ = H, = D(l-a)T. ...(37) To the same order of approximation the strain-energy per unit of area is given by the formula [Eh/(1 - a')} [(e, + e,y - 2 (1 - C,- T% . . .(38) To get a closer approximation in the case of a plane plate we may regard the strain in the slender prism as varying uniformly over the cross-sections. Then we know from Article 306 that X^ and Fj do not vanish, but the third of (34) and the formulae (35) still hold, and therefore also (36) and (37) are still approximately correct, while iVi and N^ are given according to the result of Article 806 by the formulae N, = -D^{k, + Kb), N^ = -D^{k, + k,). These values for Nj, N'^ could be found also from (12) of Article 296 by omitting the couples L', M' and substituting for G^, G2, Hi from (37). From this discussion of the case of a plane plate we may conclude that the approximate expressions (33) and (34) for the components of strain are adequate for the purpose of determining the stress-couples ; but, except in cases where the extension of the middle plane is an important feature of the deformation, they are inadequate for determining the stress-resultants. The formulae (37) for the stress-couples are the same as those which we used in Articles 313, 314. The results obtained in Articles 307, 308, 312 seem to warrant the conclusion that the expressions (37) for the stress-couples are sufiScient approximations in practically important cases whether the plate is free from the action of body forces and of tractions on its faces or not. In the case of a curved plate or shell we may, for a first approximation, use the formulae (33) and the theorem of Article 306 in the same way as for a plane plate. Thus equations (34) and (35) are still approximately correct. We may obtain from them the terms of lowest order in the expressions for the stress-resultants of the type T, S and the stress-couples. On substituting in the formulae (31) and (32), we find, to the first order in h, (39) 534 STRESS AND STRAIN IN A [OH. XXI\ and, to the third order in h, This first approximation includes two extreme cases. In the first the extensional strains ej, £2, ■^ are small compared with the flexural strains SATi, ZK.i, zr. The stress-couples are then given by the formula G, = -D{k, + (7K,), G, = -D{k,+ ,7K,), -H, = H, = I){l-(T)T,...{S1bis) and the strain-energy per unit of area is given by the formula which we found by means of a certain assumption in Article 317, viz. : but the stress-resultants are not sufficiently determined. In the second extreme case the flexural strains zk-^, zk^, zt are small compared with the extensional strains 61, e^, ot. Then the stress-resultants of type T, S are given by the formulae (36), and the stress-resultants of type N and the stress-couples are unimportant. The strain-energy per unit of area is given by the formula [Ehjil - a-)} [(6,+;,)=- 2(1 -Si and S^ by means of (31) and (32) of Article 328, and in this calculation we may replace \/Ri and I/R2' by l/Ri and l/R,. We find S.= ^.+i)(l-.)-^^-ii>(l-.).(i- + i-), ^ In calcuTating a second approximation to Ti and T^ we may not assume that Z^ vanishes. As in the case of the plane plate, we take the shell to be free from the action of body forces and of tractions on its faces. We observe that the axes of x, y, z specified in Article 323 are parallel to the normals to three surfaces of a triply orthogonal family. This is the family considered in Article 327, and the parameters of the surfaces are a, (3, z. We write temporarily y in place of z, and use the notation of Articles 19 and 58. The values of hi, h^, A3 are given by the equations 1 . /, y\ 1 n/, y\ 1 We write down an equation of the type of (19) in Article 58 by resolving along the normal to the surface y. This equation is + ^{jB(l-i)(l-^^} -5(-jx'J(-('-a-f('-r^{H'-a=»' * London Math. Soc. Proc, vol. 20 (1889), p. 372, or Scientific Papers, vol. 3, p. 280. 536 EQUATIONS OF EQUILIBRIUM Returning to our previous notation, we write this equation [CH. XXIV -f(>-4)---x!('-i) '■-=»■ To obtain an approximation to Z,, we substitute in this equation for X^, ... the values given by the first approximation, and integrate with respect to z. We determme the constant of integration so that 2, may vanish at z^h and z= -h. We must omit the terms containing X, and Y^ and use the approximate values given in (35) for X^ and 1 „. Further we may omit the factors 1 - z/flj and 1 - zlR^ and such terms as eiz//Ji. We thus find the formula _ ,.(43) Now we have A':,= ^3p(«;,^ + ^v = f372(«w + °'^»:) + rr^^^' and hence, by means of the formulse for e„, e„„, Z,, we calculate approximate values for Tx , 7*2 in the forms* ~ Y^a- \ 'R^ ^' Ri ). ...(44) ~r^\ Bi ^ Ri The formulae for the stress-couples are not affected by the second approximation, so far at any rate as terms of the order Dki are concerned. 331. Equations of equilibrium. The equations of equilibrium are formed by equating to zero the resultant and resultant moment of all the forces applied to a portion of the plate or shell. We consider a portion bounded by the faces and by the surfaces formed by the aggregates of the normals drawn to the strained middle surface at points of a curvilinear quadrilateral, which is made up of two neighbouring arcs of each of the families of curves a and /S. Since the extension of the middle surface is small, we may neglect the extensions of the sides of the quadrilateral, and we may regard it as a curvilinear rectangle. We denote the bounding curves of the curvilinear rectangle by a, a + 8a, /9, /3 + S/3, and resolve the stress-resultants on the sides in the directions of fixed axes of * The approximate forms of Si, S2 , Tj , T2 obtained in this Article agree substantially with those found by a different process by A. B. Basset, loc. cit., p. 534, in the cases of oyllndrioal and spherical shells to which he restricts bis discussion. His forms contain some additional terms which are of the order here neglected. 330, 331] OF A THIN SHELL 537 X, y, z which coincide with the tangents to ^ and a at their point of intersection and the normal to the strained middle surface at this point (Fig. 73). Fig. 74 shows the directions and senses of the stress-resultants on the edges of the curvilinear rectangle, those across the edges a + Ba and /3 + S/3 being distin- guished by accents. The axes of the stress-couples H^, G, have the /*' same directions as T^, S^; those of -02, ffa have the same directions as T^,8.^. Fig. 73. Fig. 74. The stress-resultants on the side a of the rectangle yield a force having components -T^BB/3, -S^BSI3, - N^BB^ parallel to the axes of x, y, z. The corresponding component forces for the side a -I- Sa are to be obtained by applying the usual formulae relating to moving axes; for the quantities Ti, Si, iVj are the components of a vector referred to moving axes of x, y, z, which are defined by the tangent to the curve /9 = const, which passes through any point and the normal to the strained middle surface at the point. In resolving the forces acting across 538 EQUATIONS OF EQUILIBRIUM [CH. XXIV the side a + Ba parallel to the fixed axes, we have to allow for a change of a into a + Ba, and for the small rotation (p/Sa, g/Sa, )\'Sa). Hence the components parallel to the axes of x, y, z of the force acting across the side a + So. are respectively T,5S/3 + Sa I- {T,BB^) - S,BS^ . r.'Sa + N,BS^ . q.'Sa, da S,Bh^ + Bal {S,BBI3) - N,BS^ .p,'Boi + T,BS/3 . r.'Sa, 00. N,BS^ + Sa I- {N,BS^) - T,BB^ . q/Ba + S,BB^ .p.'Sa. oa In like manner we write down the forces acting across the sides /3 and l3 + 8,3. For 13 we have S^ABa; - T,AB«, -JSf^ABa; and for /3 + 8/3 we have - S.ABa - S/3 ^ (S.ABa) - T.ABa . r^B^ + N.ABa . q,'B0, T.ABa + 8/3 ^ (T.ABa) - N,ABa . p,'B0 - S.ABa . r.'B/S, op N,ABa + B0^{N,A Ba) + 8,ABa. q,'B^ +T,ABa. p,'B^. Let A", Y', Z' and L\ M', denote, as in Article 296, the components, parallel to the axes of x, y, z, of the force- and couple-resultant of the externally applied forces estimated per unit of area of the middle surface. Since the area within the rectangle can be taken to be ABBaB^, we can write down three of the equations of equilibrium in the forms ^^"d'cT " ^W''* -^''^'^^^ + ''^'^^^) + (^^'^1^ + ^-^^'^^ + ABX'-^O, ' ^^^ + ^-^^ -iPi'N.B +p.;N,A) + (n'T.B- r,'S,A)+ ABY'= 0, J- (45) ^^la~ + ^-^W^ ~ ^^''^^^ ~ ^"''^^^^ + {p^S,B+p.;T,A) + ABZ' = 0. Again the moments of the forces and the couples acting across the sides of the rectangle can be written down. For the side a we have the component couples - H^BB^, - G,BBI3, 0, and for the side a + 8a we have the- component couples H,BB0 + Ba^^ iH,BBl3) - G, BS/3 . n'Ba, G,BB^ + Ba I- (G,BB0) + H,BB^ . r.'Ba, oa - H,BB^ . q.'Ba + G,BB0 .p^Ba ; 331, 332] BOUNDARY-CONDITIONS 539 for the side /3 we have the component couples G^ASa, - B^ASa, 0, and for the side /3 + S/3 we have the component couples - G.ASa - S/8 ^ (G.ASa) - H.ABa . r,'B^, H.ABa + Sy8 ^ (HUBa) - G.ABa . r.JB^, G.ABa . q,'B^ + H.ABtt . p,'B^. Further the moments about the axes of the forces acting across the sides tt+ Ba and ^ + B^ can be taken to be 58^ . N^ABa, - ABa . N'.BB^, ASa . S,BB0 + BS^ . S.ASa. The equations of moments can therefore be written in the forms ^-^ - ^-^ - (G.Br,' + H,Ar,') + (N, + L') AB = 0, '-^^'^HH.Br^-G.Ar:)-iF.-M')AB=0, \ -^''^ G,Bp,' + G,Aq,' - {H,Bq,' - H,Ap,') + (S, + S,) AB = 0. Equations (45) and (46) are the equations of equilibrium. 332. Boundary-conditions. The system of stress-resultants and stress-couples belonging to a curve s drawn on the middle surface can be modified after the fashion explained in Article 297, but account must be taken of the curvature of the surface'. Regarding the curve s as a polygon of a large number of sides, we replace the couple HBs acting on the side Bs by two forces, each of amount H, acting at the ends of this side in opposite senses in lines parallel to the normal to the surface at one extremity of Bs ; and we do the like with the couples acting on the contiguous sides. If P'PP" is a short arc of s, and the arcs P'F and PP" are each equal to Bs, these operations leave us with a force of a certain magnitude, direction and sense at the typical point P. The forces at P and P", arising from the couple on the arc PP", are each equal to H, and their lines of action are parallel to the normal at P, the force at P being in the negative sense of this normal. The forces at P' and P arising from the couple on the arc P'P are each equal to H— BH, and their lines of action are parallel to the normal at P', the force at P being in the positive sense of this normal. Now let iJi'^', iJa''' he the principal radii of curvature of the strained middle surface at P, so that the equation of this surface referred to axes of f, rj, z which coincide with the principal tangents at P and the normal is approxi- mately ^-H^V^i'^' + '77-^2 '"1 = 0. .540 BOUNDARY- CONDITIONS [OH. XXIV Also let (f) be the angle which the tangent at P to P'PP" makes with the axis of f. The point P' has coordinates — 8s cos , — 8s sin <^, 0, and the direction-cosines of the normal at P' are, with sufficient approximation, 8scos^/J?i<", Sssin^/E,"', 1. The force at P arising from the couple on PP has components jffSscos^/ii,"', HSs sin 4>/R,^'\ H - BE parallel to the axes of f , Tj, z. Hence the force at P arising from the couples on P'P and VP" has components parallel to the normal to s drawn on the surface, the tangent to s and the normal to the surface, which are iTSs sin ^ cos ^(l/-RiW-l/-R2''0> SIsIB:, ~BH, where R', = [cos2^/ii!i''i + sin' <^/i?2'"]""S is the radius of curvature of the normal section having the same tangent line as the curve s. Hence the stress-resultants T, S, N and stress-couples H, G can be replaced by stress- resultants T+^H sin 2ct>{1/P,'"-l/R2''^}, S + H/R', N-dH/ds, ...(47) and a flexural couple G. The boundary-conditions at an edge to which forces are applied, or at a free edge, can now be written down in the manner explained in Article 297. The formula (47) are simplified in case the plate or shell is but little bent, for then the radii of curvature and the position of the edge-line relative to the lines of curvature may be determined from the unstrained, instead of the strained, middle surface. They are simplified still more in case the edge is a line of curvature*, for then N does not contribute to T. 332 a. Buckling of a rectangular plate under edge thrust. The theory of the equilibrium of thin shells will be applied to particular cases in Chapter XXIVa. Special problems for plane plates have already been discussed in Chapter XXII. As exemplifying the application of the equations of Article 331 we shall consider here the question of the stability of a plane rectangular plate under edge thrust parallel to its plane. The problem will be treated by the method used in Article 267a supra. If the thrusts are not too great, the plate simply contracts in its plane in the manner indicated in Article 301. We take the centre of the rectangle as origin, and lines parallel to the edges as axes of x and y, and take the edges in the unstrained state to be given by the equations x=±a,y=+b. Let Pi and P^ be the values of the thrusts along these pairs of edges respectively. Then in the simply contracted state Ti= -P^ and T2= - P^ every- where and the remaining stress-resultants and the stress-couples vanish. We now suppose that Pj and P2 are such that a very small transverse displacement w can be maintained by them in the plate so contracted. Using x and y in place of a, /3, we have A =B = \, and l//fi = 1/7^2 = 0, and the formulae of Article 326 give * The result that, in this case, H contributes to S as weU as to N was noted by A. B. Basset, loc. cit., p. 534. See also the paper by H. Lamb cited on p. 506. 332, 333] BUCKLING OF A RECTANGULAR PLATE 541 Equations (46) of Article 331 with the formulse (37) of Article 329 give 9% D{\- d^w 1 (cfw dv\ 1 d fdw \ "■' dx' ' "- " a^ [dii" '^d4>)' ■^ adx \d The displacement being periodic in (f> with period 27r, and' the shell being supposed to vibrate in a normal mode with frequency p/2ir, we shall take u, 'd, w to be proportional to sines, or cosines, of multiples of (^, and to a simple harmonic function of t with period 27r/p. The equations of vibration then become a system of linear equations with constant coefficients for the deter- mination of u, V, w as functions of x. We shall presently form these equations ; but, before doing so, we consider the order of the system. The expressions for El, 62, ■5T contain first differential coefficients only; that for «i contains a second differential coefficient. Hence G^ and Gs contain second differential coefficients, and iVj contains a third differential coefficient. The third equa- tion of (49) contains d^w/doc* in a term which is omitted when we form the equations of extensional vibration. Thus the complete equations of vibration will be of a much higher order than the equations of extensional vibration. It will be seen presently that the former are a system of the 8th order, and the latter a system of the 4th order. The reduction of the order of the system which occurs when the equations of extensional vibration are taken instead of the complete equations is of fundamental importance. It does not depend at all on the cylindrical form of the middle surface. * The difBculty arising from the fact that inextensional displacements do not admit of the satisfaction of the boundary-conditions is that to which I called attention in my paper of 1888 (see Introduction, footnote 133). The explanation that the extension, proved to be necessary, may be practically confined to a narrow region near the edge, and yet may be sufficiently important at the edge to secure the satisfaction of the boundary-conditions, was given simul- taneously by A. B. Bassett and H. Lamb in the papers cited on pp. 534 and 506. These authors Hlustrated the possibility of this explanation by means of the solution of certain statical problems. 333, 334] CYLINDRICAL SHELL ^ 547 (as) General equations. In accordance with what has been said above, we take M = J/'sin m0 cos (p< + f ), w = F cos «(/) cos (pt + e), w=Wsmncj) cos (pt+e), ... (50) where U, V, W are functions of x. Then we have dO . ^ , , W+nV . f 1 = ^ sin »i<^ cos [pt + e), 62 = sin ?i<^ cos {pt + e), °^^(^ "*""«) °0S»K/)C0S(p< + f), ^^^^ ■ J. / . , N nV+n^W . _, , K\ = -j^ Sin n

coa{pt + €){^cr-^ ^ — ], ^. = i)cos»,^cos(p* + e) ^-—I{n^+ ^)=-H.. The first two of equations (48) become ' 3^ a 30 ' ^ a 3<|) 3.J? ' i\^.= -i)sin»0cos(^« + e)|^ -^ ^^ii-^ + ^^JI, N,= -D cos «,^ cos (^< + .) j- -^ - -3 " +^ ^^2 - ^s ^) • T nT^/ , , , 2-2o--3 '^ ^ ex a o

F)l=0, (51) 548 VIBRATIONS OF A THIN [CH. XXIV A3 h L 2 3D A3 n4S-s <^^'-^s-i-h '"' The boundary-conditions at x=l and x= —I are a a o(p and all the left-hand members can be expressed as linear functions of U, V, W and their differential coefficients with respect to x. The system of equations for the determination of u, v, w as functions of x, has now been expressed as a linear system of the 8th order with constant coefficients. These coefificients contain the unknown constant p^ as well as the known constants h and n ; and n, being the number of wave-lengths to the circumference, can be chosen at pleasure. If we disregard the fact that h is small compared with a or I, we can solve the equations by assuming that, apart from the simple harmonic factors depending upon ^ and t, the quantities u, V, lu are of the form ^e™^, rje™', fe™*, where ^, ij, ^, m are constants. The constant m is a root of a determinantal equation of the 8th degree, which is really of the 4th degree in m^ for it contains no terms of any uneven degree. The coefificients in this equation depend upon p". When m satisfies this equation the ratios ^ : v '■ ^ a^re determined, in terms of m and p', by any two of the three equations of motion. Thus, apart from and t factors, the solution is of the form r=l r = l r = l in which the constants ^r, fr' are arbitrary, but the constants r;,., ... are expressed as multiples of them. The boundary-conditions at a; = ? and x = — I give eight homogeneous linear equations connecting the ^, ^' ; and the elimination of the ^, ^' from these equations leads to an equation to determine p'. This is the frequency equation. (6) Extensional vibrations. The equations of extensional vibration are obtained by omitting the terms in equations (51)— (.53) which have the coefficient Djh. The determinantal equation for m^ becomes a quadratic. The boundary-conditions aX x=±l become Ti = Q, Si = 0, or dU_ W + nV_ dV nU_ dx '^ a ~ ' dx a ~ ' 334] CYLINDRICAL SHELL 549 Since h does not occur in the differential equations or the boundary-conditions, the frequencies are independent of h. In the case of symmetrical vibrations, in which u, v, w are independent of cannot vanish at any particular value of x unless -B„=0. Thus the boundary-conditions cannot be satisfied by the assumed displacement. The correction of the displacement required to satisfy the boundary-conditions would appear to be more important than that required to satisfy the differential equations. (e) Nature of the correction to be applied to the inextensional displace- ment. It is clear that the existence of practically inextensional vibrations is connected with the fact that, when the vibrations are taken to be extensional, the order of the system of equations of vibration is reduced from eight to four. In the determinantal equation indicated in (a) of this Article the terms which contain m' and m' have A^ as a factor, and thus two of the values of m' are large of the order 1/A. The way in which the solutions which depend on the large values of m would enable us to satisfy the boundary- 334] CYLINDRICAL SHELL 551 conditions may be illustrated by the solution of the following statical problem* : — A portion of a circular cylinder bounded by two generators and two circular sections is held bent into a surface of revolution by forces applied along the bounding generators, the circular edges being free, in such a way that the displacement v tangential to the circular sections is proportional to the angular coordinate <^ ; , it is required to find the displacement. We are to have v = c, where c is constant, while u and w are independent of J{hla) of the potential energy due to bending*. In the case of vibrations we may infer that the extensional strain, which is necessary in order to secure the satisfaction of the boundary-con- ditions, is practically confined to so narrow a region near the edge that its effect in altering the total amount of the potential energy, and therefore the periods of vibration, is negligible. 335. Vibrations of a thin spherical shell. The case in which the middle surface is a complete spherical surface, and the shell is thin, has been investigated by H. Lambf by means of the general equations of vibration of elastic solids. All the modes of vibration are extensional, and they fall into two classes, analogous to those of a solid sphere investigated in Article 194, and characterized respectively by the absence of a radial component of the displacement and by the absence of a radial component of the rotation. In any mode of either class the displace- ment is expressible in terms of spherical surface harmonics of a single integral degree. In the case of vibrations of the first class the frequency pjlir is connected with the degree n of the harmonics by the equation p=aV//x = (w-l)(?i + 2), (54) where a is the radius of the sphere. In the case of vibrations of the second class the frequency is connected with the degree of the harmonics by the equation p'aY fo?9 (n^ -I- ?i -f- 4) -i°^ + (w^ -(- w - 2) i — O" + 4(n2 + „_2)(l±^'l=0. (55) If ;( exceeds unity there are two modes of vibration of the second class, and * For further details in regard to this problem the reader is referred to the paper by H. Lamb already cited. t London Math. Soc. Proc, vol. 14 (1883), p. 50. 334, 335] SPHERICAL SHELL 553 the gravest tone belongs to the slower of those two modes of vibration of this class for which n = 2. Its frequency p/iir is given by p = VWp)a-Mri76), if Poisson's ratio for the material is taken to be I. The frequencies of all these modes are independent of the thickness. In the limiting case of a plane plate the modes of vibration fall into two main classes, one inextensional, with displacement normal to the plane of the plate, and the other extensional, with displacement parallel to the plane of the plate. See Articles 314 (d) and (e) and 333. The case of an infinite plate of finite thickness has been discussed by Lord Rayleigh*, starting from the general equations of vibration of elastic solids, and using methods akin to those described in Article 214 supra. There is a class of extensional vibrations involving displacement parallel to the plane of the plate ; and the modes of this class fall into two sub-classes, in one of which there is no displacement of the middle plane. The other of these two sub-classes appears to be the analogue of the tangential vibrations of a complete thin spherical shell. There is a second class of extensional vibrations involving a component of displace- ment normal to the plane of the plate as well as a tangential component, and, when the plate is thin, the normal component is small compared with the tangential component. The normal component of displacement vanishes at the middle plane, and the normal component of the rotation vanishes every- where ; so that the vibrations of this class are analogous to the vibrations of the second class of a complete thin spherical shell. There is also a class of flexural vibrations involving a displacement normal to the. plane of the plate, and a tangential component of displacement which is small compared with the normal component when the plate is thin. The tangential component vanishes at the middle plane, so that the displacement is approximately inextensional. In these vibrations the linear elements which are initially normal to the middle plane remain straight and normal to the middle plane throughout the motion, and the frequency is approximately proportional to the thickness. There are no inextensional vibrations of a complete thin spherical shell. The case of an open spherical shell or bowl stands between these extreme cases. When the aperture is very small, or the spherical surface is nearly complete, the vibrations must approximate to those of a complete spherical shell. When the angular radius of the aperture, measured from the included pole, is small, and the radius of the sphere is large, the vibrations must approximate to those of a plane plate. In intermediate cases there must be vibrations of practically inextensional type and also vibrations of extensional type. I * London Math. See. Proc, vol. 20 (1889), p. 225, or Scienti/k Papers, vol. 3, p. 249. 554 VIBEATIONS OF A THIN [CH. XXIV Purely inextensional vibrations of a thin spherical shell, of which the edge-line is a circle, have been discussed in detail by Lord Rayleigh * by the methods described in Article 321 supra. In the case of a hemispherical shell the frequency pj2Tr of the gravest tone is given by ^ = V(Wp)(/i/aO (4-279). When the angular radius a of the aperture is nearly equal to tt, or the spherical surface is nearly complete, the frequency p/iir of the gravest mode of inex- tensional vibration is given hy p = VWp) [h/a" (tt - aY} (5-657). By supposing TT — a to diminish sufficiently, while h remains constant, we can make the frequency of the gravest inextensional mode as great as we please in comparison with the frequency of the gravest (extensional) mode of vibration of the com- plete spherical shell. Thus the general argument by which we establish the existence of practically inextensional modes breaks down in the case of a nearly complete spherical shell with a small aperture. When the general equations of vibration are formed by the method illus- trated above in the case of the cylindrical shell, the components of displace- > ment being taken to be proportional to sines or cosines of multiples of the longitude 4>, and also to a simple harmonic function of t, they are a system of linear equations of the 8th order for the determination of the components of displacement as functions of the co-latitude 6. The boundary-conditions at the free edge require the vanishing, at a particular value of 6, of four linear combinations of the components of displacement and certain of their differential coefficients with respect to 6. The order of the system of equa- tions is high enough to admit of the satisfaction of such conditions ; and the solution of the system of equations, subject to these conditions, would lead, if it could be effected, to the determination of the types of vibration and the frequencies. The extensional vibrations can be investigated by the method illustrated above in the case of the cylindrical shell. The system of equations is of the fourth order, and there are two boundary-conditionsf. In any mode of vibration the motion is compounded of two motions, one involving no radial component of displacement, and the other no radial component of rotation. Each motion is expressible in terms of a single spherical surface harmonic, but the degrees of the harmonics are not in general integers. The degree a of the harmonic by which the motion with no radial component of displace- ment is specified is connected with the frequency by equation (54), in which a is written for n ; and the degree /3 of the harmonic by which the motion • London Math. Soc. Proc, vol. 13 (1881), or Scientific Paperg, vol. 1, p. 351. See also Theory of Sound, 2nd edition, vol. 1, Chapter X a. t The equations were formed and solved by E. Mathieu, J. de VEcoU poly technique, t. 51 (1883). The extensional vibrations of spherical shells are also discussed in the paper by the present writer cited in the Introduction, footnote 133. 335] SPHERICAL SHELL 555 with no radial component of rotation is specified is connected with the frequency by equation (55), in which /3 is written for n. The two degrees a and /S are connected by a transcendental equation, which is the frequency equation. The vibrations do not generally fall into classes in the same way as those of a complete shell ; but, as the open shell approaches completeness, its modes of extensional vibration tend to pass over into those of the com- plete shell. The existence of modes of vibration which are practically inextensional is clearly bound up with the fact that, when the vibrations are assumed to be extensional, the order of the system of differential equations of vibration is reduced from 8 to 4. As in the case of the cylindrical shell, it may be shown that the vibrations cannot be strictly inextensional, and that the correction of the displacement required to satisfy the boundary-conditions is more important than that required to satisfy the differential equations. We may conclude that, near the free edge, the extensional strains are comparable with the flexural strains, but that the extension is practically confined to a narrow region near the edge. If we trace in imagination the gradual changes in the system of vibrations as the surface becomes more and more curved*, beginning with the case of a plane plate, and ending with that of a complete spherical shell, one class of vibrations, the practically inextensional class, appears to be totally lost. The reason of this would seem to lie in the rapid rise of frequency of all the modes of this class when the aperture in the surface is much diminished. The theoretical problem of the vibrations of a spherical shell acquires great practical interest from the fact that an open spherical shell is the best representative of a bell which admits of analytical treatment. It may be taken as established that the vibrations of practical importance are inex- tensional, and the essential features of the theory of them have, as we have seen, been made out. The tones and modes of vibration of bells have been investigated experimentally by Lord Eayleighf. He found that the nominal pitch of a bell, as specified by English founders, is not that of its gravest tone, but that of the tone which stands fifth in order of increasing frequency; in this mode of vibration there are eight nodal meridians. * The process is suggested by H. Lamb in the paper cited on p. 506. t Phil. Mag. (Ser. 5), vol. 29 (1890), p. 1, or Scientific Papers, vol. 3, p. 318, or Theory of Sound, 2nd edition, vol. 1, Chapter X. CHAPTER XXIV a EQUILIBRIUM OF THIN SHELLS 336. Small displacement. When a thin plate or shell is held deformed by externally applied forces, the strained middle surface must, as we observed in Article 315, coincide very nearly with one of the surfaces that are applicable upon the unstrained middle surface. We may divide the problem into two parts : (i) that of determining this applicable surface, (ii) that of determining the small displacement by which the strained middle surface is derived from this applicable surface. This is the procedure adopted byClebsch* in his treatment of the problem of finite deformation of plane plates. There is some degree of indefiniteness about this division of the problem, because any one of the surfaces, which are applicable upon the unstrained middle surface, and are derivable one from another by displacements of the order regarded as small, would serve equally well as a solution of the first part of the problem. Greater precision could be imparted to the procedure if we regarded the two steps as (i) the deter- mination of an inextensional displacement, which need not be small, and (ii) the determination of an additional displacement, involving • extensional strains at least of the same order of magnitude as the additional flexural strains, and possibly large in comparison with them. The first step would then be analogous to the determination of equilibrium configurations of a thin rod, discussed in Chapters xix. and xxi. ; but, unless the displacement is small, little progress can be made. When the displacement is small, a different method is more effective. The equations of equilibrium (45) and (46) of Article 331 are a set of six equations connecting the six stress-resultants T^, ... and the four stress-couples Gi, ... with the displacement (m, v, w) ;^ for the six quantities of type »/, which occur in these equations, have been connected with u, v, w by equations (24) and (25) of Article ^26. If the first approximations to the stress-resultants and stress- couples are regarded as sufficient, four of the six stress-resultants and all the stress-couples are expressed in terms of the quantities e^, e^, •nr and k^, k^, r by equations (36) and (37) of Article 329 ; and these quantities are expressed in terms of u, v, w by equations (21) and (26) of Article 326. We have there- fore a set of six differential equations to determine the five quantities * ElasHcitiit, § 70. 336, 337] SMALL DISPLACEMENT 557 Ni,N2, u, V, w. Apart from the apparent redundancy of the set of differential equations, the method consists in simplifying the equations by omitting all terms which are of order higher than the first in N^, N^, u, v, w, and then solving the differential equations as if they were exact and not merely approximate. The excess of the number of equations above the number of quantities to be determined would constitute a serious difficulty if the equations were exact; but the difficulty disappears when the approximate character of the equations is taken into account. The apparently redundant equation is the 3rd of equations (46) of Article 331. Now the expressions for the stress- couples Gi, ... are of the first order in u, v, w, and the expressions for p^, ... in terms of u, v, w contain terms which are of the first order in u, v, w, and some of them also contain terms independent of u, v, w, but p^ and g/ contain no such terms. When terms of order higher than the first in u, v, -(o are omitted, the 3rd of equations (46) of Article 331 takes the form and this equation cannot in general be reconciled with equations (36) and (37) of Article 329. When the more exact equations (42) of Article 330 are employed to express Si, S^ in terms of w, v, w it becomes an identity. On the other hand, when Bi and -ffj may be neglected it is satisfied identically by the simpler expressions given in (36) of Article 329. In either case there are sufficient equations to determine the unknown quantities in terms of the variables a, /3, by which the position of a point on the middle surface is specified ; and they are a system of differential equations of a sufficiently high order to admit of the satisfaction of arbitrary boundary-conditions of stress or displacement at the edge of the shell. 337. The middle surface a surface of revolution. The case where the middle surface is a surface of revolution, including technically important problems relating to cylindrical, spherical, and conical shells, will chiefly occupy us.. We choose as, the variable the angle, which the axial plane, passing through a point of the unstrained middle surface, makes with a fixed axial plane. This will be denoted by 4>. The correspond- ing B is the distance of the point from the axis of revolution, and it is independent of tp, but is, in general, a function of the other variable a. The quantity A and the principal curvatures (l/Ri and I/R2) of the unstrained middle surface are also functions of a and independent of 0. When, as we shall generally assume, the edge of the shell consists of two circles of latitude (a = const.^, the forces acting upon the shell must either be independent of 4, or periodic functions of ^ with period 27r, and we may suppose them expanded in Fourier's series of sines and cosines of integral multiples of ^. The 558 THE MIDDLE SURFACE [CH. XXIV A components of displacement and the stress-resultants and stress-couples may be supposed to be expanded in similar series. Whenever the thickness 2h is independent of (f>, we may treat separately the terms of these series which contain sines or cosines of n or cos n it is convenient to introduce a new notation. In doing this we shall take occasion to avoid also the constant repetition of the quantity 2Ehj(l — a''). We shall write M = P" cos (n^ -t- e), v=Vs\n{n^ + e), w = W cos (w^ -I- e), '1\= \-a^ s,= 2 Eh l-a^ N,= 2Eh (?,= 2Eh tiCOs{n(f) + €), Ti = -rz — ^ *2 cos (w0 -I- e) 2Eh ?ii cos (w -I- e) = - H^. The equations show that no other arrangements of sine and cosine are permissible. The equations connecting U, ... with t^, ... are A da 1 dV R^^'^KAB da^ B R W\ JJ^dB nV_W A da Rj'^ABda'^ B R,' ^^'^AdaKA da 9. 9. = -W \ (T d f\ dW U\ nV A da\A da'^Rj ^ BR^ B' ^ AB da [a da "*■ R, '^ Rj^^KBR^ B' J'^ABda [a da ^ Rj] ' AB da [a (2) together with an additional equation involving s^ or s^ or both. For a first approximation we have ,,- , /I dV nlT V dB\ but we know that this approximation is not always sufficient. In such problems as that of a shell strained by uniform pressure applied to one face, or rotating about its axis of figure, we have to take w = 0. In 337, 338] A SURFACE OF REVOLUTION 559 such a problem as that of a cylindrical tube with a horizontal axis strained by its own weight, we have to take n = l. We shall refer to problems of this latter kind as problems of " lateral forces." The half-thickness h will be taken to be independent of both a and It is clear that, when the boundary-conditions are suitable, we can obtain solutions, which are sufficiently exact for practical purposes, by omitting gi, 92, h, as being small of the order (h^la)(t^., 4, Si), and adopting equations (3). We shall describe such solutions as " extensional." Equations (4) show that in working out these solutions we may put Wj and n^ equal to zero. When all the conditions are symmetrical about the axis, so that n = 0, and we are not dealing with torsion, we put F = 0. We also put 6 = 0. Equations (3) then show that Sj = S2 = 0. Equations (4) then give s^'2if ^'=»- '—"w^- w The first of these gives ii+^wrj -^'<=^* = °°"'^*- = <^i s*y (^) The second of equations (8) combined with the second of equations (2) then gives ^=-f+'^^w^'' (1^) and equation (9) combined with the first of equations (2) gives, on sub- stituting from (10), dU _ «! L-f^YV __f^ 7' dx^l-a' 2EhJo 2Eli ' so that U= a.+ ^^-^Jl [\ydx] dx-^jy'dx, (11) where a^ is an arbitrary constant. It appears that by means of the extensional solution we can eliminate the external forces, but we cannot in general satisfy the boundary-conditions at the edges. For example, we could not solve the problem of a tube strained by external pressure and plugged at the ends. In such problems the stress- resultant of type Ni cannot vanish. 339] SYMMETRICAL CONDITIONS 561 (b) Edge-effect. Putting n and V zero we see from the fifth of equations (2) that fh vanishes. Then, whether we use equations (3) or the more complete equations (42) of Article 330, we may conclude that s^, Sg vanish. We have seen that the extensional solution and the solution for torsion peiinit us to satisfy the equations (4) containing external forces, and we may now simplify the system of equations by omitting X', Y', Z'. The equations (4) become, on omitting those which are satisfied identically, ^ = 0, «, = 0, p + *B = 0,p-rH = 0, (12) dec ' ax a ax and the relevant equations of (2) become ^ dV W ^ dU W ,,J'W ,.„- The first of equations (12) gives ij = const. If we retained this constant we should merely reproduce the results of the extensional solution, so far as these depend upon the arbitrary constant fflj. For our present purpose it is sufficient to put «i=0, , (14) dU W and consequently Ij^ ~ "' — ' ^ > W and t^ = -{\-a-')~ (16) On eliminating n^ between the 3rd and 4th of equations (12) and sub- stituting for g^ and t^ from (13) and (16), we find the equation -*'^^S-(l-^)5=«' (^^) the complete primitive of which can be written W = e«*"* (^1 cos qxja + Bj sin qx/a) + e"**'" {A^ cos qx/a + B^ sin qx/a), . . .(18) where Ai, Bi, A 2, B.^ are arbitrary constants, and ^ = (a/2/i)^ {3 (1 - <7^)}* ,. (19) It will be observed that this quantity is the same as that which was denoted by q in the special problem discussed in Article 334 (e). The integration of equation (15) introduces an additional arbitrary constant. If we retained this constant we should merely reproduce the results of the extensional solution, so far as these depend upon the arbitrary constant a^. For our present purpose it is sufficient to put f = -^ [ew» {(^1 - 5i) cos qxja + (^1 + B^) sin qxja] 'Zq - e-«*"^ {(^2 + -Ba) cos qx/a - (A^ - B^ sin qxjaW (20) L. E. ^" 562 CYLINDRICAL SHELL [CH. XXIV A The results here obtained indicate a state of stress existing in the parts of the shell that are near the edges, and such that the stress-resultants and stress-couples diminish to very small values at a distance from the edges comparable with a mean proportional between the thickness and the diameter. We describe this special state as the " edge-effect." It will be observed that the combination of the solution in Article 338 with the two solutions in this Article permits us to assign to the values of . a a a '^ ] When X is near zero the most important term is approximately {B^+Br^e''"!'', and when X is near I it is approximately (B1+B2) e-''(''-'^)K 341. Stability of a tube under external pressure. Before proceeding with the integration of the equations of Article 337 for a cylindrical shell, under conditions answering to values of n, which exceed zero, we shall turn aside to discuss the technically important probleui of the conditions of collapse of a thin cylindrical tube, subject to external pressure Po. To simplify the problem as much as possible we shall assume that the conditions, which hold at the ends of the tube, are such that the state of the tube, when the pressure is too small to produce collapse, is expressed with sufficient approximation by the extensional solution of Article 339 (a), and in this solution we shall further take X' and Oj to vanish. Thus this state is given by the equations T, = S, = S, = N, = N, = 0, G, = G, = H, = H, = 0, and T, = -apo (28) It may be observed that these results can be deduced easily from those of Article 100 by taking r^ - t\ to be small and adjusting e so that zz = 0. We have now to suppose that a small additional displacement (u', v', w') is superposed upon the displacement answering to this state. Let Ti, . . . G^, . . . be the stress-resultants and stress-couples calculated from this displacement. 36—2 564 STABILITY OF A TUBE [CH. XXIV A Then the stress-resultant of type T^ is — ap„ 4- T2. In forming the equations of equilibrium, we must omit all quantities of the second order in u, v', w' but we must not omit the products of jOo and quantities which are of the first order in «', v, w'. In particular we must replace A and B by A(\ + ej, B{1 + e,) in any terms that contain pf,. In regard to the calculation of S-! and Sn we shall use the equation and replace equations (3) by the equation Eh (dv' a d ' On substituting from these in the remaining equations we find the set of three equations dT, 1 9*80' , _ dx a dtp s^i' ^ a 82V _!_ 1 aff/ 1 ae; dx a d(j) a dx ^-" "■' d ' d'Gi' dx'^ a dxd(f) 2 d^H ' 1 d^G ' T ' In these equations we are to put (29) 82 = - 2Eh f3M' a /a?/ 1 — o-^ [9a; a \d(f> ■w T' = 2Eh 1-, Eh 1+a \dx Eh [dv dv' 1 du' h^ I aw + ; dv' du' 1 idv a\d(j> dx dx a d a d(p ' 3a'^ \dxd(j) ' dx ldu__hP^ /aw dv' "" ' 3a^ [dxd^ "^ a^ 2_Eh?_ 3 1 -(7= g,^_^ Eh' ff/ = 31-0^= 2 Eh' aw a /d'^w' dv'\ dx" "^ a^ V a^ "*" d^J a^' ^ /aw dv'- "" daf "^ a? va^ "•" a^. 1 /aw dj/\ a \dxd(b dx) ' 3 1 -h ' ■w 341] UNDER EXTERNAL PRESSURE 565 We assume as expressions for the components of displacement the forms ■u' = Ucoa mx sin n^, v' = Fsin mm cos n^, w' = Tf sin mx sin n^, . . .(30) (31) and write Po- 1- a' a Equations (29) become, on omitting terms of order higher than the first in U, V, W, (•'-^-i-^3--C-|----^%^.)--- *|--'^-^|.-)>-». 1 + (T mn ~~2 ffin a V /n 3-0- h; + [-„ + -^^ ^^„ m' (-■ 2 3a^ » + |-„.)r-o, a Va^ 3 a^ 3a* J /I n,2 - 1 ,^, hP , 2A' mW k' , ,„ - The elimination of U, V, W from these equations leads to a determinantal equation determining '^ in terms of a, h, m, n. This equation may be sim- plified by observing that '^ must be small of the order (a/h) (po/E), and we may therefore approximate by omitting terms of the orders '^^ or '^h^. We may also omit terms containing h*. Further, m must be of the order y , where I is the length of the tube, and we may omit terms which contain h^m. The determinantal equation is then m^ + 2 a'' 1 + a mn ~^ ~a ' am, a 1+. ■*)•?■ 1-0- „ »2 ^2 „ a^"''3a' + — ,n% (0- + ^)- a n h^ •- ;; — n a^ 3a' 1 n'-l hV a" a^ 3a* 0, and, when we evaluate the determinant, omitting terms of the orders indi- cated, we find 1 2a= or approximately C /»,2 \2 -f- a-m*\ + \{1- a-") m* + g-^ n* {li" - 1)' = 0, 3a=^ ^^ ^w^(«^-l) 566 STABILITY OF A TUBE UNDER PRESSURE [CH. XXIV A The condition that the tube may be unstable in respect of a small dis- placement of the type specified by m and n is therefore _-2Eh{(y-\)h' mW ) . ^'^ a (3(l-a')a- n'{n'-l)} ^ ^ The forms (30) would admit of the satisfaction of boundary-conditions of the type Ti = 0, v = 0, w = 0, at the ends x = and ^ = ^ of the tube, if m. is an integral multiple of tt/?. Also the pressure required to produce collapse, given by (32), increases as m increases. With these boundary-conditions the least pressure, for which collapse is possible, is given by putting in = tt/I. When the tube is very long equation (32) becomes so that a very long tube cannot collapse unless the pressure exceeds 3D/a' ; and, when this value is but slightly exceeded, it collapses by flattening the section to an elliptic form (// = 2). But, for a shorter tube, collapse into a form given by n = 3 may occur for a smaller value of j^o than collapse into a form given by m = 2. The condition that this should happen is -^"^ {^W^) - 3^ (sb)} > W^^^ K3^ - 1) - (2^ - 1)!, . . 432 h' or mW > 5(1- a') a' When this condition is satisfied the tube collapses so that its section becomes a three-lobed curve of the form given by the equation r=a + b cos 3<^, where b/a is small. This statement holds provided m is not too great ; but, as m increases or I diminishes, a value will be arrived at, which is such that the value of j3„ given by n = 4 is smaller than the value given by n = 3. Then the tube collapses so that its section becomes a four-lobed curve. If m becomes large the validity of the approximations which led to (32) becomes doubtful. It follows from the nature of the quantity called Z' that, when there is internal pressure pi as well as external pressure po, the left-hand member of equation (32) should be po — pi- The result that, for a very long tube, the pressure required to produce collapse is ZDja? or {-lEjil -o^)}(A/a)^, was obtained by G. H. Bryan*. The analogous result for a ring is given in Article 275 supra. The fact that tubes sometimes collapse so that the section becomes elliptic, sometimes so that it becomes a three-lobed curve, and so on, had been observed long before by W. Fairbairnt, but it remained without explanation until the problem was attacked by R. V. Southwell J. The problem was discussed by Southwell in * Cambridge, Proc. Phil. Soc, vol. 6 (1888), p. 287. t London, Phil. Trans. R. Soc, vol. 148 (1859), p. 389. J London, Phil. Tram. R. Soc. (Ser. A), vol. 213 (1913), p. 187. 341, 342] OYLINDRICA.L SHELL. LATERAL FORCES 567 this memoir as an example of the application of his theory of elastic stability, ami the solution, including the result expressed in equation (32), was there obtained by him with- out using the theory of thin shells. In another paper*, written later although published earlier, the same writer obtained the results by a method (based on the theory of thin shells) which has been followed to some extent in this Article. The reader, who wishes for further information on the experimental as well as the theoretical aspect of the subject, may refer to a valuable Report by G. Cook in Brit. Assoc. Rep., 1913, and to papers by Cook and Southwell in P/til. Mag. (Ser. 6), vol. 28 (1914) and vol. 29 (1915). It should be noted that the theoretical solution has been obtained by assuming rather exceptional terminal conditions. The application of it to the problem of the stability of a boiler flue, which is strengthened to resist collapse by " collapse rings " placed at intervals along the length of the flue, is discussed by Southwell in the papers cited. 342. Lateral forces. (a) Eoitensional solution. We return to the integration of the equations (4) for a cylindrical shell in the case where /i = 1, and begin with an extensional solution in which ffi, 9a, j"*!. «i, «2 are omitted, and Sj and s^ are given by (3). The relevant equations included in (4) are d«, Si 1 - <7= „„ ^ ^' _ ^ -1- ^JT °^ Y" = n dx a '2Eh t, 1 a ■2Eh Z" = 0, .(33) while equations (2) and (3) give dU dU ' da- V-W a V-W 7 a 'dV U\ .(34) The 2nd and 3rd of equations (33) give 1 I give (7"+Z") = 0, dsi I — a- dd'^^Eh from which we get Si = «n 1 -ff= 2Eh {Y" + Z")dx = 0, (35) where a^ is a constant of integration. The 1st of equations (33) then gives ♦ Phil. Mag. (Ser. 6), vol. 25 (1913), p. 687. 568 CYLINDRICAL SHELL [CH. XXIV A where a^ is a constant of integration. Thus s^, ti, U are known, for U is given by the 3rd of equations (33). The 1st and 2nd of equations (34) give dU _ U — O'ta dx so that U dx,. .(37) where a^ is a constant of integration. Thus U is known. The 3rd of equa- tions (34) gives so that dV U . 2s, 'U . 2s, dx a 1 — ff ■ dx,. .(.38) where a^ is a constant of integration. The 1st and 2nd of equations (34) then give W = V-a to — (rt. .(39) in which V is known, so that W is determined. The extensional solution involves four arbitrary constants, and answers to the two solutions given in Article 338 and 339 (a) for the case where « = 0. It enables us to eliminate the external forces from the equations of equilibrium. (6) Edge-effect. In investigating the edge-effect we shall omit X", Y", Z", and take s, and s^ to be given by the equations Sl + «2 = — . A s,-s,= (l-o-)f-r---y .(40) \dx ot / ■ . the 1st of which is one of the equations of equilibrium. This procedure is equivalent to adopting equations (42) of Article 330. The remaining five of the equations of equilibrium become dx a CLSi t^ tiz /\ dx a a ' ^ + !h + ^ = V dx a a ' ' -r-' + ^ + n^ = 0, dx a dx a .(41) 342] LATERAL FORCES. EDGE-EFFECT 569 The 2nd and 3rd of these equations give s, + n^ = const. If we retain this constant, we shall only reproduce the results of the extensional solution, in so far as they depend upon a,. We shall therefore take Si+Wi = (42) When we substitute for n^ from tfiis equation in the 5th of equations (41), multiply the left-hand member of the equation so obtained by 1/a, substitute -Sa for Si - hja, and subtract from the left-hand member of the 1st equation of (41), we find that t^ — g^/a = const. If we retain this constant, we shall only reproduce the results of the extensional solution, in so far as they depend upon o^. We shall therefore take *i-? = (43) a\ a I a I a\ a I .(44) In the 1st of equations (41) substitute s^ — h^ja for - s^, and from the 2nd and 4th of equations (41) eliminate ri^. We obtain the two equations dt, ^l ( ^ K\ dx — is dx \ Equation (42) gives n^ when s^ is known, one of the equations of (41) can be used to find n^ when the other quantities are known, and there remain three equations, which can be taken to be (43) and (44). Equations (2) are in this case dU ^ _V-W ^ __dU , V-W ^_ Z\dx^ h' / d'W r-W\ . A^ „ . d rV-W\ dx a 9: h'' (d'^W V-W dx' ^-^^-)^J y In these equations and equations (40) we shall put .(45) V-W K= dW_ dx U a and find dx^ d^W era di; dri ., + .,= 3(l--')f- <*«' On introducing these values of t, and h into the 1st of equations (44), we find the equation ¥ ((Pi, 1 - (-£)}!. =-)<'-)(i-s^) -<>-')• as .(52) The integration of equations (49) and (52) presents no difficulty. The primitives are of the forms where i^,. (l - £) ^ (m - 1^) 4,.. (. = 1, 2, 3, 4). 342, 343] GENERAL UNSYMMETRICAL CONDITIONS 571 and the four values of m,. approximate to the complex fourth roots of - 3 (1 - a^)/kV. When ^ and Sj are known all the stress-resultants and stress-couples are known, and 7] also is known, without any additional integration, but such integration is required in order to determine the displacement w. Thus two additional arbitrary constants will be introduced. They may, however, be omitted as they can only reproduce the results of the extensional solution, in so far as tihese depend upon the constants a,, at, and the displacement determined by these constants is a rigid-body displacement. Apart from the constants ftg, Oj, which do not affect the stress-resultants or stress-couples, the solution obtained by combining the extensional solution and the edge-effect, contains six arbitrary constants a^, a,, A^, A^, A-,,, A^. These are sufficient to secure that Ti, Sj -I- iTi/a, iVj — a~'^dB'i/d4>, G^ shall have any values at the edges which they can have, these values being necessarily restricted by the conditions of rigid-body equilibrium, expressed by the two equations Wi -I- Si = const, and ti — g-^ja = const. The way in which H^ enters into the expressions for the quantities, which can be given at the edges, has been explained in Article 332. An example which repays detailed investigation is afforded by a long cylindrical tube, held so that at one end ^ = its axis is horizontal, and bent by a load concentrated at the other end sc=l, the load being applied by forces of the types Ni, Sy proportional to cos<^ and sin (f>, where (j) vanishes in the vertical plane passing through the axis. It can be proved that, apart from a local perturbation, the curvature of the axis is precisely that given by the ordinary theory of flexure (Chapter X.V.). The local perturbation is the edge- effect near the loaded end, and the stress answering to it diminishes, as the distance from the loaded end increases, nearly according to the exponential law e-sC-^)/", where q is the quantity so denoted in Article 339 (i). 343. General unsymmetrical conditions. When the conditions correspond with values of n which exceed unity two novel circumstances arise. One of them is concerned with the isolation of the edge-effect. In previous solutions this isolation was brought about by equating to zero the arbitrary constants which occur in two of the integrals of the equations of equilibrium, these integrals expressing conditions of rigid- body equilibrium. When n > 1 all the conditions of rigid-body equilibrium are satisfied identically, and recourse must be had to a new method. The other novel circumstance is the geometrical possibility of purely inextensional displacement. When w = 0, or 1, the only possible inextensional displacements are rigid-body displacements, and arbitrary constants expressing such displacements have duly made their appearance in previous solutions. They do not affect the stress-resultants or the stress-couples. When w > 1, the place of such rigid-body displacements is taken by purely inextensional displacements, which affect the values of the stress-couples, but not those of the stress-resultants, in so far as these can be expressed by equations (2) and (3). 572 CYLINDRICAL SHELL [CH. XXIV A (a) ExteTisional solution. We consider first the extensional solution. As before, we are to omit gi, g^, ih, n^, hi, and take equations (3) as giving a sufficient approximation to Sj, s^. The relevant equations are with r nV-W dU n V-W _1 -"■ /'<^^_ '»J?\ : + °"— ^^' '2 = '^^+ a ' *'~ 2 \d!C a)- dU ^^'^di' The 2nd and 3rd equations of the former set give and from this we obtain the equation where a^ is an arbitrary constant, and x^ an appropriate fixed value of x, e.g. the value at one edge. Thus .?, is known and the Ist equation of the same set gives where 02 is an arbitrary constant. Further t^ is given by the 3rd equation of the same set, and thus h,h, »i are known. Then 11 is to be found from the equation dU _ tx — irti, di ~ \^^ ' which gives J xa I— a-' where 4„ is an arbitrary constant. Thus U is known, and V is to be found from the equation which gives dV_nl7 2si dx a 1 - 0- ' _. „ nx , n where 5„ is an arbitrary constant. Thus V is known and W is given by the equation (1 - ■) - j3- cot ^ r— -j ) . ^ \a dO a a sm a J .(55) s,= i(l-<7) The third of equations (54) combined with the 1st and 2nd of equations (55) gives 2^ = 1(^ + 1^ cot^ + 4^)+ai^Z", (56) a a\dd sin oJ 21ih and, on eliminating W by means of this equation, the 1st and 2nd of equations (55) become ^ ^ ,, AldU Ucosd+nV l.+ o- „,; «i = 1(1 - which may be written d'U . , „dU l-cot^^-^ sin' Qj sm^ Eh \ ^ d0 I -^+cot6l-j^+ 1-cot^^-^^ F-2n^^p^i7=-a'''^r-( F' + s- ■ a do^ do V sm^ oJ sitfp Eh \ 28in0 Z" When « = the second of these equations can only lead to the solution of Article 338. To solve the first of them for U we observe that a particular integral of the homogeneous equation obtained by equating the left-hand member to zero is i7 = sin 0. On putting 11= U,sm0, 344] EXTENSIONAL SOLUTION 577 we find the equation __4 3cot0^ = -a^-^^Z --— jcosec^, from which -^ = a,cosec3^-a^^^ cosec'^J^^sm^^ (Z -^^)de, where a, is a constant of integration, and 6^ is any particular value of 6, e.g. the value at one edge of the shell. Hence we obtaia the complete primitive of the equation for U in the form U=hiSi.n6 — \a^ (cot 6 — sin 6 lege tan ^6) -a^^^smdfcosec'dU\m'e(x"-^~)de\de (58) When rt > we put the equations for U and V give ;^ + cote5|+ l-cot^^--^-^ U-2ri-^-^^ , 1 + o- / ,,„ 1 d^" . „„ . n = -a' Eh ^"-2W + ^"+2^^")' = -a^ X"-t'^ _ F"___;^^^' ^A r ^ de 2 sine When n=l particular integrals of the homogeneous equations obtained by equating the left-hand members of these equations to zero are | = 1 - cos 6, r) = \ +cos6. We find as before ^ = 6, (1 -'cos e) + \cH {(1 - cos 6) loge tan 1 ^ - f^^^j \ ^^"'°'^^/l(l-cos^)^sin0 x{/V-cos^)sin^_(z"-|f +F" + ^^")c^^}ci^, f ^s , 1 /, 2 + cos ^"1 7; = 6, (1 + cos 6') + i a J (1 + cos 61) log, tan i6l + ^-^^^^J -^^^^^+'^°'^^/l(l + cosVsm^ ><[\\i^<^ose)^r.e[x''-\%-Y''-^^z')de]de.} \ L. .(59) 37 578 • SPHERICAL SHELL [CH. XXIV A When n > 1, particular integrals of the equations, obtained from the equations determining f and rj by equating their right-hand members to zero, are f = sin ^ tan" 1 6 and r) = sm0 cot" 1 0. The complete primitives of the equa- tions for f and r] are f = 6, sin d tan" i0 + a,smd cot" A ^ f - + — -: cot^ ^d + ^ tan" ^ (9 ) \ - a' ^^ sin 61 tan" u( cosec» 61 cot» i 61 x{/;sin".tan"i^(z"4-F"-^f'.^^^")<^^}<^^, /2 1 1 " 7; = 62sin0cot"i0-|-ci.,sin6'tan"^0 -4-— -, tan^i^ + ---r cot"^0 \7t/ lb ~J~ X It J. -a'^i±^ sine cot" ^di cosec' (9 tan''" ^ y In each case W is given by (56) when U and V are known. The method of integration here explained permits of the elimination of the external forces in all cases*- It may be noted that, when w = and there are no external forces, equations (54) and (55) give either U=0, l, the corresponding terms represent an inextensional displacement, exactly as in Article 343. The terms containing b^ and 62 in equations (60) should therefore be removed, and made the basis of an approximately inextensional solution. This solution can .be found for any value of n. 345. Edge-eflfect. Symmetrical conditions. In a spherical shell the sixth of the equations of equilibrium (4) is satisfied identically by adopting equations (3), and there is no necessity to have re- course to the second approximation for Sj and Sj. So we take these to be expressed by (3). When the conditions are symmetrical, so that w = 0, we either have U =0, and then we have the solution for torsion, or else we have * The method will be found in a paper by the writer in London, Proc. Math. Sac, vol. 20 (1889), p. 89. 344, 345] EDGE-EFFECT. SYMMETRICAL CONDITIONS 579 F= 0, and then Sj, s^,hi all vanish. We may suppose the external forces to be eliminated by means of the extensional solution, and then the relevant equations become d dd (ti sin 0) - t^ cos 0-niamd = 0, -^ (wi sin 6) + (ti + 4) sin de = 0, = 0, with a . . .. ^ \gi sin 6') - g^ cos d-niasmd= 0, .(64) dU de .(65) + ?7cot6'-(l + o-)F|, From the 1st and 3rd of equations (64) we eliminate ti, obtaining the equation t^ (^i sitf + n-, sin cos 0) = 0, so that we have ti sin^ + 11^ sin cos = const. If we retain this constant we shall merely reproduce results included in those obtained in the extensional solution. We therefore write ti = — Ui cot 0. Then the 1st of equations (64) becomes «a=- dni .(66) .(67) Now put -j^ + U =^. Then the 3rd and 4th of equations (65) give and the 4th of equations (64) becomes or |.[3+«°"i-("»'-^+''>f = - n, (68) 37—2 580 SPHERICAL SHELL [CH. XXIV A Again the 1st and 2nd of equations (65) give de l-a^ dd^ ' dd^ = cot0(^-?7cot0)-f=cot^^^-r, and from (66) and (67) this becomes -^^4-cote^-(cot^^-.)n.=-^ %^^^^^e"^-{=:3zi9i'>-) + tircoty. dr Again, we have the equations .(83) U W r r tan y _ :_ __ ^ ^ ~ dr \r r tan -y dU . fU 9i _ h? fd'W adW h? ( ^^ = -3 1 3 V dr'' "^ r do dyW IdW dr^ r dr )■ .(84) In these we write f for dW/dr. The 2nd of equations (83) becomes, on substituting for g^, gz from the 3rd and 4th of equations (84) and using (82), h' fd'P id^ ^\ 3 \dr^ ' r dr r- Xow the 1st and 2nd of equations (84) give U=Wcoty + r*-i^, .(85) l-o-^ dU _ti— ati dr~ l-o-" We eliminate U from these equations, thus obtaining the equation (1 - a=) ^ cot 7 + ^ {r {t, - at,)} - {t, - at,) = 0, which becomes, on substituting for t^ from the 1st of equations (83) and for <, from (82), !"• f Tr ^"^''^} - «! - (1 - '^O f cot^ 7 = 0, d^n, Sdrii 1 - a-2 ' + dr'' r dr r^tan^y or On putting n,r = r), r = \a?, equations (85) and (86) become 1 = 0. dx^ xdx x"^ h'''^' = 0, ^ 1^_4 2(1-0-^ ) dx' X dx x^ tan'' 7 ^^ = 0. .(86) .(87) .(88) 347] EDGE-EFFECT. SYMMETRICAL CONDITIONS 585 To solve these equations put then the integrals of the equations are Bessel's functions of kx of order 2. There are four complex values of k, but two of them are irrelevant, as the Bessel's functions are not altered by changing the sign of k. The two relevant values of k are (1 ± i) {3 (1 - sin 7 + „, rX" sin 7 = 0,' .(92) ^ (Sir) sin y-t, + s, sin 7 + -^^^ rY" sin 7 = 0, t-^ + ^-i£rZ"t^ny = 0.) On multipljdng the 1st of equations (92) by sin 7, the 2nd by — 1, and the .3rd by — cos^ 7, and adding, we obtain the equation J- {t,r) sin'' 7-3-, (Sir) sin 7 + -^y r {X" sin 7 - F" - Z" cos 7) sin 7 = 0, which can be integrated in the form ^i£ f ' '' ^^"'^^^ 7 - 1^" - Z" cos 7) c^r (93) iiTsin 7 — Sir = ai — The elimination of t^ between the 1st and 2nd of equations (92) gives ^ (T -J- («! r) smy + n, + t2 cos ■y=0, ar -J- (hir)ainy+ffi + hismy+n2rainy=0, ar -i-(ffir)smy-hi-g2smy-nirsiny=0, - hi coay+isi+Si) r sin y=0. .(99) 588 CONICAL SHELL [CH. XXIV A We multiply the eft-hand members of the 1st, 2nd, and 3rd of these equations in order liy siny, - 1, and —cosy, and add, obtaining the equation ^. ih '■) sin^ y - ^ («i »•) S'° y - ^ ('^1 '■) «i" 7 cos y = 0, which is immediately integrable. As in" previous examples no constant of integration need be introduced, and we may write the integral in the form % cos y = ii siny — Sj (100) We eliminate <2 from the 1st and 2nd of equations (99), obtaining the equation -J- {ti r) sin y - T- (Si r) sin^ y — «2 cos^y + % sin y cos y=0. We eliminate gi2 from the 4th and 5th of equations (99), obtaining the equation -y- (ffi r) sin y+-T-, (^i »") sin^ y - Ai cos^ y + ihr sin^ y-niv sin y = 0. From these two equations we eliminate Wg, obtaining the equation '■ j". (^1 *■) s"'^ y~'3~idi^) sill y cos y — i'j-{h r) sin^ y - rs^ sin y cos^ y — T- (^1 )') sin^ y cos y + Ai cos^ y + raj?- siny cosy = 0. In this equation we substitute for wi from equation (100), and for n^ from the 6th of equations (99), obtaining the equation *■ J-, ('i 1") sin^ y - T- (9i J-) sin y cos y - r -=- (si >•) sin' y + (rai sin y - Ai cos y) cos^ y - ^ (Ai r) sin2 y cos y + /ii cos' y + r sin y ( rfj-2 rf = tan y ( o-'' ^j^ + >; ) + o" ;r». ('■^) + ^' c^/-2 c^?-' , f f sin2y /c^v d^W-s\, .,-., = (l-0-.-£r>3^. {>H ^'''^ The same relation may also be used to express g^ in terms of {h-gilir tan y)] and f. In the same way from the 2nd and 5th of equations (106) we can obtain the equation 1 + 31^,) = ('^ - r^^^^r^tr^^y f ^^^^^^4 ' ^'''^ and use this relation to express g'a in terms of {t^-g^lir is-ny)} and |. Then, since {t^- g„l{r ia,ny)\ is connected with {h-giKri&ny)] by (103), we have h, h, gi, g^ expressed in terms of [h-gilirtany)} and ^. Again the 3rd and 6th of equations (106) with the 6th of equations (99) yield a single relation between s,, /h and |, and this relation can be put in the form .(109) and thus by (101) s, is expressed in terms of {ti-gj{rta.n y)} and f. The same relation avails ti) express hi also in terms of the same two quantities. (c) Formation of two linear differential equations. It remains to obtain two equations for the determination of ^ and {)-H^+^^-'^^)} = ^^'(^ + '^) + ^^^- (110) In proceeding now to form the two equations for determining ^ and {ti - gil{r tan y)} we shall permit ourselves a certain approximation, depending upon the circumstance that hjr is .small. In equations (107) — (110) there are terms which contain factors of the form {a + {h^/r^) p\, and other terms which do not contain such factors. In any term, containing a factor of this form, an approximation may be made by omitting {h^jr^)p as small in comparison with a. We shall not alter any expression which does not contain such a factor. For example we shall not alter the expression 7(2 Id , ,, .] "■~ / \ Sr^ tan^ y/ cos y \dr dr'^ ) ' 349] EDGE-EFFECT. LATERAL FORCES 591 which occurs in the right-haud member of equation (107). According to this plan we have, as approximate equivalents of equations (107)— (109), the three equations ''=('>-.-t!ir^)+OT^,{i('-^)+4' '^=(^-.i?r>3-^f|('-^H4,[ (Ill) \ r tan y/ 3 r^ %\v? y and these give, to the same order of approximatibn, ■^' Srtany "' 3r tan y , A2£cosy Now write Z==ti—^ , (113) }• tan y ^ ' and express =^+3^fi('-^)+'^4' I '^=ii(^^^)+3^.i^H('-^)+4-| Also by the 3rd of equations (111), combined with equation (101) and the 3rd of equations (112), we have ^1 L „, A^lcos^y «i 1 = -. s(l-o-) , . „ ' (115) I r tan y sm y r^ sm^ y Now substitute from (114) and (115) in (102). We obtain the equation =l(i-J—) \ sm'' yj m*\fA^<-^*"-^mhV'- <"" This is the first of the required equations. It has, as premised, been formed without using equation (110). We next form an approximate equivalent of equation (110) according to the plan adopted for forming equations (111). It is dh_h_ (dh h\ 2^1 (1 + 0-) _ 1 - T% /^ ;^\ .jj^s In this equation we substitute for or 3 592 CONICAL SHELL [CH. XXIV A The coefficient of JA^cot^y in the left-hand member of this equation is {? (1 - (r^)li^} ('-3 + 2 cosec^y), and the equation becomes d^{Lr)\d{Lr) /„ ,. l + o-l Z l-'^^f, AM2-3sin^y) ] or, by om- method of approximation, ^(MH.l'^l^_{2(l-.) + 2y^l:^=i::^^ (119) dr^ r dr [^ ' sm^yj r rtan^y^ This is the second of the required equations. (d) Method of solution of the equations. We shall now transform equations (116) and (119) by putting Lr^hC, r = \hx^ (120) The equations become dH\dC ( .^ + <^\ i 2(1-'^') . I di^^xdx ^\ '"'^sinV^' tan^y *' ^ Between these equations we eliminate f, obtaining for | the linear dififerential equation of the fourth order (122) where for brevity we have written ai = 8{l + (r + (l-o-)cosec2y}, a2 = 8{l-(r (-(1 + 0") COSec^y} (123) If f were found from equation (122), f or Lrjh would be Icnown from the 1st of equations (121), and then, as we have seen, all the stress-resultants and stress-couples would be known. To determine the displacement a further integration would be necessary, becausfe the expressions (106) for (126) obtaining the equation + [2426-6(4nH-23)24 + 2{6OT2 + 6m2-(l+aiH-a2)22}z3 - {4m? + 6m22 - 2m (1 + ai + aa) 22 + (1 + 3m - qj) e^} z^] ~ CtiZ + W+ — ^i^^+2ni32-Mi2(l + ai + a2)22+TO(l + 3ai-a2)23 + (aia2-4ai)2*|z=0 (127) We seek a solution of this equation in series in the form X=2»(Co + Ci2 + C22^ + C323+...), (128) and find that such a solution is formally possible if, and only if, '"-!^=«.-- <-> and There is no difficulty in obtaining as many of the coefficients Cj, ca, ... as we wish. The coefficient o^ is arbitrary, ^inoe there are four complex values of m, which satisfy (129), there are four independent solutions of this kind, and they suffice to determine the edge- effect. It is known that, in general, the series such as that in the right-hand member of (128) are not convergent, but avail for the approximate numerical calculation of the functions such as X. In this property they resemble the well-known "semi-convergent expansions " of Bessel's functions, these expansions being, in fact, particular .examples of asymptotic expansions of integrals, which are analogous to normal integrals, of linear differential equations. 350. Extensional solution. Unsymmetrical conditions. In a conical shell, under Conditions answering to values of n which exceed unity, we can have an extensional solution. For this we neglect gi, g^, h^, n^, n^ and use equations (3) to express s^, s^. The relevant equations among equations (4) become -T- (tir) sin y + nsi — U sin 7 -1- g„, r sin 7X" = 0, d . \ — + ^-(^" + ^"t-n7)^cir = 0, 2Eh .(132) where Oj is a constant of integration. Thus t^, Sj, t^ are known. To determine the displacement we have the equations W ^ dU (U nV t,= + 0- — + — -. — r \r r sin 7 r tan 7 dU (U nV W dr \r r sm 7 r tan 7, \ s, = Kl-<^)(' dV V nU .(133) dr r r sin 7, The first two of these equations give dV _ti — <7t2 which canT)e integrated in the form U=An+jy^^dr, (134) where An is a constant of integration. The 3rd of equations (133) gives d iV iV\_ nU \r ) r' sin ' + 2si dr\r J r'sin7 ' (1 - a-)r which can be integrated in the form V = + -S«r+-3— -r -J ^ ~dr\dr sm7 sm7 r'' i \ 1 — n-2 + 2 r^s, 1 - o" Jr.r .(135) where B^ is a constant of integration. Now that U and 7 are known, W is determined by the equation 1^= P'tan7 + nFsec7 - - — ^r tan7, where the terms of W which contain An and 5„ are .(136) An tan 7 — -. + nBn,r sec 7. \ sm7C0S7/ ' 350, 351] APPROXIMATELY INEXTENSIONAL SOLUTION 595 351. Approximately inextensional solution. The terms containing 4n, -S™ represent an, inextensional displacement. We have thus determined, as functions of r, (n^ - 1) An w(w'-l) B, 5^2 = ■ 3 [sin* 7 cos 7 r^ sin''' 7 cos 7 r X = -i{l-a)^ .(139), sui''7C0S7 r' The quantities n,, n^ are given by the 4th and 5th of equations (4) in the forms n^r sin 7 = - -3- {hiv) sin y-ng^- h^ sin 7, MiT sin 7 = -T- (5^1 r) sin y-nh^- gz sin 7, * Cf. Lord Eayleigh, London,_Proc. Math. Sob., vol. 13 (1882), p. 4, or Scientific Papers, vol. 1, p. 551. 38—2 596 CONICAL SHELL [CH. XXIV A which become on using (139) _hU n' (n' - 1) An n^n"-!) B^ 3 |sin^ 7 cos 7 r^ sin' 7 cos 7 r" .(140) 6 (sitf 7 cos 7 r- sm 7 cos 7 r The solution which we seek is independent of externally applied forces, as the effects to which these give rise can be found from the extensional solution, so we omit X", Y", Z" from the first three of equations (4). On substituting for Wi, 7i2 from equations (140), these three equations become d -J- (tiX) sin 7 — ns^ — t^ sin 7 = 0, h^ fn' (n' - 1) An «' (n^ - 1) Bn J- (Siv) sin 7 - w<2 - S2 sin 7 - -5- <,cos7 + -^ -^> ^ Kt--* -f -'^ -(^^ 3 (sin^ 7 cos 7 Vsm^ 7 / r' sin 7 cos 7 Vsin^ 7 = 0, (141) In these we are to calculate ^1, ^2. *ij «2 from the formulae found in Article 330. In the case of a conical shell these formulae give dU . /U ?iF-Fcos7\ t. dU fU nV - W cos7\ dr \r r sin 7 / + h^ 2 - 2(7 - Sa-^ d'^W a-(2 + a) /nVcosry-n'W 1 dW Sr tan 7 I 2 (1 - a) dr^ 2 (1 - o") dir . fU . nV- Fcos7N r' BVd' 7 r dr ■)}• dU (U nV- lrcos7\ dr \ r r sin 7 / A' { (r{\ + 2a) d'W 2 + q- / nFcos7-w''Tf \dW\\ ' 3r tan 7(2(1 -0-) dr'' "^2(l-a)V ''-■--•' ^^ r' ava.' 7 r dr.)\ ' s =i-(l_a-)-[/'— -— _^?^^+— ^1— ^ f Vcosy-nW \) \\dr r r sin 7/ 'Sr tan y dr\ r sin 7 /J ' {\dr r r sin 7 \ _ A^ d_ /Vcosjy--nW\[ J 3r tan y dr\ r sin 7 / j Now we write for brevity ^ ,, dV' fU' nF'- lf'cos7\ \ h = -J h or I 1 : '- ] ctr \r r sin 7 / , dU' fU' nF'-If' cos7\ dr \r rsmy /' \ar r rsmyj I (142) 351, 352] APPROXIMATELY INEXTENSIONAL SOLUTION Then we find 597 t^ = h? t( 6 (1 — 0-) I sin* 7 r' sin'7 ^,_ 2 + tr { r,^n-'-l) An n (n' - 1) B^ 6 (1 — cr) ( sin* y r* sin' y r^ ^, l-<7n(n'-l) An 6 sin' 7 r ( sm' 7 r' J .(143) and equations (141) become ^ («i r) sm 7 + ?w/ - «/ sm 7 + — -— r- l+ /^ m^ — 008^7 A h = ^ ('■I) + <^ | + mi;seoyH — ^ '- C), dr \ smyoosy /' <^ / >s /* . . i^-cos^y \ 92 = *.4"<'-'>S- .5j+-J«>}]. .(149) Then we put , .-..f.,, ^ r=ph, (150) and substitute from (149) in (148) obtaining three equations from which h is explicitly absent. The condition that h is small is now replaced by the condition that a solution is required for large values of p. A little simplification is effected by putting |i=^P, Vi = VP, fi = fp (1^1). Then the three equations become [P^^dp pr^ 2siny p »^a-<^)^^ + ..tany5? + »^i^tany(|.-^^) dp !_„■ I fdiji 1x\ „t„„.^li+/'„22+£L_^oosv')^-ii2f|-^-cosv')^=0, .(152) 600 CONICAL SHELL [CH. XXIV A l + ^, ii=Ze"^, (160) where m is at present undetermined. Then X, Y, Z satisfy the three equations ,l- 2_o, (162) and »2_cos2y^^Q (^g3) smy (c) Solution of the equations. We wish to obtain solutions of these equations in series proceeding by powers of x K We write and assume for X, Y, Z expressions of the form I (164) X=^(a„+a,. + a,.^ + ...), Y=^ {h + hz + h,z^ + ...), ^=.' (c„ + Ci^ + «.^^ + . •)• 602 CONICAL SHELL [CH. XXIV* A Then we have, for example, dx (PZ dz -2<' + l{CoC + Ci(c+l)0+C2(c+2)22+.. dz \dx) z" + 2{coc(c + l) + c,(e + l)(c + 2)z + C2(c + 2)(c + 3)22 + ... and so on. We substitute the serial expressions for X, dXjdx^ ... d*Zldx* in the left-hand members of equations (161), (162), (163) and equate to zero the coefficients of the various powers of z, beginning with the lowest. We thus obtain sufficient equations to determine m, the exponents a, b, c, and the coefficients Oq, ai, ... , bg, b^, ... ci, c^, ... in terms of Cq, which remains arbitrary. It will appear that there are four values of to, and so there are four solutions of this type. If this process is carried out it will very soon appear that a = b = c= -^. So it is simpler to assume the forms X=z"i{\dx J] , An 1 ,/rfir + »iZ 3 tany ifi \ dw^'""^ )} -™^smy tany L - -^ |(^_ + 4m^ + 6m^^ + 4m3^+.^4^j ^g^^-cos^y^n (169) smy J ^ ' In future we shall use equation (169) instead of equation (163). If from equations (161), (162), (169) we select the terms containing e""*, and equate them to zero, the first two will contain a,,, bn, c,, but the third will not contain any a's, 6's or c's with suffixes higher than k — 1. For example, taking k = 1, equation (169) will yield a new equation connecting ao, bo, Cq, with which we shall deal presently, and equations (161) and (162) will yield two equations containing a^, bi, Ci. A^ain, taking k = 2, equation (169) will yield a new equation containing a^, bi, ci, and equations (161) and (162) will yield two equations containing a2, 621 "2- _Thus the equations, by *hich a,, 6,, c„ are to be determihed, are obtained by equating to zero the coefficients of 2*"* in (161) and (162), and the coefficient of 2"+^ in (169) • 604 CONICAL SHELL [CH. XXIV A In particular by equating to zero the coefficient of z^ in equation (169) we obtain the equation -^(TTO^sin^v. on-irfiam^yCi-iT) ao + ^o'(l+<'')™« -ba — ^a (3-o-) mn ^60 + 2(2 + 0-) (1 -iT)m'n ^ 6„ + io-ni sin 7 ( n^ 2(7C0Sv Icn ^ ' cos y " -^ ' \ cos y '/ -2(T»isiny in'' 2 cos 7) Ca+'imrfi (% + a-){\ -(r)tany.Cj=0. On substituting for (% and 65 from equations (167) this equation becomes an identity. It may be observed that, if we had not assumed the exponents a, b, to be equal to — ^, but had left them undetermined, it would still have been necessary that the coefficient of z'*^ in an equation formed in the same way as (169) should vanish identically, and this condition would have led to the determination of a, b, and c. We now equate to zero the coefficients of 2^ in equations (161) and (162), obtaining the equations sm y and m^(ii-\ — — n ta,ny {mbi + ^b„) —n (3-cr) tan 7. b^+ln^ = tr cosy) («iCi+|flo) - (n^ 2COS7 I Co=0, \ cosy ij 1 1 — i Om? 61 + 2»i6a) H — n tan 7 (m^ e-, + 2mc„) = 0, 4 cos y "'4 I \ i u/; which, combined with (167), yield 2o" 4-0- , n „„_ ai=—7 Ci J- Co, bi=--. — ci (170) mtany m^tany "' ' smy ^ ', We also equate to zero the coefficient of # in equation (169), obtaining the equation - am 8in2y / 2»Mi!, +_ a„ j _ 2«2 (2 + 5 Nat. Phil., Part 1, p. 220. '« Berlin Hint, de I' Acad., t. 11 (1755). " Excrcicei de matMmatique, t. 2 (1827), p. 42. Cauchy's work dates from 1822, see Intro- duction, footnote 32. " The fluctuation of scientifio opinion in this matter has been sketched by Maxwell in a lecture on "Action at a distance," Scientific Papers, vol. 2, p. 311. 19 Electricity and Magnetism, 2nd edition (Oxford 1881), vol. 1, Part 1, Chapter V. Cf. Article 53 (vi) supra. ^ " De la pression ou tension dans un systems de points mat^riels," Exercices de matUmatique t. 3 (1828), p. 213. STRESS 609 A third way is found in an application of the theory of energy. Let us suppose that a strain-energy-function exists, and that the equations of equilibrium or vibration of a solid body are investigated by the method of Article 115, and let the energy of that portion of the body which is contained within any closed surface /S be increased by increasing the displacement. Part of the increment of this energy is expressed as a surface integral of the form Now in the formulation of Mechanics by means of the theory of energy, " forces " intervene as the coeflBicients of increments of the displacement in the expression for the increment of the energy. The above expression at once suggests the existence of forces which act at the surface bounding any portion of the body, and are to be estimated as so much per unit of area of the surface. In this view the notion of stress becomes a secondary or derived notion, the fundamental notions being energy, the distinction of various kinds of energy, and the localization of energy in the medium. This method appears to be restricted at present to cases in which a strain-energy-function exists. CauGhy's investigation of stress-strain relations in a crystalline body. The body is supposed to be made up of a large number of material points, or particles, which act upon one another at a distance by means of forces directed along the lines joining them in pairs. The force between two particles of masses m, m' at a distance r apart is taken to be an attraction of amount mm'x (f), and the function ^ (r) is supposed to vanish when r exceeds a certain finite value R, called by Cauchy the " radius of the sphere of molecular activity." The particles are supposed to form, when in equilibrium under no external forces, a " homogeneous assemblage." By this it is m^OTtrthat all of them have the same mass, and that, if three of them are situated at points P, P', Q, and a line QQ" is drawn from Q, equal and parallel to PP'and in the sense from P to P\ there is a particle at Q'. Let x, y, z be the coordinates, and M the mass, of any particle P. We draw a closed curve s round P in the plane (p) which passes through P and is parallel to the plane of (y, z), so that aU the radii veutores drawn from P to s exceed R. Let S be the area within this curve. We shall suppose that all the linear dimensions of jS are small compared with ordinary standards. The statiTial resultant of all the forces whose lines of action cross p within s is a force, of which the components parallel to the axes are denoted by XxS, TxS, Z^S, where X^, T^, Z^ are the components of the traction across the plane at P. But these components are also the sums of such expressions as where OTj denotes the mass of a particle situated on that side of the plane for which x is greater than the x of P, rrij denotes the mass of a particle situated on the other side of the plane, r-y denotes the distance between these particles, X^y, ^„, Vij denote the direction- cosines of the line drawn from m/ to OTj. The summation extends to all pairs so situated that the line joining them crosses p within s and the distance /-y does not exceed R. From the assumed homogeneity of the assemblage it follows that there is a particle Q of mass m (equal to M or mj or to/), so situated that the line PQ is of length )■ equal to Tij, and is parallel to the direction (Xi,, fHj, "«). Thus the terms of the above sums may be replaced by L. Mmx{r)'K, Mm,x{r)ii, Mmx{r)v, 39 610 NOTE B where r is the distance of a particle m from M, and X, /i, v are the direction-cosines of the line drawn from M to m. The summation may be effected by first summing for all the pairs of particles (?n,-, ra/) which have the same r, X, ^ j/ and are so situated that the line joining them crosses jo within «, then summing for all the directions (X, /*, v) on which sUch pairs of particles occur, and lastly summing for all the pairs of particles on any such line whose distances do not exceed R. The first summation is effected by multiplying the expressions such as Mmx (f) X by the number of particles contained in a cylinder of base 5 and height rX. This number is pSrKjM, where p is the density, or mass per unit of volume, of the system of particles. Thus we require the sums of such quantities as If the summation is extended to all directions (X, p., v) round P in which particles occur, any term will be counted twice, and therefore the required expressions for the component tractions X^^, ■■■ are X^ = lp2mrX2x()-), r^ = ip2mr\ij,x{r), Z^=^^p2mr\vx{r), in which the summations extend to all particles whose distance from P does not exceed R. If there is no initial stress the six sums of these types vanish, or we have l.mfK^x ('') = 0i ■•• I 'S,mrkfi.x{r) = 0, ... ; but, when there is initial stress, the values of the six components of it at any point are JT/), ..., where X^m^ip2mr\^X W. - , X^m = ipSmrX^x « The stress-strain relations are obtained by investigating the small changes which are made in the above expressions when the system undergoes a small relative displacement. As in Article 7, we may take the unstrained position of M to be given by coordinates X, y, z, and its strained position by coordinates x + u, y+v, z + w. At the same time m is displaced from (x+s., y + y, z+z) to {x + x + u + u, ...), where u, ... are given with sufficient approximation by such formulae as 80 that r\ Bbcomes r\ + 8 (rX), where du du du ^ ' \ ox '^ ay oz) and we have similar formulae for 8 {rp), 8 irv). Also r becomes r (1 -I- e), where and p becomes p , where '_ (^ 3" ^ 3^\ " " \ 'bx dy azj' The effect of these changes is to give us for .^j., ... such expressions as Z,=ip'2[m ~^^ {x {r) + ery{ (r)} {r\ + b (rX)^], X,=ys. j^m -^^ {x (r) + erx' {r)) {rX -|- 8 (rX)} {rp. + S {rp)}]^ . When there is no initial stress, these equations give us the stress-strain relations in such forms as X^ = i/)2 [mr {rx: (r) - x (r)} X^ {e^V- + eyyp? + e,y + ey.pv ^e^v\ + e:^\p}'], Xy=\pl,[mr {rx' [r) - x {r)} \p. {e^X^ -1- eyyp.^ -\- e.y + By.pv + e,^v\ -X- e^^X/x}] ; STRESS 611 and the elastic constants c^, ... are expressed by sums of the types cu =ip2 [mr {rx' (r) - x (r)} X*], ci^=cn=^pS [mr {rx' (r) - x W} XV'], "14 = cse =ip2 [mr {rx'{r)-x (r)} XV"], (he = ip2 [mr {rx' (r) - x {r)} X^]- There are 15 of these, Green's 21 coefficients being connected by the 6 relations which have been called Cauchy's relations (Article 66). When there is initial stress we have to add to the above expressions for A'^ and Xj, the terms with similar additions to the remaining stress-components (Article 75). The above investigation is given as an example of the kind of methods by which the elements of the theory of Elasticity were originally investigated. A modification by which the resiilt.s may be made to accord with experiment, at any rate for isotropic solids, has been proposed by W. Voigt, Ann. Phys. (Ser. 4), Bd. 4 (1901). NOTE C. Applications of the method of moving axes. The theory of moving axes may be based on the result obtained in Article 35. Let a iigure of invariable form rotate about an axis of whicii the direction-cosines, referred to fixed axes, are I, m, n, and let it turn through an angle Sd in time $t. At the beginning of this interval of time let any point belonging to the figure be at the point of which the co- ordinates, referred to the fixed axes, are x,y,z\ then at the end of the interval the same point of the figure will have moved to the point of which the coordinates are x+{mz- ny) sin h6 -{x-l (Ix+my+nz)} (1 - cos b6), .... Hence the components of velocity of the moving point at the instant when it passes through the point (a;, y, z) are d6 dd ,de dd dd . ,de -^''di+'"''di' -'^dt^'^'^di' -'''^dt^y^^f We may localize a vector of magnitude dBldt in the axis {I, m, n), and specify it by com- ponents a^, a>y, a>„ SO that m^c^ldejdt, .... This vector is the angular velocity of the figure. The components of the velocity of the moving point which is passing through the point {x, y, z) at the instant t are then -ya^-\-za>y, - Zttt^-^ Xa,, —Xayh-yaij,. Let a triad of orthogonal axes of {ocf, y', 2'), having its origin at the origin of the fixed axes of {x, y, z), and such that they can be derived from the axes of {x, y, z) by a rotation, rotate with the figure ; and let the directions of the moving axes at the instant t be jspecified by the scheme of nine direction-cosines. X y z x' h IHi »i y h mg % z' h ms »3 39—2 .(1) 612 NOTE C Let 5], 6i, di denote the components of the angular velocity of the rotating figure parallel to the axes of x', y', 2*, so that and let a point {3/, y', z') move so as to be invariably connected with the figure. The co- ordinates of this point referred to the fixed axes are, at the instant t, lix' + liy' + hz', ..., and we may equate two expressions for the components of velocity of the point. We thus obtain three equations of the type ^(lix' + l2i/'+l3z')=-{m.iX' + m2y' + m3i') (jii^i+nj^a + ^s^s) Since the axes of (x', y', 0') can be derived from those of (x, y, 2) by a rotation, we have such equations as TO1M2— »l2Mi = ?3. The above equations hold for all values of x\ y', 2', and therefore, x', y', z' being independent of the time, we have the nine equations dm, . . dm,i . . dm^ . -^ =7112,63- msBi, -^ =171301- miBs, -^=711116^-171^61, dn-i . „ dTio „ . dn, . . dt ='*2^3~"3^2' -^=»3Pl-%^3, -^ = mi52-«2ei- j Now let a, v, w be the projections on the fixed axes of any vector, u', v', w' the projections of the same vector on tjfie moving axes at time t. We have such equations as du d ,,,,,, ,. = h(^-v'63 + w%^ + k(^^^-w'6r + u'e3) + hQ^-u'6,+v'6,^ (^^ Hence the projections on the moving axes of that vector whose projections on the fixed axes are du dv dw di' ~di' di du' ,. ,. dv' ,. ,. dw' ,n ,„ are -^-v63 + vJ6 ■••i ^ij ••• are linear functions of dajdt, d^jdt, ..., and the coefficients of dajdt, ... in these functions are known functions of a, ft y, .... Thus we have such equations as dx da dxdfi ^ , i , , /dx^ da , dx' d^ \ .^ , ,. 1 , f , fd^ da 8a' dff \ j„ ,„] We may equate the coefficients of dajdt, d^jdt, ... on the two sidps of these equations, the quantities itg', ..., 6i, ... being expressed as linear functions of dajdt, In like manner, if u, v, w and u', v', w' denote the projections of any vector on the fixed and variable axes, equations (2) give us formulse for calculating ZujZa, .... In applications of the method it is generally most convenient to take the fixed axes to coincide with the positions of the variable axes that are determined by particular values u, ft y, ... of the parameters, then in equations (2) we may put Ii = m2=n3 = l and 1^= ... =0. When this is done the values of dujda, ... belonging to these particular values of a, ... are given by formulse of the type du da du dp du dy , fdu' da du' dfi du' dy \ d'adi+rpit+d^^t'--=[d^di+Wdt + rydt+-)-''^^+^^'' (') The above process has been used repeatedly in Chapters xviii., xxi., xxiv. As a further illustration we take some questions concerning curvilinear orthogonal coordinates. The coordinates being a, ft y, the expression for the linear element being {{dalh,f + (dpjh,f+{dyjh,f}^, and the variable axes being the normals to the surfaces, we have , 1 da , I dp , Idy '^=h,dt' "'^h.dt' '"'>-h,di- To determine the values of 5i, 6^, 6^ we have recourse to Dupin's theorem cited in Article 19. It follows from this theorem that the tangents drawn on a surface y, at points of its intersection with a surface ft to the curves in which the surface is out by two neighbouring surfaces of the family a, say a and a4-8a, ultimately intersect when ha is diminished in- definitely, and the point of ultimate intersection 7 is a centre of principal curvature of the 614 NOTE C surface ;i. In Fig. 75 the point F is () 57 + 3^ ^''"''') W + 3^ ^""^'^ J- a3A,53+7aA,52, 3 (ia 3 „ , (^/3 '3 rfy 3^ (^^Aa) ^ + g^ (^.Aa) ^^ + g^ (Z,A2) ^ 3 , „ , , aJa . 9 , .. . , 0?/3 , 9 , „ . rfy 9^ (^vAa) ^, + g^ (A V A3) ^^ + g- (X,A3) -j^ = la (^"^^) Tt + 3^^ ^" ^3) f + g^ (9.As) J -?y A3^3+?7 A3^. where ^2, 6^ are given by (6). Equation (19) in Article 58 can be written down at once. 21 Cf. B. B. Webb, Messenger of Math., vol. 11 (1882), p. 146. INDEX. AUTHOES CITED. [The numbers refer to pages.'i Abraham, M., 44 Adams, F. D., 118 Airy, G. B., 17, 86 Alibrandi, P., 107 Almansi, E., 22, 107, 250, 362 Amagat, E. H., 103 Aron, H., 29 Atcheiley, L. W., 211 BacL, C, 110, 351 Barth^ltoiy, A., 497 Basset, A. B., 24, 403, 460, 534, 536, 540, 546, 549, 551 Bauschinger, J., 110, 111, 112, 113, 114, 115 Beltrami, E., 49, 86, 133, 306 Bernoulli, Daniel, 3, 4, 5 Bernoulli, James, 3 Bernoulli, James (the Younger), 5 Betti, E., 16, 44, 47, 171, 172, 230 Binet, J., 28 Blanchet, P. H., 18 BoUe, L-., 581, 585 Boltzmann, L., 116, 143 Borchardt, C. W., 53, 106, 246, 274 Born, M., 417, 419 Boscovitch, B. J., 6 Boussinesq, J., 16, 20, 24, 26, 29, 87, 183, 186, 188, 191, 209, 238, 241, 319, 321, 356, 437, 441, 447, 465 Biaun, F., 116 Bresse, M., 24, 454 Brewster, D., 87 Bromwich, T. J. I'A., 290, 305, 31B, 437 Bryan, G. H., 30, 415, 490, 491, 541, 566 Burgess, R. W., 412 Burkhardt, H., 7, 15 Butcher, J. G., 116 Calliphronas, G. C, 369 Canevazzi, S., 23, 385 Cardani, P., 102 Cauchy, A. L., 8, 9, 10, 11, 12, 14, 18, 19, 27, 35, 57, 72, 79, 81, 105, 108, 398, 497, 608 Cerruti, V., 16, 236, 239, 241, 246, 308 Cesiro, E., 55, 219, 221 Chladni, E. F. F., 5 Chree, C, 122, 126, 144, 173, 250, 2S1, 258, 262, 263, 268, 270, 274, 275, 279, 281, 290, 291, 293, 431, 435, 448, 449 Christoffel, E. B., 18, 303 Cilley, F. H., 108 Clapeyron, B. P. E., 22, 380 Clausius, E., 9, 10 Glebsoh, A., 14, 17, 21, 24, 27, 28, 29, 35, 177, 178, 205, 273, 393, 394, 402, 403, 474, 656 Codazzi, D., 518 Coker, E. G., 87, 118, 370 Cook, G., 567 Cornu, M. A., 102, 129 Cosserat, E. and F., 126, 250, 251 Coulomb, 0. A., 3, 4, 119 Coulon, J., 308 Cowley, W. L., 380 Cox, H., 447 Culmann, K., 23 Daniele, E., 275 Darwin, G. H., 16, 119, 268, 271, 272 Davidoglou, A., 448 Dougall, J,, 184, 283, 235, 241, 244, 374, 375, 492 Dubois, F., 585 Duhamel, J. M. C, 106 Duhem, P., 47 Dunkerley, S., 449 Eddy, H. T., 454 Edwardes, D., 251, 275 Estanave, E., 496 Euler, L., 3, 4, 5, 407, 411, 608 Everett, J. D., 103 Ewing, J. A., 32, 84, 110, 111, 112, 113, 118, 351 Ewing, J. A. and Eosenhain, W., 110 Fabr4, 367 Fairbairn, W., 566 Pilon, L. N. G., 87, 135, 212, 279, 321, 333, 369, 373, 374 Flamant, 110, 209 INDEX 617 Foppl, A., 110, 117, 120, 331, 351 Forsyth, A. R., 592 Fourier, J. B. J., 447 Fredholm, I. , 230, 250 Fresnel, A. , 7 Fuohs, S., 196 Fuss, P. H., 3 Galileo, 2 Garrett, C. A. B., 448 Gauss, C. F., 501 Gehring, F., 27 Germain, Sophie, 5 Goldsehmidt, V., 155 Grashof, F., 22, 343, 351, 352 Green, G., 11, 15, 18, 57, 83, 109, 304 Greenhill, A. G., 143, 319, 324, 325, 425, 430, 431, 432, 448, 449 Griffith, A. A., 327 Guest, J. J. , 120, 456 Hadamard, J., 29, 57, 308 Halphen, G. H., 430 Hamburger, M., 198 Harnack, A., 63 Hausmaninger, V., 25 Heppel, J. M., 22 Herglotz, G. , 272 Hertz, H., 16, 26, 191, 196, 215, 311 Hess, W., 24, 407, 423 Hicks, W. M., 230 Hilbert, D., 170 Hilton,. H., 148 Hobsou, E. W., 581 Hooke, E., 2 Hopkins, W., 45 Hopkinson, B., 115, 447 Hopkinsou, J., 25, 106, 144 Hoppe, E., 25, 458 Hough, S. S., 272 Ibbetson, W. J., 17, 134 Jaerisoh, P., 18, 281, 288 Jeans, J. H., 268, 272, 278, 290 Jellett, J. H., 511 Jouravski, 22 Karman, Th. y., 119, 503, 606, 607 Kelvin, Lord, 12, 15, 40, 54, 55, 59, 77, 93, 97, 106, 107, 109, 116, 117, 170, 181, 247, 256, 271, 298, 303, 311, 606 Kelvin and Tait, 14, 20, 24, 29, 35, 57, 67, 69, 119, 130, 188, 259, 270, 271, 27S, 319, 387, 421, 422, 427, 444, 446, 465, 468, 496, 606, •607, 608 Kerr, J., 87 Kirehhoff, G., 13, 23, 27, 28, 30, 49, 50, 97, 164, 168, 306, 387, 393, 394, 399, 402, 405, 420, 448, 465, 497, 532, 606 Klein, F. and Sommerfeld, A., 419 Knott, C. G., 496 Kohlrausoh, E., 107 Konig, W., 311 Korn, A., 230, 250 Kriemier, C. J., 415 Kiibler, J., 415 Lagerhjelm, P., 97 Lagrange, J. L., 3 Lamarle, E., 415 Lamb, H., 18, 29, 55, 79, 171, 219, 241, 245, 270, 273, 281, 288, 290, 313, 314, 374, 455, 459, 469, 477, 496, 506, 516, 540, 546, 551, 552, 555, 606 Lam6, G., 13, 15, 17, 51, 55, 81, 87, 89, 119, 139, 142, 274, 606 Lam6 and Clapeyron, 11, 15 Laplace, P. S., 6 Larmor, J., 165, 167, 263, 272, 307, 321, 406, 417 Laura, E., 311 Lauricella, G., 119, 230, 241, 244, 246, 250, 308 Le Eoux, J., 57 Levy, H. , 380 L6vy, M., 22, 211, 380, 385, 430, 431, 496 Lewis, W. J., 155 Liebisch, Th., 147, 160, 608 Liouville, J., 305 Lipsohitz, E. , 97 Lorentz, L., 309 MacCullagh, J., 167, 304 Maodonald, H. M., 303, 325 MaUock, H. E. A., 102, 129, 142 Marcolongo, E., 241, 247 Mariotte, E., 2 Masoart, M. E., 87, 147 Mason, W., 118 Mathieu, E., 29, 497, 554 MaxweU, J. C, 17, 82, 86, 87, 106, 116, 144, 219, 230, 608 Mesnager, A., 209 Meyer, O. E., 116 Michell, A. G. M., 425,427 Michell, J. H., 17, 21, 24, 29, 86, 89, 133, 199, 201, 208, 209, 210, 213, 215, 217, 219, 241, 275, 278, 331, 359, 362, 366, 369, 450, 458, 459, 471, 472, 493, 494 Miers, H. A., 155 Miller, J. W., 422 Minchin, G. M., 67, 133 Mohr, 0., 22, 119, 120, 382 618 INDEX Morera, G., 86 Morrow, J., 102, 448 Miiller-Breslau, H. F. B., 23 Navier, 6, 22, 25, 375 Neumann, C, 170, 246 Neumann, P., 14, 87, 106, 153, 174 Newton, I., 6, 196 Orlando, L. , 241 Ostrogradsky, M., 18 Pearson, K., 22, 117, 211, 366, 369, 380, 606. See iiiin Todhunter and Pearson Perry, J., 353, 422, 444 Perry and Ayrton, 385 Peschka, G. A. V., 456 Phillips, E., 26 Poehhammer, L., 16, 18, 25, 275, 291, 293, 371, 435 Poincare, H., 230, 246, 415 Poisson, S. D., 6, 8, 10, 11, 12, 17, 18, 25, 27, 290, 298, 305, 465, 493, 497 Pollard, A. F. C, 211 Poncelet, J. V., 12, 119, 121 Poynting and Thomson, 142 Praudtl, L., 327, 415, 425, 427 Prescott, J., 425 Purser, F., 145, 280, 323, 333 Rankine, W. J. M., 22, 110, 304, 351, 353, 449, 606, 607 Rayleigh, Lord, 19, 26, 29, 96, 106, 171, 177, 272, 292, 293, 296, 304, 805, 309, 311, 434, 435, 436, 437, 444, 447, 448, 471, 493, 497, 503, 504, 506, 514, 515, 535, 544, 645, 549, 553, 554, 555, 595, 607 Eeissner, H., 581 Reeal, H., 456 Ribifere, C, 373 Richardson, L. F., 211 Ritter, A., 23 Ritz, W., 496, 497 Routh, E. J., 389 Saalfichiitz, L., 407, 417 Saint-Venant, B., 13, 14, 17, 19, 20, 21, 22, 23, 25, 26, 27, 29, 49, 57, 109, 119, 121, 127, 129, 159, 162, 316, 322, 324, 329, 334, 344, 345, H49, 390, 399, 401, 422, 437, 438, 444, 447, 450, 455. 456, 481, 493, 496, 606 Salmon, G., 43, 52, 501 Schneebeli, H., 197, 198 Schoenflies, A., 148, 155 Sohweydar, W., 272 Scoble, W. A., 119 Searle, G. F. C, 32, 94, 118 Sears, J. E., 115, 446 S^bert and Hugoniot, 26 Somigliana, C, 241, 243, 244, 246 Southwell, E. v., 415, 416, 566, 567 Stokes, G. G., 10, 11, 12, 18, 26, 38, 47, 96, 102, 306, 308, 310, 322, 372, 447 Tait, P. G., 197 Taylor, G. I., 327 Tedone, 0., 230, 241, 246, 276, 308, 311, 333, 375 Terazawa, K., 273 Thomson, J. J., 55 Thomson, Sir W. See Kelvin, Lord Timosohenko, S., 414, 415, 432 Timpe, A., 219, 333 Tissot, M. A., 63 Todhunter and Pearson, 7, 13, 27, 57, 97, 119, 121, 415, 447, 454 Tresca, H. , 114 Tutton, A. E. H., 496 UnwiD, W. G., 78, 110, 111, 115, 117 Verdet, E., 7 Vioat, L. J., 114 Voigt, W., 14, 21, 25, 44, 79, 97, 116, 119, 155, 157, 159, 161, 330, 354, 446, 606, 608, 611 Volterra, V., 119, 219, 225, 244, 308 Voss, A., 500 Walker, G. W., 314 Wangerin, A., 276 Warburg, E. G., 116 Webb, R. E., 69, 275, 380, 615 Weber, H., 241 Weber, W., 114 Wehage, H., 120 Weierstrass, K., 170 Weingarten, G. , 219 Weingarten, J., 87 Wertheim, G. , 13, 97, 107 Weyrauch, J. J., 23, 375 Whittaker, E. T., 419 Wilberforce, L. li., 422 Willers, F. A., 331 WUliams, G. T., 118 Willis, E., 26, 447 Wilson, Carus, 209, 372 Winkler, E., 454 Wohler, A., 117 Young, A. E., 428 Young, T., 4, 7 Zimmermann, H., 380 INDEX. MATTERS TEEATED. [The numbers refer to pages.] Additional deflexion, of beams, 353 JEolotropy , 108, 147 ; of inertia, 304 ; curvi- linear, 162 ; produced by permanent set, 116 JEolotropic solid. Constants and modulusea of, 103 — 105, 159 ; propagation of waves in, 18, 302—305 Afterstiain, 114 Angle, Transmissiou of stress from an, 209 Anorthic crystals, 156 Anticlastic curvature, 21, 129, 130, 345, 367 Applicable surfaces, 500 Applied Mechanics, Treatises on, 110; criticisms of some methods used in, 351 — 353 Arches, 454 Average strains. Determination of, 172 Axes, Principal, of strain, 37, 60; of symmetry, 148 ; Principal torsion-flexure, of a rod, 388 ; Moving, 611—615 Barriers : see Continuity, multiple Bars : see Beams, Rods, Torsion, &o. Barytes, Constants for, 162 Beams, ^olotropio, 160, 350 Beams, Bent, Curvature of, 343, 365, 368, 869; Deflexion of, 345, 858, 377—380; Twist pro- duced in, by transverse load, 845, 864 ; Stress produced in, by transverse load, 136, 334—336, 351, 352, 856—859 ; Shearing stress in, 137, 386—338, 347, 351, 352; Displace- ment in, 388—340, 843—347 ; Obliquity of cross-sections of, 138, 344, 345, 353 ; Distor- tion of cross-sections of, 346 ;, Strength of, 349 ; Extension produced in, by transverse load, 367, 870. See also Bending of beams. Bending moment, Bernoulli- Eulerian theory. Loading, Neutral plane, Ac. Beams, Continuous, 22, 375 ; Single span, 876 —880 ; Several spans, 380—386 Bells, Vibrations of, 5, 555 Bending of Beams, History of theory of, 2, 3, 20, 21; by couples, 127—129, 160; by terminal load, 186—138, 334—358 ; by uni- form load, 859- 370; by distributed load, 371 — 375 ; of particular forms of section (narrow rectangular, 186, 851, 369 ; circular. 340, 347, 348, 368 ; elliptic, 841, 347, 353 ; rectangular, 342, 345 ; other special forms, 343, 352) ; naturally curved, 456 Bending moment, 334, 365; Relation of, to cur- vature, 127, 343, 365, 371 Bernoulli- Eulerian theory, 19, 371, 374, 875 Beryl, Constants for, 161 Blade : see Stability Body forces, 73; Particular integrals for, 181, 227, 243, 257, 310 Boiler flues, Collapse of, 567 Boundary -conditions, 98, 132, 165 ; in torsion, problem, 319 ; in flexure problem, 835, 337, 348 ; in plates, 27, 464—469 ; in shells, 29, 539, 545 ; in gravitating sphere, 264 ; in vibrat- ing sphere, 284; in vibrating cylinder, 294, 296 Bourdon's ga,uge, 503 Brass, Constants for, 13, 103 Breaking stress, 112 Bridges, Travelling load on, 26, 447; Suspen- sion, 367 Buckling, Resistance to, 414. See also Stability Gannon : see Gun construction Cantilever, 21, 874 Capillarity, 6 Cartography, 63 Cast iron, 107, 112, 113 Cast metals, 95 Gauchy's relations, 14, 98, 161, 607 Central-line, of prismatic body or curved rod, 128 Chains, Links of, 454 Circuits, non-evanescible and reconcilable : see Continuity, multiple Circular arc : see Arches and Circular ring Circular cylinder. Equilibrium of, under any forces, 273, 275 ; bent by its own gravity, 368 (see also Beams) ; strained symmetrically, 278 ; Vibrations of, 18, 291—296 Circular disk. Rotating, 145 ; Equilibrium of, under forces in its plane, 215 — 217 Circular ring. Equilibrium of, 408, 430, 454 — 457 ; Stability of, 431 ; Vibrations of, 457—460 Clamped end, of a rod, 376 620 INDEX Collapse : see Stability Collision : see Impact Colour fringes : see Light, polarized Combined strain, 349 Compatibility, Conditions of, 49 Complex variable : see Conjugate funetions Compression, Modulus of, 12, 101, 104, 142 ; of a body under pressure, 102, 104, 140, 142, 162, 173, 174 ; of a sphere by its own gravitation, 140 ; of a body between parallel planes, 173, 279 ; Centre of, 185, 310 Cones, Equilibrium of, 201 Conformal transformation : see Conjtigate func- tions Conical shells, thin, Extensional solution for equilibrium (under symmetrical conditions, 582, under lateral forces, 586, under unsym- metrical conditions, 593); Edge-efiect (under symmetrical conditions, 583, under lateral forces, 587, under unsymmetrioal conditions, 597) ; of variable thickness, 585 ; Inexten- sional deformation of, 595 ; Approximately inextensional solution, 595 Conjugate functions. 46, 213, 217, 275, 318, 324, 341 Conjugate property, of normal functions, 178 ; of harmonic functions, 228, 251 Constants, Elastic, Controversy concerning, 13 —15 ; Definition of, 97 ; Magnitude of, 103 ; Thermal variations of, 107 ; of isotropic bodies, 100 ; of crystals, 161 ; Experimental deter- mination of, 22, 102, 142, 161, 496 Continuing equation, 438 Continuity, multiple, 218, 221 . Coordinates, Curvilinear orthogonal, 51 ; strain In terms of, 53, 615 ; Stress-equations in terms of, 165, 615 ; General equations in terms of, 138, 165. See also Cylindrical coordinates, Polar coordinates, Conjugate functions Copper, Constants for, 103 Crushing, of metals, 114 ; of cylindrical test pieces. 279 Crystalline medium: see JSoiotropie solid and Crystals Crystals, Symmetry of, 153 ; Classification of, 155; Elasticity of, 14, 157; Elastic constants of, 161 ; Neumann's law of physical behaviour of, 14, 153 Cubic capacity, of a vessel, 122 Cubic crystals, 155, 160 Curl, of a vector, 46 Current-;, electric, 223 Curvature : see Beams, Rods, Plates, Shells, Surfaces Curve of sine.i, a limiting case of the elastica, 3, 411, 412 Cylindrical bodij nf any form of section. Equi- librium of, under tension, 101 ; under gravity, 125, 173 ; under fluid pressure, 126 ; in a state of plane strain, 275; in a state of stress uni- form along its length, 354 ; in a state of stress varying uniformly along its length, 356 ; in a state of stress uniform, or varying uniformly, over its cross-sections, 481. See also Beams, Plane strain. Plane stress. Bods, Plates Cylindrical coordinates, 56, 89, 141, 162, 275, 276, 291 Cylindrical flaw, in twisted shaft, 121, 321 Cylindrical shells, thick. Equilibrium of, under pressure, 141 Cylindrical shells, thin, Inextensional deforma- tion of, 506 ; Vibrations of (inextensional, 514, 549 ; general theory, 546 ; extensional, 548) ; Extensional solution for equilibrium (under symmetrical conditions, 560, under lateral forces, 567, under unsymmetrioal con- ditions, 572) ; Edge-effect (under symmetrical conditions, 561, under lateral forces, 568, under unsymmetrioal conditions, 574) ; under pressure, 562 ; Stability of, under pressure, 563 — 567 ; Bending of, 571 ; Approximately inextensional solution, 572 Dams, Stresses in, 211 Deflexion : see Beams and Plates Density, Table of, 103 Diagrams, of plane stress, 86 ; Stress-strain, 111 ; unclosed, 118 Dilation, cubical, 41, 59; uniform, 44; Average value of, 173 ; in curvilinear coordinates, 54 ; Waves of, 18, 297, 302, 304 ; Centre of, 185 ; Determination of, 281 Discontinuity, connected with dislocation, 221 ; Motion of a surface of, 299—301 Disk : see Circular disk Dislocation, theory of, 219 — 223 ; of hollow cylinder with parallel fissure, 223 ; of hollow cylinder with radial fissure, 225 Displacement, 35; effectively determined by strain, 50, 220; many-valued, 219 ; one- valued in torsion and flexure of hollow shaft. 325, 338 Distorsion, Waves of, 18, 297, 302, 304; of cross- sections of twisted prism, 325; of cross-sections of bent beam, 346 Divergence, of a vector, 46 Double force, 185, 244, 245, 310 Dynamical resistance, 26, 121 , 437 — 444, 447 Earth, In a state of initial stress, 107, 140 ; Strained by its own gravitation, 140, 263; Stress produced in, by weight of continents, 268; EUipticity of figure of, 268; Tidal effective rigidity of, 270—272 ; Period of spheroidal vibrations of, 289, 290 INDEX 621 Earthquakes, 313 Ease, State of, IH Edge-effect, in thin shells, 551. See also Cy- lindrical shells, thin, Conical shelh, thin, Spherical shells, thin Elastica, 3, 24, 407—411 ; Stability of, 417, 427 Elastic after-working, 114 Elasticity, defined, 90 ; Limits of, 113 Ellipsoid, Solutions of the equations of equi- librium in, 250, 251, 275, 276 Elliptic cylinders, Solutions of the equations of equilibrium in, 275 ; under torsion, 322 ; under flexure, 341 ; coufooal, 325, 342 Energy, intrinsic, 91. See also Potential energy &ndi Strain- energy -func tion Equilibrium, General equations of, 6 — 12, 83, 98, 131, 132, 138, 167, 227; of bifurcation, 415 Equipollent loads. Elastic equivalence of, 130 Existence theorems, 170, 230, 493 Experimental results. Indirectness of, 94 Extension, 32, 40, 44, 59 ; of a beam bent by clistributed load, 367 ; of a plate bent by pressure, 485 — 491 Extensional vibrations, of rods, 434, 437 — 446 ; of a circular ring, 457 ; of plates, 497 — 499, 553; of shells, 545; of cylindrical shells, 548 ; of spherical shells, 554 Extensions, principal, 42, 60 Extensonieter, 94, 111 Factors of safety, 120, 121 Failure : see Rupture Fatigue, 117 Fissure : see Dislocation Flaws, 121, 257, 321 Flexure : see Beams Flexure functimis, 337, 348, 358 Flexure problem, 337 Flexural rigidity, oihe&m, 316; of rod, 394; of plate, 470 Flow : see Plasticity Flue : see Boiler jiues Fluid, 115 Fluor spar, Constants for, 161 Flux of energy, in vibratory motion, 175 Fourier's series, 323 Fracture : see Rupture Frameworks, 23 Frequency equation, 177 FresneVs wave-surface, 304 Funicular polygon, 382 Galileo's problem, 2 Geophysics, Applications of theory of Elasticity to, 272 Girders : see Beams Glass, Constants for, 13, 103 Graphic representation, of stress, 86 ; of the theory of torsion, 325 ; of the theory of tlexure, 346 — 348 ; of the solution of the problem of continuous beams, 382 — 386 Gravitation : see Compression and Earth Gravity, Effect of, on vibrations of sphere, 290 ; on surface waves, 313 Green's functions, 229 Green's transformation, 83 Groups, of transformations, 69, 149, 155 Gimconstriiction, 143 Hamiltonian principle, 164 Hardening by overstrain, 113 Hardness, 16 Harmonic function, 228 Harmonics, spherical, 228, 247, 278, 282 ; gene- ralized, 580 Heat : see Thermal effects Height, consistent with stability, 431 Helix : see Rods, thin and Springs Hereditary phenomena, 119 Hertz's oscillator. Type of waves dueto, 311 Hexagonal crystals, 156 Hooke's law, 2, 9, 12 ; generalized, 95 ; excep- tions to, 95, 110 Hydrodynamic analogies, to torsion problem 319 Hysteresis, 116 ; in special sense, 117 Identical relations, between strain -components : see Compatibility, conditions of Impact, 16, 196 — 198 ; of spheres, 198 ; longi- tudinal, of bars, 25, 444 — 446 ; of metal rods with rounded ends, 446 ; Initiation of vibra- tions by, 447 ; transverse, of bars, 447 Incompressible soJid, Equilibrium of, 263 ; Vibra- tions of, 287, 289; Waves on surface of, 813 ' Inextensional displacement, in thin rod, 452 ; in thin shell, 506, 524 Initial stress, 107, 263, 268, 611 Integration, Methods of, for equilibrium, 15, 228, 230, 245, 250; for vibratory motion, 177, 305 Integra-differential equations, 119 Intermolecular action, 6, 7, 9, 10, 608, 609 Invariants, of strain, 43, 60, 100; of stress, 81 Inversion, of plane strain, 213 ; applied to plate, 494 Iron, Constants for, 103 ; Elastic limits for, 114 ; Yield-points for, 114. See also Cast iron Isostatic surfaces, 87, 89 Isotropic solids, 100 Isotropy, transverse, 158 Kinematics, of thin rods, 387—392, 450—454 ; of thin shells, 520—527 622 INDEX Kinftic analogue, for naturally straight rod, 404; for naturally curved rod, 406 ; for elastica, 407 ; for rod bent and twisted by terminal forces and couples, 419 ; for rod subjected to terminal couples, 423 Kinetic moduluses, distinguished from static, 96, 97 Lamina, 4 Lead, Constants for, 103 Length, Standards of, affected by variations of atmospheric pressure, &c., 122 Liqht, polarized. Examination of stress-systems by means of, 87, 372,874' Limitations of the mathematical theory, 110 Limits, elastic : see Elasticity, Limits of Lines of shearing stress, in torsion, 320, 326 ; in flexure, 347 Lines of stress, 87 ; for two bodies in contact, 196 ; for force at a point, 200 Load, 95 ; Sudden application or reversal of, 121, 179, 443; travelling, 26, 447 Loading,* eSect of repeated, 117; asymmetric, of beams, 348 ; surface, of beams, 372 Longitudinal vibrations, of rods : see Exten- sional vibrations, of rods Longitudinal waves, 8, 11 Magnetometer, deflexion-bars, 122 Matter, Constitution of, 6 Maxwell's stress-system, 82, 85, 133, 608 Membrane, Analogy of, to twisted shaft, 327 Middle third, Eule of, 84, 210 Modes of equilibrium, permanent and transitory, 375 Modulus, 104, 608. Seealso Compression, Modulus of. Rigidity, Young's modulus Molecular hypothesis, 6 — 10 Momentary stress, 115 Mommts, Theorem of three, 23, 380—382 Monoclinic crystals, 156 Multiconstant theory, 13 Xeutral axis: see Galileo's problem Neutral plane, 344 Neutral surface, 367, 370 Nodal surfaces, of vibrating sphere, 288 Normal forcex. Bod bent by, 429 Normal functions, of vibrating systems, 177 Normali ntegraU, odine&r differential ec[uations, 592 Notation, 606 Nuclei of strain, 184, 196, 206, 310; Solution of the problem of the plane in terms of, 238, 240, 241 Optics, Influence of theories of, in stimulating research in Elasticity, 7, 8, 11, 30 Orthogonal surfaces, 52 Perturbations, local, 187 Pendulum, Analogy to elastica, 407 Photo-elasticity : see Light, polarized Piezo-electricity , 147 Piezometer, 94, 142, 163 Plane, Problem of the, 15, 189—191, 201, 235 —243 Planes, principal, of stress, 79 Plane strain, 45, 135 ; Displacement accom- panying, 202 ; Transformation of, 213 Plane stress, 81, 135 ; Displacement accom- panying, 204 ; in plate, 473 PZam«sti-«ss,Gc«craZt2ed, 136,206; in bent beam, 136, 369 ; in plate, 477 Plasticity, 114 Plate, elliptic. Bending of, by pressure, 490, 491 ; vibrations of, 497 Plate, thick. Stretching of, 473— 476; Bending of, 130, 476—492 Plate, thick circular, Bending of, by central load, 481; byuniform pressure, 487, 490; by variable pressure, 491 Plate, thin, bounded by straight edge, 495 Plate, thin circular, 493 — 495 Plate, thin rectangular, 495 — 496, 497 Plate, thin. Subjected to forces in its plane, 206 —218; Boundary-conditions for, 27,461—469 ; Bending of, 492; Vibrations of, 496—499, 542, 553 ; General theory of, 532 ; Stability of, 540 Plates, History of theory of, 5—6, 27—29 Poisson's ratio, 13, 101, 105 Polar coordinates, 56, 89, 139, 162, 199, 274, 290, 509 Potassium chloride. Constants for, 161 Potential, Newtonian, Theory of, 170, 228 Potential energy, of strained body. Minimum, 169 ; Theorem concerning, 171 Potentials, direct, inverse, logarithmic, 190, 191 Pressure, hydrostatic, 79 ; does not produce fracture, 120 ; between bodies in contact, 191 — 196. See also Compression Prism, Torsion of : see Torsion Prismatic crystals, 156 Punching, of metals, 114 Pyrites, Constants for, 161 Quadric surfaces, representing distribution of strain, 37, 41, 60, 62, 65 ; representing dis- tribution of stress, 79, 82 Quartz, Constants for, 161 Oblique crystals, 156 Radial displacement, 139, 141, 162 INDEX 623 Radial vibrations, of sphere, 289 ; of spherical shell, 290 ; of cylindrical shell, 549 Mari-constant theory, 13 Bayleigh-waves, 313 Rays, equivalent, 147, 154 Reality, of the roots of the frequency equation, 178 Reciprocal strain-ellipsoid, 37, 60 Reciprocal tfteoraii, Betti's, 16,171,231, 233, 243 Refraction, double, due to stress, 87; to unequal heating, 106 Resilience, 171 Restitution, Coefficient of, 25, 446 Rhombic crystals, 156 Rhombohedral crystals, 156 Rigid-body displacement, snperposable upon dis- placement determined by strain, 50 ; or by stress, 167 Rigidity, 101, 105. See also Flexural rigidity. Torsional rigidit'^, Tidal effective rigidity Ring : see Circular ring Rock-salt, Constants for, 161 Rods, naturally curved. Approximate theory of, 402, 450 ; Problems concerning, 406, 427, 430, 454—457 ; Vibrations of, 457 — 460 Rods, thin. Kinematics of, 387 ; Equations of equilibrium of, 392 ; Strain in, 395 ; Approxi- mate theory of, 395 — 401 ; slightly deformed, 401 ; Problems of equilibrium concerning, 405—432 ; Vibrations of ,433— 437 ; of variable section, 431, 448 ; Problems of dynamical re- sistenoe concerning, 437 — ^448 Rotation, of a figure, 67 ; Strain produced in a cylinder by, 144; Centre of, 185, 310; Strain produced in a sphere by, 262, 268 Rotation, Components of, 39, 55, 71 ; Determina- tion of, 234, 242 Rotationally elastic medium, 167 Rupture, Hypotheses concerning conditions of, 119 Safety : see Factors of safety and Rupture Saint-Venant's principle, 129 Scope of the mathematical theory, 121 Screw-propeller shafts, 120 Secondary elements of strain, 57 Semi-inverse method, 19 Set, 111 Shafts: see Rotation, Torsion, Whirling Shear, Pure, 33; Simple, 33, 68, 69; Used by Kelvin and Tait to denote a strain, 607; by Eankine to denote a stress, 607 Shearing strain, 45 Shearing stress, 80; Cone of, 82. See also Beams and Lines of shearing stress Shells, thin, Inextensional displacement of, 500—516; General theory of, 29, 517—546; Vibrations of, 514—516, 541—555; Equi- librium of, 536—540, 556—559. See also Cylindrical shells, thin, Conical shells, thin. Spherical shells, thin. Torsion Simple solutions, 183, 188 Sound waves, 96 Sphere, Problem of Ithe, 15 ; Equilibrium of, 140, 247 — 272 ; with given surface displace- ments, 249; with given surface tractions, 251; under body forces, 257, 259; Vibrations of, 281—290 Spherical cavity, in infinite solid, 256 Spherical shell. Equilibrium of, under pressure, 140, 162; under any surface tractions, 256, 274 ; under body forces, 259 ; Vibrations of, 290 Spherical shells, thin, Inextensional defol'ma- tion of, 508; Inextensional vibrations of, 215 ; Extensional and other vibrations of, 652; Extensional solution for equilibrium, 575; Approximately inextensional solution, 578 ; Edge-effect, 578, 582 Spheroidal vibrations, 289, 290 Springs, spiral, 23, 421 Stability, General criteria, 30, 97, 412 ; Strength dependent on, 122 ; of strut, 3, 411 — 415 ; Southwell's method for, 416, 567; of elastica, 417, 427; of rod subjected to twisting couple and thrust, 423 ; of flat blade bent in its plane, . 425 ; of ring under pressure, 24, 430 ; Height- consistent with, 431 ; of rotating shaft, 448; of plate under thrust, 540; of tube under pressure, 563 — 567 Statical method, of determining frequencies of vibration, 448 Steel, Constants for, 103; Elastic limits and yield-points for, 114 Strain, Cauchy's theory of, 8; Examples of, 32, 33; Homogeneous, 36, 64 — 71; determined by displacement, 39; Components of, 40, 69 ; Transformation of, 42; Invariants of, 43, 44, 60 ; Types of, 44, 45 ; Eesolution of, into irrotational dilatation and equivoluminal dis- tortion, 47 ; Identical relations between com- ponents of, 49 ; Displacement effectively determined by, 50, 220; referred to curvi- linear' coordinates, 54, 615; General theory of, 57 — 64 ; Composition of finite homoge- neous, 69 ; appropriated by Eankine to denote relative displacement, 606 Strain-ellipsoid, 37, 61 Strain-energy -function. Introduction of, 11 ; ■ Existence of, 12, 92; Form of, 96; in iso- tropic solids, 99, 153 ; in SBolotropic sohds, 98 ; in crystals, 157 ; in bodies exhibiting various types of seolotropy, 158; Generali- zation of, 97 624 INDEX Strength, Ultimate, 112 Stress, Cauchy's theory of, 8 ; Notion of, 72, 608; Specification of, 75; Measure of, 77; Transformation of, 78; Types of, 79; Eeso- lution of, into mean tension and shearing stress, 81 ; Uniform and uniformly varying, 84, 99, 101, 124 ; Graphic representation of, 86 ; Lines of, 87 ; Methods of determining, 98; Direct determination of, 17, 133, 134, 471 ; appropriated by Bankine to denote in- ternal action, 606 Stress-difference : see Rupture Stress-equations, 83 ; referred to curvilinear coordinates, 87, 166, 615 Stress-functions, 17, 85, 134, 202—205, 276—279 Stress-resultants and stress-couples, id rod, 392; in plate, 28, 461 ; in shell, 531 Stress-strain relations, 94, 95, 97, 100, 101 Strut : see Stability Supported end, of a rod, 376 ; edge, of a plate, 468 Surface of revolution, Equilibrium of solid bounded by, 276 ; as middle surface of shell, 557 Surface tractions, 73 Surfaces, Curvature of, 517 Suspension bridges, 367 Symbolic notations, 305, 607 Symmetry, geometrical, 148; alternating, 148; elastic, 149; of crystal, 153; Types of, 158 Tangential traction, 77, 82 Tenacity, 112 Tension, 73; Mean, 81 Terminology, 606 Testing machines, 111 Tetragonal crystals, 156 Thermal effects, 93, 106 Thermodynamics, Application of, 91, 93 Thermo-elaitic equations, 106 Tidal deformation, of solid sphere, 270 Tidal effective rigidity, of the Earth, 271 Tides, fortnightly , Dynamical theory of, 272 Time-effects, 114 Topaz, Constants for, 162 Tore, incomplete. Torsion and flexure of, 450 Torsion, History of theory of,. 4, 19 — 21 ; of a bar of circular section, 126; of prisms of isotropic material, 315 — 329; of prisms of aaolotropic material, 329 ; of prisms of special forms of section, 322—325, 830 ; Stress and strain that accompany, 44, 315, 320 ; Strength to resist, 321 ; of bar of varying circular sec- tion, 330 — 332 ; Distribution of terminal force producing, 332; of a thin shell, 559 Torsion function, 317 Torsion problem, 317 Torsional couple, in rod, 392; in plate, 461; in shell, 531 Torsional rigidity, 317 ; Calculation of, 327 Torsional vibrations, of cylinder, 292 ; of rod, 435 ; of circular ring, 459 Tortuosity, Measure of, in central-line of rod, 387, 403, 420, 451, 454 Traction, 72 ; used by Pearson in sense of ten- sion, 607 Transmission of force, 181, 209. See also Plane, Problem of the Trees, 432 Triclinic o-ystals, 156 Tubes : see Cylindrical sheTU Twinning, of crystals, 161 Twist, of a rod, 387, 388 Typical flexural strain, 503 Uniquene-is of solution, of the equations of equilibrium, 168; of the equations of vibra- tory motion, 174; Exceptions to, 30, 415 Variational equation, 164 ; Difficulty of forming, for thin shells, 534 Variations, Calculus of, 170 Vibrations, General theory of, 176 — 179 Viscosity, 115 Watch-spring, 117, 496 Waves, Propagation of, in isotropic media, 11, 18,297—302,308—311; in seolotropic media, 18, 302—305; in infinite cylinder, 291—296; due to variable forces, 308 — 311; due to forces of damped harmonic type, 311 ; over surface of solid, 311 — 314 ; in a plate, 314 Wave-surfaces, 303 Wedge, Pressure on faces of, 210 Whirling, of shafts, 448 Winding ropes, in mines, 444 Work, done by external forces, 91 Yield-point, 112 Young's modulus, 4; in isotropic solids, 101; in seolotropic solids, 105, 159; Quartic surface for, 105, 160 CAMBRIDGE : PRINTED BY J. 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