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THE MATHEMATICAL THEOEY OF PERFECTLY ELASTIC SOLIDS AN ELEMENTARY TREATISE ON THE MATHEMATICAL THEOBY OF PERFECTLY ELASTIC SOLIDS WITH A SHORT ACCOUNT OF VISCOUS FLUIDS BY WILLIAM JOHN IBBETSON, M.A., FELLOW OF THE KOYAL ASTRONOMICAL SOCIETY" AND OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, MEMBER OF THE LONDON MATHEMATICAL SOCIETY ; LATE SENIOR SCHOLAR OF CLARE COLLEGE, CAMBRIDGE, IKonban : MACMILLAN AND CO. AND NEW YORK. 1887. [All rights reserved.] Eh PRINTED AT THE UNIVERSITY PRESS, BY ROBERT MACLEHOSE, 153 WEST NILE STREET, GLASGOW. PREFACE. In the present Volume I have attempted to present to the English student a continuous and fairly complete analysis of the Mathematical Theory of Elasticity, as it stands at present, together with a brief account of the physical basis on which the theory rests, and of the considerations which limit its practical application to natural materials. It would, of course, have been impossible to exhaust so wide a subject within the limits of an elementary text-book, and my endeavour has rather been, after giving a very full and clear account of the properties of Strain and Stress, considered separ- ately and in their relations to one another, to indicate to the student as many as possible of the various modes of further advance, in order that he may be able to read without difficulty any of the more specialised memoirs, both theoretical and practical, that! constitute the already enormous literature of Elasticity. The labour involved in the collection and arrangement of the materials for such a work can only be appreciated by those who have fully studied the subject for themselves, and it would have been largely increased had I undertaken to acknowledge in foot- notes the sources from which each theorem or formula was derived. My original intention was to complete the Volume by a Bibliographical and Historical Chapter, but during the twenty- one months that this book has been in the press the announce- ment of the late Dr. Todhunter's great work on the history of the VI PKEFACK. subject, and ultimately the appearance of its first volume under the editorship of Prof. K. Pearson, have led me to abandon that design, though very unwillingly. Such references as have been inserted are intended chiefly as guides to further reading. A portion of the projected scheme has however been retained as Appendix III. (pages 162-168), on the history of Hooke's Law, and this perhaps suffers from its isolation. It must be under- stood that all the statements and remarks contained in it refer exclusively to its subject, and not at all to the general question of Green's Theory and the minimum number of Elastic Coefficients, on which I hold the orthodox opinion, though I cannot regard the matter as finally closed to discussion. I have adopted the notation of Thomson & Tait's " Natural Philosophy" for Strain and Stress, in spite of its obvious theo- retical deficiencies, partly because it is the one most familiar to English readers, and partly because it is so eminently readable and speakable. I am inclined personally to prefer the double- suffix notation on all other accounts, and I would suggest the following system as the most generally useful (the symbols in parentheses being those employed in the present work, and the suffixes referring to the generalised coordinate notation of Chapter V.) :— Strains, e^(e), e^f), e^y), s^a), s^{b), s^c), efa), e.,( 24, „ i ,, -pi „ 273, , , 20, „ I " L ' ., 283, , , 34, „ ■sjyfVdx&ydz „ ^fffVOxdydz. „ 283, , , 36, ,, S(I 7 ) „ Z(K«) „ 302, , , 24. „ = $ ,, = 4» „ 310, , , 21, „ - „ -0 „ 331, ei id of last line, for r _ "- ,, i— '-« ,, 350, li ie 27, for 20 „ 21 ., 372, F by a mistake, igure 43. The upper dotted curve was inserted ind has no connection with the problem of § 320. CHAPTER I. PROPERTIES OF ELASTIC SOLIDS. Solid Matter as it really is. 1.] Molecular Structure of Matter. To superficial observation matter presents innumerable gradations of " coarse- ness " or " fineness " of structure, from the obvious " granularity " of sandstone and other rocks to the apparently perfect "con- tinuity " of crystals, jellies and liquids. A little reflection, however, will show that these terms have a purely relative application, depending upon the magnitude of the smallest constituent portions of matter which are perceptible to our senses as individually distinct. The paper upon which these words are printed appears to the eye perfectly smooth and uniform ; but under a microscope of even moderate power it is seen to be really a more or less compact mass of tangled linen threads. If a mass of sandstone as large as the earth could be reduced to the size of a cricket-ball, each of its parts shrinking in the same proportion, it would possess no more real uniformity than before, yet none of the optical means at our disposal would enable us to detect any granularity of structure. Similarly, if a pint of water, or a globe of glass of equal volume, could be magnified in the same way to the size of the earth, it is quite possible that it might appear to us vastly more coarse-grained in structure than a mass of sandstone or gravel as we see it in Nature. That is to say, the smallest constituents which we could then distinguish from one another might be larger than the crystals in the sandstone, or even than the pebbles in the gravel (see § 36, below). 2.] We have, in fact, strong reason to believe that all kinds of matter, however apparently continuous, are ultimately granular in structure, being composed of very minute (but not infinitely small) material particles or molecules, which perform incessant motions so long as the matter contains any heat ; their capacity for relative motion, as well as their size, mass and closeness of aggregation, varying considerably in different kinds of matter. A ■2 PROPERTIES OF ELASTIC SOLIDS. [3. 3.] Intermolecular Forces. These molecules exert upon one another mutual forces, to which the cohesiveness of matter is due. Of their nature little or nothing is known with certainty, except that their intensity in the natural arrangement of the molecules varies within very wide limits for different kinds of matter, while, if the molecules be artificially separated by appreciable distances, it is impossible to detect their existence by the most delicate instruments.* It appears, therefore, that we are justified in assuming their sphere of action to be exceed- ingly limited. 4.] Impressed Forces. The molecules are also liable to be influenced by external "impressed" or "applied" forces, such as Gravitation and other natural forces of attraction and repulsion. 5.] Natural State. When matter is entirely free from the action of such external forces, it is said to be in its " natural state." This term does not imply that matter is ever found, or can even be conceived to be in this state under natural con- ditions ; but that in this state, and in this only, it may be supposed isolated from all co-existing matter, so that all the phenomena it presents depend only on its individual nature. 6.] Solid Matter. In the kind of matter called solid each molecule performs small vibrations about a Tnean position, which, so long as the body is in its natural state and maintained at constant temperature, may be regarded as fixed. Under the same conditions the vibrations of each molecule may be assumed strictly periodic, and the mean value of the amplitudes of the vibrations of any considerable number of molecules may be supposed constant. 7.] Homogeneity and uniform density. If, when the matter is in its natural state, and at any uniform temperature, the mean positions of the molecules are uniformly distributed, and if their masses and the periods and mean amplitudes of their vibrations are the same throughout, the matter is said to be " naturally homogeneous." It follows that a closed surface of given volume, but of any form, whose least dimension is very large in comparison with the greatest mean distance of two adjacent molecules, will, if drawn anywhere within the substance of homogeneous matter, always include the same number of molecules, — and therefore the same total mass. The mass thus enclosed by a surface of unit volume is called the Density of the matter, in any given system of units. * See, however, Note .it. end of volume. 8] PROPERTIES OF ELASTIC SOLIDS. 3 8.] A Homogeneous Solid Body is a continuous portion of homogeneous solid matter, bounded by a surface consisting of one or more completely closed sheets, each of which has at every instant a definite form and volume. The form of the bounding surface, and the volume enclosed by it (or between its sheets) in the natural state of the body, at any given uniform tem- perature, are called the natural form and volume of the body at the given temperature. The relative arrangement of the mean positions of the molecules under the same conditions will be called the natural configuration of the molecules at the given temperature. 9.] We have seen in § 3 that the only property which we can ascribe with certainty to the intermolecular forces is that they depend in some way on the molecular configuration; the law of dependence varying for different kinds of matter. Since, however, when the body is free from the action of external forces, we can hardly conceive of their being affected by any other consideration, we shall assume that, in the natural state, they depend solely on the configuration of the molecules, and on the temperature. It is obvious that the form and volume of the bounding surface, which is merely the envelope of the external layer of molecules, must in all states of the body depend solely and entirely on the molecular configuration. 10.] Definition of Strain. Any departure from the natural configuration is called a Strain. Thus a Strain may be defined as any change in the relative arrangement of the mean positions of the molecules from that which is natural to the body at the given temperature. 11.] Elastic Properties. It is found by experiment that all solid bodies possess, in a greater or less degree, the properties included under the title of Elasticity. These may be summed up, in general terms, as follows : — (i.) The natural form and volume of the body (and therefore also the natural configuration of the molecules) are always the same when the body is at the same uniform temperature, through whatever cycles of gradual changes of temperature {within certain limits) the body may be brought, so long as it is not subjected to external force. (ii.) Hence it has a perfectly definite and characteristic natural or "unstrained" configuration at each temperature ivithin these limits, which cannot be altered, while the tem- perature remains the same, except by the application of external force. 4 PROPERTIES OF ELASTIC SOLIDS. [11. Or, in other words, it always requires the application, of external force to produce strain. (Hi.) Given the type of the external forces applied, the greater they are the greater will he the strain produced ; and, conversely, the greater the strain to be produced, the greater the external forces which must be applied. (iv.) If the applied forces and the consequent strain be con- fined within certain limits, the body offers continuous resistance to the strain, so that it requires the continued exertion of external force to maintain the body in a given state of strain; and when this force is removed the body tends to return to its natural state at its ultimate temperature. 12.] Limits of Elasticity. All these elastic properties are exhibited in very different degrees, and subject to many limita- tions, by different classes of natural solids. Short of the strain required to produce absolute rupture (called the proof-strain of the material) there is always a limit to the elasticity of every natural substance. So long as the applied forces are such as to produce a strain well within this limit the resistance increases steadily with the strain, while it always requires sensibly the same force to maintain the same strain at the same temperature ; and on the removal of this force the body returns to a state sensibly identical with its natural state. When, however, the strain exceeds the elastic limits of the material the properties of the body undergo a marked change, and it passes into what is known as the ductile state. In this condition the resistance still increases with the strain, but much less rapidly than before the limit was passed, and the tendency to return towards the natural state is much diminished, so that, when the external force is removed, the body is found to have acquired a " set " or permanent strain, 13.] Ductility and Brittleness. Those materials whose elastic limit is separated by a considerable interval from the point of rupture, and whose state of ductility therefore has a distinct range, are called ductile, malleable, or plastic. To this class belong most of the natural metals, as well as steel gradually cooled. Thus under the enormous pressures applied in the Mint, the density of gold is permanently altered from 191558 to 19"367, and that of copper from 8o3o to 8916. At the bottom of this class are various soft solids (of which putty or tallow may be taken as a familiar example) whose elasticity is almost imperceptible, and which are for all practical purposes wholly ductile. 13.] PKOPERTIES OF ELASTIC SOLIDS. O On the other hand, crystalline bodies, glass (when cold), jellies, and steel suddenly chilled from a red heat have extremely little ductility, so that, practically, breakage is the first intimation we receive of having reached the elastic limits. Such materials are called brittle. The two classes, however, are not separated by any hard and fast line, the various gradations of tempered steel, for example, forming a series of connecting links. 14.] Elasticities of Shape and Bulk. The elastic resist- ances of a solid may be roughly divided into resistances to Distortion, Expansion, and Compression respectively. The limits of these are often very different in the same solid, the first having generally a very small range. 15.] Tempering. The reason for the limitation imposed in § 11 (i.) on the changes of temperature to which the body may be supposed subjected, is that, by sudden and violent changes of temperature, many substances, and notably metals and glass, may be entirely altered in all their elastic properties. The brittleness of glass (Prince Rupert's drops) and of steel (glass-hard steel) when heated to redness and suddenly chilled in water is proverbial. But glass may be " toughened " by gradual cooling in hot oil, and steel by gradual and cautious reheating may acquire a vast number of degrees of " temper " intermediate between brittleness and ductility. All these different qualities of steel must be regarded as distinct materials, none of whose elastic properties are absolutely identical. The change produced in a metal by tempering is obviously analogous to that produced by straining it beyond its elastic limits ; and some very striking results have been obtained in the way of tempering wires by giving them a permanent strain. Mr. J. T. Bottomley has shown that the tensile strength of soft- iron wire may be increased more than 25 per cent, by prolonged tension ; while Messrs. A. & T. Gray find that the power of copper wire to resist twisting about its axis may be reduced to J of its natural value by giving it a permanent twist. 16.] Viscosity and Fatigue. Besides the above well- known restrictions, two remarkable irregularities have been dis- covered by Sir William Thomson in the elasticity of metals, strained within their elastic limits, which are probably common to all natural solids. In the first place the resistance to strain is found to vary with the rate at which the strain is imposed. This proves the existence of a property of solid matter analogous to the "viscosity" of fluids, in virtue of which the latter oppose to change of shape a resistance proportional to the 6 PROrERTIES OF ELASTIC SOLIDS. [16. rapidity of the change. The law by which the increase of resist- ance in the case of solids depends on the increase of the rate of straining is certainly not so simple, but the analogy justifies the application of the term solid viscosity to this property. Secondly, it was found that wires which had been frequently and recently strained, well within their elastic limits, exhibited less marked tendency to elastic recovery, and much greater viscosity than when they had been left at rest in the natural state for some days before the experiment. This result shows that the elastic properties of a natural solid may suffer diminution or Fatigue by frequent exercise, and that these properties may be more or less fully restored by repose. 17.] All these limitations and imperfections in the Elasticity of natural solids present insurmountable difficulties in the way of an analytical theory ; and for the purposes of a first approxima- tion they must be eliminated. If we class the more or less "imperfectly elastic" substances, which we find in nature, according to the range of their elasticity and the degree of perfection in which they exhibit its character- istic properties within these limits, they are seen to form an ascending scale suggesting an ideal summit which is never actually reached in nature, but only more or less closely approximated to under favourable circumstances. This ideal, which we shall adopt as the subject of our investi- gations, we define as a Perfectly Elastic Solid. Real Matter with ideally perfect Elasticity. 18.] A Perfectly Elastic Solid is characterized by the following properties up to the point of breakage : — (i.) In its natural state at any temperature the molecular configuration, together with the form and volume of the bounding surface, are perfectly definite, and characteristic of that tempera- ture. (ii.) If the temperature (supposed always uniform through- out the body) be changed, the solid passes continuously to the natural state for the new temperature, through all the inter- mediate states natural to the intermediate temperatures. (Hi.) It requires the application of external force to produce a strain at any temperature ; and it requires the continued application of the same force (or system of forces) to maintain the strain. (iv.) It always requires the same force (or system of forces) to maintain the same strain at the same temperature, through whatever intermediate states of temperature and strain it may 18. J PROPERTIES OF ELASTIC SOLIDS. 7 have been brought to the given state, and at whatever rate these intermediate changes may have been passed through. (v.) When all external forces are removed it returns to its natural configuration for the temperature at which it is left. 19.] Approximation of natural solids to Perfect Elas- ticity. Under very small strains which do not approach the Mastic Limits of the material ; produced so gradually and maintained for so short a time as never to call Viscosity into play, or to produce Elastw Fatigue ; and subject to changes of temperature too limited and gradual to impart a Temper: the metals, crystals, glass, jellies, indiarubber, etc., may all be said to have approximately perfect elasticity, as defined in the last article. 20.] Intrinsic Energy in the Natural State. Assuming, as we shall do throughout, that the body is free from all influences due to electrification, magnetisation, etc., it is obvious that the energy possessed by it in its natural state, at any given temperature, must consist of two parts : — (i.) The intrinsic potential energy, due to the configuration of the mean positions of the molecules under the intermolecular 'forces; and (ii.) The kinetic energy due to the vibrations of the mole- cules about these mean positions. The intermolecular forces being supposed governed by fixed laws which, in a given body in its natural state, depends solely (§ 9) on the configuration and temperature, it follows that the potential energy in the natural state also depends only on the configuration and temperature. But by § 18 (■£.), the natural configuration depends only on the temperature, lience we may state, in mathematical language, that, in the natural state, the potential energy is a function of the temperature only. Again, according to the most modern theory of gases, the kinetic energy due to the motion of their molecules is simply proportional to the absolute temperature. The relation is probably not so simple in the case of elastic solids, but we are justified in assuming, as in the theory of Thermodynamics, that the kinetic energy of the molecules is some function of the absolute tem- perature, so that neither can be altered without altering the other; the form of the relation being such that both increase or diminish together. 21.] Stability of the Natural State. According to § 18 . (Hi., iv.) it requires the application of external force to disturb the body from a given natural state, and to hold it in any given state of strain, the temperature remaining unchanged: and when the force is removed, the body returns to .the state from which it was disturbed. Hence the natural configuration 8 PROPERTIES OF ELASTIC SOLIDS. [21. at each temperature is one of stable equilibrium for straining disturbances without change of temperature. And since, by the last Article, the kinetic energy of the molecules is the same in every state at the same temperature, it follows by a well-known theorem in Statics, that the natural configura- tion at each temperature is such that the potential energy has its least possible value for that temperature under the given law of intermolecular force. Hence it follows that if the body be strained in any manner, while the temperature is kept constant, the potential energy will be increased. And since in this case the kinetic energy remains constant, the increase of potential energy is necessarily equal to the work done on t)ie body by the external forces in producing the strain. If now, the temperature still being maintained constant, the body be allowed to work against the external forces, it will, in returning to its natural state, lose all the additional potential energy which it acquired by the strain. This then must be the exact measure of the work done by it against the external forces, which is thus equal and opposite to the work done upon it by them in producing the strain. This result may obviously be extended to a body starting from equilibrium in any given state of strain, and passing, at constant temperature, through any cycles of strain back again to its initial state, the total sum of work done on or by the body being identically zero. Thus a perfectly elastic body maintained at constant tempera- ture forms with any system of external straining forces a perfectly conservative system, the excess of the body's potential energy over that natural to the temperature being a function only of the strain and of the temperature, and vanishing with the strain. 22.] Temperature free to vary. In general, when the temperature of the body is left free to vary, energy communi- cated to the body, either in the form of heat or of mechanical work done by external forces, will be distributed in both forms. Thus, the primary effect of the addition of heat is to raise the temperature of the body, and thus to increase the molecular kinetic energy. But, since no external forces are applied, we know by § 18 (ii.) that the configuration of the molecules must change to that natural to the new temperature. Hence, if the law of intermolecular force be such that the potential energy of the natural configuration increases with the rise of temperature, some of the heat will be expended in produc- ing this increase, so that the resultant rise of temperature will be that due to an increase of kinetic energy less than the full 2.2.] PROPERTIES OF ELASTIC SOLIDS. 1) energy-equivalent of the added heat. On the other hand, if the natural potential energy diminishes with the rise of tempera- ture, the effect of adding heat will be to convert some of it into kinetic, and the resultant rise of temperature will be in excess. Again, if mechanical work be done on the body by external forces, so as to produce a strain, the mode of the molecular vibrations (which must depend upon the configuration) may be altered, and consequently the kinetic energy and the temperature may suffer change. Here again the increase of potential energy due to the strain will not be the exact equivalent of the work done in straining the body. 23.] Dissipation of Energy. The availability of heat for conversion into mechanical work, or potential energy, depends entirely on its distribution as to temperature. Every reversible conversion of mechanical work into heat is accompanied by the removal of a proportional quantity of heat from a body or portions of a body at a lower temperature to a body or portions of the same body at a higher temperature ; and if the distribu- tion so produced could be maintained indefinitely, the process could at any time be reversed without the addition or removal of heat from the body as a whole, the work done by the body in recovering its original thermal and mechanical state being pre- cisely equal to that done on it by external forces in producing the first change of state. But, under natural conditions, it is impossible to maintain, for any length of time, a non-uniform distribution of temperature without constantly supplying heat to some portions of the surface, and constantly removing it from others. If the body be supposed guarded from loss or gain of heat by radiation, the process of gradual conduction is constantly tending to equalize the tem- perature throughout its mass, and thus to dissipate its intrinsic energy for mechanical purposes, by rendering its heat unavailable for reconversion into potential energy or mechanical work. Now, by § 22, Strain produces a change of temperature which varies with the strain; hence a non-uniform straining of the body will produce a non-uniform distribution of temperature, and consequently the energy of a body so strained will be liable to dissipation by means of conduction. 24.] Conditions for a Conservative System. In order, therefore, that a perfectly elastic solid may form with external mechanical forces a perfectly conservative system, we must assume one of two conditions : either (i.) That the body is perfectly guarded from loss or gain of heat by radiation or surface-conduction, and that all the stages of strain and recovery are passed through so rapidly as to prevent all possibility of dissipation by conduction in its interior ; or 10 PROPERTIES OF ELASTIC SOLIDS. [24. (ii.) That the straining is so gradually performed, that heat may be constantly communicated to or taken from the different parts of the body, by suitable means, in such a manner as to maintain every portion uniformly at the initial temperature. 25.] The two cases are perhaps of equal practical importance, and the former is certainly the more interesting theoretically, but the relations between temperature, kinetic energy, and inter- molecular force are at present so hopelessly obscure that but little can be done towards its development.* It may be observed that, even if conditions (i.) were exactly fulfilled, a natural solid would still be found to dissipate energy irrecoverably by reason of its viscosity ; (see the second condition of § 19). 2C] Theory Adopted. To simplify our theory, and elim- inate as many unknown physical relations as possible, we shall assume that the conditions of § 24 (ii.) are always satisfied. We may observe that all the conditions of § 19 will be satisfied at the same time, if the strain be small ; so that results obtained for small strains on this assumption will be very approximately true for many natural solids. The body is then to be supposed always maintained at one constant temperature, uniform throughout, and thus the results of § 21 may be accepted as rigorously true. The kinetic energy of the molecules will be constant, and so also will the natural potential energy, or that possessed by the body when free from strain. 27.] Energy of the Strain. Since we are only concerned with the Strain and its effects, we leave these constant terms in the energy of the strained body altogether out of account ; and it is the excess of the potential energy of the strained body over its potential energy in the natural state which we shall in future refer to indifferently as the Potential Energy of or due to the Strain or of the strained body, or, more briefly, as the Energy of the Strain. By § 21, the Energy of the Strain is in all cases equal to the mechanical work done on the body by the external forces in producing the strain. Now, by § 18 (iv.), the same system of external forces, applied to the body in its natural state, invariably produces the same strain. Hence, if the strain be given, the forces to be applied, and also the displacements of their points of applica- tion are fully specified. Thus the Energy of a given Strain, being equal to the work done in producing it, is completely determined when the strain * See Sir W. Thomson's Reprinted Papers, Volume I., pages 293-313. 27.] PROPERTIES OF ELASTIC SOLIDS. 11 is specified ; or, in mathematical language, is a function solely of the given strain, and absolutely independent of any intermediate states of strain through which the body may have been brought. 28.] Stress defined. The effect of Strain, or change in the relative positions of the molecules, is to call into play Stress, or change in the mutual forces between the molecules. In the natural state, the molecules form a system of bodies performing small oscillations about mean positions under purely mutual forces. It follows, not only that the resultant of all these mutual forces acting within the body is identically zero, but also that if all the molecules were placed at rest in their mean positions, the resultant of the intermolecular forces acting on any one molecule would be identically zero. The system of intermolecular forces in the natural state must therefore be regarded, whether with reference to individual molecules or to the body as a whole, as an equilibrating system. 29.] Similarly, when the body is held in equilibrium in a given state of strain by suitable applied forces, the intermolecular forces — altered from their natural values by the change of configuration, but still purely mutual — together with all the applied forces on the several molecules must form an equili- brating system on the body as a whole. And the altered inter- molecular forces on any individual molecule, together with the applied force on that molecule, likewise form an equilibrating system. Now the altered intermolecular forces, being still purely mutual, must, as before, have by themselves a null or zero effect on the body as a whole. Hence it follows that any system of applied forces capable of holding an elastic body in equilibrium in a given state of strain must be such that its component forces, acting at the points to which the strain has displaced their original points of application, form by themselves an equilibrating system. 30.] Defining then, in accordance with § 28, the stress between two molecules as the change in their mutual action due to the change in their relative position, we see that the effects of applying any system of forces to an elastic solid are : — (i.) To produce such a strain that the external forces acting on the molecules in their new positions shall satisfy the ordinary conditions of an equilibrating system, such as would hold the body in equilibrium if the molecules were to become rigidly connected in their new positions ; and (ii.) In so doing, to call into play stresses between the molecxiles, such that the resultant force on any one molecule due to stress is equal and opposite to the applied force. The stresses, therefore, always resist further strain, and on any \2 PROPERTIES OF ELASTIC SOLIDS. [30. relaxation of the applied forces tend to restore the body to its natural state, diminishing continuously as the potential energy of the strain is expended in the process, and finally vanishing together with the strain. 31.] Work done by Stress. Since the stress on each molecule is always equal and opposite to the applied force, while the displacement of their common point of application is neces- sarily the same, it follows that all work done by the applied forces may be reckoned as work done against the stresses, and vice vevsd. Thus, in passing from a state of strain in which the potential energy (§ 27) is W, to a second state in which it is increased to W+ SW, the work done on the body by the applied forces in opposition to the stresses is S W ; while, if the stresses be allowed to restore the body to its original state, they will do work S W against the applied forces. 32.] Strain-Coordinates. Let us suppose that any changes in the relative configuration of the molecules may be represented by variations of a certain number of independent coordinates 9, , x> V'v ■ ■, the word being used in its generalised Lagrangian sense. Then, since the Potential Energy of the strain depends only on these changes, it must be capable of being expressed as a function of the strain-coordinates. Similarly, if V be the mutual potential energy of any two molecules, due to the stresses they exert upon one another, V must be a function of the differences between the actual values of 9, $,..., defining their relative positions, and their initial values in the natural state. If then SV be the small increase of V due to a small increase of strain, which changes 9, ,-.., to 6 + 86,

V'>--- °^ anv one P a i r * The symbol 3 is used throughout this work to denote partial differenti- ation ; d being reserved exclusively for total differentiation. The usual flux- notation is also frequently employed for partial differentiation as to time. 3-2.] PROPERTIES OF ELASTIC SOLIDS. 13 of molecules, the work done against stress in producing a small increase of strain in the relative positions of that pair is {B.86 + .8ct> + X. S x + }. Hence the whole work done on the hody against stress is £S2{e.80 + 'p.8<£ + X.<5 x + }=SIF, by §31. Thus it follows that = 3*730; €> = 3r/3; 33.] Simple Strains. A strain which consists in the variation of only one of the coordinates, such as 6, is called a Simple Strain of the type defined hy 6. Similarly, is called the simple stress of the same type. In the case of a simple strain we have evidently '= /Odd, e ede. 6 being the value of 9 in the natural state for the pair of mole- cules to which V belongs. A complex strain, in which more than one of the coordinates suffer change, is said to be compounded of, or to have for its components, the simple strains Two complex strains are said to be of the same type when their simple components differ only by a constant factor. Thus, if the first strain changes the coordinates to Q v v Xv-> an d t°e second to 6 2 ,

---> the conditions that they may be of the same type are A] - 0p _ #i - ftp- Xi - X o _ ^2 - s o 4>i~ o X2 ~ Xo 34.] Summary. We have now shown that if a perfectly elastic homogeneous solid body be strained by external forces, while always maintained at the same temperature — (i.) The potential energy of the strain will always be equal to W, the work done by the external forces in producing the strain. (ii.) The strain calls into play internal elastic forces or Stresses, which are of the nature of purely mutual reactions between the molecules ; the stresses between any pair of mole- cules having for their potential the mutual potential energy of the pair, and consequently tending to resist further strain and to restore the body to its natural state ; while the resultant of all the forces on any one molecule due to stress is always equal and opposite to the applied force. 14 PROPERTIES OF ELASTIC SOLIDS. [34, (ii-L) The potential energy and the stresses are functions solely of the actually existing state of strain, and absolutely independent of all intermediate states through which the body may have been brought. (iv.) As the external forces are relaxed, the stresses experience less and less opposition, so that they diminish continually as they restore the body to its natural state, expending on that process precisely the amount W of work which was done against them in straining the body, and finally vanishing with the strain. (v.) It is obvious that when the molecules are in motion under external forces the effective force to which the motion of the mean position of any molecule in the direction opposed to the stress is due, together with the resultant stress on that molecule, is equal to the applied force. Ideal Continuous Matter with Perfect Elasticity. 35.] Difficulty of further developing the Theory. We have thus deduced, from what we believe to be the true proper- ties of matter, the laws of equilibrium and motion of the mole- cules of a perfectly elastic solid. In order to develop our Theory analytically, we must be able to follow the movements of each molecule throughout the strain, and to discover all the mechanical conditions to which it individually is subjected. For this purpose we require to know the absolute mass and dimensions of the molecules of the body under consideration ; the law of distribution of their mean positions in the natural state ; the law of intermolecular force — the manner in which it depends upon, and varies with, both the configuration and the tempera- ture ; the limits of its sphere of action ; and, lastly, the connection between mean configuration, period and amplitude of vibration and the temperature. 36.] Unfortunately, on almost all these points, our ignorance is at present absolute ; and where we have any means of forming an opinion, the conclusions arrived at are so vague as to be value- less for our purpose. For instance, as to the dimensions of the molecules the latest conclusions of science are summarised as follows by Sir William Thomson * : — " The four lines of argument which I have now indicated lead all to substantially the same estimate of the dimensions of mole- cular structure. Jointly they establish, with what we cannot but regard as a very high degree of probability, the conclusion that, in any ordinary liquid, transparent solid, or seemingly opaque * Lecture on the Size of Atoms, Royal Institution, February 3, 1883, 36.] PROPERTIES OF ELASTIC SOLIDS. 15 solid, the mean distance between the centres of contiguous mole- cules is less than the five-millionth and greater than the thousand- millionth of a centimetre. " To form some conception of the degree of coarse-grainedness indicated by this conclusion, imagine a globe of water or glass, as large as a football (or say a globe of sixteen centimetres diameter), to be magnified up to the size of the earth, each constituent mole- cule being magnified in the same proportion. The magnified structure would be more coarse-grained than a heap of small shot, but probably less coarse-grained than a heap of foot-balls." 37.] Boscovitch's Theory. As to the law of intermolec- ular force, we are in still more complete ignorance. The most natural assumption is that the action between any two molecules is reducible to a single force acting between their centres of mass, and varying only with the distance between these points. This theory is always referred to as Boscovitch's, after the Jesuit Father who first formally stated it. It was adopted by all the earlier workers in Elasticity, who drew from it deductions leading to a simple and consistent theory, which however, was unable to bear the light of experiment, It was first disproved by Stokes, and has come to be regarded as an absurdity by all living physicists of any eminence (see § 208, below). Sir William Thomson has however quite recently shown * thab the points in this theory which have had to be rejected are not legitimate deductions from Boscovitch's principle. 38.] As to the greatest distance at which the intermolecular forces are appreciable, Cauchy deduced from his Theory of the Dispersion of Light that it must be comparable with the length of a luminous wave — the mean value of which may be taken as about one fifty-thousandth of a centimetre; and, although his theory was based upon Boscovitch's hypothesis, yet this result seems to hold good. Here we practically reach the limits of our knowledge of solid matter.-|- 39.] Conventional Theory substituted. Our ignorance of its intimate dynamical properties placing it out of our power to deal analytically with matter as it really is, it becomes necessary to substitute a hypothetical substance which will lend itself to mathematical treatment : attributing to it such arbitrary properties as will approximate the results of our analytical theory * Lectures on Molecular Dynamics, John Hopkins University, Baltimore, U.S.A., pages 124-132 of the papyrograph reprint. t See however Quincke's and Plateau's results, quoted in Tait's " Proper- ties of Matter," Article 293, 10 PROPERTIES OF ELASTIC SOLIDS. [39. to the deductions we have drawn from experiments on real matter. Our theory will then take its place as the last in the series formed by the various branches of Dynamics, which must be regarded as successive steps, each approaching nearer than the preceding to the true state of things, but none of them actually realised in nature. 40.] Dynamics of a Particle. The smallest " element of volume " which the refinement of analysis can reach must still, for the purposes of that very analysis, be held to have three linear dimensions, so that if it be occupied by an " element of mass " subject to forces which vary from point to point throughout space, this mass must in general be acted upon both by a force and by a couple ; both of them elementary, of course, but yet measurable by analysis. Hence we have recourse, for our first and simplest conception of dynamics, to the purely abstract idea of a "Material Particle," which we define as a very minute but still finite mass, so condensed that its linear dimensions are inappreciable to our analysis, and therefore infinitely small, even when com- pared with out smallest " element of volume." Such a particle cannot, of course, be subjected to couples, and therefore Dynamics is reduced to its simplest form. 41.] Dynamics of a Rigid Body. We next advance to the conception of a " Rigid Body," which we regard as an aggre- gation of such particles, so connected as to be entirely incapable of relative motion. The particles are supposed to be uniformly distributed, and, in the case of a homogeneous body, to be all of equal mass. Since the particles of the body remain in an invariable state of relative equilibrium, the mutual forces exerted by them upon one another, must under all circumstances of equilibrium or motion of the body as a whole, form by themselves an equili- brating system (D'Alembert's Principle). They consequently cannot possibly do any work, and there- fore do not enter into the equations of energy. In fact, we only owe to them the Kinematical or Geometrical equations which express in various analytical forms the fundamental fact that the body always moves as a whole, without relative motion of its particles. Moreover the external action on each particle takes the form of a single force, and these various forces can always be com- pounded into a single Resultant Force and a single Resultant Couple, which may be regarded as acting upon the body as a whole. Thus for all mechanical purposes the supposed structure of 41.] PROPERTIES OF ELASTIC SOLIDS. 17 the body may be, and is, altogether left out of consideration, and it may indifferently be regarded as a portion of perfectly con- tinuous or structureless matter. 42.] Continuous Elastic Matter. We now take our last step in advance, and recognise the possibility of relative motion between the constituent parts of a body. Replacing the molecules of our perfectly elastic solid by the abstract particles, of which an infinite number are contained in the smallest element, we transform it into a portion of con- tinuous elastic matter, capable of experiencing, and of offering a certain resistance to alterations of its form and volume. 43.] Homogeneity of Continuous Matter. In accord- ance with this view we now define a Homogeneous Body as such that any two equal and similar portions, similarly situated in the body, are precisely identical in all physical and mechanical properties, however small they may be taken within the limits of analytical refinement. Thus we quite abandon the idea of granular or molecular structure, and, by diminishing the size of our particles in- definitely, extend that perfect degree of homogeneity which in nature is common to many substances, taken in bulk (see § 1), to the smallest volumes which we can conceive. 44.] Points, Lines, and Surfaces in the Body. Our body being now composed of perfectly continuous and naturally homo- geneous matter, we must, for Geometrical purposes, replace the molecules by recognisable " points in the body " which are to be taken as necessarily coinciding with material particles, or in- finitely small portions of the continuous matter. Similarly, we define a "line in the body" as a cord of continuous matter, of any form and any length, and of infinitely small transverse dimensions ; while a " surface in the body " is to be regarded as a sheet of continuous matter of any form, and of infinitely small thickness. Points, lines, and surfaces in the body must, of course, when once chosen, be supposed to maintain their identity throughout all changes of form and position. 45.] Heat-vibrations neglected. In this transformation we entirely ignore the heat-vibrations of the molecules, because (i.) The molecules being replaced by particles infinitely close together, either their amplitudes will be reduced to the vanishing point, or they will have the same phase, period, and amplitude for all points of' the body, in which case they will not enter into the strain (see § 48) : (ii.) The temperature being constant, the kinetic energy is B 18 PROPERTIES OF ELASTIC SOLIDS. [45. also constant, and may be left out of consideration together with the constant part of the potential energy proper to the natural state (see § 27). Thus every point in the body is to be supposed at rest, except in so far as its motion is due to change of strain. 46.] Course of our Analysis. Strain will now consist in relative displacements of points in the body, and consequent distortions of lines and surfaces, and changes in the form and volume of portions of the body enclosed by the latter. Our analysis of Strains will therefore have for its aim to dis- cover a simple system of independent strain coordinates, the variation of any one of which will constitute a Simple Strain ; and to learn how to express any change of form or volume in terms of these as standard types. We shall next investigate the corresponding Simple Stresses, (which will be of the nature of resistances offered by the body to the respective Simple Strains), and the relations which must exist between them and the applied forces, in order that the body may be held in equilibrium in any given state of strain by these two opposed systems. To complete our general theory we shall then only require to know how to express stress in terms of the strain to which it is due. We shall then be able to calculate the potential energy due to any given strain, and the external forces required to produce it; or, conversely, the strain produced by any given system of applied forces; so that the solution of any problem will be reduced to a mere matter of analysis. We shall, for the next five chapters, confine ourselves to the consideration of bodies whose dimensions are at least finite in all directions. CHAPTER II. ANALYSIS OF STRAINS. 47.] We have defined Strain as any change in the relative 'positions of points in the body, produced by external forces. For purposes of analytical treatment it is of course necessary to assume that the relative displacements of points in a body undergoing strain follow some definite law depending on their relative positions in the natural or unstrained state. In other words, the displacement of any point Q in the body, relative to a given point P, must be some function of the initial position of Q relative to P; and it is further necessary to suppose, in order that the strain may produce no breach of continuity in the sub- stance of the body, that the relative displacement is a continuous function of relative position, for by this limitation we secure that the increase of the distance between any two points is, at the most, of the same order of magnitude as their initial distance. 48.] Secondly, it is to be observed that Strain, as defined, depends solely on such relative displacements. Any displacement — whether linear or angular — which is the same for all points, and which therefore produces no alteration in their relative arrangement in the body, amounts merely to a translation or rotation of the body as a whole, such as might be suffered by any perfectly rigid body ; and since such motions do not call into play any elastic forces, they are not included under the head of Strains. Although, however, a rotation of the body as a whole does not by itself constitute a strain, and can add nothing to the energy of any true strain that may accompany it : yet, since in discussing strains which vary from point to point we only con- sider a small portion of the body at a time, and since a rotation which varies from one portion of the body to another does constitute a strain, we make a point of recording rotations, and only ignore such displacements as are parallel, equal, and of like sign for all points in the body. 20 ANALYSIS OF STRAINS. [49. 49.] Now, let us take an unstrained body, and refer the positions of all points in it to a system of rectangular axes, fixed in space, whose origin coincides with any point M in the body. If the body be now strained in any manner the point M will in general suffer a displacement from its initial position at 0, the amount and direction of which will depend upon its situation in the body. But it follows from the last Article that, without modifying in any manner the effects of the Strain, we may impress upon all points of the body displacements equal, parallel, and opposite to that of M ; the effect of which will of course be to move the body back, parallel to itself, until M once more coincides with 0. 50.] Thus we may, whenever it will simplify our analysis,* suppose that point of the body which coincides with our arbitrarily chosen origin to be absolutely fixed in space, without in the slightest degree restricting the perfectly general character of the strain. Although, however, the origin may be regarded as fixed both in space and in the body, the axes are only fixed in space. That is to say, the straight lines in the unstrained body which coincide with the axes will no longer do so after the strain ; and, in fact, they will in general be no longer straight lines, but continuous curves intersecting more or less obliquely in the origin. Assuming then that the point of the body chosen as origin is fixed, the absolute displacement of any point in the body (and therefore also its component displacements parallel to the fixed axes) must be a continuous function of the absolute coordinates of the point ; and these absolute displacements now constitute the strain. Theory of Small Strains. 51.] Equations of Displacement. Let P be any point in the unstrained body, whose coordinates referred to the fixed axes are (x, y, z). Let the body be subjected to a very small strain, and let P in consequence be displaced to P" (x+u, y+v, s+w). Then u, v, w are the component displacements of P, parallel to the fixed axes, and we must have u = (x,y, z)j V = X fa y»*)r, w = 4>fa y,z)\ where u, v, w are supposed very small, and , x, ^ are arbitrary * See Appendix I., at the end of this Chapter, on the advantage of regarding a point in the body as fixed. 51.] ANALYSIS OF STRAINS. 21 functions, continuous throughout the body. We shall assume that all their partial derivatives as to x, y, and z are also con- tinuous, so that none of them can become infinite. 52.] Let another point of the body, initially at Q (x+ h, y + k, z+l) ; very close to P so that h, k, I are small quantities of the first order in comparison with x, y, z : be displaced by the same strain to Q' (x+h+vf, y + k+v', z + l+w'). Then, as before, u' = (x + h, y + k, z + l)) v = x (x + h, y + k, z + I) [■ . w \j/ (x + h, y + k, z + l)) Since u = u + h^~ + «~- + %- ox oy oz -- v + fc + *_ - + %- ox ay oz ow -■ w + h^- + ox /dw jCho o"y ~dz j The coordinates of Q, relative to P, have been changed from (h, k, I) to (h+v! — u, k + v' — v, l+w'— w) ; so that if Sh, 8k, SI be the increments of these relative coordinates, due to the strain, 3u ,3it ,3m 6/t = h^r + k~- + % ox oy oz Sk = h^~ + fc- + f 5t- ox oy oz SL -■ h^- + *~— + fc— ox oy oz , (1) 53.] Elongation. If L be the length of any line in the unstrained body, and if this length be altered to L' by the strain, the ratio (£' — L)/L, or the increase of length per unit of initial length, is called the Elongation of the line. If the line is diminished in length by the strain, it is said to suffer negative elongation, and the positive ratio (L-L')IL is otherwise called its Contraction. Let PQ=p, P'Q'=S+Sp; then p is of the same order of magnitude as h, k, I, and Sp is of the same order as Sh, Sk, SI. And since p i =h 2 +k 2 + l i , we have to the order of approximation adopted, P Sp=hSh+kSk+lSl. 22 ANALYSIS OF STRAINS. [53. But, if e be the elongation produced by the strain in the elementary straight line PQ, e=Splp. h Bh k Sk I SI /o\ «=-._ + -. — +-.— \*l P P P P P P Now if X, fji, v be the initial direction-cosines of PQ, hl\ = kj l L = ljv = p (3) Hence, substituting from (1) and (3) in (2) we find -i\ ou 3it 3*A dx 'dy ~oz' ~dw 3jo /\3w 3jo , 3uA /a\ or, re-arranging terms, ,.,3w. ..3u .,3u> Cow 3«\ 3a; 3vy 3s v 3jr 3z' ♦<♦£)♦*<£♦£) < 5 > 54.] From the form (5) we see, by writing successively (X=l, ^=0, V =Q), (X=0, M =l, i/ = 0), (X = 0, M = 0, v =l), that du/dx, dv/dy, dw/dz are the elongations of elementary straight lines drawn from (x, y, z) parallel to Ox, Oy, Oz, respectively. Again from the form (4-) it is easily seen that e may be written in the form £ = ( A— + fJ.~ - + v— )(ku + txv + w>), v ox oy oz' if we assume X, /u, v constant as to x, y, z ; that is, if we suppose the elementary straight line to be drawn in the given direction (X, /u, v) from different points of the body. Now if p be regarded as a current coordinate, giving the initial distances from (x, y, z) of points situated in the given direction (X, /x., v), and if U be the displacement of (x, y, z), resolved along this line in the positive direction of p, we have U= Xu + fiv + vw ? =x^+ ? + i' 3 r 3p "ox ~dy 3j) Thus ( = dU/d P (6) which gives the elongation of an elementary straight line drawn in any direction from any given point of the body. 55.] ANALYSIS OF STKAINS. 23 55.] Change of Direction. Again, if (X', n', v) be the direction-cosines of P'Q', \' = (k + Sh)l(p + S P ) h Sh h Sp — I — . — j CtC.j P P P P therefore substituting from (1) A.' = ,(1- ox' ay az P ■/dv 'dx K 1 - ~ov\ v. /dw 'bvo ~bx 'dy f v(l -e-t 3i> )) •(7) Since X, fi, v, as well as h, k, I and p, for any elementary straight line in the body, are of the same order of magnitude after as before the small strain, it follows that all lines and surfaces in the body preserve not only their continuity, but also the continuity of their curvature, throughout the strain. 56.] Permanence of Intersections of Lines and Sur- faces. It is easy to show that the points of intersection of lines, and the curves of intersection of surfaces, in the unstrained body become the points or curves of intersection of the same lines or surfaces in their strained state. That is to say, if two curves in the unstrained body intersect in P, and if P be removed to P' by the strain, the curves will be strained into curves intersecting in P'. And similarly for the curves of intersection of surfaces. For let the coordinates of P be (x, y, z) and those of P' (x, y', z') so that x' = x + u,y' = y + v,z' = z + w. Then if u, v, w are given as functions of x, y, z, we can express x', y', z' explicitly as functions of x, y, z ; and therefore (theoretically at least) u, v, w as functions of x', y', z". Now the two equations f 1 (x,y,z) = 0) (A) J^x,y,z) = 0i V taken separately represent two surfaces in the unstrained body, and, if these surfaces intersect, the same equations, regarded as simultaneous, represent their curve of intersection. The equations of the surfaces into which these are strained are f 1 (x'-u,y'-v,z'-w) = 0) £gj f 3 (x -u, y' - v, z' -w) = 1 where u, v, w are supposed to be expressed explicitly in terms of x, y', z 1 . ">4< ANALYSIS OF STRAINS. [56. But equations (B), regarded as simultaneous, may be taken as representing either the curve of intersection of the surfaces which they separately represent, or the curve which before the strain was represented by the simultaneous equations (A). Thus the curve of intersection of any two strained surfaces in the body is the strained form of the curve of intersection of the same surfaces before the strain ; and by a precisely similar method we can show that the point of intersection of any two strained lines is the strained position of the point of intersection of the same lines before the strain. 57. General Effect of Strain. We see from equations (5) and (7) that the magnitude and direction of every elementary straight line in the body are in general altered by the strain, and that these changes are in general different for different elements. Hence the general effect of the strain is both to shift and to distort all lines and surfaces in the body. We shall reserve the exceptional cases for future discussion. 58.] Limitations of Small Strain. From equation (5) it appears that the elongation of an elementary straight line, drawn in any direction from a given point, is of the same order of magnitude as the first derivatives of the component displacements of that point with regard to its initial coordinates. In future, unless the contrary is explicitly stated, we shall confine ourselves entirely to the consideration of " small strains," implying thereby that all these first derivatives, like the displacements themselves, are small quantities of the first order, or else zero. Homogeneous Strain. 59. Definition. We shall now suppose the character of the strain restricted in such a manner that all the first derivatives, dujdx dw/dz, are independent of x, y, z. This assumption involves a relation between the displacements and initial coordinates of the form u = ex + /?j2/ + 7 X 2 | v = a&+fy + y. l z L (8) where the coefficients are all absolute constants, which for a finite strain are finite or zero, and for a small, strain are all small quan- tities of the first order or else zero. A strain of the character defined by this assumption is said to be a Homogeneous Strain. We shall now proceed to investi- gate its principal properties. 60.] ANALYSIS OF STBAINS. 25 60.] The results which we have already obtained for any small strain take the following forms in the case of small homo- geneous strain. The component displacements of the point Q(x+h, y + k,z + l) relative to the point P(x, y, z) are, by (1), Sh^eh + fiJc + yJ Sk^aJi+fk + yJ ■ (9) SI = aji + fijc + gl The elongation of the line PQ (direction-cosines X, /m, v) is given by £ = eX2 +//»« + g V * + (j8, + yjpv + (y, + « 3 )vA + (a, + ft) V (10) whence it is obvious that e, f g are the elongations of elementary straight lines parallel to Ox, Oy, Oz respectively. Lastly, the new direction-cosines of P'Q' are, by (7), A' = (1 - <• + e)A + /3j/i + y t v ti = ttjA + (1 - e +/)n + y 3 v ■ (11) v' = a 3 A + ft/x + (1 - « + g)v 61 .J Parallel Straight Lines. It is obvious from equations (10) and (11) that e, X', n', v depend entirely on X, n, v, whence we deduce that, in any small homogeneous strain, all parallel straight lines in the body, of elementary length, remain parallel and are elongated in the same ratio. But we may consider any straight line, finite or infinite, in the unstrained body, as made up of consecutive elementary straight lines, all of which are parallel to one another and meet consecutive elements. By equations (11) these will be strained into elemen- tary straight lines all parallel to one another, and by § 56 each of these will meet the consecutive elements. Hence they must all lie in a straight line ; so that a straight line in the body, of whatsoever length, will remain a straight line, though in general its direction will be changed. £1 the same way, since two parallel straight lines of any length may be divided into elements, all of which are necessarily parallel, it follows that all parallel straight lines in the body remain parallel straight lines, though in general their direction will be changed. Also, since by (10) all their elements will be elongated in the same ratio, parallel straight lines of any length are elongated in the same ratio; and, in particular, equal and parallel straight lines are strained into equal and parallel straight lines, though in general their length, direction, and distance apart are all altered by the strain. 62.] Parallel Planes. Again, since (§§ 56, 61) intersecting straight lines remain intersecting straight lines, a plane must 26 ANALYSIS OF STRAINS. [62. remain a plane; and since any two parallel planes intercept equal lengths on any system of parallel straight lines which meet them both, and since these intercepts are strained into equal and parallel (§61) straight lines, terminated (§ 56) by the strained planes, it follows that all parallel planes are strained into parallel planes, though in general their direction and distance apart are altered by the strain. 63.] Similar and similarly situated Geometrical Figures. From the two last articles it follows directly ,that every parallelogram, is strained into a parallelogram, and every parallelepiped into a parallelepiped, though both are in general distorted. Since similar and similarly situated plane figures (in the same or parallel planes) have their homologous sides parallel, it follows that all similar and similarly situated plane figures are strained into plane figures similar and similarly situated to one another, though not necessarily to the former. In fact, since all parallel chords are elongated in the same ratio, it is obvious that the strained form of any plane figure is an enlarged or diminished orthographic projection of its un- strained form upon some plane. Hence, in particular, an ellipse {including ike circle) is always strained into an ellipse or circle ; and when a circle is strained into an ellipse every pair of orthogonal radii of the circle is strained into a pair of conjugate radii of the ellipse. Again, since in similar and similarly situated solid figures all similarly situated sections are similar, it follows that all similar and similarly situated solid figures are strained into solid figures similar and similarly situated to one another, though not in general to their unstrained forms. 64.] Strain Ellipsoid. Since all the sections of an ellipsoid are ellipses (or circles), and since no other surface possesses this property, it follows from the last article that every ellipsoid (or sphere) is strained into an ellipsoid (or sphere) ; and when a sphere is strained into an ellipsoid, every set of three orthogonal radii of the sphere becomes a set of three conjugate radii of the ellipsoid. The ellipsoid into which a sphere of unit radius, described about any point P of the unstrained body as centre, is altered by the strain is called the Strain Ellipsoid at the point P. Of course, in a homogeneous strain, the strain ellipsoids at all points of the body will be equal, similar and similarly situated. 65.] Principal Axes of the Strain. Every set of or- thogonal radii of the unit sphere becomes, by § 64, a set of con- jugate radii of the Strain Ellipsoid ; and the ellipsoid has one — 65.] ANALYSIS OF STKAINS. 27 and only one — set of orthogonal conjugate radii, namely, its principal axes. Hence, in every homogeneous strain there is One — and only one — set of three orthogonal straight lines passing through each point of the body, which remain orthogonal after the strain, although their directions are generally altered. These principal diameters of the Strain Ellipsoid are called the Principal Axes of the Strain at P. 66.] Pure Strain. When the strain is such that the Prin- cipal Axes retain their initial directions it is said to be a Pure or Irrotational Strain. It is sufficiently obvious that the most general small homo- geneous strain will consist of a small pure homogeneous strain, sufficient to produce the required distortion, together with a small rotation of the body as a whole, about a suitable axis, sufficient to bring the Principal Axes at each point into their new positions. Analytical Investigation. 67.] We shall now proceed to prove these properties of Homogeneous Strain analytically. Since, by equations (8), u, v, w are linear functions of x, y, z, their partial derivatives of the second and all higher orders will vanish. Hence, equations (1) or (9) will be absolutely true, independently of the magnitude of h, k, I, so that equations (10) and (11) will hold for straight lines of any length. From this § 61 follows immediately. 68.] Initial and Pinal Coordinates. The equations giving the final coordinates (x r , y', z) of any point P in terms of the initial coordinates (x, y, z) are, by equations (8), x' = x + u = (1 + e)x + (3$ + y x z \ y' = y + v =( V s + (l+f)y + yA (12) z' = z + w = a^a + /3^y + (1 + g)z ) Hence, to the first order of small quantities, x=(l-e)x'-p i y'-y 1 z' y= -a 2 a; , + (l -/)//- y£ • (13) z= - a(x, y, z) = 0, becomes after the strain the surface given •HK 1 - e K - /V - w^ K 1 - f)v - r^'- «.*']. [(i -?K -«*«'- Atf} =o, which equation, the coefficients being constants, is clearly of the same order as the former. Thus, planes are strained into planes, and quadrics into quadrics ; and since a small (or even a finite) strain cannot possibly convert a finite line into one of infinite length, it is clear that a closed surface must remain a closed surface. Thus, an ellipsoid or a sphere is always strained into an ellipsoid or sphere. The straight line being formed by the intersection of two planes ; and the ellipse or circle, being formed by the intersec- tion of a plane with an ellipsoid or sphere, must obviously retain their original characters. Also, since equal and parallel straight lines are strained into equal and parallel straight lines, it follows that a plane bisecting a system of parallel straight lines is strained into a plane bisecting a system of parallel straight lines, so that any system of parallel chords of an ellipsoid, with their diametral plane, become a system of parallel chords and corresponding diametral plane of the strained ellipsoid. Hence it follows at once that every set of three conjugate diameters becomes a set of three conjugate diameters. 70.] Strain Ellipsoid. The ellipsoid a? y- s 2 , 2* + # + v' 1 70.] ANALYSIS OF STRAINS. 29 becomes the ellipsoid and, in particular, the sphere a? + y 1 + s? = 1 becomes (1 - 2e)x'* + (1 - 2/)y' 2 + (1 - 2g)z'> - 2(ft + y,)y'z - 2( 7l + ^ -2(0, + ^)^'--! (15) which is the Strain Ellipsoid at the origin (§ 64), referred to the fixed axes. 71.] Change of Notation. It will be observed in equa- tions (10) and (15) that the coefficients j8, and y 2 , y 1 and Oj, a„ and /Sj occur in pairs. This will frequently happen in future equations, and we shall considerably simplify our analysis, and make it much easier to interpret, by changing our notation as follows : — let us take 2s 1 =(3 s + y 2 \ 2s 2 = y x + « 3 2*3=0, + ft 20, -7,-oJ 20 3 = O,- ft/ e,f, g being retained. 72.] The equations of displacement for small homogeneous strain then take the form u = ex + (s 3 - 3 )y + (s 2 + 2 )z ) « = (« i +0 s )a:+/y + (« I -0 1 )s l (17) mi = (« 2 - 2 )cc + («! + 0,)$r + gz J the elongation becomes e = eA 2 +/ / i 2 + S'v 2 +2* l /iv + 2s 2 vA + 2s s A/i (18) and the final direction-cosines are A.' = (1 - e + e) \ + (s 3 - 6 3 )n + (s 2 + 2 )v ) /*'=(»»+ »^+ 0-'+/)/*+ (»!-»>[ (19) "' = (*2 ~ *i)* + ( s i + *> +( 1 - e + 9>) while the equation of the Strain Ellipsoid, referred to the fixed axes, is (1 - 2e)a:' 2 + (1 - 2/)y 2 + ( 1 - 2g)z'* - 4s,iy V - 4*,«V - is&'y' = 1 (20) .(16) 30 ANALYSIS OF STRAINS. [73. 73.] The direction-cosines of the principal axes of this ellipsoid are given by the equations (l-2e)A.'-2yt'-28.y = - 2s s X.' + (1 - 2 f )/i'-2s 1 v' A' M ' _ -2 g-,A'-2«i/x' + (l-2y)v' V which may also be written in the form e\' + s^' + s,v' _ s 3 A' +fix + x,v' _ a,A' + S]/u,' + gv X \i v These equations therefore give us the directions, after the strain, of the Principal Axes of the Strain. .(21) Graphic Properties of the Strain. 74] The Elongation and Compression Quadrics. If we describe about the origin a quadric surface of the form ex 2 +fy i + gz 2 + 28 l yz+2s^x + 2s : ^y = B 2 (22) (which we shall regard as fixed in space, like the axes of reference), and if r be the radius vector intercepted by the surface on a straight line in the body drawn from the origin in the direction (\, /a, v) we shall have r 2 (eA 2 +//J. 2 + gv 1 + 2«,/*v + 2*^ A + 2a 3 A/*) = W. Thus, by equation (18), if e be the elongation of this radius vector, or of any straight line in the body parallel to it « = j52/r 2 (23) This surface is called the Elongation Quadric of the strain. 75.] It follows from equation (23), the right-hand side of which is essentially positive, that every radius which meets this surface suffers a positive elongation, and conversely that every radius drawn in a direction of positive elongation will meet the surface. If therefore the strain be such that all lines in the body are elongated, the Elongation Quadric must be an Ellipsoid. If however the strain consists of elongations in some direc- tions and contractions in others, e will be negative for some radii, which therefore cannot meet (22). In fact, in this case the Elongation Quadric is an hyperboloid whose radii are the lines which suffer elongation, while those lines which suffer contraction are the radii of the conjugate hyperboloid represented by ex i +fy 2 + gz*+ 2s x yz + 2s^xc+ 2sjcy= - B 2 ....24) which is called the Compression Quadric. 76.] ANALYSIS OF STRAINS. 31 76.] In the case in which all lines in the body undergo con- traction, all radii from the origin must meet the Compression Quadric (24), which is therefore an ellipsoid; and in this case there is no Elongation Quadric. 77.] Cone of no Elongation. In the case of § 75, the hyperboloids of elongation and contraction are separated by their asymptotic cone, whose equation is ex 2 +/y' 2 + gz 2 + 2s^z + Isgac-v 2sjcy = (25) It appears from (23) that for all the generators of this cone, and of course for all parallel lines in the body, e=0. It is there- fore called the Cone of no Elongation. 78.] Cones of Constant Elongation. Lastly, the direc- tion-cosines of all lines in the body suffering a given elongation e (whether positive or negative) must satisfy (18), which may be written eA. 2 +//* 2 + gv* + 2sjjitv + 2s 2 vk + 2s 3 A/t = e(A. 2 + /i, 2 + v 2 ). All such lines must therefore be parallel to one or other of the generators of the cone (e-e)3? + (f-€)y 2 + (g-€)z 2 +2s 1 yz+2s i zx+2s^/ = (26) We thus obtain a series of Cones of Constant Elongation. 79.] It is to be observed that all the quadrics described in the last five articles form a concentric and coaxial system. If Of Or), Of be their principal axes, their equations when referred to them will respectively become where e 1; e v e s are the roots (in descending order of magnitude, let us suppose) of the discriminating cubic *-+, »„ * =0 (28) *tl *V 9 ~ the direction-cosines of Of, Or/, Of, with reference to the original axes, being given by the equations ek + s^ + y _ s 3 A +f(i + s lV ^ s 2 k + s^ + gv _ where for is to be substituted e v e„ or e 3 , according as A, /x, v are the direction-cosines of Of, Or/, or Of. .(27) 32 ANALYSIS OF STRAINS. [80. 80.] Principal Axes of the Strain. Since the elongation e of any radius of the elongation quadric varies inversely as the square of the radius, and since the squares of the least and greatest radii of the quadric are -B-/e, and -B 2 /e 3 , it is obvious that e always lies between e 1 and e 3 , and that the directions of maximum and mini- mum elongation (or of minimum and maximum contraction) are those of the least and greatest axes of the quadric. But if we consider the deformation of the unit sphere into the Strain Ellipsoid, it is clear that those radii of the sphere which are drawn in the directions of maximum and minimum elongation must become the greatest and least axes of the Ellipsoid. Thus the lines in the body which, before the strain, coincided with the principal axes of the elongation quadrics, become the principal axes of the Strain Ellipsoid. Equations (29) therefore give the initial directions of the Principal Axes of the Strain. Pure Strain. 81.] Conditions for Pure Strain. The strain is said to be pure (§ 66) when the Principal Axes retain their initial direc- tions. Now, comparing equations (29), which give the directions of the Principal Axes before the strain, with equations (21) which give the directions of the same lines after the strain, we see that they appear to be identical. We must not, however, infer from this that the Principal Axes necessarily retain their initial directions. From equations (19) it appears that the differences between the initial and final values of the direction-cosines of any line are of the same small order as the strain coefficients ; now in equations (21) and (29) the direction-cosines all appear multiplied by these same coefficients ;. so that it is quite impossible, to the order of approximation adopted, that any distinction should be made in such formulae between X and X', /x and /*', v and v. For instance, eX' + s-n' + sy ^7 = {eA + s^ + s 2 v + e(A - A) + s 3 (/i - fj.) a(— )}-A-{l- A "i- A }> and substituting from equations (19), this expression is identical, to the first order of small quantities, with eA + 8 3 /j. + s 2 " S2.] ANALYSIS OF STRAINS. 33 82.] Non-rotated Straight Lines. The three straight lines through the origin which (together with all lines in the body parallel to them) really retain their initial directions in space are to be found by putting X' = X, // = /*, v'=v in (19). Thus we get e.X + (a. _ e^fi + (s., + ft.) v _ (sj + a ) X H f /1. f ( Sl -6,)v X ft = («..-0 3 )A.4-(*, + 1 )/* + ff»' {Sft) for the direction-cosines of the non-rotated straight lines. The conditions for Pure Strain arc therefore simply the con- ditions that the equations (21), (29) and (30) may be identical ; and these obviously are 83.] Equations of Displacement. Principal Elonga- tions. The equations of displacement (17) thus become, in the case of Pure Strain, it = ex + s 3 y + s»z \ v = * s x+fi/ + s i z - (31) W - gJB + Stf + ffzj It will be observed that they only involve six independent strain coefficients. If now U, V, W be the displacements of any point P in the body, parallel to the Principal Axes 0£ Ot], 0£; if (\, fi v V] ), (X.j, u 2 , v 2 ), (X 3 , /x 3> e 3 ) be the drrection-cosines of these axes referred to the original arbitrary axes Ox, Oy, Oz ; and if {x, y, z) and (£ rj, f) be the coordinates of P referred to the two systems ; we have ^ £ = X x x + tw + v,z^ f= A 3 rc + /*;,?/ +iv?J JJ = X^u + faV + r,jr'\ V= X„u + [i,v + v.,w . W '= X r u + juv + i' : .w | Thus, from (31) U = X,(ex + s.j/ + x.z) + fii(S;p: +/// + «,s) + v,(x. r x + s^j + gz) - (A,e + /*,% + v^x + (A,s 3 + /»,/+ v^y + (A,s 2 + /x,^ + v,f/).t = A 1 e 1 a; + ftcy + v^z, by (29). Ultimately therefore we find P=VJ • (32) c :H ANALYSIS OF STRAINS. [83. The point initially at (£, y, f) is therefore displaced to and obviously the effect of any Pure Strain is simply an elonga- tion (or contraction) of the whole body parallel to each of the Principal Axes. The three Principal Elongations e^ e 2 , e 3 are the roots of the discriminating cubic (28) of the Elongation Quadrics. By comparing equations (31) and (32) it is evident that equation (18), giving the elongation e of any line in the body may be written in the form + 2s,yz + 2s.xx + 2s&y =C 2 (33) This is obviously coaxial with the elongation quadrics, and when referred to the Principal Axes takes the form (l+ £l )^+(l+^ 2 + (l +£3 )f 2 =:(^. Since e„ e 2 , e s are small, it is necessarily an ellipsoid. Let ?• be the radius vector drawn in the direction (X, p, v) and let (I, m, n), as in the last Article, denote the direction-cosines of r referred to 0£ Or/, Of. Let p be the perpendicular from the centre on the tangent plane at the extremity of r, and let (I', m', n') be the direction-cosines of p referred to 0£, Orj, Of. Finally, let e be the elongation suffered by r. By the ordinary formulae of Solid Geometry, I=prHl +tl )/C* | m =prm(l + e 2 )/C- J-. n'=prn{l + ^/C J Hence, squaring and adding, ; 3 V{l + 2(^ + c 2 m 2 + ^ 2 )} = ^; pV{\ + 2e) = C* ; .-. r(l+i) = C 2 /p. Thus the strained length of the line in the body initially coinciding with r varies inversely as p. 84.] ANALYSIS OF STRAINS. 35 Again, since equations (19) refer to any arbitrary set of axes, we may suppose them to refer to 0£ 0>j, Of; hence the new direction-cosines, referred to these axes, of the line in the body initially coinciding with r will be (1 - e + ejl, (1 - e + t 2 )m, (1 - yc«=i I, ri = (I - e + e 3 )w J or, the line in the body which initially coincided with the radius vector r finally coincides with the perpendicular p, and its final length varies inversely as p. This ellipsoid is called the Position Ellipsoid, from the fact that it gives us a graphic construction for the position and length, after pure strain, of any line in the body whose position and length before the strain are known. Rotational Strain. 85.] Returning to our arbitrary axes, let us suppose that the body, after undergoing the small Pure Strain represented by equations (31), is further subjected to a small rotation of the body as a whole, of amount Q, about any axis (a, fi, v) through the fixed origin 0. The coordinates (x', y', z') of a point P, initially at (x, y, z), will be after the Pure Strain x' = (1 + e)x + s 3 y + s 2 z\ y' = * s »+(l+/)y + * 1 *h z = s s x + s x y + (1 + g)z) and the final coordinates (x", y", z") of the same point after the rotation will be x" = x' + jui2s' — vQy' ~\ y" = y' + vilx' -Atte' I z" — z'+ XQy' — fxS.x'\ 30 ANALYSIS OF STRAINS. [85. the square and higher powers of the small quantity Q being neglected. To the same approximation we shall have for u, v, iv, the resultant displacements, u = ex + (s 3 - vQ,)y + (s 2 + /xfl)z \ v = (s 3 + vil)x +fy + (s r - AI2)z J-. w = (s 2 - /ittjx + (s 1 + AJ% + gz) 86.] Comparing these equations with (17) we deduce that the general Homogeneous Strain represented by (17) consists of the Pure Strain represented by (31), together with a small rotation of the body as a whole, the components of which about the fixed axes are 6 V 6 2 , 6 3 ; so that the amount Q of this rotation, and the direction-cosines of its axis, are given by This is the result that was anticipated in § 66. Principle of Superppsitiov . 87.] Writing equations (17) in the form u = [ex + sjy + S;z] + [ftjZ - 8jy\ ~\ V = [8jB+fy + s.z] + [6& - 6 lZ ] J-, w = [s.>a; + s t y + gz\ + [8$ - 8.x] j it is evident that the displacements due to a small rotational strain are simply the algebraic sums of the displacements due severally to the pure strain and the accompanying rotation ; and it is further evident, from § 85, that this result depends entirely on the supposition that all the coefficients involved in the suc- cessive displacements are small quantities whose squares and higher powers may be neglected. Consequently the same principle ought to apply to all small strains and rotations, whether they be homogeneous or not ; and it is easy to show that this is the case. Suppose the body first subjected to a small strain whose displacement coefficients are [e, f, g, s v s 2 , s 3 , B v 2 , 6 J. The coordinates of any point P in the body, after this strain, will be x' = (l + e)x + (*j - 6 s )y + (a, + 8 2 )z, etc., etc. Now let the body be subjected to a second small strain [«'■ /', 9> 8 i> 8 V s 's' 0'ii $' 2 > 0' 3 ]- The final coordinates of P will be given by x" = (1 + e')x' + (s' 3 - 6' 3 )y' + (s'„ + ff^y, etc., etc. ST.] ANALYSIS OF STRAINS. ot Thus the resultant displacements of P, due to the two suc- cessive strains, will be u = (1 + e'){(l + e)x + (s 3 - 6 3 )y + (s, + 6. 2 )z} + («'. - ^' 3 ){(* 8 -i- 3 )x + (1 +f)y + («! - ^)a} etc., etc., and, to the first order of small quantities, u = [(e + e> + (s 3 + s' 3 )y + (« 2 + s' 2 )z] + [(0 2 + 0' 2 )s - (0 3 + 0' 3 )(/], etc., etc. This result may "be extended to any number of small strains, so that finally we have for the resultant displacements u = [2(e) . » + 2(s 3 ) . y + 2(s 2 ) . z] + [2(0.) . z - 2(0.) . ?/] -j «-Pfc) . a + 2(/) . j, + 2( Sl ) . s] + [2(0 3 ) . a-^fi,) . «] L (34) w = [2(« 2 ) . a, + 2(0 . y + 2(g) . z] + [2(0,) . y - 2(0 2 ) . x] J 88.] Thus the resultant of any number of small strains is a small strain in which the coefiicients of pure strain and rotation are respectively the algejjraic sums of the corresponding co- efficients in the component strains. And, conversely, any small strain may be arbitrarily resolved into any number of small component strains, subject only to the condition that the algebraic sums of the several coefficients must be equal to the corresponding coefficients in the original strain. This result is called the Principle of Superposition of small strains, and is a particular case of a theorem of very general application in Mathematical Physics. Components of Pure Strain. 89.] We are now in a position to analyse equations (31) which represent the most general Pure Strain. By the last Article it may be regarded as the resultant of the six component pure strains represented respectively by u = w = 1( V .) 8.p) U = S.J v = Q w — s.p:J Each of these components only involves one strain coefficient, and they are in consequence called Simple Strains (compare 33); and since any one of the coefficients may be altered without affecting the others, these simple component strains are in- dependent 3S ANALYSIS OF STRAINS. [00. 90.] Simple Elongations. Let us consider first the strain represented by (i.), assuming e to be positive. The discriminating cubic (28) becomes Thus e, = e, e 2 = 0, e 3 = ; and (with the notation of § 83) equations (29) give X, = 1,^ = 0,^ = 0. The Elongation Quadric degenerates into the pair of parallel planes ex 2 = B 2 , the Principal Axis 0£ coinciding with Ox, while Orj, Of are indeterminate. The cone of no elongation degenerates into the plane of yz, and the cones of given elongation e are the cones of revolution (e - e)x 2 = t(y 2 + z 2 ). The strain evidently consists of a uniform elongation, of amount e, of all lines in the body parallel to Ox, all lines in perpendicular directions remaining unchanged in length, while the elongation of any other line depends only on its inclin- ation to the axis of the strain, being given by e=eX 2 . In fact, the Position Ellipsoid (§ 84) becomes the prolate spheroid It is obvious that this strain increases the volume of the body, or of any portion of it, in the ratio (1 + e). These results can easily be adapted, mutatis mutandis, to the case where e is negative (uniform contraction). 91.] Similarly (ii.) and (iii.) represent simple elongations of amounts / and g, respectively parallel to Oy and Oz. The elon- gations produced by them in the line (X, fx, v) are fa 2 and gv 2 , and they increase the volume of all portions of the body in the ratios (1 +/) and (1 +g) respectively. 92.] Simple Shears. In the case represented by (iv.) the discriminating cubic is ■4>, 0, 0-0. 0, -, «, 0, s„ - which reduces to Thus e 1 = s 1 , e. 2 = 0, e 3 = — s,, if 8 t be assumed positive. Substi tuting in. equations (29) they give K=h Pa = 0, y 2 = 0, 92.] ANALYSIS OF STRAINS. 39 Thus Or/ coincides with Ox, while Of and Of lie in the plane of yz, and bisect internally and externally the angle between the positive directions of axes Oy, Oz ; and the strain consists of a uniform elongation, of amount s v parallel to Of, together with a uniform contraction, of equal amount, parallel to Of, all lines in the body parallel to Oq or Ox retaining their initial lengths unaltered. The volume V of any portion of the body thus becomes Vil+s^l-s,); that is to say, it remains unchanged, and the strain produces distortion only. 93.] The Elongation and Compression Quadrics are cylinders, whose generators are parallel to O17 or Ox, and whose transverse sections are conjugate rectangular hyperbolas, their equations being or *«*-{*)= ±-8*. The Cone of no Elongation degenerates into the pair of asymptotic planes yz = Q, or ?-? = 0. Hence every line lying in either of the planes xy and zx retains its length and its inclination to Ox unaltered, and so of course does every line in the body parallel to either of these planes. [This may be shown directly by substitution in equa- tions (18) and (19).] Thus every geometrical figure described in any plane parallel to xy or zx will retain its initial form and dimensions unaltered. 94.] For this reason these two sets of parallel planes are called Planes of no Distortion. Since each set of parallel planes remains a set of parallel planes, and their lines of intersection maintain their identity, the strain can only consist of a relative shifting of the two sets of planes of no distortion, after the manner of jointed wicker-work, so as slightly to diminish the right angle between the positive directions of Ox and Oy (which include between them the axis of elongation Of), and to increase the supplementary angle by the same small amount. A strain of this nature is called a Simple Shear of the two systems of planes of no distortion ; or simply a shear of the planes of xy and zx. Since the strain is really in two dimensions, all the effects produced in the plane of yz being exactly repro- duced in all parallel planes, it is sometimes called a Simple Shear 40 ANALYSIS OF STRAINS. [94. in the plane of yz (in which case the positions of the axes 0£, Otj must be specified) ; and this or any parallel plane may be termed the Plane of the Shear. The Amount of the Shear is measured by the change in the right angles between the planes of no distortion, as described in the last Article. Let Fig. 1 represent the section by the plane of yz of a prismatic portion of the body, bounded by planes of no distortion which cut the plane of the section in the square ABCJD. Then Fig. I the axis of elongation Og will coincide with the diagonal AGO, and the axis of contraction Of with the diagonal BOD. The sole effect of the shear will be to change the square base of the prism into the rhombus A'B'C'P/, where A'O = (1 + s^)AG; B'D={\- 8l )BD. If the sides of the square meet the axes of reference in P, Q, R, S, and if PR, QS are strained into P'R', Q'S', the amount of the shear will be the sum of the angles QOQ' and POP', and since these angles are equal the amount is twice the angle POP'. 94.] ANALYSIS OF STRAINS. 41 Now, by equations (19), we have for the strained position P'R of the line in the body PR, initially coinciding with Oy, V = s,. Hence cos P'Oz = s„ and to the first order of approximation and 2s, is the amount of the shear. \f \b> u !»' A / ■•''' '-— ~"y]A ' Ctrl' -, ,.-"'/ p y y y c ;s' S J}\ Fig. 2 . 95.] Shearing Motion. There is another, slightly different, point of view from which we may regard the mode of operation of a small shear. Let A BCD in Fig. 2 represent the transverse section by the plane of yz of the same square prism as in Fig. 1 ; and let P, Q, R, S be as before the middle points of the sides. Suppose that, keeping fixed the plane of xy, we give to every parallel plane in the body a motion parallel to the fixed plane, and proportional to its perpendicular distance from it, those 42 ANALYSIS OF STRAINS. [95. planes lying on the positive side of xy being shifted in the positive direction of Oy, and those on the negative side in the negative direction. Since each point in OQ for (instance) moves perpendicularly to OQ through a space proportional to its distance from the fixed end 0, it is obvious that OQ is strained into a straight line OQ'; and the displacements of points in QS at equal distances on opposite sides of being equal and opposite, QfO and OS' will remain in one and the same straight line. It follows that all planes in the body parallel to zx are simply turned through a constant angle of QOQ' about the lines in which they meet the plane of xy, while by hypothesis every plane in the body parallel to the latter undergoes a bodily translation in its own plane. If the strain be of very small amount the lengths of the lines QS, etc., will not be appreciably altered, so that the result will be to strain the square ABCD into the rhombus A'B'CD' without altering the lengths of its sides. Thus it is obvious that the planes in the body parallel to xy and zx respectively form two systems of Planes of no Distortion. 96.] A strain of this kind is called a Shearing Motion of the planes parallel to xy in the positive direction of Oy. Its amount is measured by the constant ratio between the distance traversed by any one plane and its perpendicular distance from the fixed plane : that is, by the tangent of the angle QOQ'. The amount of a small shearing motion is therefore measured by the diminution or increase of the supplementary right angles between the planes of no distortion. The change of direction A OA' of the diagonal plane AC is /QA\ x J QA' - QA \ V . OQ) = tan" = oQOQ' very nearly. Similarly, the change of direction BOB of the other diagonal plane is also approximately ~QOQ'. — 97.] Comparing these results with §§ 93, 94, it is obvious that a small shearing motion of amount 2s 1 of planes perpendicular to Oz in the positive direction of Oy, is equivalent to a small Shear of amount 2s 1 of planes perpendicular to Oy and Oz, together with a small rotation of the body as a whole through the angle *j about Ox in the negative direction (i.e., from Oz towards Oy). 97.] ANALYSIS OF STRAINS. 43 In like manner the student may satisfy himself, by drawing a suitable figure, that a small shearing motion of. amount 2s, of planes perpendicular to Oy in the positive direction of Oz is equivalent to the same smaU shear, together with a rotation of amount s, of the body as a whole about Ox in the positive direction. 98.] A shearing motion is therefore a rotational strain, the shear or pure strain being the same to whichever set of planes we give the shearing motion, while the accompanying rotations are in opposite directions in the two cases. \V z \B' Q ;s* B R tf\\ —jA' 1 p:. x - C'-'' p y D S £>\ / jc" \ N. Fig. 3. This fact suggests a method of producing (as in Fig. 3) a non- rotational shear of amount 2s,, by means of two shearing motions each of amount s,, applied successively (or simultaneously) to the two sets of planes. In Fig 3, the first shearing motion takes place parallel to Oy, so as to change the square ABCD into the rhombus A'B'CD', at the same time producing the rotation g0£". The second equal shearing motion, parallel to Oz', produces the equal and opposite 44 ANALYSIS OF STRAINS. ['JS. rotation £0g, thus bringing the principal axes back to their initial positions, and at the same time shearing the rhombus A'B'CD' into the rhombus A"F'C"D", which will be seen to be identical with the A'B'CD' of Fig. 1. 99.] All these results can, of course, be shown analytically. The equations of displacement for the shearing motion represented in Fig. 2 are manifestly ■ 2« z - -0 J v = 2n, w which may be written v ■■ w = \ v = (s 1 + «,«) V. to = (*, - sjy) Comparing these with equations (17), we see that they represent a shear of amount 2s, accompanied by a rotation - s, about Ox. Similarly a shearing motion of amount 2s, parallel to Oz is represented by »=(*,-*,)4, »=(*! + *> J and is therefore equivalent to the same shear, together with a rotation +s, about Ox. Finally, the case of the last article is to be represented by superposing the two shearing motions u = Q \ m=0 \ v = s 1 zl, v = Q L w = ) iv = s^J the resultant of which is obviously the simple iriotational shear « = 1 100.] Notation for Shears. Similarly, equations (v.) and (vi.) of § 89 represent small simple shears of amounts 2s 2 and 2s s , of planes perpendicular to Oz and Ox, and of planes perpen- dicular to Ox and Oy respectively. We shall generally find it more convenient to use new symbols a, b, c for the amounts of these small shears, reserving «,, 8 t , s 3 for their component elongations and contractions. Thus we shall have a = 2« 1; b = Is.,, c = 2*3. 101.] ANALYSIS OF STRAINS. 43 101.] Finite Shear. The properties of small shear which have been discussed in the preceding Articles are only the limiting forms assumed by the properties of Shear in general, when its amount is indefinitely diminished. Consequently, although they may be accepted as rigorously true for the purposes of our analysis of small strains (§ 58), it is impossible to draw figures which shall answer with perfect accuracy to the descriptions given. The student will find in Appendix II., at the end of this Chapter, a short account of the corresponding properties of Finite Shear, which however have for us only a kinematic interest. 102.] Cubical Dilatation. Of the six component strains we have seen that (i.), (ii.) and (iii.) increase the volume of the body, or of any part of it in the ratios (1 + e), (1+/), (l + <7) respectively, while (iv.), (v.) and (vi.) consist of pure distortions without change of volume. If the volume V of any portion of the body be increased by the strain to V, the ratio (V - V)/V is called the Cubical Dila- tation of the body. This may be either positive or negative : in the latter case, the positive ratio (V- F')/Fis sometimes called the Cubical Compression. We shall always use the symbol A to denote cubical dilatation. 103.] It appears from the last Article that r/F=(l+e)(l+/)(l + «,) = (1 +e+f+g). Hence A = e+/+<7 (35) Since rotation cannot affect the volume, this relation holds equally for the general Homogeneous Strain. It is obvious that the expression for the dilatation should be independent of the directions of the arbitrary axes of reference, and we see by expanding equation (28) that this is the case, - A being the coefficient of in that equation. Hence we may write A = £,+£, + £, (36) 104] Uniform Dilatation. Dilatation is generally accom- panied by distortion ; for (putting shear aside as contributing nothing to dilatation) if e, f, g be different for any system of axes, a sphere in the body will be strained into an ellipsoid, and so on. It is however possible to produce dilatation without dis- 4(i ANALYSIS OF STRAINS. [104. tortion; for suppose the strain such that the three principal elongations are all equal, so that ei = e 2 = £ s = ^A- Any cubical portion of the body with its edges parallel to the principal axes will then have each edge elongated in the ratio (1 + JA), and will remain a cube, the effect of the strain being simply to increase its volume in the ratio (1 +A). 105.] In this case it is obvious from equations (27) that the Elongation Quadric becomes a sphere, and in order that (22) may reduce to the proper form, we must have S] = s 2 = s 3 = / ' whatever be the axes of reference. This strain is called a Uniform Cubical Dilatation of amount A, and, as we have seen, is equivalent to three equal elongations, each of amount £A, in any three orthogonal directions. The equations of displacement are v = \ky I (vii.) i« = JA« J Thus Uniform Dilatation, being expressed by a single co- efficient, is to be (§ 89) regarded as a Simple Strain. Types of Reference. 106.] Summary of Results. We have now shown that the simplest of strains — the Uniform Elongation — is the basis of all the more complex strains : that, in fact, the most general Pure Strain is the resultant of three orthogonal elongations parallel to its principal axes. Further, we have shown that equal elongations (of like or unlike sign) may be so combined as to produce two more kinds of simple strain: namely, a distortion without dilatation or a dilatation without distortion. 107.] Again, it has been proved that the most general equations (31) of Pure Strain may be regarded as expressing it as the resultant of the following six independent simple pure strains : — (I.) An elongation of amount e parallel to Ox. (II.) An elongation of amount / parallel to Oy. PLATE I. Distribution of the Standard Component Strains. {Page 47.) 107.] ANALYSIS OF STRAINS. 47 (III.) An elongation of amount g parallel to Oz. (IV.) A shear of amount a, of planes perpendicular to Oy and Oz. (V.) A shear of amount b, of planes perpendicular to Oz and Ox. (VI.) A shear of amount c, of planes perpendicular to Ox and Oy. The completeness with which these components express the most general pure strain will be realised when it is remembered that, since every set of parallel planes in the body must remain a set of parallel planes, the strain will be completely specified when we can express every possible relative motion of any set of parallel planes. Now, the axes of reference are perfectly arbitrary, and from the preceding articles we can construct the following schedule : — The Symbol denotes Relative Motion Parallel to Axis of of Planes Perpendicular to Axis of e c b c i a b \ 1 9 X X X y y y z X y X y z X y so that any small pure strain can be represented by a proper combination of these six quantities. 108.] Thus the most general equations of Pure Strain really refer it to an arbitrarily chosen system of six orthogonal standard types : namely, three elongations parallel to three arbitrary ortho- gonal axes of reference, and three simple shears of the planes perpendicular to them, the axes of the shears bisecting the angles between the axes of elongation. The most general equations of Rotational Strain (17) refer it to the same six standard types of strain, with the addition of three component rotations about the axes of reference. Plate I. represents the positions of the principal axes of the ■iS ANALYSIS OF STRAINS. [10S. component strains ; Ox, Oy, Oz being the axes of elongation, 0£ and 0£ the axes of the shear a, and so on. 109.] Referring to §§ 32 and 46, we see that these six standard strains satisfy all the requirements of the system of "strain-coordinates" which we set out to seek; they may be chosen arbitrarily, they are perfectly independent, and any small strain can be expressed in terms of them, while they possess the great advantage — in point of simplicity — of vanishing in the natural state of the body. We therefore adopt them as our standard types of simple strain, and, in order to completely specify any given small strain, we have only to enumerate its six orthogonal components in terms of the corresponding standard units. 110.] Type of Strain. When the six standard components of any two strains are to one another, each to each, in the same ratio, the strains are said to be of the same type, or of exactly opposite types, according as this ratio is positive or negative. (See §33.) The ratio of their components is called the Ratio of the Strains, and when this ratio is ±l,the Strains are said to be equal. Strains of the same and of opposite types are also called " con- current " and " contrary." Any number of small strains belonging to two opposite types compound into a strain belonging to one of these types. Two equal and contrary strains exactly annul one another. Specification of Strains. 111.] By equation (34) any number of Pure or Rotational small homogeneous strains can be compounded into one, if we are able to enumerate the standard components of each. Now, every pure strain consists of a uniform cubical dilata- tion, a uniform elongation in some given direction, a simple shear with given axes ; or is compounded of any or all of these (§ 89). We shall therefore be able to form the equations of motion for the most complex combination of pure strains, when we know how to specify each of these simple strains in terms of its standard components. The more general combination of homogeneous rotational ■strains may then be deduced by compounding the rotations separately, as in equations (34). We shall now therefore proceed to show how the specifica- tions of the various simple strains may be separately obtained. The simplest method is by consideration of the Invariants of 111.] ANALYSIS OF STRAINS. U) the Elongation Quadric, which are the coefficients of the dis- criminating cubic (28). Expanding that equation it becomes 3 - \e +/+ g) + (fg - Sj 2 + ge - s 2 2 + ef- s 3 2 ) e, s 3 , « a =0 (37) «» /, *i *2> *1> 9 OT <^ 3 -^ 2 (ei + «i! + «3) + ^( t 2 e 3 + £ 3 £ l + M2)- £ l e 2«3 = ( 38 ) Denoting these coefficients by D, J, K respectively, we have D = e +f+ g = e, + e, + e :i J = fg - s, 2 + ,9e - s./ + ef— .«.,'- = e 2 e, + e 3 €[ + *,e 2 K=\ a, s 3 , s„_ i = «!«„«„ *- (39) s 3> /> *i 112.] Uniform Cubical Dilatation of amount a. This case has been discussed in § 105. The Quadric is a sphere, and the three roots of the cubic (37) are equal. The requisite conditions are 1 9 3 L (40) *1 = So = *3 = J and the equations of displacement are « = *Ay| (41) Conversely, any strain or combination of strains whose com- ponents satisfy (40) amounts to a uniform cubical dilatation of amount A. • 113.] Simple Elongation of amount e in direction (I, m, n). In this case the roots of the cubic are respectively e, 0, 0. Hence it must reduce to 2 (tj> - e) = 0. Thus, 2>-.l ■/ = 0V (42) The two last conditions in combination are easily shown (Aldis' Solid Oeometry, § 91) to be equivalent to either of the sets of three ge-s 2 * = 0) e/--« a a = esj - SoS 3 = 1 .(43) 50 ANALYSIS OF STRAINS. [113. while the first gives us e+f+g^t (44) 111 e /ah\ Sj S 2 S3 SjS-2' V 3 by virtue of (43). Again, I, m, n are the direction-cosines of the only determinate axis (§ 90) of the strain. Hence, by equations (29), el + s 3 m + s 2 w _ s 3^ +f m + s i n _ sJ + h m + 9 n _ £ /4g\ £ m « n Eliminating e, /, g from these equations by means of (43) we get J-jl 3 ( _ + _ + _ \=Zs 1= = ms^ = ns 3 . « Vsj s 2 s 3y / Thus, s x = einn s 2 = enl J- (47) s 3 = dm J whence, by (43), /=«•*[ (48) ? = en 2 j and the equations of displacement are u = tPx + dirty + enlz | v = elmx + tm?y + emnz J- (49) w = mix + emny + erfz J Conversely, if the components of a given strain satisfy (43) it amounts to a simple elongation. Its amount is then given by (44) or (45), and its direction by ls x = »ts 2 = ns 3 = (e +f + g)lmn (50) 114.] A Simple Shear of amount la whose axes of elongation and contraction are in the directions (Z 7 , m,, %,) (Z 4 , m,,9i 2 ). In this case (§ 92) we have e^xr, e 2 = 0, e 3 = - } ( 51 ) and J= -o- 2 (52) Also, by equations (29) e h + *3 TO i + Stfii _ s 3 l x + fm l + s^ sj, y + s 1 m 1 + gn t k m 1 rii ~ = o-^Zj - n 2 l 2 )x + o-^rij — m 2 n 2 )y + ^(n^ - n£)z J Equations (54) and (55) might of course have been deduced by superposition from equations (48) and (49). Conversely, if the components of a given strain satisfy (51) it amounts to a simple shear whose amount la is given by (52), while the directions of its axes are given by (53). In these last equations o- must be taken as the positive root of (51). 115] Resultant of any number of simple strains. We can now form, with the greatest ease, the equations of displace- ment for the most complex combinations of small strains, either pure or rotational. Retaining the notation of the last three articles for pure components, and remembering (§§ 85, 86, 87) that any rotation Q of the body as a whole about an axis (X, fx, v) can be resolved into component rotations Q.\, fi/*, Q.v about the a^8 of reference, and that these rotations are to be compounded separately, the principle of superposition gives us at once for the :r2 ANALYSIS OF STKALXS. [] l.l. standard components of strain and rotation in the single equiva- lent strain e = 2(£A) + 2(6?) + 2(<7Z 1 S - 2 A 2 s 2 + 2\^.^ 3 g' ■■= X 3 2 e + ii 3 y+ v*g + 2 /v . i s 1 + 2i/ 3 A 3 s 2 + 2A 3 /i 3 s 3 »i' = KK e + /V*s/+ VaS' + (/Vs + WsK + ( V A + v aK) s 2 + (V 3 + V 2 ) S 3 < = Vi e + /y*,/+ W + (/Vi + /VsK + ("s^i + "l^K + (Aj^ + A^Sj S 3 ' = W e + /VV + W + (/Vi + AVlK + ("A + V 2 A lK ANALYSIS OF STRAINS. [123. Zu 3d ~dio N dx' /= &y 9 = 3i a = ■2.1 dw 3« ~ ~dy 3« b = = 2s 2 3m = S 4 3io 3a; c- -2s 3 3d "dx 3m "by 3u> 3d 28 l ~dy ' "S 3m 3u> ■ie 2 = Zz" 3a; 3d 3m 2d 3 ~ ~dx '3i/ c>u 3d 3io A ~ 3x 3y ~dz t .(59) and, by § 103, If we give these components their proper values at any point P (x, y, z), the strain of an element of the body described about P will possess all the properties discussed in §§ 59-121, the various surfaces involved being, of course, referred to axes drawn through P parallel to the fixed axes of reference Ox, Oy, Oz. The directions of the principal axes (§65) and the form and dimensions of the Strain Ellipsoid will of course vary from point to point of the body. The Strain Ellipsoid must now be defined as the ellipsoid into which a sphere of unit radius and centre P would be strained, if the strain-components had throughout the sphere their actual values at P. Iwotational Strain. 124.] The conditions that the strain may be irrotational, i.e., that every element may suffer pure strain without rotation of its principal axes, are, as before, 0j=O, 9. 2 =0, 6 3 =0, at every point of the body. Thus, by equations (59), .(60) ~dw 3d dy~ = Wz 3m ~dw dz"' 3a; dv 3m 3sT "^il 124.] ANALYSIS OF STRAINS. 57 These are the well-known conditions that u . dx + v . dy + w . dz may be a perfect differential of some function oix,y, z. Denoting this function by we have udx + vdy + wdz = d, and therefore ~dx' ~~dy' cte ' The function may be called by analogy the Displacement- Potential of Irrotational Strain. It may be any continuous single-valued function of the coordinates, except that (since the origin is supposed fixed) it must not contain any terms of the first degree. Equations (59) may now be written e ~?)x"f-'dy*' 9 ~?>z* B 2 ^ 3^ 3»<£ 1 cyydz' 2 ciz'dx' 3 "dxdy where, as usual, the symbol y 2 denotes the operator .(61) / 3^ j? 3*\ \dx i + dy 2 + 'dzy The condition that the dilatation may everywhere vanish, or that the strain may consist of distortions (shears) only, without change in the volume of any element, is therefore V 2 4> = (62) 125] Resultant Displacement. If we write then U is the resultant displacement of the point P (x, y, z). The direction-cosines of this displacement are u/U, v/U, w/U. But if we describe in the body the system of surfaces whose equations are formed by equating to different constants, and which are consequently called Equipotential Surfaces, the direc- tion-cosines of the normal at P to the equipotential surface passing through P are also u/ U, vj O, wj U. Hence each point of the body is displaced along the normal to the equipotential surface passing through the point. Again, if through P we draw an elementary straight line dv 58 ANALYSIS OF STRAINS. [125. normal to the equipotential surface through P, and if the coordi- nates of its extremity be x + dx, y + dy, z+dz, we have u v w dv=Yj ■ dx+ jj . dy + jj. dz; Udv = udx + vdy + wdz = d. Hence the amount of the resultant displacement at P is dd> U =l ( 63 ) 126.] If is a homogeneous quadratic function of (x, y, z) it is obvious from equations (61) that the strain is homogeneous throughout the body. The equipotential surfaces for Homogeneous Strain are there- fore concentric Quadrics. By Euler's theorem on homogeneous functions we have in this case 32 ,32<£ a 2 ^ .„ a 2 ^ „ 3><£ „ 2>'d> ^ ok 2 ^ oy' oz' J oyoz ozox "oxoy = ex 2 +fy i + gz 2 + 2s x yz + 2s 2 zx + 2s.^cy. Thus (§ 22) in pure homogeneous strain the equipotential surfaces and elongation quadrics are identical. It has already been pointed out (§ 84) that in this case the resultant displacement is normal to the elongation quadric, and this agrees with the result of the last Article. 127.] Lines of Displacement. Since in every irrotational strain the displacement of each point is normal to the equi- potential surface through the point, it follows that, if we draw a system of equipotential surfaces throughout the body, the dis- placements of all points in the body will take place along a system of curves which cut these surfaces everywhere orthogon- ally. These curves are called the Lines of Displacement. If ds be the element of arc (drawn in the positive direction of the axes) of the displacement-curve through P, we evidently have 1 dx 1 dy 1 dz u ds~ v ' ds~ w ' ds' dx __ dy __ dz ~d(f> 3$ 3<£ 'dx 'dy 'dz The function must therefore always be such that it is possible to draw a system of continuous curves cutting orthogon- ally the system of continuous surfaces defined by = constant. PLATE II. Equipotential Cylinders and Curves of Displacement in Simple Shear. (Page 59.) 128.] ANALYSIS OF STRAINS. 59 128.] As a simple example, take the case of a shear in the plane of yz. This is a strain in two dimensions, and the equi- potential surfaces are the rectangular-hyperbolic cylinders yz = constant. Thus the differential equation of the line of displacement through P is dx dy dz ~~ z ~~y' They are therefore the orthogonal rectangular hyperbolas given by = constant ) y* ~z 2 = constant See Plate II., in which the dotted lines represent the curves of displacement, and the entire lines the sections of the equipotential cylinders by the plane of the shear. The directions in which displacement takes place along the curves in the four quadrants are shown by the arrows. Strain in two Dimensions. 129.] It will be useful to collect here the forms assumed by our various results when one, and one only, of the roots of the discriminating cubic (28) is zero. One of the principal elongations (which we will suppose always e 3 ) will then vanish, and the dis- placement of every point in the body (if the strain be homo- geneous) or of every point in a given element of the body (if it be heterogeneous) will be parallel to the plane containing e 1 and e„. The elongation quadrics become cylinders, having this plane for a normal section, and the strain may be said to be wholly in two dimensions. We shall, as before, use the notation of Homogeneous Strain. Taking Oz perpendicular to the plane of the strain, the equations of displacement take the form u = ex + (s — d)y \ v=;{s + d)x+fy) (17 ) or, if the strain be pure, «-« + ^l (3 r, The elongation of the line OP lying in the plane of xy, and making an angle \}s with Ox, is given by « = « cos 2 )/' +fsm' 2 \l' + 2s sin ^ cos \p (18') The conditions that it may be a simple shear are e+f= 8 = If the strain be heterogeneous :!} <-> 3u 3« "dv "bu . 'dv ~bu 3a; 3y' ox ay If the strain be everywhere irrotational udx + vdy = d, where is the displacement-potential. 62 ANALYSIS OF STRAINS. [129. The equipotential curves are given by 0= constant, and the curves of displacement are the orthogonal system. If there be no dilatation anywhere, satisfies 3*' + 3y» = (61) EXAMPLES. N.B. — The factor a is introduced to denote a small quantity whose square and higher powers may be neglected. The expres- sion {e,f, 9, *!, s 2 > h) is used to denote the specification of a strain (§§ 111 et seqq.) • 1. Refer to its principal axes the Elongation Quadric of the strain {3a, -a, - a, 0, 0, 2a}, and hence show that it consists of a simple shear of amount 6a, together with a uniform elongation of amount a perpendicular to the plane of the shear. 2. Show that the strain {0, 0, 0, a, a, a} consists of a uniform cubical compression and a uniform linear elongation, each of amount 3a. 3. Show that the strain {a, a, 0, a, a, a} consists of a shear of amount 2ax/3, a linear contraction of amount a perpendicular to its plane, and a uniform cubical dilatation of amount 3a. 4. Show that the strain {a, 0, 0, a, a, a} is equivalent to a uniform cubical dilatation of amount a, together with three shears /— 4 4 in orthogonal planes of amounts 2a*/2, +^a, —^a; the shears having Of and 0r\, Oq and Of, Of and Of for their respective axes. 5. Prove that the strain {>/ 8 s 2 +K e— /)* in the plane of xy, the axis of elongation making an angle ta,n~ 1 [3sj(e — /)] with Ox. 7. Defining the term " areal dilatation " in analogy with linear elongation and cubical dilatation, show that in a homo- geneous strain a system of quadrics can be described with the origin as centre such that the areal dilatation of any section varies inversely as the square of the perpendicular radius vector. 8. Prove that all planes in the body suffering a given areal dilatation S have for their normals the generators of the cone (8 -/- g)j? + (8-g- e)y 2 +(S-e -f)z* + 1s$z + 2s^xc + 2s B xy = 0. 9. Prove that in the case of § 119 (Hi.), the elongation being perpendicular to the plane of the shear, and all three principal elongations being supposed positive, the direction of the elongation must coincide with the greatest axis of the Strain Ellipsoid if e 1 +e 3 >2e 2 ; and show that in this case °- = i( £ 2- £ 3) [• £ = M 2e l- € 2- £ 3)J ' 10. Show that a simple elongation e parallel to Ox may be replaced by a cubical dilatation e together with two shears, each of amount § e, having Ox and Oy, Ox and Oz respectively for their axes. 11. What is the nature of the strain represented by the equations of displacement u = — wyz ; v = t»zx ; w = 1 12. Find the volume and the moments and products of inertia of a sphere of radius R, originally homogeneous, after under- going the strain represented by u = ax + a'a; 2 v=py+py-\ w = yz + y's 2 13. Prove by combining equations (29) and (61) that if one of the principal axes at each point is normal to the equipotential surface through the point, then either = F(ax + fix + yz + 8) ) or 4> = ^) r where r* = x 2 +y 2 + z i , a, ft, y, and S are constants, and F is any function which makes dip/dx, d

/dz vanish at the origin. What strains do these forms of ^ represent ? 134 ANALYSIS OF STKAIXS. The equations may be written 1 Wl-L W- 1 '* Ui -\(, av \ 2u' ?x 2v' "ay 2w' tz v yh Thus U. dU~\ .dtp. Assume A . dU=-\ . do). Then U . dto = Ad' .'. when p *' - we also have ^ = ^=^-0. Thus *=.?». r c* ?.'/ ?2 Now ?«.£-W.to. ( and x are conjugate functions of x and y (§ 245). 15. Prove by equating the values of X', fi , v given by equa- tions (19) to X, n, v that, in any homogeneous strain, there is always one and may be three straight lines through every point of the body which retain their initial directions. Show that the elongations in these directions are the roots of the cubic e-4>, * 3 - 3 > *2 + e 2 i = °- h+ e v /-•£> «i-0j \ s s -0 2 , s l + 6 v g- Hence show that, when all the roots of this cubic are real, these three directions are orthogonal, and 0=0=0=0. • 16. Show that the integral f(udx + vdy + wdz) taken round any closed curve in the body is zero if the strain be irrotational, and is otherwise equal to ANALYSIS OF STRAINS. 65 where dS is an element of any surface drawn within the body and having for its edge the given closed curve ; A, fn, v being the direction-cosines of the normal to the element. ' 17. Show from equations (59) of § 123 that the integral ff(\e i + p6 2 + vd 3 )dS, taken over any closed surface drawn within the body, is identically zero. APPENDIX I. On the Geometry of Strains. All physical quantities may be broadly classified into two categories called respectively Scalar and Vector. A scalar quantity involves no conception but that of magnitude, but the characteristic property of all Vectors is that they involve the idea of direction as well as that of magnitude. This broad distinction includes under the head of Vectors several classes of quantities which differ from one another in their degree of definition, as we shall presently explain. They may all be assigned to one of two divisions : — li/near and angular vectors — which we shall discuss separately. Linear Vectors. (Displacement, Velocity, Elongation, Force, &c.) The most perfectly defined linear vector, which may be called a motor, involves the specification of five characteristic elements. (i.) Its magnitude, which it has in common with scalars, and which is expressed by a scalar or numerical factor multiplying its purely vector or directed factor, and denoting its ratio to an arbitrarily chosen unit vector with which it is in all its other properties identical. This factor is called its Tensor. (ii.) Its direction, or that of a family of parallel straight lines in space, along any one of which it may be supposed to act. (iii.) Its way of acting along these lines, which is analytically expressed by an arbitrary convention as to its algebraical sign, so that, if a vector acting in one way is considered positive, a vector acting in the directly opposite way is considered negative, the two vectors being otherwise identical. 66 ANALYSIS OF STRAINS. (iv.) Its position in space, or the particular line of the family along which it may be supposed to act. (v.) Its origin, or the particular point in this line from which it is to be reckoned, or at which it is to be applied. The following are good examples of motors : — (1) A given displacement of a given point in a given direc- tion. (2) A force of given magnitude and in a given direction actiDg at a given point of a body. The component displacements, parallel to arbitrary rectangular axes, of each point of a strained body are of course vector quanti- ties, but if the body be left free in space they are highly imperfect vectors ; the reason being that such a strain does not specify the absolute displacements of points in the body, but only their relative displacements in given directions. Consequently we are only given (ii.), (iv.), and (v.), while (i.) and (iii.) are quite indeterminate. Vectors of this nature, which can be taken in either way so as to satisfy the specified conditions, are called Dipolar. If, however, we determine in any abritrary way the absolute displacements of any one point in the body, it is obvious that we thereby raise the component displacements of all points in the body to the rank of perfect motors. The simplest condition to impose is of course that one point in the body shall remain fixed, and since this assumption cannot affect the strain, while the analytical advantages of increased simplicity and definition are so obvious, we shall always avail ourselves of it. As an analytical example let us take the simple case of a uniform elongation of all lines in the body in the direction Ox. If e be the amount of the elongation, and x v x 2 , x 3 ..., «,', xj, x a '. . . the initial and final abscissae of any number of points in the body, the only condition to be satisfied is that the projections (x 2 — xj, (x 3 -x^)... upon Ox of the distances between these points are to be increased in the constant ratio (1+e). We thus obtain a group of equations of the form X2-x 1 ' = (l + e)(x 2 -x 1 ) or u 2 -u 1 = e(x i -x 1 ). The solution of this group is of course u - ex = constant, or u = ex-C, where the constant C may be of either sign and of any magnitude whatever. Let x', x" be the abscissae of those points of the body which are nearest to and farthest from the plane of yz. (i.) If we take C< ex', u will be positive for every point in the body. ANALYSIS OF STRAINS. 67 (ii.) If we take ex'ex", u will be negative for every point in the body. All these solutions obviously satisfy the conditions of the strain. It is clear that (ii.) amounts to regarding a plane in the body as fixed in space — namely, that for which x = C/e. If we take this for the plane of yz, G= 0, and the equation of displacement becomes u — ex, and u is now a perfectly denned motor. The simplicity of this solution points to the advantage (much greater of course in more complex strains) of regarding one point in the body as absolutely fixed, and taking that point as the origin of our arbitrary axes of reference. Angular Vectors. (Rotation, Angular Velocity, Couple, &c). As an example of the most perfectly defined class of angular vector, which may be called a Rotor, we shall consider a simple rotation about a given axis. Its specification includes (i.) Its magnitude. (ii.) The direction of its axis, or the direction normal to the system of parallel planes in which the displacements take place which constitute the rotation. (ivi.) The way of the rotation, which is expressed by an arbitrary convention as to algebraical sign (see below). (iv.) The position of the axis, or the particular line in the body, drawn in the direction defined by (ii.) which remains at rest. (v.) Its origin, or the initial position of any plane in the body through the axis of rotation, from which we measure the angular displacement. An ordinary Couple is a good example of an imperfect Angular Vector, for it may be moved about in any manner in its own or any parallel plane without altering its effect. In fact we can only specify its magnitude, the direction of its axis, and its way. The convention as to the way of angular vectors (e.g., rotations) is as follows : — Taking the coordinate axes always in their cyclical order — xy, yz, zx — a rotation about any one of these axes in the direction 68 ANALYSIS OF STRAINS. from that axis which comes next towards that which comes last in the cyclical order is reckoned positive, a rotation in the reverse direction being reckoned negative. A positive couple is one which tends to produce a positive rotation, and so on. In all branches of Physics but one the directions in which we suppose the axes drawn, with reference to their cyclical order, is quite indifferent ; but, in order to secure uniformity of notation, it is desirable to adopt in all cases that already employed in Electromagnet- ism, in which the positive direction of rotation about either axis bears the same relation to the positive direc- tion of translation along it as does the rotation to the trans- lation in the case of an ordinary " right - handed " screw (Fig. 4). This is also sometimes called a " counter- clockwise " rotation, from the fact that if one of the co- ' B< ' ordinate axes be drawn out- wards from the centre of the clock-face, the positive direction of rotation is contrary to that of the hands. Now if a body left free in space is subjected to a strain accompanied by a rotation of given small amount fi and with its axis in a given direction (X, p., v) it follows from the purely relative character of the displacements specified in the strain that those portions of them due to the rotation will be given (like those previously discussed) by a group of equations of the form u 2 - Uj = fin(z 2 - Zl ) - vi% 2 - Vl ) I V 2 - V l = 'flfej - X l> - ^(h-Zl) [ w 2 - w, = A£% 2 - y x ) - p£t( Xi - xj) the general solution of which is u = fjSlz — v£ly + A \ v = vQx — kQz + £ V w = tely - fiSlx + C] where A, B, C are purely arbitrary constants. In other words, a small rotation of the body as a whole about any axis may be reduced to a small rotation about any parallel axis, by the superposition of a suitable linear displacement of the body as a whole. ANALYSIS OF STRAINS. 69 Such a displacement does not affect the strain, and therefore, so far as the conditions of the strain go, the position (iv.) of the axis of the rotation is completely unspecified, and with it the amounts of the component displacements. Hence, as before, in order to transform the strain-rotation into a complete Rotor, we assume the point of the body coincid- ing with the coordinate origin to remain at rest ; an assumption which clearly amounts to determining that all axes about which the body can rotate must pass through the origin. APPENDIX II. On Finite Shears. A simple finite shear consists of a uniform elongation of all lines in the body parallel to a given axis Og, accompanied by a contraction in the reciprocal ratio of all lines in a perpendicular direction Otj, lines parallel to Of retaining their initial lengths unaltered. Thus lines of unit length parallel to 0£ Or/, Of respectively become lines of lengths a, 1, 1/a, where a is a finite quantity greater than unity which is called the Ratio of the Shear. The displacements parallel to the principal axes are given by i)+ Y=t) J- Hence if the point (£, j;, f) be displaced to (f ', i/> f ) Thus the equation of the Strain Ellipsoid is £'2 and its semi-axes are a, 1, a -1 . Fig. 5 represents the principal section in the plane of f£ of the ellipsoid, and of the unit sphere from which it is derived ; the mean axis (which retains its unit length) being perpendicular to the plane of the paper. Since the radius of the sphere and the mean semi-axis of the ellipsoid are both of unit length, the common sections of the two surfaces are the Circular Sections of the ellipsoid. These are the two planes through 0>j, whose lines of inter- section with the plane of the paper are the common radii A'OC\ BOD'. 70 ANALYSIS OF STRAINS. Since these sections remain great circles of the sphere — and therefore retain their original form and dimensions — it follows from the properties of Homogeneous Strain that the two systems ? ~^Z--" Q ^\^ / 2 + f 2 , ov by £'±af' = 0. Consequently their positions in the unstrained body are given by «£±C=0. Let these cut the plane of the section in AOC, BOD. Then AO£ =D0£ = tan-ia, 4'0Jf = .D'0£ = tan-i(l/a). The effect of a simple finite shear is therefore to change that angle between the two systems of undistorted planes which is bisected by the plane of fy from 2tan _1 a to 2tan - Vl/a). ANALYSIS OP STRAINS. 71 The angle AOA' through which any one of these planes is turned is obviously tan -1 £(a — a' 1 ). It is clear that any rhomboidal prism, such as PQRS, bounded by undistorted planes, is strained into an equal and reciprocally — similar rhomboidal prism FOUR'S', by a simple interchange of the angles and diagonals of its transverse section. To represent the effect of a finite shear by a Finite Shearing Motion we must there- fore take any such v \(' rhomboidal prism, and X^ \ y ; — holding fixed one of its mesial planes BOD — cause all the undis- torted planes of the same system to move parallel to it, each through a distance pro- portional to its perpen- dicular distance from the fixed plane, until each angle of the rhom- bus has been changed into the supplementary angle. Let PQRS, rQ'R'S' (Fig. 6) be the initial and final forms of the rhombus, and let AOO, A'OC be the initial and final positions of its other mesial plane ; ON being perpendicular to PQ. We have angle AOB = 2 tan _1 (a _1 ), AON= A'ON= tan-^a - a-'). Now if A be the Amount of the shearing motion (or the ratio of the displacement of any sheared plane to its perpen- dicular distance from the fixed plane), A=AA'/0N=2. AN/ON. Thus A= a -a~\ Again, angle POP = angle QOQ' = tan -1 (a) - tan~*(a~ ? ) = tan -1 J(a - a -1 ). Thus, finally, we see that a simple irrotational shear of ratio a may be replaced by a shearing motion of amount A = a— d~ l , together with a backward rotation of the body as a whole through an angle tan-»(|4 ) = tan"^(a - o" 1 ). Fig. e. 72 ANALYSIS OF STRAINS. To apply these results to the limiting case of an infinitely small shear (§§ 95-98) we have only to write ) y"' = xS sin 2 + y(2 - S cos 2) ) ' x" = x'" cos 8 - y"' sin S ) _ y" = x'"sm8 + y'"cQsSr And, finally, - a;" = a:[2cos8 + ^cos(2^ + S)]-2/[2sinS-5sin(2(/. + S)] ) y" = a:[2 sin S + S sin (2<£ + 8)] + y[2 cos S - S cos (2 + 8)] I ' In order that these two values for (x", y") may be identical for all values of x and y we must have 2 cos 8 = z\ \Z^-\dx .^)dydz all acting in the negative directions of the axes (§ 137). The arrowheads in Fig. 8 denote the directions of the component forces at the centre of each face. These force-components on the two opposite faces perpen- dicular to Px' or Ox together amount to component forces -~-p . dxdydz 37, -~- . dxdydz dZ. -~- . dxdydz on the element in the positive directions of the axes, and com- ponent couples Z x . dxdydz in the negative direction about Py', and Y 1 . dxdydz in the positive direction about Pz'. F 82 ANALYSIS OF STRESSES. [140. 140.] Similarly, if the components of the stress across the small plane area A 2 B 2 C 2 D 2 , drawn through P perpendicular to the axis of y, he X 2 , Y 2 , Z v the total stresses across the pair of opposite faces EHJK and FGML together amount to com- ponent forces -~-^ . dxdydz 3F 2 j j, j -^— . dxdydz 3 ^2 j j j -~-^ . dxdydz in the positive directions of the axes, and component couples X 2 . dxdydz in the negative direction about Pz', and Z 2 . dxdydz in the positive direction about Px'. Lastly, if the components of the stress across the small plane area A^B Z CJD S , drawn through P perpendicular to Oz or Px', be X 3 , F 3 , Z B , the total stresses across the pair of opposite faces EKLF and HJMG together amount to the component forces -~— 3 . dxdydz i. dxdydz dZ s -rr— . dxdydz in the positive directions of the axes and the component couples Y 3 . dxdydz in the negative direction about Px', and X s . dxdydz in the positive direction about Py'. 141. J Conditions for Equilibrium of the Element. It is sufficiently obvious that when the body is in equilibrium in any given state of strain, any portion of it may be supposed to become rigid in that state [compare § 30 (i.)] without affecting its own equilibrium, or that of any other portion of the body. Thus the conditions for equilibrium of the element under consideration must be precisely the same as if it were a rigid body at rest under the actual stresses and applied forces. 141.] ANALYSIS OF STRESSES. 83 Now if p be the density of the body at P, and X, Y, Z the intensities at P of the component applied forces per unit mass, it follows from § 138 that the components of the applied force on the element may be taken to be pX . dxdydz\ p Y . dxdydz j- pZ . dxdydz) and that they may be supposed to act at its centre P. Collecting the results of the last three Articles we see that the element is subject to component forces • pX \dxdydz Yldxdydz parallel to the coordinate axes, and to component couples [Z 2 — Y^dxdydz \ a / [X s - Z^dxdydz [ ^" clc ' [Y 1 -X^dxdydz] about these axes, respectively. The conditions of equilibrium of the element are therefore expressed by the six equations 3X[ 3X 2 dX 3 -dx~ + ldy- + ^ + P X 3a3 "by ~dz L ox + "by + oz +f> -Y- iff oz oY 1 oY„ dY 3 . ox oy ' ' oz •(1) ox oy ~dz ' Z 2 -Y a = 0) (2) *8-*l = 0f Y.-X^OJ 142.] Simplification of Notation. Equations (2) will be satisfied, and our analysis much simplified, if we adopt the new notation formed by writing X, = P, r,-o, Z.^E, Z 2 = 1 3 = S, a g = Z j = T, Y^ = X 2 = U. 84 ANALYSIS OF STRESSES. [142. The general equations (1) of equilibrium then become ox ay oz ~dx 'dy ^ " ■dz ~bx ~oy oz ■(3) where X, Y, Z are the components of the applied force per unit mass at (x, y,z), p is the density at the same point, and the other symbols are best explained by the following schedule : — The Symbol denotes the Stress-Component Parallel to Axis of across a small Plane Area drawn through (x, y, z) Perpendicular to Axis of P Q R S T U X y z {I X y z 1 y> „} Z 1 y\ X ) 143.] Equations of Motion. If the body, instead of being in equilibrium in a given state of strain, be in process of strain- ing — i.e., if any relative motion of its parts is taking place, the component forces of § 141, instead of vanishing, must be equal to the components of the " effective " force on the element, which, if x, y, z be the component accelerations of P, are px . dxdydz\ py . dxdydz I. pi . dxdydz) Since the effective couples involve the Moments of Inertia in place of the mass of the element, they are always indefinitely small in comparison with the effective forces. 143.] ANALYSIS OF STRESSES. 85 Hence equations (2) are still very approximately true, and the equations of motion are (4) In these equations, since u, v, w are the variable portions of the coordinates of any point, we may obviously write u, v, w instead of x, y, z, whenever the former will be preferable. 2? T Fig.G. 144. J Resolution and Composition of Stresses. The six quantities, P, Q, -fi, S, T, U are the normal and tangential components of the stresses across the three small orthogonal plane areas drawn through any point P (x, y, z) of the body perpen- dicular to Ox, Oy, Oz respectively. The fact that these six quantities are the only stresses involved in the equations of equilibrium and of motion suggests that we may be able to adopt 86 ANALYSIS OF STRESSES. [144. them as our standard system of stress-components, and to express in terms of them the stress across a small plane area drawn through P in any direction whatever. From P draw, as in § 138, Px', Py', Pz' parallel to Ox, Oy, Oz, and cut off from the body an elementary right-angled tetrahedron PABC, having for its base any oblique plane which will cut Px, Py', Pz in points A,B,G, such that the edges PA, PB, PC are all positive in direction. Let X, p, v be the direction-cosines of the normal to the base, directed outwards, or away from P ; then X, p, v are all positive quantities. Let A be the area of the base ABC, and p its perpendicular distance from P. Then £pA is the volume of the tetrahedron, and XA, ^A, i y, «)• If F, 6, U be now taken to be the components of the Surface Traction at the point, the conditions of equilibrium of the tetra- hedron must be the same as those just investigated, and (proceed- ing to the limit in which the vertex of the tetrahedron moves up to the surface) equations (5) will represent the relations which must exist between the components of Surface Traction and the orthogonal Stress- components at each point of the surface. The general problem in the Mathematical Theory of Elasticity (see § 135) is to find solutions of equations (3) or (4), for the stress-components throughout the body, which will also satisfy equations (5) at every point of the bounding surface. The solution will not be complete until we know the relations between Strain and Stress, so that we can find the alteration of form and volume of any element of the body. These relations will be investigated in the next chapter. 146.] Equilibrium of the body as a whole. It may be observed that the solution of equations (3) and (5) converse to that just proposed — namely, given the distribution of Stress throughout the body, to find the distribution of Applied Force and Surface Traction required to maintain it — is always obtain- able. For when P, Q, R, S, T, U, p are known as functions of x, y, z, these equations give explicitly the appropriate values of X, Y, Z, F, G, H. Now we have shown (see §§ 133, 134, and compare § 29) that the applied Forces and Surface Tractions (considered together in Chapter I., under the head of " External Forces ") are the only forces which can be considered as acting upon the body as a whole, and it follows from the considerations of § 141 that, when the body is in equilibrium in a given state of strain, these two systems of forces must be so connected as to satisfy the ordinary conditions of equilibrium of a rigid body. 88 ANALYSIS OF STRESSES. [146. We ought then to be able to show that the values of these forces given by (3) and (5) satisfy the analytical conditions f/fpXdxdydz +ffFdS = O' fff P Ydxdydz+ffGdS = U - (6) fffpZdxdydz +f/HdS = fffp(yZ-zY)dxdydz+ff(yH-zG)dS =0~ fffp{zX-xZ)dxdydz+ff{zF -xH)dS=0 (7) fffp{*Y- yX)dxdydz +ff(xG - yF )dS=0 obtained by equating to zero the component forces and couples acting on the body as a whole ; the triple integrals being taken throughout the volume of the strained body, and the double integrals over the whole of its bounding surface. Now we have by equations (3) 3a; ~dy ~dz fffpXdxdydz= -Jff% + %+'%)^-> or, integrating by parts, = -ff[P]dydz-ff[lT\dzdx-ff[Tyixdy, where the square brackets [ ] denote that the enclosed term is to be taken within proper limits. Hence if X, n, v be the direction-cosines of the outward normal to the element dS f/fpXdxdydz = -ff(P^ +Up. + Tv)dS, and therefore by equations (5), f/fpXdxdydz +//FdS= 0. In the same manner it may be shown that the second and third of equations (G) are satisfied. Again, by (3), fffp{yZ ~ zY)dxdydz =ff[zU-yT]dydz +ff\_zQ - y S]dzdx +ff[zS-yli\dxdy =Jf{HzU-yT) + p.(zQ - yS) + V (zS-yR)}dS =jy{z(\U+nQ + vS) - y(\T+p.S+ vM)}dS. 146.] ANALYSIS OF STRESSES. 89 Hence, by equations (5), fffp(yZ - zY)dxdydz +//(yH - zG)dS = 0. Similarly we can prove the remaining two of equations (7). Thus the values of the components of Applied Force and Surface Traction given by equations (3) and (5) satisfy identically the conditions of equilibrium of the body as a whole. This result verifies the statement of § 130 that force is passed on through the body from layer to layer by appropriate stresses, so that the external forces are ultimately brought into opposition with one another as if the body were rigid. 147.] If the body, instead of being in equilibrium through- out, be in process of straining, then taking the values of the components given by (4) and (5) it is easy to show by a similar method that the expressions on the left-hand side of equations (6) and (7), instead of vanishing, are equal to the effective forces and couples. /jjpxdxdydz fffpydxdydz ■ Jjjpzdxdydz fffpky* - zy)dxdydz fffp(zx — xz)dxdydz ffff>(xy - yx)dxdydz Types of Reference. 148.] The Three Normal Stress Components. Reason- ing as in § 132, with the modification introduced in § 135, we see that (i.) The normal traction P, across the small plane area drawn perpendicular to Ox, tends to produce an elongation in the direction of Ox of the neighbouring portion of the body. Thus (§ 33) the function of the Simple Stress P is to produce and maintain the Simple Strain e. (ii.) The normal traction Q, across the small plane area drawn perpendicular to Oy, tends to produce an elongation in the direction of Oy. Thus the function of the Simple Stress Q is to produce and maintain the Simple Strain /. (Hi.) Similarly the function of the Simple Stress R is to produce and maintain the Simple Strain g. These statements must only be taken as pointing out the 90 ANALYSIS OF STRESSES. [14S. analogies between the components of Strain and Stress, and the primary and most obvious functions of the latter. We are not at present concerned with the exact relations of Stress to Strain, and we must not take it for granted that any one stress-com- ponent can produce the analogous component strain alone with- out producing a simultaneous change in the other components. 149.] The Tangential Stress Components. The tan- gential traction S, in the positive direction of Oy across the small plane area drawn perpendicular to Oz, clearly tends to drag in that direction the layer of matter immediately in contact with the negative side of the area relatively to the layer in contact with this one on its negative side. And if we consider that this action takes place across every small plane area drawn perpendicular to Oz in the neighbourhood of the point P it is evident that the tendency of this tangential traction is to produce and maintain the Shearing Motion described in § 95 and represented in Fig. 2 — that is, a positive shearing motion parallel to Oy of planes perpendicular to Oz. This will be perhaps ■'■ more obvious on consult- ing Figure 10, which re- presents the action in the plane of y'z' on the elementary cube having P for centre. For the sake of distinctness only the traction - couples /'about Px' are inserted, the normal components and those portions of the tangential tractions which combine to form a force on the element (§§ 139, 140) being omitted. The couple due to the traction we have just considered is marked S r Similarly, the equal tangential traction S in the positive direction of Oz across small plane areas perpendicular to Oy, gives rise to the equal and opposite traction-couple marked /S 3 ; the tendency of which is to produce a positive shearing motion parallel to Oz of planes perpendicular to Oy. Now (§ 98) these shearing motions are rotational strains, compounded of identical shears and opposite rotations. The tendency of the two traction-couples in combination is to produce these two shearing motions simultaneously, and therefore (see F'g.lO. 149.] Analysis of stresses. 91 § 98 and Fig. 3) to produce a simple irrotational shear of planes perpendicular to Py' and Pz', or to Oy and Oz. (iv.) The two tangential stress-components S at the point P are therefore, when considered in combination, called a Shearing Stress of amount 8 in the plane of y'z'. (v.) In like manner, the two tangential tractions T combine to form a Shearing Stress of amount T in the plane of z'x' ; and (vi.) The two tangential tractions U combine to form a Shearing Stress of amount U in the plane of ctfy. The primary function of these three component stresses is then to produce and maintain the three component shears a, b, c respectively. (See remarks at end of § 148). 150.] Resolution of Shearing Stress. Just as we proved that all the simple strains — and therefore any strain whatever — can be resolved into simple elongations and contractions, so we may show that every stress may be regarded as the resultant of normal or longitudinal tractions or pressures. This will follow from the preceding Articles if we can — prove it for a single shearing stress. Let us then suppose the body held in a given state of strain, such that all the standard stress- components at the point P vanish, except the shearing stress S in the plane of y'z'. Equations (5) D Fig. II. then give us for the stress-components across a small plane area drawn through P in any direction (X, fi, v) F=0 G = Sv H=Sp, Now if the stress across any such area be wholly normal we must have F/\=G/fi.=H/v. Substituting, we see that two planes can be drawn through P, such that the stress across them is wholly normal, their direction- cosines being respectively (0, 1/^2, 1/^2) •"»* (0. " V*A V*/*)i and for both these planes 92 ANALYSIS OF STRESSES. [150. Thus the shearing stress of amount S in the plane of y'z' is equivalent to a normal traction of amount S across the plane bisecting the positive angle between those of x'y' and z'x', together with a numerically equal normal pressure across the plane bisecting the negative angle between the same planes. 151.] The same result may be obtained by considering the equilibrium of the prism ABD (Fig. 11) one of the halves into which the cube of Figure 10 is divided by the diagonal plane BD. It is obvious that we may regard this prism as isolated if we suppose the existing stresses still to act across its faces. Now, if a be the elementary length of either edge of the cube, the areas of the faces AB and AD are each a 2 , while that of the face BD is a 2 /v/2. Also the forces on the prism due to the tan- gential tractions on the former faces are each S . o 2 in the positive directions of Py' and Pz'. The force across the face BD must therefore have equal com- ponents in the negative directions of the axes. That is to say, it must be a normal traction of intensity S . a* . Jtya* J2 or S. Similarly by dividing the cube along the plane A C we can show that the stress across this plane is a normal pressure of intensity S (Fig. 12). 152.] Discrepancy in the measurement of Shear and Shearing Stress. Although the methods of resolution of shear and shearing stress are thus completely analo- gous, there is a discrepancy between the measurement of shear in terms of its component elongation and contraction, and that of shearing stress in terms of its component normal traction and pressure which should be carefully Fig. 12. noted. Thus the amount of a shearing stress compounded of a normal traction and pressure S is taken to be S, while the amount of a shear compounded of an elongation and contraction 8j is taken to be 2s x or a (see § 100). The discrepancy exists solely in the nomenclature adopted 152.] ANALYSIS OF STRESSES. 93 and, though on some accounts to be regretted, is not a serious defect. The results of §§ 148, 149 are collected in the subjoined schedule for comparison with that of § 106. tends to produce of Planes The Component Relative Motion Perpendicular to Parallel to Axis of Axis of P X X U X y T X z u y X Q y y s y z T z X S z y K z z 153.] Small Stresses. All the properties of stress hitherto proved are equally true, whatever be the magnitude of the stress. Our analysis is however to be applied to the theory of bodies suffering infinitely small strains, and we shall therefore from this point restrict it to such stresses as are required to produce and maintain these strains. Now we cannot suppose anything short of an absolutely rigid body to offer infinite resistance to a finite strain ; and since it is a matter of experimental observation that the straining of the more perfectly elastic natural solids increases continuously with the stress applied, within the limits of their elasticity, we may safely assume that, if we adopt a finite unit of force per unit area, the numerical measures of the stress-components will always be of the same order of dimensions as those of the components of the strain which they serve to maintain. For the purposes of our theory we may therefore always assume the components of stress to be small quantities of the first order (§ 58) whose squares and higher powers, together with their products with each other or with the strain-components, may be neglected in comparison with their first powers. Now, strictly speaking, equations (3) of § 142, (4) of § 143, and (5) of § 144 express the relations between the forces across the faces of an element of the body which, in the given state of strain, is either a rectangular parallelepiped with its edges parallel to the fixed coordinate axes, or a tetrahedron with three edges parallel to the same axes. But, from what has just been said, •>4 ANALYSIS OF STRESSES. [153. it follows that if the strain be a " small strain," and consequently the stress a " small stress," these relations will, to our order of approximation, take precisely the same form if we suppose the stress-components holding in equilibrium, in a given state of strain, elements of the body which have these shapes in their natural state. 154.] For instance the element of the body which, in the natural state, is the rectangular parallelepiped of § 138, becomes when strained an oblique parallelepiped, the areas of whose faces are (1 +/+ g)dydz, (1 + g + e)dzdx, (1 + e +f)dxdy, while its volume is (L + A)dxdydz, and its density is (1 — A)p. Hence it is easily shown that the first of equations (3) § 142 would in this case become aP ?>U dT ^(l + f + g) + ^(l + g + e) + ^- z (\ + e + f) + P X = and by the last article this reduces to ■dP dU -dT v n ox ay oz r as before. Again, the element of the body which in the natural state is the tetrahedron of § 144 becomes when strained an oblique-angled tetrahedron, the areas of whose faces are A(l + e +/+ g - e)) AA(1+/ +S ) H&(l+g + e) vA(l+e+/) where A here denotes the unstrained area of the face ABC, and e is the elongation in the initial direction (A, ft, v) of the normal to this face, given by equation 18 of § 72. The first of equations (5) § 144 ought therefore, on this assumption, strictly to be F(l+e +/+ g - e) = PA(1 +f+g) + U l i(l+g + e) + Tv{\ + « +/), which, to our order of approximation, is identical with F=PX+U/ji. + Tv. We may therefore, in all cases, take the system of standard stress-components which we have adopted as acting normally and tangentially across small plane areas through the point (x, y, z) which, in the natural state of the body, are perpendicular to the 154.] ANALYSIS OF STRESSES. 95 fixed rectangular axes of coordinates: and though they are always taken parallel to the fixed axes, yet the strain they are capable of producing is so small that they may be considered to be normal or tangential throughout the process. 155.] Principle of Superposition. Since the stress-com- ponents are simply the components, resolved in fixed directions, of force per unit area, it is at once obvious that any number of stresses, applied simultaneously, must have for their resultant a single stress whose components are the algebraic sums of the corresponding components of the constituent stresses. Moreover, it follows from the last Article that the application of any small stress will have the same effect, whether the body is in its natural state or has already been strained by another small stress. This principle may be extended to any finite number of small stresses, the strain produced being still small (§ 87), and finally we see that the resultant of any number of small stresses, applied simultaneously or successively, is a single small stress whose components are the algebraic sums of the corresponding com- ponents. And conversely (compare § 88) any small stress may be arbitrarily resolved into any number of small stresses, subject only to the above condition as to the sums of their components. This result is called the Principle of Superposition of Small Stresses. 156.] Type of Stress. When the six components of any two stresses are to one another, each to each, in the same ratio, the stresses are said to be of the same type, or of exactly opposite types, according as this ratio is positive or negative. (Compare § HO). The ratio of their components is called the Ratio of the Stresses, and when this ratio is ± 1 the stresses are said to be equal. Stresses of the same and of opposite types are also called " concurrent " and " contrary." Any number of small stresses belonging to two opposite types compound into a stress belonging to one of these types. Two equal and contrary stresses annul one another. 157.] Homogeneous Stress. When the components of the stress have the same values at all points of the body, it is said to be homogeneous ; and the body is said to be homogeneously stressed. It is obvious from equations (3) that a body cannot be in equilibrium under homogeneous stress if there are any Applied Forces. 96 ANALYSIS OF STRESSES. [157. Homogeneous stress must therefore be produced by Surface Tractions only, the components of which must satisfy a certain condition at every point of the body. For if we write P, U,T U,Q,S (8) T, S, R and denote by p, q, r, s, t, n, the minors of the determinant St corresponding to P, Q, R, S, T, U, we get from equations (5) the relations t*=(rxF+qG + BH)l-&\ (9) v = (tF+ s G + vH)f§i\ to be satisfied by the direction-cosines of the normal at each point of the surface. Squaring and adding, we eliminate A, fi, v, and obtain ( V F+nG + tH^ + {aF+qG + eHf+(tF+sG + tH) 2 = '§, i (10) Since in the case of homogeneous stress all the coefficients are absolute constants, equation (10) represents a definite relation existing between the components F, 0, H of the Surface Traction at each point of the bounding surface. 158.] Stress to be treated as Homogeneous. In the following investigation into the graphic and other properties of Stress, we shall for the sake of simplicity treat it as if it were homogeneous ; because then, its character being identical through- out the body, we can confine ourselves to the consideration of its properties at the origin. Of course it will be understood that, as in the case of Strain (§ 122), the results obtained will be equally true for an elementary portion of a body under heterogeneous stress, described about any point P, if that point be taken as the origin of relative coordinates, and the stress-components be given their proper values at P. These applications will be pointed out as occasion requires. Graphic Properties of Stress. 159.] Change of Axes of Reference. Let P, Q, R, S, T, U be the components at the origin of a given stress, referred to the arbitrary system of rectangular axes Ox, Oy, Oz: required, in terms of these, the corresponding components P', Q', R', S', T, V of the same stress referred to any other arbitrary system of rect- angular axes Ox', Oy', Oz'. 159.] ANALYSIS OF STRESSES. !>7 Let the direction-cosines of the new axes be given by the schedule : — Then P', U', T are the components parallel to Ox', Oy', Oz' of the stress across the small plane area drawn through the origin perpendicular to Ox'. But by equa- tions (5) the components parallel to Ox, Oy, Oz of this stress are PA X + tf/ij + 2Vj UX 1 + Qfr + Sv x TX 1 + Sfr + R Vl x' y 1 z' X K \ K y h. /*2 Ms z v i "2 V S .(11) Thus .(12) P = Aj(PA, + U lh + 2V X ) + l i. 1 (UX J + Q^ + SvJ + v l {TX 1 + S^ + Rv r ) V = X 2 (PX X + Ui^ + TvJ + ti 2 (U\ + Q^ + Svj) + v 2 (TX 1 + Sft + BvJ etc., etc. Thus we finally obtain, by rearrangement of terms F = PX l * + Qp* + Rv* + 2Swi + 2? Vi + Z^Vi' Q' = PA/ + Qtf + R v * + 2Sfi 2 v 2 + 27V 2 A 2 + 2 UX.^ R' = PA 3 2 + Qfi^ + Rv 3 2 + 2Sfi 3 v 3 + 2Tv 3 A 3 + 2 UX il i 3 S' = PA 2 A 3 + Qptp s + Rv 2 v 3 + SQt 2 v 3 + /x 3 v 2 ) + ^("2 X 3 + V 3 A 2) + U ( Vs + V 2 ) T' = PAjAj + Qp sfh + Rv sVl + S(ji s v 1 + ^i/g) + ^("3 A l + "A) + U ( Vl + Vs) V = PAjA 2 + Q/ij/ig + Rvjv 2 + S^vg + /ijvj + ^VjAj + v 2 A,) + ^(Aj/n 2 + Aj/Bj) Adding together the first three of these equations, we get F + Q' + R' = P+Q + R (13) and since both systems of axes are completely arbitrary, this proves the perfectly general theorem that The sum of the normal components of stress across any three small orthogonal plane areas drawn through a given point of the body is absolutely constant for that point, however the planes be turned about it; and in the case of homogeneous stress, this constant has the same value at every point of the body. 160.] Resultant Stress. Let A,B,G denote the resultant stresses across the small plane areas drawn through the origin perpendicular to Ox, Oy, Oz ; and A', B, C the resultant stresses across those perpendicular to Ox', Oy', Oz". The components of A, B, C parallel to Ox, Oy, Oz are of G 98 ANALYSIS OF STRESSES. [160. course P, U, T; U, Q, S ; T, S, B, respectively; while the com- ponents of A' parallel to the same axes are given by equations (11), and those of B' and C by similar formulae. Squaring and adding, we get A* = P*+m+T*\ B^W + Q^ + WX ( u ) it'* = (\P + H U + Vl Tf + (\U + frQ + v^ + (\T + fLjS + v^W E* = {\P + n 2 U+ v 2 Tf + (X 2 U + N Q + v 2 5) 2 + (X 2 T + fi 2 S + v 2 Rf I. . . (1 5) C' 2 = (A 3 P + n s U+ v t T)* + (\ 3 U+ H Q + v ,Sf + (\ s T + n a S+v 3 B)*\ Expanding (15) and adding them together, we get A' i + £' i +C't = P 2 + Q 2 + R i + 2(S 2 + T i + W). Henee, by (14), A* + B* + C'* = A* + JP + C t (16) from which we deduce that The sum of the squares of the resultant stresses across any three orthogonal plane areas drawn through a given point of the body is constant, however they be turned about the point ; and when the stress is homogeneous, this constant has the same value at every point of the body. 161.] Reciprocal relation between Stress-components. Since (X , /*,, v,) are the direction-cosines of Ox' referred to Ox, Oy, Oz, and since P, TJ, T are the components of A parallel to these axes, it follows that the component of A parallel to Ox' is PX 1 + Ufr + TV,. But we have already seen (11) that this is the component of A' parallel to Ox. Hence, since the directions of Ox, Ox' are quite arbitrary, we deduce that If any two small plane areas be drawn through any given point of the body, the component perpendicular to the first area of the stress across the second is always equal to the component perpendicular to the second of the stress across the first. 162.] First Stress Quadric. We now proceed to give these theorems geometrical significance. Describe the quadric Px i + Qy* + Rz !i + 2St/z + 2Tzx+2Um/=l (17) Let r be the length of the radius vector in the direction (X, /*, v) and let p be the perpendicular from the centre on the tangent plane at the extremity of r, (I, m, n) being the direction cosines of p. 162.] ANALYSIS OF STRESSES. 99 Then, if F, G, H be the components, parallel to the axes of reference, of the stress across the central section perpendicular to t, F, G, H will be given by equations (5). But we have PX+Up + Tv UX+Qp + Sv Tk + Sfi + Bv 1 I m n ~ pr ^ ' Thus Fjl=Glm = H/n = \/pr (19) The resultant stress across the central section perpendicular to r therefore acts in the direction of p; its amount is 1/pr ; and the amount of its normal component is 1/r 2 . 163.] Principal Axes of the Stress. It is obvious from the last Article that if the section coincide with any one of the principal sections of the quadric, the stress across it will be wholly normal. It is thus always possible to draw through each point of the body three orthogonal plane areas across which the stress is wholly normal. These are called the Principal Planes of the stress at the point, and their normals are called the Principal Axes. The normal tractions across the principal planes are called the Principal Normal Stresses. We shall denote them by iV,, i\ r 2 , A r 3 . Let 0£, Or/, Of be the Principal Axes of the stress at the origin ; then they are also the principal axes of the quadric (17). It also appears from § 162 that the squared reciprocals of its principal semi-diameters are N v N 2 , JV 3 . Hence the equation of the quadric referred to the principal axes is ^ 2 + ^2'7 2 + ^ 2 = 1 (20) Of course N lt N v N s are the roots (in descending order of magnitude, let us say) of the discriminating cubic IP-*, U, T | J U, Q-4>, S =0 (21) T, S, K— while the direction-cosines of Og, Or/, Of are given by the equations PX+UpA-Tv UX+Q/i + Sv Tk + Sn + Ev X /* v ~ {""' where N is to receive successively the suffixes 1, 2, 3. These equations (22) might of course have been deduced directly from equations (5). p, u, T v, Q, S T, s, R 100 ANALYSIS OF STRESSES. [164. 164.] Invariants of the Stress. Expanding the cubic (21), and employing the notation of § 157, it becomes <£ 3 - + R 1 + 2(S ! + T- + V) = A- + £* + C 2 (28) Thus the theorem of § 160 simply states that $ 2 - 2 . J is an invariant. 105.] Traction and Pressure. We saw in § 162 that the normal component of the stress across the plane perpendicular to the radius vector r was 1/r 2 . Hence if the stress be such as to produce a traction across every small plane area drawn through the origin, the quadric (17) is an ellipsoid, and iV,, N v N 3 are all positive, and so therefore is £L If the stress across every plane be a pressure, the quadric represented by equations (17) and (20) will be imaginary ; N v N v N r It will all be negative, and the pressures will be given by the ellipsoid Px- + Qif + Ez" + 2Syz + 2Tzx + Wmj = -1 (29) or N^ + N 2 yf + N 3 C-= -1 (30) 166.] Normal Cone of Shearing Stress. If the stress at the origin be a traction or a pressure according to the direction of the plane across which it is measured, equations (17) and (29), or (20) and (30), will represent two real conjugate hyperboloids, radii which meet the first being normals to planes across which there is a traction, while radii which meet the second are normals to planes across which there is pressure. 166.] ANALYSIS OF STRESSES. 101 These two hyperboloids are separated by their asymptotic cone Px?+Qy i + Bz' + 2Syz + 2Tzx+2Uxy = (31) or NtF + ITtf + irf-O (32) Since any radius vector lying in this cone is of infinite length, the normal component of the stress across the plane perpendicular to it vanishes ; whence we see that all planes whose normals are generators of this cone suffer only tangential stress. It is there- fore called the Normal Cone of Shearing Stress. 167.] Second Stress Quadric ("Director Quadric"). Let us now construct the reciprocal quadric, whose equation referred to the principal axes is S- + 1L+ C=l (33) If r be the radius vector drawn from the centre to the point (£> "?> on the surface, and p the perpendicular from the centre on the tangent plane at (£ rj, f), the direction-cosines of p referred to the principal axes are pil^v 2>n/& 2 , Pi/^s- Now equations (5) give for the components parallel to Og, Otj, Of of the stress across a plane the direction-cosines of whose normals referred to the same axes are (\ , /i , v ) Hence the component stresses across the plane perpendicular to p (that is, the section conjugate to r) are given by G =pr,i. Hence the resultant stress across the section perpendicular to p (or conjugate to r) acts in the direction of r ; its amount is pr ; and the amount of its normal component is p 2 . 168.] Tangent Oone of Shearing Stress. By considering the sign of the normal component, as in § 166, we see that if the stress at the origin be a traction in every direction (33) is an ellipsoid ; if a pressure in every direction we have the alternative ellipsoid ii+ '>1+J: = _i (35) 102 ANALYSIS OF STRESSES. [168. while if it is a traction across some planes and a pressure across others, we have the pair of real conjugate hyperboloids (33) and (35). These are separated by their asymptotic cone £i + ?L + IL = o (36) and it is easy to see that this cone envelopes all those planes through the origin which suffer only tangential stress. It is therefore called the Tangent Cone of Shearing Stress. 169.] Third Stress Quadric ("Stress Ellipsoid")- It is obvious that equation (10) may be regarded as an equation to be satisfied by the components, parallel to the arbitrary axes, of the stress across any plane through the origin. Hence if we construct the quadric (px + ay + tz) 2 + (ax + s\y + ez) 2 + (tx + sy + xz) 2 = §t 2 . (37) the radius vector r drawn to the point (x, y, z) on the surface will represent in magnitude and direction the stress across the central section whose normal is in the direction given by writing x, y, z for F, G, Mm. equations (9). Transforming to the Principal Axes, we find that the radius vector r of the quadric W 2 + W* + N* =l < 38 ) represents in magnitude and direction the resultant stress across the central section whose direction-cosines referred to Og, Or\, Of are given by h» = lW,\ (39) where (£, r\, f) is the extremity of r. This quadric is of course always an ellipsoid. K (iv l v m)> (£s> Vj?*)i (£" n » & De tJ ? e coordinates of the extremities of three radii r v r a , r v representing in magnitude and direction the resultant stresses across any three orthogonal central sections of this quadric, it follows from (39) that they must satisfy the relations Cssg . Ws , dig n \ .(40) 169.] ANALYSIS OF STRESSES. 103 which are the well-known conditions that r v r 2 , r 3 may be con- jugate semi-diameters of the quadric* Hence we deduce that any three conjugate radii represent in magnitude and direction the resultant stresses across three ortho- gonal central sections. This also follows directly from equation (39), the geometrical interpretation of which is that r represents the stress across the section whose normal is the radius of the " auxiliary sphere " corresponding to r. 170.] Relation between the Second and Third Quad- rics. If any radius vector from the common centre meet the Third Quadric in (£ v t) v f x ) and the Second in (£ 2 , •?„, f 2 ) and if r t be the length intercepted on it by the Third, we see by (39) that r, represents in magnitude and direction the stress across the plane t+l4;-» <«> which is the same as that is, the central section of the Second Quadric conjugate to 7 1 ,. Hence if r v r 2 , r 3 be the lengths intercepted by the Third Quadric on any three conjugate radii of the Second, each represents in magnitude and direction the stress across the plane containing the other two. Thus the Third Quadric may be regarded as giving a graphical construction for the magnitudes of stresses, and the Second for the directions of the planes across which they act. We shall therefore distinguish them as the Stress Ellipsoid and the Director Quadric. 171.] In the cases where the Principal Stresses are of differ- ent signs, and there is consequently a real Tangent Cone of Shearing Stress (36), each generator of this cone represents three coincident conjugate radii, and the plane conjugate to any generator is the tangent plane to the cone along that generator. Thus if r be the length intercepted by the Third Quadric on any generator of the Tangent Cone of Shearing Stress, then r represents in magnitude and direction the shearing stress across the plane which touches the cone along that generator. 172.] Fourth Stress Quadric. Finally, let us describe that reciprocal of the Third Quadric whose equation is (Px+Uy + TzY+(Ux+Qy + Szy+(Tx + Sy + Ezy=l (42) or Ntf* + N 3 W + N B *e=l (43) This is likewise always an ellipsoid. 104 ANALYSIS OF STRESSES. [172. It is obvious, by squaring and adding equations (5) or (34), that if r be the radius vector of this quadric perpendicular to any given central section, the amount of the resultant stress across that section is 1/r, and its components parallel to the principal axes are Nitfr Nfilr> N dl>- ( 44 > g, rj, f being the coordinates of the extremity of r. Hence if (£, r, v Q, (£,, q v Q (£,, *,„ Q be the extremities of three radii, perpendicular to three central sections the resultant stresses across which act in three orthogonal directions, we have from (44) the relations which are the conditions that the three radii may be conjugate. Hence the resultant stresses across any three central sections whose normals are conjugate radii act along three orthogonal radii. 173.] Relation between the First and Fourth Quadrics. We see from (44) that if (£,, rj v £) be the extremity of the radius vector r, of the Fourth Quadric, then the resultant stress across the central section perpendicular to r, acts along the normal to the plane A",£i + -tfiTOi + -N'««i = <> (46) which is the section of the First Quadric conjugate to r r Hence if r v r 2 , r, be the intercepts by the Fourth Quadric on any three conjugate radii of the First, the resultant stress across the central section perpendicular to either acts in the direction perpendicular to the plane containing the other two ; while the amounts of these resultant stresses are 1/r,, l/r 2 , 1/r, respectively. Special Forms of Stress. 174.] Hydrostatic Pressure. All the preceding theorems apply to the most general form of the Stress, when N v N v JV are all unequal and of any sign, but none of them vanish. The cases in which two of the principal stresses are equal are not worth working out in detail, as the results already obtained may be easily modified to suit them, if we remember that all the 174.] ANALYSIS OF STRESSES. 105 Stress Quadrics become surfaces of revolution. The necessary and sufficient conditions are p TU Q US ff ST } P -HT = Q -^ = R --U' \ (47) or J'^-iJJ-llK^-isjj-aTit^o) [Compare § 120 (Hi.).] The case where all three of the principal stresses are equal is however remarkable. If N x = N t = N t = -II, the discriminating cube (21) or (24) must reduce to Hence we must have (<£ + II)3 = 0. .(48) •§= -311 $ = 3112 »=-m The stress-quadrics become spheres, the Third in particular becoming and the Second f 2 + -f + £ 2 = n ) ' Since any radius of a sphere coincides with the perpendicular from the centre on the tangent plane at its extremity, it follows that the stress across every central section is normal, and that it has the same value for each. Thus if II is positive the stress at the origin consists of a normal pressure II across every plane area which can be drawn through it. This is the nature of the stress which exists at every point of a fluid at rest under any forces, and it is therefore called Hydrostatic Pressure. If n is negative, we have simply to replace the pressure by a traction. In order that the Stress Quadrics may be spheres we must obviously have S =T=U=0 S (49) Thus it is evident from equations (5) and § 157 that a homo- geneous hydrostatic pressure can only be maintained by a uniform normal pressure of like amount appbed over the whole bounding surface. We may here notice another discrepancy in the numerical reckoning of Strain and Stress (see § 152). Three equal orthogonal contractions e compound (§ 104) into a uniform com- pression of amount 3e, while three equal orthogonal normal pressures II compound into a hydrostatic pressure of amount n. 106 ANALYSIS OF STRESSES. [175. Stress in Two Dimensions. 175.] Plane of the Stress. The remaining important types of stress are characterised by the vanishing of the third invariant ^, and therefore also of one at least of the Principal Normal Stresses. For the present we shall confine ourselves to the case in which only one of them, say N 3 , vanishes. There is then no stress whatever across the small plane area drawn through to coincide with the plane of &. The Stress Quadrics become cylinders with their generators parallel to 0£ ; and since in the third of equations (39) v cannot be greater than unity, it follows that at the extremity of every radius vector which represents the resultant stress across a real plane through the origin we must have f=0. Hence the directions of the resultant stresses across all plane areas drawn through lie in the normal section by the plane of £17 which contains N 1 and N 2 . The stress is therefore said to be entirely in two dimensions, and the plane of fy is called the Plane of the Stress at 0. 176.] The Stress Conies. It is obvious that all the graphic properties of the stress will depend upon curves in the plane of the stress, and especially on the normal sections of the Stress Cylinders by that plane. These curves we shall call the Stress Conies. 177.] Case iD which JTj and JT 2 have the same sign. In this case Jf is positive, while |P has the same sign as N t and N 2 . Assuming this sign to be positive, the Second and Third Stress Conies become the ellipses f _ rf \ .(50) .(51) The first of these is the Director Conic, replacing the Director Quadric of §§ 167, 170. If (£ t)) be the extremity of the radius vector representing in magnitude and direction the stress across the plane whose direction cosines referred to the principal axes are (X, /x, v) we have from equations (39) A = "N, i" : V = N >: ^ 2 (52) and therefore W* + W*~^ ~^ (53) 177.] ANALYSIS OE STRESSES. 107 Thus the resultant stresses across all planes through whose normals lie on a circular cone with axis Of and semi-vertical angle a, are represented in magnitude and direction by the radii of the ellipse jfTi + J^sin'a (54) To each such cone belongs one of these ellipses, the whole system being similar to and coaxial with the ellipse (51), which is the largest of all. In the limit when a=0, the ellipse (54) vanishes into the origin, so that, as we already know, the stress across the Principal Plane fi/ is zero. As a increases, so does the size of the ellipse, and therefore the magnitude of the stress, until the limit is reached in which a = \ir, when the ellipse (54) coincides with (51). The radii of (51) therefore represent the stresses across planes whose normals lie in the plane of the stress : — that is, across planes drawn through Of. This system of Stress Ellipses replaces the Stress Ellipsoid of §§ 169, 170. 178.] Again the trace of a given plane on the plane of the stress (or the line in which the two planes intersect) is given by and the projection of its normal on the plane of the stress by [lg - X.7I = 0. Hence if (£, y^ be the extremity of the radius vector representing the stress across this plane, we get from equations (52) — for the trace of the plane N^ N % ~ u ( 5o > and for the projection of the normal ivrivr < 56 ) These equations afford us two geometrical constructions for deter- mining the direction and magnitude of the stress across a given plane. First Method. If (£, 17,) be the point on the stress-ellipse (54) whose radius vector represents the stress, and if (£, q a ) be the point on the auxiliary circle of that ellipse corresponding to (£, ^), then (assuming that ^ > NJ ' thus (56) may be written £%-^ 2 = 0. Thus the projection of the normal to the plane is that radius of 108 ANALYSIS OF STRESSES. [17S. the auxiliary circle which corresponds to the radius of the ellipse representing the stress. Conversely, if the plane be given, we can construct the stress- ellipse (54) and its auxiliary circle £* + y- = AY sin- a ; (57) if we then project the normal to the given plane, and find the point on the ellipse corresponding to the extremity of that radius of the circle which coincides with the projection, the radius vector of this point represents in magnitude and direc- tion the stress across the given plane. We have seen in § 165 that when iV, and i^ 2 are both positive this stress is always a traction. It is well known that ;1 if 0P V 0P 2 be two con- jugate radii of an ellipse, the corresponding radii 0Qi> OQ t of its auxiliary circle (Fig 13) are at right angles ; and conversely. Thus OP i represents the stress across a plane whose trace is 0Q , and the projection of its normal OQ 1 ; while OP^ represents the stress across a plane, making *--.... _..-■-■' the same angle with Of, I whose trace is 0Q V and the projection of its nor- F '6-I3- mai 0Q 2 . Thus, conversely, if through any two perpendicular radii UQi> OQi planes be drawn so that their normals may make the same angle a with Of, the resultant stresses across them will be represented by conjugate radii of the ellipse (54) ; and, in particular, the stresses across any two orthogonal planes through Of are represented by conjugate radii of the ellipse (51). Second Method. If (£, >/,) be as before the point on the ellipse (54) whose radius vector represents the stress across the given plane (X, /u, v), and if this radius (produced if necessary) meet the director-ellipse (50) in the point (f 2 , jj 2 ), we have & : & : : Vi '■ v* Hence equation (55) giving the trace of the plane may be written y, JV 2 ' o, which represents the radius of (50) conjugate to the first. 178.] ANALYSIS OF STRESSES. 109 Conversely, if the radius of the director-ellipse (50) conjugate to the trace of any given plane be drawn, the intercept on this by the stress-ellipse (54) represents in magnitude and direction the resultant stress across the given plane. Since two conjugate radii of an ellipse never lie in the same quadrant, no plane through the origin is subject to simple Shearing Stress. 179.] If J is positive and |9 negative, JV, and N^ are both negative. All the theorems proved in the last two Articles will be equally true, the only change necessary being to substitute for (50) the equation fri.— 1 (58) The stress across every plane through the origin will be of the nature of a pressure. 180.] Case in which JV 1= = i^. If ft = 0, and J = \W, then N x = N t = N (say) ; N having the same sign as IB. The results of the previous Articles may be modified to suit this case, by writing everywhere " circle " for " ellipse," and " orthogonal " for " conjugate." Thus the stress across any plane whose normal is inclined at an angle a to Of is represented in magnitude and direction by the radius of the circle ^ + r, 2 = iV" 2 sin 2 a (59) which is perpendicular to the trace of the plane. In other words the stress across every such plane is iVsina, and acts along the projection of its normal on the plane of the stress. Every plane through the axis Of suffers a normal stress N. The stress is obviously symmetrical about Of, and the directions of 0£ and 0r\ are indeterminate. 181.] Case in which i\ r , and N t have opposite signs. If 11 = 0, and J is negative, one of the principal normal stresses will be a traction and the other a pressure, the sign of the greater of the two being the same as that of jB ; we shall suppose A\ to be positive and i\T a negative. Instead of the ellipse (50) we now have the pair of conjugate director hyperbolas JM = 1 1 < 60 > and ! + * 2 =- lJ - < ki > 110 ANALYSIS OF STRESSES. [181. separated by their asymptotes = 0, .(62) The system of stress-ellipses (54) will of course remain unaltered. To modify the results obtained by the first method of § 178, we must remember that, since N 2 is now negative, the coordinates of the point on the auxiliary circle corresponding to the point di> fi) on ^ ne ellipse (54) are now t-t ** C2-C11 V2 ft • Vi if N, be numerically greater than JV 3 ; Or | 2 = -]y .£,, r) 2 = Vi if J\ r be numerically less than N r The analogous construction for the present case is then as follows : — To find the resultant stress across a plane whose normal makes an angle a with Of, project this normal on to _._[1_. the plane of the stress : let OQ be the radius of the auxiliary circle of the ellipse (54) with which this projection coincides. Find P, the point on the ellipse cor- responding to Q, and draw the radius OP' of the ellipse, equally in- clined to the major axis on the opposite side. Then OP' will represent in magnitude and direc- tion the stress across the given plane — which may therefore be normal, tangential, or oblique. Fig. 14. Again, modifying the results obtained by the second method of § 178, we see that the intercept made by the ellipse (54) on any radius which meets (60) represents in magnitude and direction the resultant traction across the plane drawn through the con- jugate radius of (61) so that its normal makes an angle a with Of. Similarly the intercept made by (54) on any radius which meets (61) represents the resultant pressure across a plane 181.] ANALYSIS OF STRESSES. Ill drawn at the same inclination to Of through the conjugate radius of (60). SB J 8 Either asymptote of the hyperholas represents a pair of coincident conjugate radii ; hence a plane drawn at any inclination a through either of the asymptotes (62) suffers a shearing stress represented in magnitude and direction by the intercept cut off on that asymptote by the ellipse (54). 182.] Case in which N^—N r The last important case of stress in two dimensions occurs when ^=0, 3B=0; Jf being negative. We have then N, = — JV r Assuming that JV", is positive, and denoting it by Jy, the system of stress-ellipses (54) reduces to the system of circles f + j' = N'sw'a (59) as in § 180. Also the director hyperbolas (60) and (61) become the rectangular hyperbolas £W = JV-) (63) f-f=JV) (64) If OP be the radius of the circle (59) which coincides with the projection of the normal to any one of the corresponding system of planes, and if OP' be the radius making the .same angle as OP with 0(, on the opposite side of it, OP' represents in magnitude and direction the result- ant stress across the plane. The amount of this stress is therefore ± N sin a, and it may be normal, tangential, or oblique. To determine its sign we must remember that every radius which meets (63) represents a trac- tion, and every radius which meets (64) a pressure. Let OP, OQ tie conjugate radii of (63) and (64), and let Y, 0Z be perpendicular to them. By the properties of the rectangular hyperbola, the asymptotes bisect the angles between the principal axes Of 0^; also OP and 0Q are equally inclined to the asymptote which lies between them, and consequently OP and 0Z are equally and oppositely inclined to Of, and 0Q and 7 to Or). Fig. 15. 112 ANALYSIS OF STRESSES. [182. Now OP is the direction of the traction across any plane drawn through OQ, and therefore having the projection of its normal along OZ; and, similarly, OQ is the direction of the pressure across any plane drawn through OP, and therefore having the projection of its normal along OY. In this case, therefore, the trace of a plane and the direction of the stress across it make equal angles with an asymptote ; while the angle between the stress and the projection of the normal is bisected by Og (the axis of principal normal traction), or by Or/ (the axis of principal normal pressure), according as the stress is a traction or a pressure. Fig. 16. 183.] Let us now follow the changes in the stress across a plane through 0, as it moves round in such a manner that its normal describes a cone of semi-vertical angle « about Of. The numerical magnitude of the stress will of course always be the same — namely, N . sin a. Let OQ (Fig. 16) represent the trace of the plane in any position, OZ the projection of the normal, and OP the direction of the stress. Then the angle P0£ is always equal to either of the angles QOq and Z0£. 183.] ANALYSIS OF STRESSES. 113 Let OQ coincide first of all with Ojj ; then OZ and OP both coincide with Og. The normal component of the stress is a traction N sin 2 a, and the tangential component N sin a cos a acts along the line in which the plane is cut by that of f£ As OQ moves away from Or/ towards the asymptote, the stress becomes more and more oblique, the angle POZ constantly increasing, until, when the plane actually passes through the asymptote, the normal traction has vanished altogether, and we have only a shearing stress of amount JV sin a acting along the asymptote. As OQ passes the asymptote the normal component reappears as a pressure, OP having also passed the asymptote in the opposite direction. This normal component continually increases until, when OQ coincides with 0£, and OP and OZ with Ojj, the resultant stress acts along Oy, and consists of a normal pressure N' sin 2 a, together with a tangential component iVsin a cos a along the line of intersection of the given plane with that of r/£. This cycle of changes is then repeated in the reverse order until the trace of the plane once more coincides with Or). 184.] Position of the Plane of the Stress. Since this plane is perpendicular to the axis 0£ of zero stress, its direction- cosines referred to the arbitrary axes Ox, Oy, Oz are to be obtained by writing N = in equations (22). We thus get the equations Uk+Qn + Sv = 0Y (65) Tk + S l x. + Rv = Q) only two of which are independent, in virtue of the condition 1 = 0. Taking this into account we find that equations (65) are equivalent to either of the pairs of equations s k= ty = nv\ (66) k /l V I or ~J^ = Jq = '7c\ .(67) The equation of the Plane of the Stress in the case when 1 = 0, referred to the arbitrary axes, may therefore be written in either of the forms xjp + yjq + z Jx = j (68) ;♦[♦!-•[ < 69> H 114 ANALYSIS OF STRESSES. Stress in One Dimension. [185. 185.] We now come finally to the case in which two roots of the discriminating cubic (21) vanish. If N denote the remaining root we must have g=oJ (70) a D d «p=JV. (71) Supposing N 1 and iV 2 to he the vanishing Principal Stresses, equations (39) show that at the extremity of every radius which represents the stress across a real plane we must have £ = 0, , = 0. Thus the resultant stress across every plane that can be drawn through acts along Of. The stress is therefore said to be in one dimension, and Of is called the Principal Axis of the Stress at 0, the other two being indeterminate. The third of equations (39) shows that the resultant stress across any plane (X, fx, v) is represented in magnitude and direc- tion by the length Z=vX (72) measured along Of. If then we describe a circular cone about Of with semi- vertical angle a, the resultant stress across every plane whose normal lies in this cone is given by f= TV cos a (73) Thus the stress is zero across every plane passing through Of; and it follows that in this case no plane through can suffer pure shearing stress. If we describe about a sphere of radius JV.the projection on Of of the radius of the sphere coinciding with the normal to any plane represents in magnitude and direction the resultant stress across that plane. N is obviously the maximum stress, and the stress across every plane through is of the same sign as N or |p. 186.] Direction of the Axis. The direction-cosines of Of, referred to the arbitrary axes Ox, Oy, Oz, are given by equations (22). Now it is well known that the conditions (70), when satisfied simultaneously, are equivalent to either of the sets of three ? =q= r= 0> (74) or g = t = ti = 0i (75) 186.] ANALYSIS OF STRESSES. 115 that is, by QR - S 2 --= RP - T 1 = PQ - W = 0, or by the equivalent set TU-PS= US-QT = ST-RU=Q. Thus equations (22) may be written in either of the forms SA = 2>-*M (76) _i_ jl. v \ JP- JQ- Jli ■■■■■<") or and the equations of the Axis of the Stress are Sx = Ty=Uz\ (78) x y z y 7? = 7c == ^lJ < 79 > 187.] Heterogeneous Stress. In general the standard components of the stress will vary from point to point of the body, all of them (§ 137) being continuous functions of the coordinates (x, y, z) of the point at which they act. The Principal Normal Stresses, and the direction- cosines of the Principal Axes at each point will therefore also be continuous functions of its coordinates. All these theorems that we have proved for the Stress at the origin will be equally true (§ 1 58) for the Stress at any point P, if we refer the Quadrics, etc., to the principal axes at P, or to a system of axes through P parallel to the arbitrary axes Ox, Oy, Oz. EXAMPLES. 1. Discuss the properties of the following stresses : (i) {3a, -a, -a, 2a, 0,0}; (ii.) {0, 0, 0, «, a, a}; (Hi.) {a, a, 0, a, «, a} j (iv.) {a, 0, 0, a, a, a}; (v.) {13a, 10a, 5a, -6a, -3a, -2a}; (vi.) {3a, -a, -2a, 3a, -a, -2a}. 116 ANALYSIS OF STRESSES. Show that the principal normal stresses are respectively : — (i. ) 3a, 3a, - a ; (ii.) 2a, - a, — a ; (Hi.) (Jl + l)a,-(j3-l)a,0; (iv.) (J2 + I)a, -(v/2-l)a, 0; (v.) 14a, 14a, ; (vi.) J la, - J la., 0. 2. Prove that if through any point of a strained body a system of planes be drawn, such that the normal component of the stress across each has a given value N, the normals to these planes will generate a quadric cone. 3. If the stress be in two dimensions at the origin, and the plane of xy be made to coincide with the plane of the stress, show that (i.) The principal normal stresses N v N 2 are the roots of the quadratic (-P)( = 0. (ii.) The angles \fs v i/<- 2 which Og and 0>? make with Ox are the roots of tan 2^ = - 2U P-Q (Hi.) If PQ > U 2 , four planes can be drawn through at a given inclination a to the plane of the stress, so that the stress across each shall have a given normal component N (provided of course that N is taken within proper limits) ; and the projections on the plane of the shear of the normals to these four planes lie in the lines (P - N cosec^)^ + (Q - N cosec'a)?/ 2 + 2 Uxy = 0. Hence deduce the limits of N for a given value of a. (iv.) What is the corresponding theorem when PQ < U 2 ? (v.) Show that four planes can always be drawn at a given inclination a to the plane of the stress, such that the stress across each may have a given tangential component T (taken within proper limits). Show that the projections on the plane of the stress of the normals to these planes are the lines sin 4 a(ZV + Wxy + Qf) + T 2 (^ + ff = sin 2 a(ar ! + y*)[(Px + Uyf + (Ux + Qyf]. Hence deduce the limits of T for a given value of a. ANALYSIS OF STRESSES. 117 (vi.) Show that, for a given value of a, N is a maximum when the projection of the normal coincides with 0£, and a minimum when it coincides with Oti. (vii.) Show that, for a given value of a, T is a minimum when the projection of the normal coincides with 0£ or Orj, and a maximum when it bisects either of the angles between these axes. (viii.) Hence show that the two planes through suffering greatest tangential stress are those which bisect the angles between the principal planes gz and yz. 4. Prove that, when two of the principal normal stresses are equal, the normals to those planes which suffer maximum tan- gential stress are all inclined at an angle of 45° to the direction of the third principal stress. ' 5. Show that, in general, the normals to planes through the origin, the stress across which has a given tangential component T, lie on the cone whose equation, referred to the principal axes, is (p + tf + fl{(N* - T 2 )£ 2 + {N* - TV + (N* - T 2 )?} -(Ntf + Nrf + Njy^O. 6. Show that if r be any radius vector of the surface (JV S - iWC 2 + (N, - Ntf?¥ + (tf; - NtfFrf = 1, and T the tangential component of the stress at the origin across the section drawn through it perpendicular to r, then Tr 2 =l. Show that the sections of this surface by the principal planes are conjugate rectangular hyperbolas, having the principal axes for their asymptotes. Hence, or otherwise, prove that the maximum tangential stress is suffered by the two planes (l/*/2, 0, ±l//v/2), and that this maximum is \{N^ — i\ r 3 ) ; N v N v i\T 3 being in descend- ing order of magnitude. CHAPTER IV. POTENTIAL ENEEGY OF STRAIN. 188.] Introductory. We saw in Chapter I. (§§ 21, 26, 27) that the Potential Energy of a perfectly elastic body, due to Strain produced at constant temperature, must always be equal to the work expended by external forces (including Applied Forces and Surface Tractions) in producing the strain ; that this work (§ 31) is done against the Eesistance (§ 135) offered by the body to stress, and is therefore equal to the work done by the Stresses (§ 135) during the Strain ; and finally (p 27, 29, 34) that the Potential Energy and the Stress in any given state of the body are functions only of the actually existing Strain. It is obvious that, since our new definition of Stress (§§ 131, 135) retains its essential characteristic (§ 29) of a purely mutual action between the component parts of the body, these theorems are as true for the perfectly elastic continuous mass with which we are now dealing as for the perfectly elastic molecular structure which we considered in Section ii. of Chapter 1. The course now to be taken by our investigation will there- fore be as follows : — Regarding the six component stresses as functions only of the six analogous components of the strain which they suffice to maintain, under the given system of external forces, we shall first find an expression for the work done by them during an elementary increase in each of these components. This expression will involve the stresses and the increments of the strains, and we shall show that, in virtue of equations (3) and (5) of Chapter III., it is identically equal to the work that must be done by the Applied Forces and Surface Tractions, to produce the small increments in the displacements of their points of application which constitute the increment of Strain. We shall next employ the principle of superposition of small strains (§ 87) and stresses (§ 155) to express the six standard components of a small stress in terms of the six components of the corresponding small strain; and then, by eliminating the stresses from the 188.] POTENTIAL ENERGY OF STRAIN. 119 expression just found, we shall obtain the differential of the Poten- tial Energy of strain, expressed as the differential of a function of the six component strains. Finally, integrating this from the natural state of the body {0, 0, 0, 0, 0, 0} to the given state of strain {e,f, g, a, b, c}, we shall obtain the Potential Energy in the latter state as a function of e, f, g, a, b, c. Work done by Stress during a small arbitrary variation of the Strain. 189.] Work done in increasing a simple elongation. Let us first suppose the body to be in equilibrium in the state of homogeneous strain {e, f, g, a, b, c}. Since the components {P, Q, R, S,T, V} of the stress required to maintain this condition are functions only of the strain-components, it follows that the stress also is homogeneous. Let us investigate the work done by stress in producing a small arbitrary increment Se of the component e, all the other strain -components remaining as before. Consider a finite rectangular parallelepiped of the body, the coordinates of whose centre in the original state of strain are (x, y, a), and whose edges of lengths h, k, I are respectively parallel to the fixed arbitrary axes Ox, Oy, Oz. The stresses throughout its interior can do no work (§ 133) upon it as a whole, so that all the work done by stress is due to that which acts across its bounding surface. Again, every point (x+x') of the parallelepiped is displaced parallel to Ox, the amount of the displacement being (x+x')Se [§ 68 and § 89, (i.)]. Hence only those com- ponents of the stress across its surface can do work which act parallel to Ox. Now if we take a slice of elementary thickness, bounded by the parallel planes x and x' + dx, every point in its perimeter suffers the same dis- placement ; and, the stress-components at each point of this perimeter being as Fig. IT represented in Figure 17, the forces 120 POTENTIAL ENERGY OF STRAIN. [189. acting parallel to Ox on the four edges of the slice are respectively T.kdxT U.ldx' - T . kdx' ' -U.ldx', It is therefore obvious that these forces together can do no work in such a displacement : and, this being true for every such slice, it follows that the only forces which can do any work on the parallelepiped, in increasing the elongation e, are the normal components of the tensions acting across the ends perpendicular to Ox. Since P is a function of the strain-components, it will be altered to P + f.8e oe by the small increase of e. Hence the work done by the tension acting across the positive end lies between and P + P.M ■dP 3e (* + l) 8 * .8e\kl(x + '£\l Similarly, the work done against the tension which acts across the negative end lies between P. Mi and ('♦£•'■ (x — \Se v )«H) Se Hence, on the whole, the work done by stress lies between PSe.hM \ aud (p+^.8e\Se.hkl{ (*+%•*)>... Thus, neglecting the square of Se, the whole work done by Stress in producing the small increment Se of the single com- ponent e is P8e . hU. 190.] Strain and Stress Heterogeneous. In precisely the same manner we may show that, if the strain be not homo- geneous, the work done by stress on the elementary rectangular 190.] POTENTIAL ENERGY OF STRAIN. 121 parallelepiped dxdydz having its centre at (x, y, z), in producing a small increment Se of the elongation of this element parallel to Ox, is simply PSe . dxdydz (1) where P, e, Se are continuous functions of x, y, z. The work done on the whole body by stress in producing any continuously distributed (but otherwise perfectly arbitrary) small variation Se in the elongation e throughout it is therefore ff/PSe. dxdydz (2) and we see that, in varying a simple elongation, only the corre- sponding longitudinal traction (§ 148) can do any work. Hence P, Q, R are the Simple Stresses (§ 33) corresponding to the Simple Strains e, f, g. 191.] Work done in increasing a Simple Shear. Let us next suppose the component shear a to suffer a small incre- ment 8a, the other components retaining their initial values. If 0£ and Of be the internal and external bisectors of the angle yOz we know (§§ 92, 100) that the shear a or 2s 1 in the plane of yz is equivalent to an elongation s t in the direction of 0£, together with a contraction s v or an elongation ( - sj, in the direction of Of. The small increment Sa of the shear may there- fore be resolved into the small increment Ss 1 of the elongation parallel to 0£, together with the small increment Ss 1 of the contraction, or increment ( — SsJ of the elongation parallel to Of. Again, a shearing stress of amount S in the plane of yz (or any parallel plane) may be resolved (S§ 150-152) into a longi- tudinal traction S parallel to 0£, together with a longitudinal pressure S, or traction ( — S) parallel to Of. Hence, by superposition, we deduce from the last Article that the work done by stress on the element dxdydz, with its centre at the point (x, y, z) of the body, in producing the small incre- ment Sa of the shear of the element in the plane of yz, is S8s 1 . dxdydz + ( - S)( - Ssjdxdydz ; that is 2 'S' Ss i • dxdydz, or SSa . dxdydz (3) where jS, a, Sa are continuous functions of x, y, z; Sa being otherwise arbitrary. The work done on the whole body by Stress in producing such a change throughout it is therefore /YVSSa. dxdydz (4) and it follows that, in varying a simple shear, only the correspond- ing shearing stress (§ 149) can do any work. 122 POTENTIAL ENEliGY OF STRAIN. [191. Hence S, T, U are the Simple Stresses (§ 33) corresponding to the Simple Strains a, b, c. 192.] Work done by Stress in any small arbitrary variation of the Strain. Superposing (2) and (4), and the analogous formulae for /, g, b, c, we see finally that the work clone by Stress in producing small arbitrary and independent variations of all the strain-components throughout the body, such that the strain at any point (x, y, z) is altered from {«./. g, «. t>, c\ to {e + Se, f+ 8f, g + 8g, a + 8a, b + 8b, c + 8c} is given by 8 W=/ff[P8e + Q8/+ R8g + S8a + T8b + U8c]dxdydz (5) where {P, Q, R, S, T, U} is the specification of the stress required to maintain the body in equilibrium in its original state of strain, under the given external forces. Work done by the Applied Forces and Surface Tractions in producing a small variation of the Strain. 193.] Expression for this Work. As in the last Chapter, let X, Y, Z represent the components of the Applied Force per unit mass at the point (x, y, z) in the interior of the body, p the density at the same point, and F, G, H the components of the Surface Traction per unit area applied to the element dS of the bounding surface : these systems of forces and tractions consti- tuting the system of " external forces " which, with the distribution of stress {P, Q, R, S, T, U}, holds the body in equilibrium in the original state of strain {e, f g, a, b, c}. Let the effect of the small arbitrary variation of the strain, considered in the last Article, be to change the component dis- placements u, v, w of any point (x, y, z), in the interior of the body or on its surface, to u+8u, v + Sv, w+Sw. Then, by the principle of virtual velocities, the work that must be done by the external forces to produce this change is fffp{X8u + Y8v + Z8w)dxdydz +ff(F8u + G8v + E8w)dS (6) where the triple integral is taken throughout the volume of the body, and the double integral over the whole of its bounding surface. By reasoning as in §§ 153, 154, we may show that it is indifferent, to the degree of approximation which we adopt for small strains, whether the integrals in expressions (5) and (6) be taken throughout the volume and over the surface of the body in 193.] POTENTIAL ENERGY OF STRAIN. 123 its natural or in its strained state ; indeed the triple integral in (6) being integrated as to the element of mass, is absolutely identical in the two cases (§ 154). We shall always suppose, for the sake of simplicity, that in these and similar cases triple integrals are integrated throughout the volume and double integrals over the surface of the unstrained, body. 194.] Identity of the two expressions for Work done in varying Strain. Substituting for e, f, g, a,b,c in (5) from equations (59) of § 123, we get Se = 8 Vx = te Su ' etc - /cHo Ov\ 3 3 oa — SI w + =5- I = tt-Sio + ^-ov, etc. \3i/ 3s/ dy 3s ' and thus (5) becomes + V\ -pj&o + ^ry + T . ?r&u + S ■ %;&> + H . ~r8w > dxdydz Integrating by parts, as in § 146, 8 W =ff(Pbu +U&D+ TSu>)XdS -JJJ \^ u + Vx Sv+ tetoyafydz +ff( USu + QSv + SSw)fidS +ff(TSu, + SSv + SSw)vdS rrn?>T, ?>s„ or« \ JJJ W u + &? v + &*°)*>*yd* (7) 124 POTENTIAL ENERGY OF STRAIN. [194. Rearranging the order of the terms, 8 W =//{ (Pk+Ufi + Tv)8u + (Uk + Qfi + Sv)8v + (TX. + Sp + Ev)8w}dS "^ r {(^ + W + ? : ) SM+ (^ + l + S) Sw Hence, in virtue of equations (3) and (5) of Chapter III., we haye finally 8 W =//{F8u + G8v + H8w)dS +fff P {X8u +Y8v + Z8w)dxdydz. ... (9) Thus the work done hy Stress during an infinitely small change of Strain is always equal to the work done on the body by external forces in producing the change ; and either is of course equal to the corresponding infinitely small increase of the Potential Energy of the Strain. 195.] Case in which motion is taking place. If relative motion of parts of the body is taking place, so that the initial and final states of strain are only states through which the body passes ; as, for instance, when the body is vibrating about a stable state of strained equilibrium, maintained by suitable forces ; we may show by employing equations (4) of Chapter III., that the expression (6) for the work done on the body by the external forces is equal to the increase (u- + v* + w 2 )dxdydz ^JjfpifiM + vv + wib)8t . dxdydz, where St is the small interval of time occupied by the change. This again is equal to JJJp{u8u + v8v + w8v))dxdydz (10) Thus the expression (6) for the work done by the external forces— which must now of course be equal to the total change of energy, both potential and kinetic — diminished by the expres- 195.] POTENTIAL ENERGY OF STRAIN. 125 sion (10) for the corresponding increase of kinetic energy alone, becomes fffp{(X-u)8u + (Y- v)8v + (Z- w)8w)dxdydz +ff(F8u+G8v + H8w)dS (11) By equations (4) and (5) of Chapter III., (11) is identical with (8), and therefore with (5). Potential Energy of Strain. 196.] Energy per Unit Volume. Let W be the total potential energy of the body when held in equilibrium in the state of strain {e, f, g, a, b, c) by the distribution of stress {P, Q, B, S, T, U}. Also let V denote the measure of this potential energy per unit volume of the unstrained body, so that W=ff/Vdxdydz (12) We have shewn that the infinitesimal increment of V, due to arbitrary and independent infinitesimal increments of the strain- components, is given by 8V=P8e + Q8f+£8g + S8a+T8b+U8c (13) Now the potential energy and the components of the stress in any given state of strain are functions only of the components of that strain. Hence when the increments of the strain-components in (13) are sufficiently reduced, each side must become the perfect differential of some function V of the six independent variables e. / 9, <*> &> o- Thus we may write dV=Pde + Qdf+£dg + Sda+Tdb + Udc (14) and also 3T, 3F,„ dV 7 37, ZV ,, 3F dV= ^de + -g^/+ -^-dg + ^da+^db + ^ dc (15) whence we deduce that JdV JdV p _3F r ~de> Q ~ df dg dv dr 77 _3F S ~ da' T ~ db' U ~ dc .(16) (Compare §§ 32, 33. See Errata for those Articles.) 12G POTENTIAL ENERGY OF STRAIN. [197. 197.] Stress in terms of Strain. "Hooke's Law." .Since stress is a function only of the strain ultimately produced by it, it follows that if a single small stress {P, Q, R, S, T, U} produce the small strain {e, f g, a, b, c}, then two small stresses, each equal to {P, Q, etc.}, applied successively to the body, will pro- duce two successive small strains, each equal to {e, f etc.}. But, by the principle of superposition, the two successive small stresses are equivalent to a single small stress {2P, 2Q, 2R, 2S, 2T, 2U], and the two successive small strains to a single small strain {2e,2f,2g,2a,2b,2c}. Thus the single stress {2P, 2Q, etc.} will produce and main- tain the strain {2e, 2f, etc.}. This result may obviously be extended so long as the strain and stress remain small, so that ultimately we see that, if n be any finite multiplier, the stress, {nP, nQ, nR, nS, nT, nil} will suffice to maintain the strain {ne, nf, ng, na, nb, nc). Hence we deduce, solely from the principle of superposition of small strains and stresses, that if a perfectly elastic solid be in equilibrium in a given state of small strain, under a given small stress, and if the strain be increased in any finite ratio, the stress required to maintain it will be increased in the same rajjpo. In other words, the six components of stress arffi%i/mar func- i Ions of the six components of the corresponding strain. This law was discovered experimentally by Robert Hooke, and first made public by him in 1678. (For the various ways in which it has been arrived at theoretically, see Appendix III., below.) 198.] Coefficients of Elasticity. From equations (16) we see that the partial derivatives of V as to each of the strain- components must in general be linear functions of all the six components. And, finally, it appears that the potential energy per unit volume of a perfectly elastic solid under small strain is a homogeneous quadratic function of the six component strains. We may then assume 2 V= K n e 2 + k 22 / 2 + K330 2 + ic 44 a 2 + K 6b b 2 + KggC 2 ) + 2k 2s/9 + 2 *3i? e + 2k 12«/ + 2k 56 6c + 2« 64 ca + 2K ib ab + 2« u ea + %K{ b eb + 2K 16 ec + 2k- 24 /« + 2k 25 /6 + 2* 26 /b + 2K 3i ga + 2k 36 0& + 2x 3a gc .(17) 198.] POTENTIAL ENERGY OF STRAIN. 127 where the 21 " Elastic Coefficients " are, for a homogeneous body, absolute constants, depending only on the elastic properties of the body, the constant temperature at which it is maintained, and the directions of the arbitrarily chosen axes of reference. If the body be not homogeneous, the coefficients will be func- tions also of the position of the point in the neighbourhood of which V is given by (17). We shall, however, always suppose that we are dealing with naturally homogeneous bodies (§ 43, but see § 220, below). In general, the coefficients must be supposed all independent of one another ; and in fact we cannot with certainty attribute to them any property whatever, except that they are finite, and that for every possible form of small strain they must make V positive (§ 21). Differentiating (17), and substituting in (16), we get P= K u e + k 12 /+ K 13 g + K u a + K lb b + k 16 c Q = K 21 e + k. 22 /+ k 23 c/ + K u a + k 25 6 + k 26 c R = K 3l e + k 3 ,/+ K S3 g + K st a + k, 5 6 + k 36 c S = K 41 e + k 42 /+ K is g + K u a + k 4 .J> + k^c T= K 51 e + x 62 f+ K i3 g + k m « + « 55 6 + k 56 c U=K m e + K m J+ K 63 g + Kfi4 a + K eb b + k 66 c where k k = k 21 , etc., the doxible notation being employed solely for the sake of symmetry. 199.] Average Stress during change of Strain. Hence we find, by comparing (18) with (17), as we might have deduced directly from (16) by Euler's theorem on homogeneous functions, V=\{Pe+Qf+Rg + Sa + Tb+Uc) (19, whence by (12) W = h/f/^e + Q/+ Rg + Sa + Tb+ Uc)dxdydz (20) From (14) we have, by integration, .(18) /*\e,f,g, a, b, c j- r=/(Pde+Qd/+ J { 0, I). 0, 0. 0, j Rdg + Sda + Tdb + Udc). Hence the interpretation of (19) is that the average value of the stress, while the body is being brought from its natural state to the state of strain {e, f, g, a, b, c} is {£P, \Q, \R, \8, \T, \TJ\; that is, one-half of the stress required to maintain it in the specified state of strain. This might also have been deduced directly from the principle of superposition. For in each of the intermediate states of strain the stress (being always a function only of the actually existing 128 POTENTIAL ENERGY OF STRAIN. [199. strain) must be such as would keep the body in equilibrium in that state ; but, by the principle of superposition, if we have any number of states of strain, and the corresponding stresses given, the average of all these stresses will suffice to maintain equilib- rium in the state of strain which is the average of all the given states. Now, the path by which a perfectly elastic solid is brought to a given state of strain being without effect on the stress required to maintain it in its final state, the average value of the strain may be taken to be simply {%e, £/, \g, \a, £6, Jc}, and the stress corresponding to this is, by § 197, Up, IQ, iR, hs, rr, in}. This latter expression therefore represents the average value ot the stress during the change. 200.] Strain in terms of Stress. By elimination between equations (18) we can obtain the six component strains as linear functions of the six component stresses e = K n P + KuQ + K n R + K U S + K K T + K 16 U a etc., etc. I a = K„P+K 4s Q + K m R + .K u S + K a T+K ie U f ^ ' etc., etc. ' where K u = K. 21 , etc. Substituting in (19) we obtain V as a homogeneous quadratic function of the stress-components. 2 7 = K n F* + K„Q* + £„& + KJ3* + K U T* + Z m U* + 2K 23 QR + 2K 3l RP + 2K U PQ + 2K m TU + 2KJIS + 2K a ST + 2K U PS + 2K 1& PT + 2K W PU + 2K U QS+ 2K m QT+ZK„QU + 2K M RS + 2F 35 RT + 2K 3e RU. (22) whence, by differentiation and comparison with (21), _dV JdV dV] e = -dP'f = dQ' 9 =dR _3T JdV _3F .(23) 201.] Asymmetrical Elasticity. We have defined a homogeneous body (§ 43), in the most general terms, as being such that any two equal and similar portions, similarly situated in the body, possess identical elastic properties. In the most general case of homogeneity we may therefore suppose the elastic proper- ties of the body to vary in different directions ; that is to say, the specification of the stress required to maintain a given strain 201.] POTENTIAL ENERGY OF STRAIN. 129 will depend not only on the specification of the strain but also on the directions of the axes of reference. The equations of the last three Articles are applicable to this most general case of Asym- metrical Homogeneity ; the 21 elastic coefficients, and also the 21 reciprocal coefficients K ir ..K m (which are functions of the former), being taken to be all independent of one another and of the position of the origin, but varying with the directions of the axes of reference. Crystalline Symmetry. 202.] Planes and Axes of Rectangular Symmetry. Many natural solids are found to possess different degrees of symmetry in their elastic properties. Such solids are in general called crystalline, and their elastic symmetry is found to be in invariable relation to certain lines and planes connected with their constant external form of crystallisation. We now proceed to investigate the analytical conditions for various degrees of elastic symmetry, confining ourselves to the cases in which the lines and planes of symmetry are rectangular. 203.] One Plane of Symmetry. Let us suppose the elastic properties of the body symmetrical about the plane of xy, or any parallel plane (see § 201) : so that, for example, the specifi- cation referred to Ox, Oy, Oz of the stress required to maintain a given uniform elongation in the direction (X, «, v) will be the same as the specification referred to Ox, Oy, and Oz reversed, of the stress required to maintain an equal elongation in the direction (X, ft, — v). This latter becomes (X, /x, v) when Oz is reversed ; and since we know by § 113 that the specification of the elonga- tion depends only on (X, n, v), it follows that the condition that xy may be a plane of elastic symmetry is that the reversal of Oz leaves unaltered the specification of stress in terms of the specifi- cation of strain. Consequently the expression (17) for the potential energy in terms of the strain-components must also remain unchanged when Oz is reversed. Now the effect of reversing Oz is to change the signs of z and w ; hence by § 123 the signs of a and b are reversed, the other components remaining as before. Thus if xy is a plane of sym- metry all those terms in the expression for V which contain odd powers of a and b, except their product ab, must vanish. K 24 = 0, Kj3 = I 130 POTENTIAL ENERGY OF STRAIN. [203. And, finally, 2 7= K n e? + K a p + k^ + k u o? + k m & 2 + k^? + 2(«nf9 + K si9<> + K vfif) + 2k ««6 + 2(K 16 ec + /c, 6 /c + Kz&c) (24) Thus the number of elastic coefficients is reduced to 13. 204.] Three Planes of Symmetry. By three successive applications of the results of the last Article, we may show that, if all three of the coordinate planes are planes of elastic symmetry, all the terms in V involving odd powers of a, b, or c must dis- appear. In addition therefore to the above conditions we must now have «* = ) K 16 ~ K 26 ~ K S6 -"■' Thus we may write 2 7= K n e 2 + 10a/ 2 + Kjjj 2 + /c^a 2 + k m 6 2 + k^c* + l^fg + K 3i ge + « 12 e/) (25) and the number of the elastic coefficients is reduced to 9. This may be called complete rectangular symmetry; it belongs to the " tessaral " class of crystals whose form of crystallisation is a rectangular parallelepiped, the planes of elastic symmetry being parallel to the pairs of opposite faces. Equations (18) become P = K n e + k 12 /+ K 13 ^~ T=K K b by which we see that the relations between the elongations and normal tractions perpendicular to the "principal planes" (or planes of symmetry) of the crystal, and between the shear in each of these planes and the corresponding shearing stress form four independent systems. 205.] One Axis of Symmetry. Let us next suppose that there is one direction in the crystal about which its elastic properties have a certain degree of symmetry. Any line Oz drawn in this direction may be called the Axis of the crystal, and its elastic properties will be arranged with more or less symmetry in the plane of xy, or any other plane perpendicular to the axis. There are two principal degrees of such symmetry, which we will consider separately. (26) 205.1 POTENTIAL ENERGY OF STRAIN. 131 (i.) Uniaxial Crystalline Symmetry. In this case, which is common to Iceland Spar, and other crystals, called in Optics "uniaxial," there are two orthogonal planes through the crystal- line axis, such that the elastic properties of the body are not only symmetrical about each (or about any planes parallel to either), but they also bear exactly the same relations to one as to the other. Thus these two planes (which we shall take for the planes of yz and zx) may be interchanged without affecting the form of the Potential Energy, or the relations of Stress and Strain. It is thus obvious that V must involve e and / symmetrically, and also a and 6. Thus we may write 2 V= Kn { U=K ee C Such crystals may be said to have square symmetry about their axis. (ii.) Complete Circular Symmetry about an Axis. In this case, which does not occur in any natural crystal, but which is artificially brought about in wires drawn from masses of metal naturally possessing the highest degree of symmetry (see § 207, below), the elastic properties of the body are absolutely symmet- rical in all directions perpendicular to the axis ; so that, if this be Oz as before, it is absolutely indifferent in what directions we take Ox and Oy. It is obvious that in this case the expression (27) for V must retain the same form when Ox and Oy are turned through any angle a> in their own plane. Let us take as so small that its square and higher powers may be neglected : the effect of rotat- ing the axes will then be to change x, y, u, v into x + wy, y - wx, u + tev, v — am, ; z and w remaining unaltered. The effect on the strain-components will be to change e, /, g, a, b, c into e + wc, f— vac, g, a-aib,b + ma, c — 2o>e + 2&>/, respectively. Hence, neglecting the square of as, the expression (27) for 2V is transformed into 2 V + 2(o(k u - 2k k - Ki3 )(ec -fc). The term involving to must vanish for all values of as, and there- 132 POTENTIAL ENERGY OF STRAIN. [205. fore, since the strain-components must be assumed independent, we must have Thus 2 r= «„(«• +/•) + "xs* + *«(«' + * 2 ) + «*? + **JJg + g») + 2( Kn -2 KK )ef. (28) and the number of the elastic coefficients is reduced to 5. 20C] Three interchangeable Planes of Symmetry. Let us now start afresh from the case of § 204, and suppose that the elastic properties of the body are not only symmetrical about the three coordinate planes, but that they also bear precisely the same relations to each of these planes. It is then evident that the coordinate axes may be interchanged in any manner without affecting the form of V — that is to say, the expression (25) must be so modified that it may involve e, f, g symmetrically, and also a, b, c. Thus we must have K ll = K M = K 33 j K U = K B5 = K B6 f • K '23 ~ K 31 ~ K 12j And, finally, 27= Kn (e* +f+g") + ^(a 2 + 6 2 + c 2 ) + 2 Kn (fg + ge + ef) (29) Thus the number of elastic coefficients is reduced to 3. This may be called complete cubical symmetry. It occurs in Rock Salt. It is obvious that if any cubical portion of the body could be removed and replaced with any pair of its faces occupy- ing the positions originally belonging to any other pair, and then made once more continuous with the rest of the body, the elastic properties of the whole would be absolutely unaffected. Equations (18) become in this case P = « 11 e + K 23 (/ + ^)" ©= K ll/+K23(.? + «) R = K u g + Ka (e +/) and, finally, for an isotropic solid, ■2r=K u (e i +f s + g i ) + K ii (ar + ¥ + ^) + 2(K n -2 Ku )(/g + ge + ef)...(30) Thus the number of the elastic coefficients is reduced to 2. Equations (18) now become P = K n e + (k u - 2k4,)(/+ + ef- s 8 2 ) = P = - 4k 44 - The Elastic Moduli of an Isotropic Solid. 210.] Modulus of Rigidity. We see at once from equa- tions (31) that a shear in either of the coordinate planes requires only the corresponding shearing stress (§ 149) to produce and maintain it, and that this stress bears to the shear the constant ratio k u : 1. Since the directions of the axes of reference are perfectly indifferent, it follows that k u represents the shearing stress that 210.] POTENTIAL ENERGY OF STRAIN. 135 must be applied in any plane to produce and maintain the unit of shear in that plane ; analogically speaking, that is to say. A shearing stress which would produce such an enormous distortion of the body as the unit shear (see Appendix IL) would certainly not obey the proportional law, except perhaps in one or two sub- stances of exceptionally perfect elasticity. The units of strain (and all finite strains) really lie altogether outside our theory, and it is only by direct experiment that we can determine the degree of approximation to which it represents their laws. Such statements as the above must always be understood to be made under this reserve. (See Appendix IV., below.) This quantity is usually called the Modulus of Rigidity, or simply the Rigidity of the body ; it is also known as its Elasticity of Figure. We shall in future denote it by the symbol n. 211.] Modulus of Compression. Let us now suppose that the strain throughout the body is a homogeneous cubical compression, uniform in all directions of amount A. Then (§ 112) we shall have everywhere a=b =c =0 }■ and by equations (31) S = T=U=0 )' By § 174 we see that the stress at every point of the body will be a homogeneous hydrostatic pressure, of amount n = (K 11 -| ( c 44 )A; and to maintain this strain and stress we must apply a uniform normal pressure II over the whole bounding surface of the body. Thus the quantity (/c n — £*c M ) represents the uniform normal pressure which must be applied over the surface to produce the unit of cubical compression throughout the body. This quantity is usually called the Modulus of Compression, the Bulk-modulus of Elasticity, or the Elasticity of Volume. "We shall in future denote it by k. Of course a uniform normal traction over the bounding surface, of amount kA, will in like manner produce a uniform cubical dilatation A throughout the body. The reciprocal modulus 1/k is often called the Compressibility of the body, denoting as it does the cubical compression produced by a uniform surface pressure of unit magnitude. .(32) 136 POTENTIAL ENERGY OF STRAIN. [212. 212.] The New Notation. Writing then in equations (30) and (31), they become r = {k+%n)e + {k-%n){f +g y Q = (k+%n)f+{k-%n){g + e) B=(k + %n)g + (k- §)i)(e +/) S = na T = nb U=nc and 2F= (k + f n)(e 2 +/ 2 + <7 2 ) + 2(£ - \n)(f0z 138 POTENTIAL ENERGY OF STRAIN. [213. Denoting these by e, /, g we have P = Oka 3k +- n ' f=9 = 3k -2n ~2(3k + n) ■ is if q denotes Young's Modulus ? = 9kn 3kn 3k + n m .(38) In all known solids k >f n, so that there is always a lateral contraction in the directions perpendicular to that of the applied tension. If we employ the symbol a to denote the ratio {—fie) of lateral contraction to longitudinal elongation in this case, we shall have 3k — 2n m — n '2(3k + n) 2m .(39) 214.] Strain in terms of Stress. If we solve equations (32) for the strain-components, we find 3k + n 3k- 2n e ~ 9kn - F ~ \Un W + R) etc., etc., or, substituting from (38) and (39), e = \-P-\(Q + R) f=\.Q-- q {R + P) g = \.R-- q (P+Q) a = -.S n b J-.T n cJ-.U n Thus, by (19), (40) 2 V = ~ % P + « + *>* + YrS^ + & + **) + l( S2 + T2 + U2 )~<- (« ) 215.] POTENTIAL ENERGY OF STRAIN. 139 215.] Principal Axes of Strain and Stress. If the principal axes of the Strain at any point of the body are parallel to the axes of reference, we have at that point, by equations (32) of § 83, e = «i>/=«2, g = e 3 ; a = b = c = 0. Thus, at the same point, P = (to + ?i)cj + (m - n)(£ 2 + « 3 ) \ Q = (m + 9i)e 2 + (to - «)(«, + £j) It = (to + ra)« 3 + (to — n)(«! + e,) 5 = r=o £T=0 Thus the stresses across small plane areas drawn through the point, perpendicular to the axes of reference, are wholly normal, and by § 163 the axes of reference are also parallel to the principal axes of the Stress at the point. Conversely, it may be shown that, if the axes of reference are taken parallel to the principal axes of the stress at any point of the body, they must necessarily be parallel to the principal axes of the strain at the same point. Hence we deduce that, at every point of an isotropic body, the Principal Axes of the Strain and of the Stress are coincident, and that the principal elongations e v e a , e s and the principal normal stresses N v N~ s , N~ s are connected by the equations If 1 = (tn + «.)«! + (to - w)(e 2 + e 3 )\ i\r s = (to + n)% + (to -n)(€5 + CJV (42) N 3 = (to + n)c s + (to - »)(£! + tj) J ^ = -i?;-'(A r , + Jir i ) \ (42a) or by q ' q The corresponding formulae for V are 2F= (to + n)(e, s + e, 2 + e 3 2 ) + 2(m — n)(e i € s + £ 3 e, + e,e,) j = (to - n)A 2 + 2r»(e, a + e," + c s 2 ) j ' ..(43) or 2F= -fa + Ni + NsY+^N' + N' + W) (43a) These results might of course have been deduced directly from § 209, coupled with the corresponding theorem that V must also be an invariant of the stress. 140 POTENTIAL ENERGY OF STRAIN. [215. They evidently apply also to the crystalline forms of §§ 204- 206 (since in them also the shearing stress and the shear vanish together independently of the elongations and normal stresses), but not to any lower degree of symmetry. 216.] Lines and Tubes of Stress. Tie Lines and Strut Lines. Principal Surfaces of the Strain. The com- ponents of strain and stress being supposed continuous functions of the coordinates throughout the body, so also will be the direction-cosines of the Principal Axes at each point, given by equations (29) of § 79, or by equations (22) of § 163. Hence if we draw the principal axis Pg at any point P, corresponding to the continuous elongation e, and the continuous normal stress i\ 7 ,, and if an elementary length PP' be taken along Pg, and the corresponding principal axis P"£' be drawn at P', the change in direction from Pg to P"g will be a small quantity of the same order of dimensions as the elementary length PP. If this process be continued we get a broken line pp , p"P"' } composed of elements PP', P'P' , each of which coincides with the principal axis for e, and N 1 at one of its extremities. Proceeding to the limit, in which the lengths of these elements are indefinitely reduced, we have a curve such that the tangent to it at any point P is the principal axis Pg for e x and JV, at that point. It is thus possible to draw a system of continuous curves in the body enveloping the principal axis P£ at every point through which they pass. The differential equations of this system are edx + s 3 dy + s 2 dz s 3 dx +fdy + s^z s.,dx + s y dy + gdz dx dy dz 1 ' dx + Udy + Tdz Udx + Qdy i Sdz Tdx + Sdy + Iidz dx ~ dy dz = N * where of course for e, and A r x are to be substituted the proper functions of x, y, z. Since e, is a root of equation (28) § 79, and iV, of equation (21) § 163, only two equations of each set are independent. We get a second system of curves enveloping all the principal axes Pri, corresponding to e 2 and N v at the points through which they pass, and a third system everywhere enveloping Pf. It is obvious that these three systems of curves cut everywhere orthogonally ; and that the strain at each point consists of an elongation of each of the three curves which pass through it (with or without rotation), while the stress consists of a normal traction across each of the three elementary plane areas which can be drawn through the point to touch two of the curves. These curves are called Lines of Stress. 216.] POTENTIAL ENERGY OF STRAIN. 141 Fig.19- Let us take two consecutive f-lines, and also two consecutive j?-lines which intersect the former ; these four curves will enclose an elementary figure which is ultimately a plane rectangle. If now we draw the ^-curves through every point of the perimeter of this area, we shall form a tube of elementary section, called a Tube of Stress (Figure 19.) The normal section of the tube at any point is an approximately plane rectangle bounded by consecutive , Stress lines of the r\ and f systems, while each of its sides may be looked upon as composed of approximately plane rectangles bounded by the edges of the tube and by two consecutive curves of the q system or of the f system. The stress across every section of the elementary fibre of the body bounded by the tube is wholly in the direction of its length ; and the stress across any element of its surface (the tube of stress) is wholly normal to the element. It is thus obvious that the body may be supposed divided in three different ways into systems of curvi- linear fibres, which transmit stress through the body in the direction of their length, while the action between adjacent fibres is, at every point, wholly normal to their common surface. We shall adopt the terms used to denote the functions of beams in engineering structures, and call these fibres Ties when they transmit a tension, and Struts when they transmit a thrust in the direction of their length. The Stress lines which form the walls of the tubes will accordingly be called Tie-lines or Strut-lines. Thus equations (44) are the differential equations of a system of tie-lines or strut-lines according as N t is positive or negative. If .#, = we have a system of lines of zero stress. If we draw several adjacent Tubes of Stress (of the ^-system, let us say) as in Figure 20, it is obvious that any set of conter- minous normal sections of these tubes will form adjacent elements of a continuous surface. Each such surface will contain a complete system of the jj-curves, and also a complete system of 142 POTENTIAL ENERGY OF STRAIN. [216. the f-curves, and will everywhere be cut normally by the /-curves. Thus we can construct three orthogonal systems ot surfaces throughout the body, such that (i.) The curves of intersection of the three surfaces which pass through any point P are the Lines of Stress at P, and there- fore have for their tangents the principal axes of the strain P£,P n ,Pl (ii.) The tangent planes to the three surfaces at P are the principal planes of the strain. (Hi.) Each of the elements of volume (ultimately rectangular parallelepipeds) into which the body is divided by consecutive Fig. 20. surfaces of the three systems, is subjected only to elongations in the directions of its edges (with or without rotation) and suffers no sltear whatever (consequently remaining rectangular). These surfaces may be called the Principal Surfaces of the Strain or of the Stress. We shall return to them in the next Chapter. If one of the principal stresses vanishes, each of the system of principal surfaces which is cut orthogonally by the lines of zero stress envelopes the Plane of the Stress (§ 175) at every point through which it passes. The differential 216.] POTENTIAL ENERGY OF STRAIN. 143 equation of this system is therefore, by equations (66) and (67) of § 184, ■Jp. dx + Jq. dy + «/r. dz = Q} dx dy dz \ (45) and those of the lines of zero stress are dx dy dz \ J$ Vq VrJ- (45a) or sdx = tdy = jxdzj When two of the principal stresses vanish (§§ 185, 186) only one principal axis at each point is determinate. Thus we have only one determinate system of lines of stress, given by equations (76) and (77) of § 186, namely, dx dy dz ~\ ~JP = ~J$ = U%1 (46) or Sdx = Tdy = Udz) and only one determinate system of principal surfaces, given by JPdx+ 4Qdy+ jBdz = ti\ dx dy dz \ (46a) is + i + u= } { } In this case any two systems whatever of surfaces which cut these and each other orthogonally may be taken as the other two systems of principal surfaces, and their curves of intersection with the determinate system will give two systems of lines of zero stress. In homogeneous stress and strain, the Lines of Stress are straight lines, and the Principal surfaces are orthogonal systems of parallel planes. Equations of Equilibrium and Motion. 217.] In terms of the Component Strains. Substituting for the component stresses in equations (3) of § 142 their values (32) in terms of the component strains, we get for the equations of equilibrium -de ftf -dg\ (-de 36\ Y A (r» + rOte + ( m - n ){dx + dx) + n \dy- + dz-) + pX=0 ■dg , V3e 3A m da\ „ .(47) 144 POTENTIAL KNERGY OK STRAIN. |217. The equations of motion (4) of § 143 similarly become 3e fdf dg\ fbc db\ , ^ .. v " <—>!♦<-« >(14;M^SH^>=* ■■■■<«) 3<7 /3« 3A /36 3a\ , „ v „ (m + n)^ + (« - «)(^ + ^ j + ^ + ^J + ,»(* - «) = lastly, the boundary conditions (5) of § 144 take the form X[(ni + n)e + (in - n)(^/+ g)~\ + fxnc + vnb = F\ knc + [j.[(m + n)f+ (m - n)(g + «)] + vna = G , knb + /ma + v[(m + n)g + (m - n)(e +/)] = H) (49) where F, 0, H are the components of surface traction, and (X, n, v) the direction-cosines of the outward normal. 218.] In terms of the Displacements. Substituting for the component strains in these equations their values from equations (59) of § 123, we get for the equations of equilibrium 2fu '■ ^ '\dxdy 'dzdx/ dg\dx "by J 3 /3u 3i»\ y — (\ 3z\32 3a;/ etc., etc. Rearranging the order of the terms, these equations become 3 Rm 3v 3aA m ^c + dg + d^) + n ^ + P X=0 3 /3m dv ckv\ 3 /3m 3u 3mA m dz\te + Vy + dz~) + n ^ w + P Z=0 or, since a _ 3u 'dv 3u> dx 3?/ dz ' .(35) 3A dx 3A i~— - 32/ 3A to 3^ + w V 2m + P a '=0 m ty +n V 2 v + pY =0 1 3s + «V 2 w + p^ = .(51) 218.] POTENTIAL ENERGY OF STRAIN. 14-: .(52) .(52a) From these we may at once deduce the equations of motion 3A ^ A «t-s- + ny^u + p( A - ii) = 3A 3A «i ~ + 7j^- 2 io + p(2 - to) = It is equally easy, by a slightly different transformation, to throw these equations into Lam&s form 3A A30, ?6».\ where Q v &,, 6 3 are the component rotations. If we substitute for them their values from equations (59) of § 123, this form is at once seen to be identical with (52). The boundary conditions are *[('» + »)^ + ( m - '*>(! + S)] + m^ §S) ) C3» , Veto ?>v\-\ „ .(53) + V Equations of Motion and Equilibrium obtained from the Potential Energy. 219.] We obtained equations (47-53) by substitution in the equations of Stress given in Chapter HI. These relations how- ever have not been elsewhere assumed in the present Chapter, except in §§ 194, 195, to prove the equality of the small total increase of energy and the corresponding small amount of work expended on the body by the external forces. K 146 POTENTIAL ENERGY OF STKAIN. [219. We now propose to show, by an application of the principle of Virtual Velocities which is strictly the converse of that of s§ 194, 195, that, assuming the expression (34) for the Potential Energy per unit volume, the equations of motion and equilibrium (47, 48, 49) can be immediately deduced. Introducing the symbol m into (34) we have 2 V= (m - «)A 2 + 2rc(e 2 +/ 2 + g-) + n{a 2 + b 2 + c 2 ). Thus if W be the total Potential Energy of the Strain 2 W=fff{{m - n)A 2 + 2n(e 2 +/ 2 + g 1 ) + n(a 2 + V 2 + c")}dxdyds. . .(54) by equation (12), § 196. We shall consider the most general case, in which motion is taking place, for the case of statical equilibrium can always be deduced from it by making all the velocities and accelerations zero. The kinetic energy of the motion is then <3E =J//hp(i l2 + & + w^dxdyd:.. Let us now suppose that the Applied Forces and Surface Tractions are allowed to do work on the body by producing a very small variation of the strain. Let Su, Sv, Sw be the consequent small increments of the dis- placements of any point (x, y, z), in the body or on its surface, from its natural position. These may be supposed quite arbitrary and independent (but each must be a continuous function of the coordinates). Let <5e, Sf, Sg, Sa, 6b, Sc, SW and + ^,S« I /3, 3 \ /3 3 \ ) + nil k & + ~-jbw\ + ncl -, to + 5-81*1 \dxdydz ■M\ "7\ /^ ^t [(•w - ?i)A + 27te]~-8« + nc~-8« + rc&~ S/w 3 3 3 + 'IC57 8m + [(»« - «.)A + 2nf\<- Sv + Ttarr-Sc; + nb^-Su + ««.■> 8e + [(»« - m)A + 2»w/K S«> > dxdyds. Integrating by parts as in §§ 146 and 194, Sir =/Y{\(m - n)A + 2ne]8w + nc . Sv + nb . Siv}MS - 1 1 1 \ o-.[(» l - m )A + 2jte] . 8tt + -rjne) . Sv + ^ ,(' 1 ^) • ^ ( dxdyds + ff{na . Su + [(mi - n)A + 2»t/]8y + na . Sw}/ul,S - IJJ \ -,-(ne) . Su + •*-[(»» - w)A + 2n/]So + .-, (vta) . Sw !- dxdtjdz + ff{ni> ■ Su + na . Sv +• [(m - n.)A + 2ng]8io} vdS - /// -- o.(»6) • Sm + ^- y ('*a) • 8» + =c[( m - ?'*)A + 2?ty]8to J- dxdyds. Rearranging the order of the terms, 8 W = /T{X[(m - ?i)A + 2ree] + fine + vnb}Su . dS + ff{ kne t fi[(m - n)A + 2nf] + vna}Sv . dS + rf\\'nb + fina + v[(m - n)A + 2ny]}8w . d)) + p(X - u) [ 8u . dxdydz \ =r;(iic) + ~r [(m — h)A + 2nf] + ~r ('"') + P(^~ <-") (' ^ w • dxdydz -Iff\ Iff f 3 , 3 r vl: .(51)) , _. (jii>) + ~ - (?ia) + -s [(« - V, z). 221.] Absolute Moduli, Weight Moduli, Length Moduli. The moduli k, n, q, being stresses, are of course, like all other stresses, measured by the force applied per unit area. The numerical measure of a modulus and its physical dimensions therefore depend both on the unit of length and the unit of force which we adopt. (i.) The gravitation measure of force is the one most com- monly adopted. On this system a modulus represents the weight that must be applied per unit area to produce unit strain of the corresponding type. Thus the moduli may be given in pounds or tons per square inch, or preferably in grammes per square centimetre. (ii.) Since what we call the vjeight of a gramme is simply the force exerted by the earth on a given mass known as a oramme, the gravitation measure of force, and therefore also of the moduli, varies from point to point of the earth's surface, which the resistance of the body to stress of course does not. In order therefore to make these measurements available in all coun- tries the forces ought to be reduced to absolute measure, which (since the absolute unit of force, called in the British system the pounded, and in the metric system the dyne, is that force which produces the unit acceleration in the unit mass) is done by multiplying the gravitation measure by the numerical value of the acceleration produced by gravity at the spot where the measurements are made. Each modulus will then represent the number of absolute units of force to be applied to the unit of area :— say the number lfiO POTENTIAL ENERGY OF STRAIN". [221. of poundals per square inch, or of dynes per square centimetre. It should be mentioned, however, that the discrepancies in our experimental data — due to variation of material, etc. — far more than cover the small variations of gravity. Rules for reducing stresses and moduli from one system to another will be given with the tables at the end of Appendix IV., below. {Hi.) A very convenient method of measuring the moduli is to express them in terms of the length of a bar of the material, of unit section, whose weight is equal to the force per unit area that is to be applied. This, like (i.), is a local measure. When the moduli are expressed in this system they are called Length-moduli, and their numerical measures are called the lengths of the moduli. Thus if k, k denote the weight-modulus and length-modulus of a given material, expressed in the C.G.S. system of units, a force equal to the weight of k grammes, or to the weight of a bar of the material one square centimetre in section and k centimetres long, distributed uniformly over each square centimetre of the surface of any body of the same substance, will produce in it the unit of cubical compression. And similarly for the other moduli. Thus k = pk, etc., where p is the density of the body in grammes per cubic centimetre. 222] Resilience. Strength. Tenacity. Modulus of Rupture. When a given elastic body is brought to a given state of strain, and then set free, the work which it is able to do in virtue of its elasticity, in returning to its natural state, is called the Resilience of the given body for the given strain. This we know (§ 31) to be equal to the work done on the body in straining it, or to its potential energy in the given state of strain. When we speak simply of the Resilience of a material for a given type of strain, we mean its potential energy per unit volume when strained to its elastic limit (§§ 12-14) for that particular type. For a brittle substance (§ 13) with comparatively narrow limits of elasticity, within which the proportional law of § 197 may be taken as very approximately true (see Appendix IV., below), the resilience will be given at once by substituting in any of the expressions obtained for V in this Chapter the limiting values of the strain-components, any increase of which would produce rupture or marked permanent set. For example if E, A represents the limits of elongation and shear for a brittle isotropic solid, its resilience for elongation is i(w + n)E'i, and its resilience for shear is 222.] POTENTIAL ENERGY OF STRAIN. ].il The limits of safety for linear elongation and contraction, as well as for cubical dilatation and compression are generally different. Thus (the sign of a shear being a purely geometrical conven- tion, devoid of physical meaning) a natural isotropic solid has Jive principal resiliences. The resilience thus defined is measured as energy per unit volume. This might be expressed in terms of an absolute unit (e.g., ergs per cubic centimetre, or foot-poundals per cubic foot 1 , but in practice the (local) gravitation measure of energy is adopted, the unit of which is equal to the work done in raising the unit of mass through the unit height against gravity. Thus the resilience is usually expressed in gramme-centimetres per cubic centimetre, or in foot-pounds or foot-tons per cubic foot. There is also a length measure of resilience, defined as the height through which unit mass of the body must be raised to do work against (local) gravity equal to the resilience per unit volume. Thus if V be the resilience of a given material for a given type of strain in gramme-centimetres per cubic-centimetre, and T9 in centimetres per gramme ; and if n, k, q, n, k, q be the weight moduli and length moduli on the C.G.8. system : T7^=£/k = «/n = (62) where p is the density of the body in grammes per cubic centi- metre (or its specific gravity) ; whence we should have, if tl e proportional law held up to the point of rupture, for torsion, and so on. The Strength of a material for a given type of strain is measured by the stress which will produce the limiting or extreme strain of that type. Tlras we have an Elastic Strength, corresponding to the limit of perfect elasticity, and an Ultimate Strength corresponding to the point of rupture, for each type of strain. The Tenacity is the ultimate strength for elongation — that is the value of the longitudinal stress which produces the extreme elongation E. If the proportional law held up to this point, the tenacity would be qE, in grammes- weight per square centi- metre. The Length-modulus of rupture is the tenacity expressed as a length. On the same supposition this would be qE centi- metres. (See ilie table in Appendix IV.) 152 POTENTIAL ENERGY OF STRAIN. [223. Possible Discontinuity of Strain and Stress. 223.] Limitations. We have hitherto confined ourselves to the consideration of those cases of strain in which not only the displacements but also the strain-components themselves (§ 52) are perfectly continuous functions of position throughout the whole body. And in accordance with this limitation the Applied Forces and Surface Tractions (§ 136) and consequently also the Stress (§ 137) have also invariably been taken as continuous, and therefore (§ 197) suitable to maintain such a strain. Discontinuity in the Applied Forces never occurs in actual structures to any important extent, but the consideration of discontinuous Surface Tractions and Pressures is of the utmost practical importance, since, for obvious reasons, the component parts of a complicated structure must necessarily bear upon one another by definite and circumscribed portions of their bounding surfaces. Let us now therefore consider how far our theory, as at present developed, can take account of such discontinuity. We must first investigate the nature and extent of the discontinuity (if any) permissible in the displacements and the components of strain and stress, and hence deduce the characteristics of the discontinuous systems of external forces with which we are able to deal. 224.] The component displacements. To begin with, we may observe that, even though in passing from one region of the body to another the displacements may become discontinuous in form, they cannot in any case present discontinuity of magni- tude. For if it were otherwise, two points immediately in contact with the separating surface (and practically coincident with one another in the natural state) would suffer displacements differing by quantities of a higher order of magnitude than their initial distance, and rupture of the body would take place over portions or the whole of the surface of discontinuity. If then with the notation of § 51 we suppose the component displacements in any one region of the body given by u i = i(x, y, s)\ v i = Xi( a- > V, ») k "'i = M x > y» z )) and in any contiguous region by u 3 = 3 (x, y, z)\ l '-2 = Xofo V, *) [> w, = ^(x, y, z)) 224.] POTENTIAL ENERGY OF STRAIN. 153 we must have, at every point of the surface of discontinuity, "£i = *.-'(*!'-MK( 'l* ! a? + etc., etc. and those of P, are 2 ' /.3*j 3+ 4 3 v fa, and their derivatives may be taken as those which they have at P, so that fa = fa,, etc. If, as in § 52, strained lengths of these projections be denoted by h+8h, etc., we shall have — .--D(S&-2)^-f)"<£*2)] + W-t)* ]♦ with similar expressions for Sic and SI. 1 .")4 POTENTIAL ENERGY OF STIJA1N. [225. Comparing these with §§ 52-54-, we see that the elongation of any elementary straight line which crosses the surface of discon- tinuity is simply to be taken as the^g» of the elongations of the two portions into which it is divided by the surface ; and if the component displacements satisfy the conditions of § 52, each in its own region and up to its bounding surface, — that is to say : if Xi' V'i' an( l a ^ their derivatives be finite, infinitely small, or zero for all points whose displacements they represent, and also ,, x,. V r 2 — * ne strain- components at every point of the body, including those which lie on the surfaces of discontinuity, will, as before, depend entirely on the first derivatives of these functions, and the form of our theory of small strains will not be altered. It must be particularly observed that the conditions of § 52 are only imposed on each strain-component-function within and up to the boundaries of its own region, and (so far as the con- ditions of strain are concerned) no relations need be assumed between the values which any two distinct forms take at the dividing surface. In other words : while the displacements may be discontinuous in form from one region of the body to another, but must be continuous in value throughout the whole body; the strain- components admit of discontinuity both of form and value from one region to another, provided always that the discontinuities of value only occur coincidently with the discontinuities of form. 22C] The Stress-Components. Since the stress across any element of a surface of strain-discontinuity must, like that across any other surface in the body, be a purely mutual action between the two portions of matter immediately in contact with it on either side, it is obvious that even if the stress becomes discontinuous as to its form in crossing the surface, it must be to a certain extent continuous in value. For if we take an element (IS of the surface (practically coinciding with an element of the tangent plane at its centre) and form two discs of elementary thickness, bounded by two elementary plane areas parallel to dS on either side, the theorem of § 137 must apply rigorously to each of these discs, so that the components of the stresses across their further faces can only differ by quantities of the same order of magnitude as the thickness of the two discs combined. That is, if we draw a small plane area very close to a surface of discontinuity, and parallel to the tangent plane at its nearest point, the components of the stress across this area will preserve continuity of value while the area moves parallel to itself across the surface of discontinuity. The analytical conditions can easily be deduced from § 144. Let ABC in Figure 9 represent an element of the surface of dis- 226.] POTENTIAL ENERGY OF STJiAIN. 1 of) continuity separating region (I.) from region (II.). Let be any point (x v y v zj in the first region. Complete the rectangular parallelepiped of which ABC is a diagonal plane, and let 0' ( x 2> Vi> z %) be its opposite angle in region (II.) The stress at will have for its components P v Q v R v S v T v l\, continuous functions of (x v y s,) throughout region (I.) ; and similarly the components P 2 , Q 2 , &» s & T » U i of the stress at °" will be functions of (x„, y 2 , z.) continuous throughout region (II.) If the signs of (X 12 , 'n 12 , v n be taken as in § 144 they will represent the direction-cosines of the normal drawn towards region (II.) We can then show, as in that Article, by diminishing indefinitely the distance 00', that the components, in the positive directions of the axes, of the stress exerted across ABO by the tetrahedron O'ABC on the tetrahedron OABC must be r^ + u^^iwA ?' 1 A 12 + A S> 1 „ + J S 1 r 12 J And similarly, the components, in the negative directions of the axes, of the stress exerted across ABC by the tetrahedron OABC on the tetrahedron O'ABC must be Since these stresses must be identically equal and opposite, the conditions to be satisfied at a surface of stress-discontinuity are W, - A) + IhJPi - V.) + vj. T, - T t ) = 0\ ^(U 1 -U,)+i h2 (Q 1 -Q,) + y ia (S 1 -S 2 ) = o\ (G3) M 1\ - T 2 ) + h2 ( S, - S,) + v 1 „(R 1 - E 2 ) = 0) If therefore the law of § 197 is to hold throughout, and if the body be everywhere homogeneous and isotropic, the relations which must exist between the strain-components at a surface of discontinuity are A 12 [(m - w)(Ai - A^ + 2n{e 1 - &)] + fhM c i ~ c a) + v iMf>\ - h) = ®\ A 12 ra(c! - c„) + fts f(i» - ra)(A, - A 2 ) + 2w(/ -/,)] + Vj 2 ot(o, - a.) = ... .(64) A 12 w(6! - 6,) + fii/nld! - a 2 ) + v 12 [(»i - ra)(A, - A„) + 2n(g' 1 - g„)] = J These, with the relations % = m 2 , v 1 = v. 2> w 1 = iv„ (65) between the component displacements, are the conditions to be satisfied at every point of a surface in the body at which the strain and stress become discontinuous in form. l.'ili POTENTIAL ENERGY OF STRAIN. [227. 227.] The External Forces. Our next object is to discover the systems of discontinuous Applied Forces and Surface Trac- tions which are capable of producing strains which satisfy these conditions. For this purpose we shall employ the method of §219. We shall suppose for simplicity that there is only one surface of discontinuity in the body, and to make this assumption as general as possible we shall suppose that it cuts the bounding surface of the body. Let us call the two portions into which the volume of the body may thus be supposed divided regions (I.) and (II.), and let the suffixes 1 and 2 be used to distinguish all quantities belonging to them. Let 2,, 2, be the portions of the external surface of the body belonging- to the two regions, and 2,, the surface of discon- tinuity. We shall write the direction-cosines of the normal to the latter at any point (\ 12 , fi^, v i2 ) when drawn towards (II.) and (\ 2] , fn, t , v a ) when drawn towards (I.), so that we have identically A i> + Ki = f-ii + fhi — v n + v -2i - (66) If the body be isotropic and homogeneous, its total potential energy is W, where 2 W =f/A( m - "A 2 + 2 ' 1 K 2 +/i 2 + o- + e.i)}dxjy,dz, and its kinetic energy (if in motion) is ^ where 2% =fffp(u^ 2 + v* + w*)dx x d Vl dz x +jffp{u£ + v 2 2 + w 2 2 )dx 2 di/ 2 dz.,. Let us now suppose, as in § 219, that small arbitrary variations Su, Sv, Sw of the displacements are produced through- out the body. Then the work done by the external forces will In: fffpiX^u, + 7,8b, + JZJwJda^dftdz, +fffp{Xfiu„ + Yfiv. 2 + Z. 2 Sw„)dx„dy„dz, +ff{F$u l + Gfa + HJwJdZ, +ff(F£u i + <7 2 Si> 2 + HJ&wJdZ,. [It is hardly necessary to remark that no work can be done on the body as a whole by stress across 2 .] 227.] POTENTIAL ENERGY OF STRAIN. 157 The increment in the kinetic energy is 8 u l + v-$v x + WiSw^dXidy^lz! + /Yfp(ii.,8u» + i3 2 8y. 2 + w % 8w. 2 )dx. 2 dy. 2 dz. i . Thus the increment of the potential energy must be 8 IF =-fffp{(X x - i^Sitj, + (F, - «,)««! + (Z x - iv^Bw^dx^dz, +fffp{{X. 2 - ii 3 )5M 2 + (Y. 2 - v. 2 )Bv, + (Z. 2 - w. 2 )8w i }dx. i dy 2 dz, +//(F 1 8u 1 + GM + H 1 Sw 1 )d2 1 +//{F 2 8u,+ GM. + H.,5w.>)d?-2 (67) But we also have, on substituting the values of the strain- components, m= JJJ \ [(m - n)A1 + 2 " e > ] ^ + ,?c ^ +w6 ^ + »"i- 3y ; + [(»» - »A + 2 «/i] ^ + *»r^ 38m, 38);. _,28io. ) + ho x -^- + »«*!"-.— + L(«t - ")^! + 2?i■ dx^ly^L^ /77* f r/ ia <, -,38m, 38'ij., 38iu., + ./y i [{m ~ n)A * + 2ne2] w + ™ 2 ^; + ni ^; 38m 2 r/ \a *-fi^ v * ClSw., + nc 2 Bi/ ' + [( m - ' l ) A 2 + 2 W /J a^-* + n.a„ 32/ 2 LV ' 2 y2j ay» 2 fyj 38w, 38b, -38io„ ) + "Vjfc" + mt i^r' + [(»» ~ n ) A 2 + 2n 9iTfa J dx 2 dy 2 dz.,. Let us first integrate by parts the first line of each of these integrals. We thus get _//~{[(m - n)^ + 2»eJ8it 1 + nc l &v l + nhfiuo^ } \d^ x + //{ [(«» - w)Aj + 2ne 1 ]Su 1 + ncfiv^ + nbfiw x ) A I2 c/2 12 3 + y^{ [(»» - «.)A 2 + 2»e 2 ]SM 2 + nc28« 2 + m.& 2 8w> 2 ] X 2 ^2 2 + /T{[(m - »»)^2+ 2ne 2 ]8w 2 + ra<; 2 S« 2 + m& 2 8m; 2 } A^c/i^ + dx^ (nb.,) . Sio-2 > dx'idyidz-i. 1.5S POTENTIAL ENEECiY OF STRAIN. [227. Treating each line in the same way, the integral to be taken over 2,.. is found to he If L i A ia[(" 1 - M ) A i + 2 " e i] + / t i2 ,lc 'i + "l-i'^i } & 'i + {A 2l [(w- n)A 2 + 2ne 2 ] + /i 21 ?tc 2 + i< 21 »k: 2 }Sj( 2 + [ X I2 ri.Cj + f 12 [( m - rc)^ + 2»i/"j] + v 12 )KXj}8i-j + { AjjWCj + /%[(»» - n)A 2 + 2«/ 2 ] + v. n na 2 }8v. 2 + { A 12 n6j + /ij-jTOdj + v ]2 [(m - ?t)Aj + 2w^j] } 8m> x + {A 21 n6 2 + fi. 2l na 2 + v. n [(m - n)A 2 + 2?i# 2 ]}S«? 2 U2 12 ; and in virtue of equations (64), (65), (66) this vanishes identically. We have then, finally, 8 "=_//[ {\[( m ~ n )\ + 2«ejJ + /ij»c, + VjreijJSwj + { AjjtCj + /ij[(m - ji)Aj + 2ra/"j] + v 1 M« 1 }6»i -t- { Ajiiij + /Uj-rcrtj + ^j[(«t - n)Aj + 2)i.9 1 ]}Sa- 1 lt/i;, + ff I A;j[(»h - '0^2 ''" 2ne. 2 ] + ,u.,?(c., + v.pib.,} 8« 2 ^L + { A 2 «c._, + /i 2 [(»i - «)A 2 + 2n/" 2 ] + iyt; P = qe; A' = A(l+/+g) = A(l--2, <&, It represent the components of the total tractions at any point of the body in the strained state e, f, g, a, b, c, and let JP , \ Whatever hypothesis we adopt as to the nature and origin of the mutual reactions between contiguous portions of the body, it is obvious that (the temperature being constant and uniform) we must assume that they depend solely on the configuration of the body, and that they vary in a definite and perfectly continuous manner throughout all continuous changes of state, within the limits of perfect elasticity. Since, for all we know to the contrary, the total traction components may be capable of assum- ing either sign, it is possible that they may pass through the value zero for particular values of the strain-coordinates. But it is not possible, under the assumed conditions of continuity, that the rate of variation of any traction component with any strain coordinate which it involves can change sign, or vanish, for any value of that coordinate. For example, if any traction com- ponent $ be continuously increased, within the limits of perfect elasticity, and if at any stage of the process any strain coordinate a be found to increase with $, we cannot suppose that in any other stage — however limited — the value of a can decrease or even remain stationary. Hence, taking any one component $, we may assume a rela- tion of the form # = is some continuous function of the independent strain coordinates ; such that, if the first derivative of $J as to any one of these coordinates vanishes for any value of the coordinate, it must vanish for all values : — that is to say, $ must be altogether independent of that coordinate. 168 NATURAL MATERIALS: — In the natural state 5o = b , 4>)> and by Taylor's theorem, s=iu«-„[!>[ll + 2 1- Thus, substituting for the differences, + M"G1> >♦ • By what has just been said, if the coefficient of the first power of any difference vanishes, that difference does not occur at all in the expansion of $. Hence it follows that if the co- efficient of the first power of any strain component vanishes, that strain component does not appear at all in the expansion of P. In other words, the expansion of any stress component contains the first powers of all those strain components of which it is a function. Ultimately therefore, when the strain is very small, each stress component must be a linear function of all those strain components upon which it depends. Thus Hooke's law is demonstrated, independently of any hypothesis as to the origin of stress. APPENDIX IV. Elastic Properties of Natural Materials. We have already indicated (§§ 12 and 13) a rough subdivision of solid materials into the brittle, whose range of elasticity is practically coextensive with their power of resisting rupture, and the malleable, plastic, or ductile, capable of enduring stress which very greatly exceeds their elastic limits, and distinguished by their ability to acquire a permanent set under such stresses. We now proceed to a more detailed account of the behaviour of natural materials under stresses varying from zero to the point PLASTIC SOLIDS AND FLUIDS. 169 of rupture, and we shall find it convenient to subdivide the malleable bodies into two classes, namely — A. The Plastic, which acquires a set whenever subjected to stress exceeding a certain definite limit, characteristic of the material ; but whose mechanical qualities are in no way modified thereby. This class, which includes the so-called " soft solids " (such as clay and wax) as well as lead amongst the metals, has for us an almost purely theoretical interest. We shall find that it merges insensibly into the class of Fluids, and we shall naturally be led to give under this head a fuller account of that property of viscosity which, although it is manifested to a small extent (§ 16) by all solids under small elastic strains, is in its fuller development confined to fluids and to malleable solids strained beyond the limits of their elasticity. B. The Ductile, the limits of whose elasticity are extended by every stress which produces a set, and whose hardness or resistance to further set depends in consequence upon the greatest stress to which they may have already been subjected, as well as upon the intrinsic qualities of the material. This class is by far the most interesting and important from a practical point of view, including as it does nearly all those metals that are most frequently employed in structures. My general authority for the experimental facts on which the following account is based is Prof. Cotterill's Applied Mechanics, Chapter XVIII., where many references to the original memoirs, etc., will be found. Figures 23, 25, 27, 28 are taken from the same source. The account of the behaviour of a bar of ductile metal, when very cautiously elongated to the point of rupture, is in substance reproduced, together with Figures 24, 24 A, 25 A, and 28, from two letters by Prof. Alex. B. W. Kennedy, published in Mature, vol. xxxi., p. 504, and vol. xxxii., p. 269. I have also drawn freely from the very interesting discussion on Mr. Hackney's paper "On the Adoption of Standard Forms of Test- pieces for Bars and Plates," reported in the Proceed- ings of the Institution of Civil Engineers, vol. lxxvi., pp. 70-158. Figures 24 and 26 are reproduced, on a more convenient scale, from that report, and are due to Prof. Kennedy. A. — Plastic Solids and Viscous Fluids. A " perfectly plastic " solid — which is as much an abstraction as a "perfectly elastic" or "perfectly rigid" solid — is defined by the following properties : — (*.) It possesses perfect elasticity of bulk (§ 14) under purely Hydrostatic Pressures (§ 174) whether positive or negative ; that is, for uniform cubical dilatations or compressions unaccompanied by distortion (§ 211). This bulk-elasticity is limited in the direction of dilatation only by the tenacity (§ 222) of the material, and on the side of compression is theoretically without limit. (ii.) Its elasticity of form (§ 14) is perfect for all distortions 170 NATURAL MATERIALS: — within a certain perfectly defined and usually very narrow limit, which is characteristic of the material. Plasticity. The utmost resistance S which a perfectly plastic material can offer to distorting stress coincides with the limit of its elasticity of form, and it follows from § ] 35 that it is im- possible to maintain in the interior of such a body a shearing stress exceeding S by however little. The excess is, in fact, entirely unbalanced by any resistance on the part of the body, which in consequence relieves itself by continuous change of shape without change of volume, until — if the circumstances permit — the maximum shearing stress is reduced to the limit S. A perfectly plastic solid may therefore be distorted to any extent, however great, by the continuous application of a shearing stress exceeding S by any amount however small. Moreover, since S is the limit of elastic resistance to distortion, the resilience (§ 222) of the body is precisely the same as if it were only strained to its elastic limits, and consequently the large distortion produced by the excess of shearing stress is not recoverable, but remains as a permanent set (§ 12) when the stress is removed. Flow. This continuous and permanent change of shape, without change in the volume or density of any part, is called Flow ; and the tendency to flow, withm.it modification of any ■mechanical property, under continuously applied and constant distorting stress, however little in excess of a definite elastic limit S, is called Plasticity. Fluidity. Those substances for which S is absolutely zero, but which nevertheless possess perfect elasticity of volume, are called Perfect Fluids. A perfect fluid is therefore totally devoid of rigidity (§ 210), and offers no resistance whatever to shearing stress : and a purely hydrostatic pressure is the only form of stress that can be maintained within it, even for an instant. The characteristic property of perfect fluids is therefore their tendency to flow freely under any distorting stress however small : and this property is called Fluidity. Solidity. Since the quantity S — which we may call the measure of solidity — may be indefinitely small, it is obvious that no strict line of demarcation can be drawn between fluids and plastic solids, but that a series of the latter arranged in descending order of solidity may be supposed to pass insensibly into the former group. Even if the fluids and plastic solids with which we have to deal in nature were free from viscosity {see below), as we have hitherto supposed, the universal and unavoidable presence of the shearing stress due to gravity would render it difficult practically to distinguish a perfectly plastic solid of quite conceivably small solidity from a perfect fluid of the same density and compressibility. PLASTIC SOLIDS AND FLUIDS. 171 It should be observed that the solidity S is a limit, and not a modulits. Thus it is quite possible for a plastic body of very small solidity to possess very large moduli of compression and rigidity. In such a case the limiting shear S/n which the body can suffer without flow, is of course very small. The exact nature of Flow will be better understood by working out a simple example. Let a right circular cylinder of perfectly plastic material, the solidity of which is S, be placed with its base upon a perfectly smooth and rigid horizontal plate, and let another smooth and rigid plate be laid on the top of the cylinder and loaded until the total weight applied is W. Let h be the initial height of the cylinder, and A the initial area of its base. We will assume for simplicity that its ends can slip freely over the surfaces of the plates, and that the action of gravity upon it may be neglected. The load W will then be uniformly distributed over the upper surface, and the stress throughout the cylinder will be a homogeneous longitudinal pressure "W/A . There will be no longitudinal stress in any horizontal direction, and therefore the principal normal stresses will be at every point Hence it follows from Example 6 on Chapter III. that at every point there exists a shearing stress of amount "W/2A n , tending to diminish the height and increase the diameter of the body. If W/2J. be less than S, the whole stress will be within the elastic limit of the material, and the strain (wholly elastic) will be, with the notation of § 213, «! = - W/l ?, ^ = e 3 = + oW/A q. If now W be increased to the value 2-A S, the cylinder will be strained precisely to the limit of its elasticity of form, and we shall have £l =-2S/?, * 2 = * 3 = +2trS/?. The resilience per unit volume is then by equation (43) of § 215, V= \ {(m - n)(4' From the symmetry of the conditions the centre of the base will remain at vest, and if e,', e 2 ', e 3 ' be at any moment the additional elongations due to flow, we must have Thus '2=*' 3= -K'i and by § 126 the equipotential surfaces will be the hyperboloids of revolution 2^2 = ^+ ?±C\ 0£ being vertical. The lines of displacement (§ 127) — which in this case preserve their form during the whole time that flow is taking place, and are called Lines of Flow — are therefore the system of curves satisfying the differential equations The solution is to be symmetrical as to rj and f, and therefore the equations of the Lines of Flow are £(V 2 + H = constant \ v/C = constant ) PLASTIC SOLIDS AND FLUIDS. 173 Every point of the body will describe that line of flow which passes through its initial position (see Figure 22), and this process will continue until the maximum shearing stress at each point has been again reduced to the limiting value S by the expansion of the surface over which the constant load W is applied. If h', A' be the final height and sectional area, we have A'h' = Ah and therefore A': W '2S v _2SV r i+ (4solidity I Figure 23 represents an experiment of Tresca's on the flow of lead. A series of flat circular plates of lead were placed in a rigid cylinder, having a small orifice in the centre of its base, and VISCOSITY. 175 forcibly compressed. The lead issues as a jet from the orifice, and the lines of flow indicated by the distorted boundaries of the plates bear a striking resemblance to the corresponding lines in water issuing from an oritice in a horizontal plate. Tresca found lead to be very fairly plastic, and ascertained its solidity to be about S = 200,000 grammes per square centimetre = 2850 pounds per square inch. The quality of manufactured lead is however more variable than that of any other metal. Viscosity. The properties which we have ascribed to "perfectly plastic" solids and "perfect fluids" are modified in actual materials by the universal presence of more or less viscosity (i.) According to the theory of fluid viscosity advanced by Stokes in 1845, and subsequently extended and verified by him- self, Clerk Maxwell, Poiseuille, O. E. Meyer, Helmholtz and Piotrowski, and others, this property consists in a kind of sliding friction between layers of molecules, only called into play when relative motion of the layers is taking place in a direction tan- gential to their common surface. Viscosity and Shearing Motion (§ 95) must therefore be regarded as inseparable : and, since flow is merely continuous shear, it follows that flow is always opposed by viscosity. On the other hand, a uniform cubical dilatation or compression (§§ 104, 105, 112) — whether homogeneous or not— is specially characterised by the absence of shear, and this form of strain consequently possesses the unique property of being absolutely unaffected by the existence of viscosity. (ii.) The amount of viscous resistance of a given solid or fluid, at a given uniform temperature, depends only upon, and increases continuously with the rate at which shear takes place, and invariably vanishes with this rate : — or, in other words, infinitely small resistance is offered by viscosity to infinitely slow flowing. The existence of viscosity in a material does not therefore affect the conditions of equilibrium under stress, but only resists and modifies the process (other than simple dilatation or compres- sion) by which a body passes from one state of strain to another ; a relation being introduced between the magnitude of the stress producing the change of state, and the time occupie> 14° 7,920 51-248 Alcohol, 0° 12,100 78-295 )> 15° 11,100 71-824 Carbon bisulphide, ■ 14° 1 6,000 103-531 Water, o°.o 20,200 130-707 J) 1°.5 19,700 127-473 » 4°.l 20,300 i 10°. 8 21,100 j 13°.4 21,300 137-826 » 18°.0 22,000 , 25°.0 22,200 j 34°. 5 22,400 » 43°0 22,900 > 53°.0 23,000 148-825 Mercury, - 15°.0 542,000 3507092 Authority for water — Jamin, Cours de Physique, 2nd ed., t. i., pp. 168, 169: for the other liquids — Amaury and Descamps, Comples Kenilua, t. lxviii., p. 1564. NUMERICAL TABLES. 201 TABLE (C). Elastic Constants op Solids. Material. P tfxlO- 6 51x10-° A-*10- 6 E r*io- 4 i no- 4 Flint glass, 2-942 615 243 437 ; »J s» 2-935 585 240 347 Steel bare, 7-849 2120 834 1878 •00324 1310 809 Steel, cast, drawn, j 7717 1955 838 Steel wire, drawn, - 7-718 1881 ■0050 22730 925 Steel piano wire, 7-727 2049 0115 135995 2362 Iron, cast, 7-235 1375 542 938 ■00116 879 147 Iron, wrought, 7-790 2040 784 1484 ■00224 5120 457 Iron, wire, 7-553 1861 •0034 10952 633 Copper, cast, - 134 Copper, drawn, 8-893 1245 456 1717 0033 6613 410 Copper, annealed, - 8-936 1052 •003 4745 310 Copper wire, - 8-900 1185 445 1172 0036 7480 422 Brass, cast, 645 ■00198 1256 127 Brass, drawn, 8 471 1096 373 1063 Brass wire, 1001 410 597 •00344 5905 343 Gun metal, 696 •00362 4562 252 Gold, drawn, - 18-514 813 281 2538 0034 4629 275 Silver, drawn, 10-369 736 270 895 0041 5962 296 Platinum, fine wire, 21-166 1593 622 1210 •0022 3852 350 ' Tin, cast, 7-400 417 001 207 41 ! Zinc, drawn, - 7-100 873 360 506 ■0018 1448 158 ! Lead, 11-215 177 0012 135 22 Ash, 113 0106 6370 120 Teak, 169 00621 3262 105 Oak, 750 103 ■0102 5352 105 Red pine, 500 118 ■00771 3510 91 Spruce, 113 0077 3347 87 Larch, 79 •00861 2927 68 In the above table, p denotes the density in grammes per cubic centimetre; q, n, k the moduli in grammes weight per square centimetre ; E the " practical " limit of elastic elonga- tion (point C in Figure 24) ; V the resilience (§ 222) for longi- tudinal extension in gramme-centimetres per cubic centimetre ; and T the tenacity in grammes per square centimetre. This table is for the most part quoted from Sir William Thomson's article " Elasticity " in the Encyclopaedia Britannica : the experimental authorities are Wertheim, Rankine, Everett, and Sir William Thomson. 202 NATURAL MATERIALS :- Example. — For drawn copper : Density Young's modulus Rigidity Mod. of compression = 8 "8 9 3 grammes per cubic cent. = 1,245,000,000 gr. per sq. cent. = 456,000,000 = 1,717,000,000 Elongation at " breaking-down point " . = -0033 Resilience under tension = 66,130,000 gramme-centi- metres per cubic cent. Tenacity = 4,100,000 grammes per square cent. The absolute measures of the moduli, etc., can be deduced by reducing grammes to dynes, or multiplying the above values by 9814. The length-moduli and resilience in centimetres can also be deduced by dividing by 8-893, the density. (See §g 221, 222.) TABLE (C bis). Practical Table in English Measure. Material. Elastic Strength. Young's modulus and riuidity in tons per square inch. Resilience under Tension. Stress in tons per square inch. Strain. Foot-pounds per cubic foot. ? Height per £g 1 pound in feet. T. 3 C. 9 S. T. C. s. 1 8000 n Iron, cast, •000375 •001125 185 lion, wrought, - 9 9 7 0007 0007 0014 13000 5000 1060 22 ' Steel, soft, 15 15 12 •0012 •0012 ■0024 13000 5200 2900 6 Steel, hard, 25 25 20 •002 •002 004 13000 5200 8000 165 Steel wire, strongest, 150 •0115 13000 276000 577 Fir, li •0021 700 35 2150 58 Oak, 2 ■0028 700 35 4300 86 The above table is quoted from Prof. Cotterill's Applied Mechanics. The first six columns of figures give the " practical " elastic limits of stress and strain for tension (T.), compression (C), and shear (S.). NUMERICAL TABLES. 203 TABLE (D). Ultimate and Working Strength. Material. Ultimate Strengtl . Working Strength. Tons per square inch. Ulti- mate Elonga- tion. Tons per sqnare inch. T. 0. S. T. C. Iron bars, Iron plates, 25 22 22 19 18 16 •20 ■10 1 4-5 4-5 Steel, soft, 30 22£ •25 7 7 Steel, medium, 35 27 15 Steel, hard, 45 •80 Iron, cast, 7* 45 12 1-5 4-5 Lead, n Copper, sheet, 13* Copper, cast, 8 i Copper -wire, 4 Steel wire, common, 13 Oak, - H 1 0-75 045 Fir, H 027 0-5 03 This table also is taken from Prof. Cotterill's Applied Mechanics. The "working strength" of a material is the maximum statical stress to which it is subjected in practice ; the ratio which the full elastic strength bears to this constitutes the "factor of safety" always allowed to provide against un- foreseen contingencies. 204 NATURAL MATERIALS. TABLE (E). Effect on Young's modulus of cluinge of temperature. Material. Young's modulus iu millions of grammes per square Density in centimetre, at grammes per | cuoic centim. 15° 100° 200° Lead, 11-232 173 163 Gold,- 18-035 558 531 548 Silver, 10-304 715 727 637 Copper, 8-936 1052 938 786 Platinum, 21-083 1552 1418 1296 Steel, drawn, English, 7-622 1728 2129 1928 Steel, cast, - 7-919 1956 1901 1792 Iron, Berry, 7-757 2079 2188 1770 Wertheim, Annates de Chimie et de Physique, torn. xii. (1844). TABLE (F). Effect on rigidity of change of temperature. According to Kohlrausch n = n {\ -at- fit 2 ) where t is temperature Cent. i Material. a P 1 Iron, Copper, 1 Brass, 0000447 0-000520 0-000423 0-00000052 C -00000028 0-00000136 CHAPTER V. CURVILINEAR COORDINATES. 230.] Definitions and Notation. Let any three orthog- onal systems of surfaces in the body be defined by giving successive constant values to the parameters £, r\, \ in the equations Xi(*. y, «)=f| 0) Xz&y, ») = i> ( 2 ) x-ii x > y> Z )=C) ( 3 ) where \ v x 2 > X3 are continuous functions of the rectangular Cartesian coordinates x, y, z. The position of any point P (x, y, z) in which these surfaces intersect will then be fully determined by the values of the parameters ; and £ >?, f may be called the Curvilinear Coordin- ates of P. We shall also speak of the three systems of surfaces denned by them as the corresponding coordinate surfaces. Let (Xj, fi v vj, (\, m 2 > "2). (\i> M 3 . " 3 ). b e * ne direction-cosines, referred to Ox, Oy, Oz, of the normals to the three coordinate surfaces which meet in P — drawn in the directions in which the values of £, r\, f increase. Then x . . .,..3X1.3X1.3X1. //?X_iV + /?XiV + /2XiV where the proper value of f at P is to be substituted for ^ after differentiation. We shall consequently always write these derivatives 3£ 3| 3£ ~dx ~dy ~dz and so for r\ and f ; and we shall also assume that x, y and z have been eliminated from them by means of (1), (2) and (3), so that they are expressed as functions of £ tj, f 200 CURVILINEAR COORDINATES. 230.] If now we write V v= (dmdms)' 1 <*> -(gw-d)' we shall have A, =11 h x dx' ^ 1 3r, h 2 dx'^~h 2 ^i/ "■■>'' I 5.1 /tj 3z 1 cty A "*. 1 3f ~ A. dx' ^ = A, 3y' A 2 3z ' Ik 3z .(5) taking for \ y h. 2> h t the positive roots of (4). orthogonalism are by (5) 'dx ~dx dyd-y' 3« ?)z~ 3£3£ 3f3£ 3£3f 3a; 3x 3y 3y 3« 3z ^- = 3£3tj 3a; 3a; 3|3^ 3?y 3y 3|35 3« 3z ' The conditions for .(6) If these conditions be satisfied (as we shall always Buppose the case), equations (5) will also give the direction-cosines of the tangents at P to the three curves of intersection of the pairs of coordinate surfaces defined by 17 and £, f and £ £ and >/. We know by Dupin's theorem (Frost's Solid Geometry, § 603) that the curve of intersection of any such pair of surfaces is a Line of Curvature on each. Let ds v ds v ds 3 be the elements of these curves, measured from P in the directions of increase of £, y, f. Then in proceed- ing from P along s, we remain always on the same surface of system (2), and also on the same surface of system (3), describing a line of curvature on each ; that is to say, £ alone varies along s v Similarly tj alone varies along s 2 , and f alone varies along s 3 . The elementary (ultimately straight) lines ds v ds v ds s are in fact the three edges meeting in P of the element of volume (ultimately a rectangular parallelepiped) bounded by the surfaces whose parameters are £, v, Li + d^v + d'n, C+dC 230.] C0RVILINEAR COORDINATES. The Cartesian coordinates of the further extremity of x + Ajrfsj, y + fads v z + Vjf/Sj ; whence, by Taylor's Theorem, 207 ds, are lr dx = 7i 1 ds 1 by (5) and (4). Thus we find rfgj : ds 2 - dv ds 3 (') for the lengths of those three edges of the element of which meet in P(£ jj, £). Ultimately, therefore, when this element approximates to a rectangular parallelepiped, its volume is d£drid£ K h A and the areas of the three faces which meet in P are dr)d£ d£d£ d^dtj V«2 volume in form h 2 h s ' h A f 1*5*1 i*K — * ■(») .(9) In Figure 29 the six coordinate surfaces are of course really drawn for finite differences of £ »?, £ in order to exhibit the curva- 20s CURVILINEAK COORDINATES. [231. tures of the edges. The small figure at the corner (£ n , f ) repre- sents more approximately the rectangular parallelepiped into which it degenerates, when d£ cfy, oZ£ are truly elementary. •231 .] FormulEe of Differentiation. Since ds 1 is an elemen- tary line drawn in the direction (A,, /u i; i/,), we have of course 3* 3$ 3$ 3* where $ is any function of .>; y, z, and therefore also of f, r,, £ Similarly Now by (7) Thus 3$ ?4> , 3* 4 3$ = \ - + A 2 a T + A^ , etc. d.c 1 3s, ^o«., ""os, i3s t 3$ 2 3*\, 3* -/, — -. etc. 3s t yi i3f 3* 3* 3* 3* /i 1 3| = A ^ + ^% +v ^ 3* 3* 3* 3* 3* 3$ 3* 3$ j 3f 3* 3* 3* ■(•« = Mi^ + M-2 3 , + ; *A 3f ?4> 3x 3* 3* 3* 3* ■fy = Vl 3| + h ^»W, + Vs"3f 3* 3* 3* , 3* Writing x, 37, = successively for # in the first three of these equations we find 3a; , 3;v , 3a \ 7 ^ 7 ^ 7 ^ 237,' X 3 = "3g> ^3 : 23lj' 3.V 9 C .(11) '3 3f ' Thus the conditions (G) for orthogonalism may also be written 3x3a; ~dy ~dy 3z ~bz 3r,3f + 3r, 3^ + 3r,3f-° 3a: 3a; Tty "dy 3z 3z 3f3l + 3|3^ + 3C3| = 3a; 3x 3y 3y 3z 3z 3g 3.? + 3£ 3>. + 3£ 3»j .(6a) 232.] CURVILINEAR COORDINATES. 209 We also deduce from (11) A, 2 fdxV /dyY /cte\ 2 l I = f^Y f^Y (^Y v W + W + W ' 1 fdx\* /3«V /3sY .(12) It frequently happens that, while equations (1), (2), (3) express £ q, £ as more or less complex functions of x, y, z, they admit of very simple solutions for the latter coordinates as explicit functions of £ q, f. In such cases the formulae (11), (6a), (12) may he used in preference to (5), (6), and (4). Equa- tions (11) and (12) possess the further advantage that they admit of the elimination of x, y, z from the expressions for ft,, ft 2 , h 3 and the direction-cosines before differentiation. Lastly from (5) and (11) we have v 3a; v 3a; v= 3a; 3a; 3£/3y 3£ / 3a 3f = = ^/3f "?>z\ 3f 3a; 3ij / 3?/ "3y/ 3>j" 3)7 / 3a 32/ 3»j V dx 3f joy af /a» sr 'a«/ 3f .(12a) The transformation of y 2 $, where $ is any continuous func- tion of position, from Cartesians to curvilinears, is most easily effected by an application of Green's theorem, by which we know that iif^^-m dS, where the triple integral is taken throughout any volume V, and the double integral over the whole of its surface S, dn being the element of outward normal. Let us take for Fthe element of volume (8). The left-hand side of Green's equation then becomes simply 2<£, d£dr)d£ h i h A The right-hand side will be the sum of six terms, supplied by the six faces of the element. The three first terms, due to the faces which are elements of the surfaces £ y, £ are respectively 3* dt)d{ 3«j AgA 3 3* rfftfg. _3* dgdr) 3« 2 Ag/tj 3.«. s Aj/'j 210 CURVILINEAR COORDINATES. [232. or, by (7) .A .A .A h x h 2 3* ^ if -~rz ■ dr)d( 3* 3$ .d£d$ . d£drj Consequently the terms due to the opposite faces, which are elements of the surfaces £+d£, rj + dq, f+c?f, must be [ Ms 3£ 3£\ 3$ L*a 3f adv. ^/J * Thus the right-hand side of Green's equation becomes Equating the two expressions thus found, we have finally V*=V* { |(4f ) + |(i £) * |(i I) } ...„„ Hence, in particular, v 2 £ = M ! ^(w .(14) *3£ whence it follows that, if g, y, f are solutions of Laplace's equation v 2 *=o, hjh^i 3 must be independent of £ h^jhji^ independent of tj, and hjhji 2 independent of f. 232.] Principal Curvatures of the Coordinate Surfaces. It has been remarked in § 230 that, by Dupin's theorem, each curve of the s x system is a line of curvature on each of the surfaces (belonging respectively to the r/ and f systems) which intersect along it ; and so for the other systems. The lines of curvature, at any point P, on the g surface pass- 232.] CURVILINEAR COORDINATES. 211 ing through it are therefore the curves of the £17 and g£ systems which intersect in P. We shall adopt the notation 33 , 33 y f if £ t for the curvatures of the normal sections of the £ surface at P through the tangents to the gq and £g curves, with a symmetrical notation for the other surfaces. Thus 33 , £ v' ft fr fi p e Pv will denote the six principal curvatures of the coordinate surfaces at P. By § 606 of Frost's Solid Geometry we have ^W^^Si+^M, («> Now take the last of equations (6), and differentiate partially as to x : thus ■dr) 3 2 g + 3^ 3 2 g + 3g 3 2 g _ _ |~3£ ?>S + 3g 3^ 3£ 3^_~] 3a; 3a: 2 3y 3a.'3j/ 3« 3k3x |_3x 3a: 2 3?/ 3a;3y 3a 3«3a;_|' Next differentiate the same equation as to y : thus 3>7 S 2 ^ 3r? 3^ 3j? ^£ _ T3£ 3-^ 3| 3^ 3J 3^ ~j 3a; 3a:3y 3«/ ciy 2 dz ~dydz |_3a; 3a;3y 'dy "by- 3? 3y3z_|° Finally, differentiating as to 0, 3, 3^ + 3a; 3«3a; 3i? 3^ ty'&£_ |~5£ j^_ 3| 3^_ SJSV] 3y 3«/3s 3s 3« 2 |_3x 3«3a; 3y 3^3» 3« 3« 2 _|' Multiply the first of these results by 3ij/3aj, the second by 'by/by, and the third by "dyfdz, and add. Thus finally ! 3 2 £ fdvY^ /3>A 2 3 2 £ 3ai 2 + \3y/ 'dy^Xdz) 3? 2 2 3, 3, J9£ + ^ 3, 3*£ 2 3, 3, 3^ "'Sy 3a 3y3a 3z 3x 3z3a; 3a; 3y 3aj3(/ 3 2 tj brj 3 2 ij ~| 3a?3y 3a 3z3.<;_J _3|r35^3 2 j_ 55^5 3^ 3^~| 3y |_3a; 3x3?/ 3y ?)y 2 3a 3y3z_| 3£ n3ij 3 2, >j 3i? 3 2 ^ 3i/ S 21 ? - ! — ~bz L^a: 3»3a; 3y 'di/dz bz 3z 2 _]' 212 CURVILINEAR COORDINATES. [232. By (5) this may be written ( 3 2 £ W 3 2 £ 3 2 £ c> 2 £ &£ 1 = 2V^3x + 9 2 /3y + ^Wt W W \W / -K x ^ + 4 + v> Voy(4)and(5) ' Thus by (15) + h q Jh x . J3^ = Wg> . According to the ordinary convention as to sign, we consider the curvature positive when the centre of curvature is situated in what we agree to reckon the positive direction of the normal. This we have taken (§ 230) to be the direction in which £ in- creases ; so that the curvatures must be reckoned positive when the surface turns its concavity in the direction in which £ increases. The easiest way to get rid of the ambiguity of sign in the above formula for the curvature, is by the following geometrical investigation, due to Lame". It affords an independent proof of the formula, and has the advantage of absolutely determining the sign. Let PS V P f A 3 3,J (16) .(16a) Finally, if rs jt vs v © 3 be the absolute curvatures of the three curves of intersection s,, s 2 , s s , in their osculating planes at P, we have (Frost's Solid Geometry, § 581) 7TT 2 — 7TT2i w 2 233.] Surfaces in General. Let any surface whatever be represented by the equation (£, 77, f) = constant. Then $ can be expressed as a function of a;, y, z, and if we write HINIHS)* <»> the direction-cosines of the normal to the surface at any point, referred to Ox, Oy, Oz, will be S/* f A- 3 J/ h - Thus, if X, /x, i/ be the cosines of the angles which this normal makes with the elementary lines ds v ds 2 , ds 3 , drawn from the same 'point, 3<|* 3*S* 34* Hence, by (10), . A, d$) A= k3f A 2 3*. " h 3£ .(18) 2L6 CURVILINEAR COORDINATES. [233. from which we deduce that H^r^n^' <"> If dS be the element of the surface about the point (£ tj, g), its projections upon the coordinate surfaces through the point are easily seen to be - Ad£= KK .(20) 234.] Strain Components. Now let us suppose the body to suffer a small strain, and let its effect on any point P in the body be to change its curvilinear coordinates from (£ r), f) to (£+a, y + ft, f+y) ; a, ft, y being small quantities of the first order. Let e, f, g denote the small elongations of the elementary lines ds v ds 2 , ds 3 , and a, b, c the small shears (§ 94) of the right angles between ds 2 and ds s , ds 3 and ds v ds t and ds 2 , respectively. In general h v h 2 , h 3 are functions of all three of the coordinates, and by Taylor's theorem we see that the effect upon them of a small strain will be represented, (to our order of approximation), by changing them into h^Sh^, h 2 +Sh 2 , h 3 + Sh 3 , where now by (7) and therefore £7 3^9 o3^9 . 3^9 s> 7 3A„ ndh~ ~dh v ds- d Z a+eWs - d ^ + da - d i ,.^-^8/, •(21) _ 3a S/tj and so for / and g. 234.] CURVILINEAR COORDINATES. 217 Thus finally "dy 1 [~ 3A„ n'dh. I 1 n 3 + yg ■ ^ 3* a, hr 3£ '] .(22) Substituting from (16) these equations may be put in the form •<») Again, as in § 94, the small shear a is simply the cosine of the angle between the altered directions of ds i and ds 3 . Thus a = (A. 2 + SA. 2 )(A. 3 + Sk 3 ) + ( H + S/igX^g + S^) + (v 2 + Sv 2 )(v 3 + Sv 3 ) ~' by (6). Now, by (5), A 3 3a;\ 3 A 3 3a; 3 /t 3 and so for the others. Thus by (5) and (6) h l -dCh 3 d^ _ Aj 3/J h 2 da h 2 3£ A, 'drjj and finally, by (10), .(24) 218 CURVILINEAR COORDINATES. [234. Lastly, if A be the cubical dilatation at (£ 17, f ) we of coarse have A = e+/+<7; and by (22) this is easily put into the form A=/tlM3 {|(w 3 ) + l(w' 8 ) + ^(Ws) } (25) 235.] The Component Displacements. Let u, v, w be the components of the displacement of any point P (£, rj, £), resolved along the elementary lines ds v ds 2 , ds 3 : — or, more exactly, along the normals to the three coordinate surfaces which meet in the point. We proceed to find expressions for the six small component strains in terms of u, v, w. These expressions will not be so simple as in the case of Cartesian coordinates, because now, instead of resolving the displacement of each point in three fixed orthogonal directions, we resolve it along the tangents to three orthogonal curves whose directions vary continuously from point to point. We must therefore expect any expressions which involve the variations of the component displacements along these curves to involve also the curvatures of the coordinate surfaces ; and this we shall find to be the case. In Figure 31, PQ represents the edge ds 1 of the element of volume (8), represented complete in Figure 29 ; its size being supposed so reduced that its edges are practically straight lines. QR is the consecutive element of the s t curve. PC and PC are drawn in the directions of the elements ds 2 and ds 3 , and QC and QC in the directions of the corresponding elements at Q. Thus PQ, PC, PC are mutually perpendicular, and so are QR, QC, QC. Since PQ and QR are consecutive elements of a line of curvature on both the q and f surfaces through P, C will be the centre of curvature of the principal section of the r\ surface through that curve, and C will be the corresponding centre for the f surface. [The changes of direction of the elementary lines, in passing from P to Q, are of course enormously exaggerated, in order to bring C and C within the compass of the Figure.] Again, if PQ be produced onwards towards T, the plane TQR is the osculating plane of the s, curve at P ; and if we denote the absolute curvature of that curve by xs v and adopt the notation of § 232 for the principal curvatures of the coordinate surfaces, we shall have angle EQT =vs x .PQ angle PCQ = vs PQ angle PC'Q = f5^.PQ 23d.] CURVILINEAR COORDINATES. 219 vC Fig.3: 220 CURVILINEAR COORDINATES. 235.] If u', v, w' be the component displacements of Q, on the system above described, the component of its displacement in the direction QT will obviously be u' . cos RQT-v' . cos PQC-w'. cos PQC = u' . cos RQT-v' . sin PCQ - w' . sin PC'Q ; and, to the first power of the element PQ or ds 1 this is u' - (»' . CTj + w' . 33,)ds v Now U =U +1 du v = v + as,^-- 0«j w' = w + ds 1 Tis, Hence, to the first power of ds v the displacement of Q resolved along QT is i+d8 {^ 1 - v -v 7S r w -^Pi- The displacement of P in the same direction is simply u ; so that the increase of length gained by ds 1 is dsA ~ v. VSf-vi . 33 1 :]• This gain of length is of course equal to e . ds . Equating these two values, and applying the same process to cts 2 and &s v we have finally 3m e = ~ v. ZS y —V).33 f 3s 1 -n I r^ y = 5— — w . 33 -u . 33 J 08 2 rv { v 9 = —- u >-p i --°-V l - .(26) Substituting from (7) and (16), we can easily show that ^^u^yuiyiiij] <»> 235.] CURVILINEAR COORDINATES. 221 Again the displacement of Q parallel to PC is very approxi- mately w' . cos QC'P + u sin QCP, or (^ +M -r CT f) d *i- The displacement of P in the same direction being simply w, it is clear that the relative displacement of P and Q parallel to PC will diminish the right angle QPC by the small amount (1 +e)ds 1 which, to our order of approximation, is "duo Similarly, if R be the further extremity of the arc ds a , the relative displacement of P and R parallel to PQ will diminish the same right angle by the small amount du , + w -F!r The sum of these two, or the total decrement of the original right angle between ds 3 and ds v is by the last Article equal to the small component shear b. The values of a and c may be calculated with equal ease or deduced by symmetry, and finally we have '■> t ~ tV . JET + W . VS as, os, s v v f 3to 3u °s 2 3s 3 3m 3w 6 = ^7- + .57- + to . t TS,. + U . JSy ' 3s 3 3s x ' £ i" 3« 3m 3s x 3s 2 if £ 1 Substituting from (16), these may be put in the form K ^ , t ^ K ^ / , x .(28) K 3 &! 3 /*i 3 , , , A„ 3 .(29) 222 CURVILINEAR COORDINATES. [235. If we compare equations (26) with (23), (29) with (24), and (27) with (25) we see at once that .(30) which we might have inferred from (7), all the six quantities u, v, w, a, /8, y being very small. Lastly, we have seen that the edge PR rotates about ds„ towards PQ through the angle 2ni _ and that PQ rotates about ds 2 towards PR through the angle ~bw _ Exactly as in equations (59) of § 123, half the difference of these quantities measures the rotation (as distinguished from strain) of the element of volume as a whole about ds 2 . If then 6,, 2 , 3 be taken to represent the three component rotations of the element about the normals to the three coordinate surfaces through its centre, we have 2W„ = i— + W. J3,.- — -u. J3 C os 3 « » os x * * Writing down the symmetrical formulae for 0, and 0,, and eliminating the curvatures by (16), we have finally ^-WsO-Ks)]! .(31) These equations may also be deduced directly by trans- forming the corresponding Cartesian equations of § 123. Thus, with the notation of the present Chapter 6j = Ajflj + /ij&j + Vjdg, etc., etc., etc., etc. 236.] CURVILINEAR COORDINATES. 223 236.] Irrotational Strain. If the strain be pure or irrotational (§§ 124-127) there will of course be a displacement potential j J 3f and consequently by (30) 3£ 3, P-W 7-V| (32) (33) Substituting from (33) in (25), or from (32) in (27), and com- paring the result with (1 3), we see that A = v =4>- .(34) which agrees with § 124. 224 CURVILINEAR COORDINATES. [236. The conditions that a given strain may be irrotational are seen from (32) to be a/«\__3/»\ which indeed, by (31), are simply equivalent to Q^Q^Q^O. 237.] Stress and Applied Force. We shall employ the same notation for stress as hitherto, writing now .(35) for the components parallel to ds v ds 2 , ds 3 of the stresses across the elementary areas described about (£ 17, f) in each of the three coordinate surfaces through that point. The components, in the same directions, of the Applied Force per unit mass on the elementary mass of which P is the centre will be denoted by S, H, Z ; and the density by p as before. Just as in §§ 138-143, we obtain the equations of equilibrium and of motion by considering an element surrounding the point (£> V> an< i bounded by the six coordinate surfaces f±J<*£, 9±i«ftj, t±J<*£ This element is cut up by the surfaces £, r\, £ into eight such elements as that of Figure 29, having P for a common corner ; the lengths of the edges which meet in P being \ds v Jrfs 2 , %ds it respectively. Figure 32 represents this divided element (the curvatures being much exaggerated, as before), and corresponds to Figure 8 in every respect ; the three faces turned towards the eye being the concave or positive faces (§ 232). The areas of the sections A 1 B l G l D v AfifiJ)^ AfifiJD^ are ultimately given by (9), and the volume of the element by (8). Let us now resolve the total stress across each face parallel to the tangent at P to the s, curve. This component of the total stress, together with the applied force h x h 2 h 3 ' {6b > 237.] CURVILINEAR COORDINATES. 225 must be equal to the effective force pud^drjd^ in the same direction. Take first the g faces, the coordinates of whose centres are .(37) J), \c-~-:. ;;j>t'-- :: Jt^v \ 4 J \ \ \ "3 77 F"g.32. (i- i^i> v> 0- The components along the tangents to the s,, s„, s 3 curves at P of the total stress across A,B 1 C i D 1 are ultimately PefyeZf Ud-gdj TdrjdC h 2 h s h A Hence the components of the total stress across the positive £ face EFGH, along the tangents at N to PN, NA % , NA „ are [ffi +w s(«)>* Resolving these parallel to the tangent at P to the s, curve, exactly as we resolved the displacements in § 235, we have for P 226 CURVILINEAR COORDINATES. [237. the required component of the total stress across the positive £ face, -K 3 + ^4(4)] sin(i -^-^ -K + ^l(p-3] sin(i -^-^}^ The corresponding component of the total stress across the negative £ face JKLM, reckoned in the positive direction, is of course + D* - w IGs)] ™ (i ■ ^ • *■> } "'* Substituting from (7), and neglecting squares of small quanti- ties, these two faces together give a component total stress \M&- B -*&*¥»* <-> Again, the component due to the positive t) face EHJK is approximately {[^ + i <(^)] 8in(i -^-^ + K + ^1®] cos ( * • ft • **> J rf ^ and that due to the negative 17 face FGML is - K _ ¥v UM] ** {h • ft ■ **> } m - These two faces therefore contribute K(&) + %5>^ < 39) to the required component. 237.] CURVILINEAR COORDINATES. 227 Finally, the positive and negative f faces, EFLK, OHJM contribute + K + ^I(&)] cos(i -^-* 3) }^' + {K 2 -^lfe)] sin(i -^-^ "K"^l(i)] cos(| - ^ ■ * s) }****'' or, on reduction, [&£)♦;&>* < M > Equating the sum of (36), (38), (39), and (40) to (37), we deduce the first of the three equations of motion Rearranging the terms which involve T and U, and writing down the other two equations from symmetry, we have finally 3/ P \ , ?>l U \ , 3/ r \ sgvvJ 'ajVviV ^dvv 3 3"tev + 3 ^ w wj where by (30) .(41) .(42) 228 CURVILINEAR COORDINATES. [237. By means of (16) we can put equations (41) into Lame's form ■dP 3U ?>T ^ ,,_, .. A , p m _ + (Q-P). J3 % + 5(2 . ^ + p£ + 17(2 . ^ + f orp 32 '- 35 - S * + p(Z-«) = (JJ-P). J ET f 3«j 3s 2 3s g .(43) in which their analogy with equations (4) of § 143 is sufficiently obvious. 238.] Stress and Surface Traction. Precisely as in §§ 144, 145, we may show that the components of the traction across the element dS of any surface (§ 233) passing through the point (£ j/, f ) must be PX+Ufi.+ Tv\ UX.+ Qix+Sv\, Tk+Sfi + JSvj where X, fi, v are given by (18) and (19). Hence if the bounding surface of the body be represented by *(£> Vi ~ constant (44) and if S', H', Z' be the components of the surface traction at the point (£ ?7, f ) of the surface, we must have at every such point the relations 3* Ph >~ + CT »^""" ■"as of .(45) where h is given by h! :M)° + M)*« < w > It is often advisable to choose the system of coordinates so that the surface of the body shall be a coordinate surface — belonging, we will suppose, to the £ system. In this case (44) can be put in the form £= constant, and we may take $ = £. 238.] CURVILINEAR COORDINATES. 229 Thus by (19) h = h 1 , and by (18) \ = 1, fi = v = 0; as of course it should be. The boundary conditions (45) then reduce to p=;sn E7 = H'l (45a) 2 7 =Z'J The corresponding conditions, when the surface belongs to either of the other systems, may be written down by inspection. 239. Strain and Stress. Equations of Motion in terms of Strain. If the body be isotropic, the relations between Strain and Stress will of course be, as in the last Chapter, P = (w» + n)e + (m - n)(/+ g)' Q = (m + n)f+ (m - n){g + e) R = (w» + n)g + (m - re)(e +f) S = na T = nb V —iic The potential energy V, per unit of unstrained volume, is also given by formulae (33) or (34) of Chapter IV.; and the total potential energy W of the strain by (46) .jffvjte. (47) hjiz To obtain the equations of motion in terms of u, v, w, or of a, (8, y, by direct substitution from (22), (24), (25), or from (26), (27), (28) in (46), and thence in (41) or (43), is in the general case of curvilinears an excessively tedious operation, and it is not easy to put them into a symmetrical form. Lame* has shown, by direct transformation of the Cartesian equations, that they may be written <»«)|- 2 «[ 4 4(!)- i -l,©] + '' (a - ii) - <> (m + „,|- 2 «[* 1 |( e s .)-;^-@] + f (H-»).0 (48) where A is given by (27), and e i} G z , 8 by (31). In this form they present a striking analogy to equations (52a) of § 218: from which they were derived by tame\ as stated above. 230 CURVILINEAR COORDINATES. [240. Special Application of GurvUinears. 240.] Equipotential Surfaces. If the strain is pure (§ 236), the resultant displacement is at each point normal to one of a system of continuous equipotential surfaces, defined by giving constant values to the displacement potential, and the Lines of Displacement (§ 127) are a system of continuous curves, cutting these surfaces everywhere orthogonally. There is obviously no reason why we should not take the g surfaces for the equipotentials, when we can thus simplify our formulae. In this case the s, curves will be the Lines of Dis- placement ; (p will be a function of £ only, and we shall have at every point £ = y = 0|. V = W = I .(49) Also, by (31) and (32), = Ajit = zd4 ,(50) Equations (23), (24), (25) now become 1 a| l W -A, . J3 1 i v A = **K&^ ay a = - = 9.7, d< f> ft 2*,^. ST .(51) l <% v £ J from which we can substitute in (46) and thence in (41) or (43) and (42). Since the conditions (35) are in this case necessarily fulfilled, equations (48) reduce to <»«)'..|[w|(4|)] +f (a-*,|)-o '"'-""{ 2 |(^|)-^I,[4|(WJ] } + ,h-ol, 53) 240.] CURVILINEAR COORDINATES. 231 The second and third of these equations are reduced from forms symmetrical with the first, by the consideration that is independent of rj and f 241. J Principal Surfaces of the Strain. Let us next suppose that equations (1), (2), (3) represent the three systems of Principal Surfaces (§ 216). The curves of intersection, s v s 3 , s 3 , will then be the Lines of Stress. In this case £ 17, f may be called the Principal Coordinates of the Strain. Lame" applied to the coordinate surfaces under these conditions the term Isostatic. If e v e 2 , e 3 denote the principal elongations, and N v N 2 , N 3 the principal normal stresses, we must have e = «!,/= e 2 > 9 = € 3 1 a = 6 = c = : P=N V Q = N a> R = N 3 ; S=T= U=0. The conditions that £ q, f may be the principal coordinates are therefore, by (24) If these are satisfied we have by (26) .(53) and by (46) 3 dc' .(54) JVj = (m + n)ej + (m - n)(« 2 + e 3 )^ N 2 = (m + n)* 2 + (m - n)(t 3 + t x ) ] N 3 = (m + n)e 3 + (m - n)(£ x + e 3 Equations (41) reduce to W* M rx) + ^2 • f CT , + N » • j CT f + rt s - «) = .(55) .(56) 232 CURVILINEAR COORDINATES, while (43) give us Lame's standard form 3£i + p (3 -u) = (A\ - Nj) . f cr, + (tf, - N % ) ■ fs f 3s, 3iV. 3,s- 1* + p (Z - w) = (JIT, - Jjr x ) . ^Bfj + (A T 3 - ^ 2 ) . jKf^ Finally, the boundary conditions (45) become simply .(58) If the surface of the body be one of the Principal Surfaces — belonging, let us say, to the £ system — these conditions reduce further, by (45a) to [241. .(57) H' = Z' = f the Surface Traction being in this case necessarily normal. .(58a) 242.] Case in which all the Principal Surfaces remain such. There is one interesting case of the last article in which the strain is such that each of the principal surfaces is altered into another — very slightly different — member of the same family. The requisite conditions obviously are : — a independent of r) and f, |8 independent of f and £ and y independent of £ and 17. It should be noted that, although these conditions always satisfy (53), they do not in general satisfy (35), so that a strain of this character is a rotational strain — except for certain particular systems of coordinates. In fact, by eliminating a, /3, y between (34), we obtain the condition dh. 7 ~dh 3 3Aj _ dh 3 3/tj 3A,' a/ '^ ' 3f "as "^ ' at or, by (16), 13,. . 33 ', . 33 =33 . 13 f . 33 \ which is not satisfied by all coordinate systems, (59) 242.] CURVILINEAR COORDINATES. 233 Various Systems of Ciirvilinears. We now proceed to express our general formulae in terms of some of the most important systems of orthogonal ciirvilinears. As a preliminary exercise, the student will do well to convince himself that, on making they reduce immediately to the Cartesian formulae obtained in the last three Chapters. 243.] Spherical Polars. In this system we write i? = 6l = tan- 1 [(a; 2 + ^)i/«]l (60) t = z=r cos The surfaces for which r is constant are spheres with for centre and r for radius ; the 6 surfaces are right circular cones with vertex 0, axis Oz, and semi-vertical angle 6; and the to surfaces are planes through Oz making angle u> with zx. Substituting from (61) in (12), we find l h = l, h 2 = \ h,-—L. (62) r r sin and consequently by (7) ds 1 = dr, ds 2 = rdd, ds 3 = r sin Odw ; while the elements of volume (8) become r 2 sin ddrdddm. The cosines X, n, v of the angles made by the normal to the surface # (r, 6, a>) = constant with the normals to the coordinate surfaces are by (18) .(63) A.= 1 3# ~h 3r • /X = _ 1 3# ~hr 30 1 3* hr sin 6 3w 234 OUttVIUNEAB COOBDINATES. where, by (19), VW \rod) + \rsaie-d^) Formula (13) becomes [243. .(64) 32* ^4-1 3/ r 2^\ + -J_ l/sin0?*U_l_.^r....(65) Substituting from (62) in (16) CT =0 u r r CT *=- 1 „ r 30 sin 6 3w 1 3m . r d r sin sin 6) - Jf] r sm 0\_o0 3a>_J 20. rsin 6 1 rsin 6 ~3m 3(0 sin 2 26, = ■ The equations of motion (41) become r 2 or rsin 30 rsin0 3<« 32\ _i(e + J B) = P (M-S) r 3 or rsm0 30 i as rsin0 oio rsin t/ r 3 or rsin 2 30 rsii = p(i» - Z) = p(r sin 0y - Z ) as r sin 3(«- or rsin01_30 3 .(69) (70) .(71) If $ be the bounding surface of the body, the boundary conditions (45) take the form 3r r 30 r sin 3 r 3* + ^3* + _B_3$ = 3r r 30 rsin0 am HH' = hZ' .(72) k being given by (64). 23G CURVILINEAR COORDINATES. [243. The conditions (35) that the strain may be pure follow at once from (69), on making e^o, e s =o, e 3 =0; and in this case, by (32) and (33), if tj> be the displacement potential, 1 cty .(73) ■a 1 30 w = ry sin a = — : — -= - - r sin ou> If r, 6, w be the principal coordinates of the strain, we must have, by (53), 3(0 + r*«m I 3 (v\ , ~du 2»\rl 30 .(74) or the equivalent conditions Bin^ + M-o] 00 00) ~du + r'sm '■9^1-- ■dr .(75, 9 dB , du „ If these conditions be fulfilled, e, /, #, as given by (68), are the principal elongations e,, e„, e g ; and the principal normal stresses N v N v N 3 are then given by (55). Lame's equations (57) then take the form sin0 30 v ' ranfl r sm ooi j .(76) 243.] CURVILINEAR COORDINATES. 237 while the surface conditions (58) become #i 3$ 3r ! 3* 30 3* ^ s ^f = hrH' .(77) .(78) iT 3 ^ = hrsin(9Z' 0(0 244] Cylindrical Polars. In this system | = r=(a: i! + y 2 )i j »j = = tan ~\y/x) J- f- J whence a; = r cos 0, y=r sin 0. The r surfaces are right circular cylinders with Oz for axis, and r for radius ; the 6 surfaces are planes through Oz. Here we have A,-l,*,-.i,A,-l (79) T and consequently by (7) d&L = rfr, tfe 2 = rdO, ds 3 = dz ; the element of volume (8) becoming rdrdOdz. The cosines (\, /*, v) of the angles made by the normal to any surface 3>(r, 8, z)= constant with the normals to the three coordinate surfaces at the same point are, by (18), where, by (19), . 13* 1 A= k¥ 1 3* > kr 30 1 3* h 3* J (80) »=m'4m)'^' < 81 > Formula (13) now takes the form ^ = ;^) + ^p + ^ (82) 238 CURVILINEAR COORDINATES. [244. Substituting from (79) in (16), we get for the curvatures of the coordinate surfaces .(83) and by (30) .(84) Obviously on this system the s 3 and s 3 curves become straight lines, and a and y are linear, and identical with u and w. We shall therefore retain the latter symbols only. By equations (22), (24), (25), (26), (27), (28) we have 3r , 1 3i> u 3# u r 30 r 30 r g = ^ a, 1 3, , 3S ~dw A =- — (w) + -±~ + — r 3r v ' 30 3» 1 3u> , 3v 1 3io , 36 r ad 3z r 30 3s , _3w 3w 3z ~dr \ 3 tv\ 1 3« dr\rl r 30 3m 3/3 1 '< 3r r3« .(85) while by (31) 20, 1 3w 38 : _ — r-Cl r 30 3z on _ 3m _ 3w 3s 3r 2e l =!jW)-I|?t r 3r ' r dtf .(86) 244.] CURVILINEAR COORDINATES. 239 The equations of motion (41) reduce to where P, Q, E, S, T, U are given by (46) and (85). Lamp's transformation (48) becomes (87) — V w 2ra = P (5-H) .(88) If * be the bounding .surface of the body, we have for the boundary conditions (45) Or r 00 os 'Or r 90 os h being given by (81). The conditions that the strain may be pure are of course e x =o, e a =o, e 3 =o ; and in this case, if be the displacement potential, .1 30' .(89) w= -£ r 30 (90) The conditions that r, 6, z may be the principal coordinates of the strain are, by (53) 1 3io 3» ( r dt) 3s "du /dw _ „ 3s dr d(v\ 1 3w r dr\r) r d0~° J (91) 240 CURVILINEAR COORDINATES. [244. If these be satisfied, e, f, g, as given by (85) are the principal elongations e,, e 2 . e s ; and the principal normal stresses Jv,, iV 2 , JV, are then given by (55). Lamp's equations (57) reduce to r or r r W Mi ■dz -p(5-H) = p(w-Z) while the boundary conditions (58) become or .(92) .(93) .(94) 245.] Conjugate Oylindrics. In this system f=* f where t denotes *J — 1 ', -P" being any function whatever. This relation constitutes £ and ^ conjugate functions of x and ?/. Some of the most important properties of these functions will be found collected in the examples at the end of this chapter. The student will find no difficulty in proving them for himself. Differentiating the first of equations (94), we find Hence, eliminating F' (x+iy), ~dy ~by \dx ~dx)' and, on equating real and imaginary parts, 3a; 3y 3y 3k/ .(95) 241 ,(95«) 245.] CURVILINEAR COORDINATES. Similarly, by differentiating (94) as to £ and t/, we find dx _ "bij From (95) we deduce by differentiation v^=o f "" and in fact Conjugate Functions are sometimes defined as solu- tions of (96) which also satisfy (95). It further follows from (95) that the coordinate surfaces satisfy the conditions (6) of orthogonalism. The (p and tj surfaces are in fact two orthogonal systems of cylinders with their generators parallel to Oz. Again, by (95) we see that .(96) whence V-l i while equations (IS) and (19) give us , h 3*1 h3? h 3* '""it dz .(99) k, ="[®ND>(S)* <•»»> 242 CURVILINEAR COORDINATES. [245. Substituting from (97) in (16), we find for the curvatures v = 0, ja =g .ST =0, TS =0 .(101) and by (30) .(102) On this system the s 3 curves become straight lines (parallel to Oz), and y is linear, and identical with w. The strain components are now given by ■/du 3A e = h^ri - v ^r- r. /dv ~dh 9 = dz 3 (v + 3#, a = It -\ , 'du 1 'dw b= Tz +h T£ "dw .(103) and the component rotations by 26, = A 3 !?-?? 1 dr, dz oq du .3te 202 -ai-* be the surface of the body, the boundary conditions (45) become .(107) where k is given by (100). The conditions that the strain may be pure follow at once from (104), on making e 1= o, e 2 =o, e 3 =o ; and in this case, by (32) and (33), if be the displacement potential, a 7 3<& d .(108) 244 CURVILINEAR COORDINATES. [245. If £, q, z be the principal coordinates of the strain, we have by (103) .(109) f 5" = = . 4 w+ ! («A) = = The e, f, g of equations (1 03) will then be equal respectively to fj, e 2 , e 3 ; and JV 1( N t , N 3 will be given by (55). Lamp's equations (57) will take the form and the boundary conditions (58) reduce to ,3* .(110) ^ir hH ' on) a? 246.] As an example of conjugate cylinders, let £ and >/ be given by the equation x+ i-y-C cosh(£ + vtj). Then it is easily shown that x = C cosh £ . cos 17 1 y=C sinh £ . sin 17 j Thus C 2 cosh 2 £ G' 2 sinh 2 £ ' C 2 cor 2 t? C 2 sm 2 r) 246.] CURVILINEAR COORDINATES. 245 The £ cylinders have for their transverse sections a system of confocal ellipses, and the q cylinders a system of confocal hyper- bolas : the common foci of the two systems being situated on the axis of x at equal distances C on either side of the origin. These confocal conies are represented in Figure 33, § 250. From (12) we deduce Cs/cosk^-cos 2 ^ C Jwah^ + sin^ and from (101) sin v . cos 7) CT.= - - '— v * qsinh^ + sin^)* „ sinh £ . cosh £ 33 2 ^ — ' C(8inh 2 ^ + w&jfi) The geometrical interpretation of these is as follows. If A, B; A', B' be the transverse and conjugate semi-axes of the ellipse and hyperbola intersecting at any point P, (g, ij) or (x, y), the curvatures at P of the ellipse and hyperbola respectively are 33 Ali AB A'F A'E ST,= - 71 * (A*-A'*f (B*+B'*)$ 247.] Surfaces of Revolution. All the more important systems of orthogonal cylindrical surfaces (including conjugate cylinders) have one plane of symmetry through Oz, and many of them have two such planes, mutually perpendicular. It is clear that if the plane of zx be a plane of symmetry for the g and rj cylinders, Ox will be an axis of symmetry of their normal sections by the plane of xy, which we may call the £ and rj curves (strictly the gz and t\z curves). If then we suppose the plane of xy to rotate about Ox, the two orthogonal systems of curves traced upon it will describe two orthogonal systems of surfaces of revolution, having Ox for their common axis. Adding to these the system of planes through the axis of revolution, we have a complete set of new orthogonal coordinate surfaces. Let £i = xi(*, y)\ vi = X2( x >y)\ 246 CURVILINEAR COORDINATES. [247. be the original cylindrical system. Then the transformed system will obviously be •? = x 2 ( a; > 'Jy i + ^)\- C=tKa-\z/y) J Now the only quantities involved in the equations of this Chapter which depend in the least on x, y, z are h v \, h 3 : and these are supposed to be expressed, before insertion in the equations, as explicit functions of £ rj, f (§ 230). But the symmetrical form of (4) or (12) shows that x, y, z may be interchanged in any way without in the least affecting the forms of h v h v h 3 , when expressed as functions of £ tj, £ We may therefore take the axis of revolution for the axis of 2 in our new system, and take for Ox and Oy any axes whatever, perpendicular to Oz and to one another. This amounts to transforming the cylindrical system >h = X 2 (*i>2/i)[ (H2) into the system £ = *(*, Jx* + y*)) (113) V = Xt( z , n/^+V) £=ta,n-\ylx) OXj being the axis of symmetry of the old system, and Oz the axis of revolution of the new system. Similarly, if 0y 1 be an axis of symmetry, we may construct a second system of surfaces of revolution, defined by the functions r/ = X 2 (s/* 2 + y 2 ,3) ( U4 ) Suppose that equations (112) can be solved so as to give x 1 , y x explicitly in terms of £, rj 1 : let the solution be x i = F Mv Vi)) Vi = F J£vVi)\ (H2a) =. = <, ) 247.] CUEVILINEAK COORDINATES. 247 Then the solution of (113) will obviously be y = ant-*'&,v)\ (H3a) and the solution of (114) will be x = coat.F 1 (£,r,)\ ■y^smt.Ftf, v ) I (H4a) By formulae (12) we have, for the original system (112), H(|;g); c' .(115) for the first transformed system (113) imn 1 IfdF.y I ofa* ■dr,) 1 =m>v) .(116) and for the second transformed system (114) rv(l) + (t) .(117) =mv) Thus neither transformation affects the forms of h 1 and h 2 as functions of £ and »;. But, on the other hand, they both make h 3 dependent on £ and r). Considerations of symmetry alone are sufficient to shew that all three must be independent of f. 248.] As a simple example of the results of this Article, let us transform from the cylindrical polars of § 244 to the spherical polars of § 243. 248 CURVILINEAR COORDINATES. [246. The original system is 7>j = tan tyjxjfy & = *! f whence and therefore yi = £i- 8m, h'> *-l *-£ T =l This system is perfectly symmetrical about any axis perpen- dicular to 0z v and consequently the two transformed systems are identical. They are given by 7 / = tan- 1 [ v / ^TP/2]K f = tan- 1 (y/«) J whence and a: = £ . sin ?> . cos f"| y = g. sin 7/. sin (\, Z = £ . COS 1) j 1,1.1.. ST 1 ' r t - fe ^^ 8m,y - Comparing these results with the general formulae, it will be seen that they correspond in every respect. 249.] Conjugate Surfaces of Revolution. Let the original system of cylindrical surfaces be given, as in § 245, by (i + »k = f\*>i+Vi) ( 94 ) Then the transformed systems of surfaces of revolution will be given by ^ + Lt ) = F{z + iJ'x^+^) (118) and £ + ir) = F(J& + y + is>) (119) respectively, according as the axis of revolution Oz coincides with Ox, or 0y y 249.] CURVILINEAR COORDINATES. 249 If the solutions of these equations be given as before by (112a), (113a), (114a), we have by substitution in (95a) 3£ dr, 3?, 3£ : whence, by (116) and (117), 1 1 oW^NfH(lHf)- Thus if we write .(120) we have in either of the transformed systems .(121) p = Jx 2 + y' i 1 = 1 //3/a 2 m'Y | both h and h' being functions of £ and r\, but independent of f. Writing 6 for f, so that 6 has the same meaning as in § 244, we have now for the principal curvatures, instead of the values given in (101), (122) Thus if, as in § 232, w 3 denote the absolute curvature of the s 3 or fij curves, we have by (16a) by (121). ET- = — V i 3,; JZ =0, 3A' a? 1 „ 3A h m i e K 3^ 6 i) = A'* 1 0T + y" This is obvious geometrically, for these curves are circles in planes parallel to xy and having their centres in Oz, the axis of revolu- tion. 250 CURVILINEAR COORDINATES. [249. The elementary arcs are „ _<*£ ds = dr ) ds = de 1_ T' 2 T* 3 V and the element of volume is - t d£di)d6. The formula (13) becomes and equations (IS) and (19) reduce to A h 3$ ~h~3£ f- _A3$ h3>? V - A'3* = h 35 and .(123) .(124) L\3|/ \3»?, V36») also, by (30), W = a/h \ v={3/h\ w = yjh') The strain components are ,~du 3/j c = tl V — 3£ 3,, > /• ; 3t> 3A J = ' "4I,W» ^4|e*> "|(«4)t|(«i) > (125) 126) (127) .(128) 249.] CURVILINEAR COORDINATES. 251 and the component rotations ■"■-"KG)-*©]. Lame's equations (48) become (-^--K(|)-|(|)]=^- Z )J The conditions that the strain may be pure are e 1 =o,e I =o,e,=o ; and the conditions that £ »?, may be the principal coordinates of the strain are a = 0, 6 = 0, c = 0. In the latter case, equations (56) and (58) reduce, to .oj(3/iV r 1 \ , ,7-3/t , Ar A 3A' /.. /_,> (129) c)$\hh'J ' " 2 3£ ' * 3 A' 3£ h'^ = P (w-Z) .(130) and A^-h^ ,9* ^ 2 A?^ = hH' .(131) The student will find no difficulty in adapting equations (41) and (45): and, in the case of pure strain, (32) and (33). 252 CURVILINEAR COORDINATES. [249. It should be carefully borne in mind th at £and 17 are conjugate functions of r and s, where r=Jx 2 +y 2 , as in § 244, and that they only satisfy the equation corresponding in form to the equation "dx 2 ~dy 2 of g 245. That is, they satisfy dr 2 'd* 3 I ^H + ^h = 1 'dr 2 3z 2 and consequently by (82) r or r 3r They are not therefore solutions of Laplace's equation, \7 2 $ = 0, as are the conjugate cylindrics of § 245. 250.] As an example of Conjugate Surfaces of revolution, let us transform the C3 - Iindrical system of § 246 by the method of § 247. We see from Figure 33 that this system is symmetrical about both Ox 1 and Oy r We therefore have the two transformed systems z + 1 Jx 2 + y 2 = C cosh (£ + «j) and -Jx 2 + y 2 + iz = C cosh (£ + i-q). {%.) The first system gives us x 2 + y 2 z 2 _ 1 , 6' 2 sinh 2 ^ + C 2 "coih 2 ^ _ C 2 cos 2 ij <7 2 sin 2 rj the £ surfaces being conf ocal prolate spheroids, and the 1/ surfaces conf ocal hyperboloids of revolution of two sheets. These surfaces will be described by the rotation of Figure 33 about the trans- verse axis. We have and thus by (121) x 2 + y 2 = C^inh 2 ^ . sia% 1 h' G sinh £ sin 17 1 C ^/sinh 2 ! + sinV 250.] CURVILINEAR COORDINATES. 253 (vi.) The second system gives us C' 2 cosh^ C%tnh 2 £ - 1 s 2 + y 2 s 2 , T Here the £ surfaces are confocal oblate spheroids, and the q surfaces confocal hyperboloids of revolution of one sheet. These Fig.33. surfaces will he described if Figure 33 is made to rotate about its conjugate axis. We have in this case x i + y 2 = C^cosh 2 ^ . cos 2 ??, 254 CUHVILINEAR COORDINj whence by (121) k'~ l G cosh £ cos t) /,. = __ i . . C then any confocal quadric must be of the form r.2. r + ... , ... (133) A i - p B'-p Let £ and >j be the lesser and greater roots of (133), considered as a quadratic in p. (i.) When the bounding surface (132) is a prolate spheroid, B 2 >-r)>A*>£> -oo. The £ surfaces are the prolate spheroids ft^W 1 <■"> confocal with the bounding surface (132), which is represented by £ = (135) For positive values of £ these spheroids lie within (135), and for negative values of £ without it. The */ surfaces are the hyperboloids of two sheets (136) Z 2 X 2 + ?/ 2 B 2 -rj~ -q-A* i which are also confocal with (132) or (135). Taking, as before, S=e = t*n-'(y/x), it is easily shewn that ^c°^ 2 -JV 2) ] *-wM'-M>-**> V... tv-a* (137) 251.] CURVILINEAR COORDINATES. 255 _ 2 Ky-A^m-r, ) v ^ 2 -4 2 (4«-f)(,-^«)J .(138) Distinguishing the prolate system of the last Article by the suffix 1, we have evidently C^smh 2 ^ = A 2 - f| C 2 cosh 2 ^ = B*-}\ C 2 coa\ =£*- v i C*ain% = V -A*j B*-A 2 = C i \ £ = ^cosh 2 ^ - B*smh% L ■7 = /^cos 2 ^ + 5 2 sin 2 ijj ! (ii.) When the bounding surface is oblate, A*> v >W>%>- Assuming that A, B, C, and also f, rj, £ are in descending order of magnitude, we may shew that A*>t>B*> v >C*>(> - -k . The £ surfaces are the confocal ellipsoids A 2 -£ J5 2 -£ 6' 2 -£ ' ( ' the rj surfaces the confocal hyperboloids of one sheet T 2 ifl ?1 i^^-^-V 1 (146 > and the f surfaces the confocal hyperboloids of two sheets a; 2 y* z 2 252.] CURVILINEAR COORDINATES. Hence we easily deduce that y (A* - W){A* - C 2 ) (B* - C*)(A 2 - £ 2 ) (A* - C*)(B* - C*) whence 257 .(148) 1 V iv-m-a ) h _» K#-v)(*p-v)h-c*) 8 V (RIR As in the last Article, the boundary conditions are .(149) .(150) when £=0. EXAMPLES. 1. Show that if £,, >?, ; £, j; 2 ; be any number of pairs of conjugate functions of x and y, g and >/ will also be conjugate functions, where ^A + 2( Pl ^)-^( qiVl )) V = B + 2( Wl ) + 2(^) J -4, i?, £>,, p 2 >%? are conjugate functions of x and y, then a; and y (or i/ and — x) are conjugate functions of £ and »;. 3. Show that £ and »/,, any one of the pairs in Example 1, are conjugate functions of each of the other pairs, or of any pair compounded of them, like g and rj. 4. Prove that £ and r\, as given by each of the following pairs of equations, are conjugate functions of x and y : and find the value of h for each pair, [r and denote the cylindrical polars of § 244.] R 258 CURVILINEAR COORDINATES. [252. {l - } \ n = P e; («.) /£ = (•? +C 2 r-*) cos p0, \. v = (C 1 r* -Of-*) sin p6; §$ = (C l e pz + C 2 e- M ) cos py, { r) = (C^e" 1 - C 2 e - ") sin py ; ^ x = (C 1 coapri + C 2 amp7))coshp£, { y = (0^^ sin prj- C 2 cos prfisinhpg ; *{iii.) (v.) ( f. , sJy i + (x±AY £=p\og vy ^ L, ^tan"'-^; ( (x + A coth £) 2 + j/ 2 = 4 2 cosech 2 £, I x 2 + (y - A cot ?;) 2 = A 2 cosec 2 rj. 5. Transform the cylindrical surfaces of the last Example into surfaces of revolution by the method of § 249, and trace them geometrically. 7. If P be on one of the common generators of the conjugate cylinders £ and q, and if P/S i; P£ 2 be normals drawn in the direc- tions in which £ and r\ increase, show that their relative position is always such that to make P.S', coincide with P$ 2 we should have to turn it through a right angle in the positive direction of rotation about Oz. 7. If any system of orthogonal surfaces be inverted as to any centre of inversion, show that the new system thus obtained is also orthogonal. 8. Show that the pure strain defined by = F^r) + rF 2 (6) + r sin 6F 3 (u>) has the spherical polars r, 0, w for its principal coordinates. 9. Show that the pure strain defined by <]> = F 1 {r) + rF 2 {d) + F z (z) has the cylindrical polars r, 0, z for its principal coordinates. 10. Show that the pure strain defined by = F^, n ) + F 2 (z) * Here e denotes — as elsewhere in this work — the base of the Napierian logarithms. 252.] CURVILINEAR COORDINATES. 259 will have the conjugate cylindrics £ 17, z for its principal coordin- ates, if F^ satisfies the differential equation &"$)♦«)-* 11. Show that in the corresponding case for the conjugate surfaces of revolution of § 249 where F^ satisfies the same differential equation as in the last Example. 12. Show that in the corresponding case for the spheroidals of § 251 - ^(f. V) + J(A*-£)(A*- V ) . F 2 (d), or * - F&, ,) + J(AZ-()(r,-A*) . F^d), according as the £ surfaces are oblate or prolate : F x being any root of the differential equation 13. Prove that the conditions of § 242 are satisfied by the following forms of irrotational strain — (i.) For spherical polars 4> = F(r). (ii.) For cylindrical polars = F 1 (r) + F 2 (z). (Hi.) For conjugate cylindrics $~F^, V ) + Flz), where K*'fH (iv.) For conjugate surfaces of revolution 4> = F(£, v ), where F satisfies the same conditions as F } in the last example. 260 CURVILINEAR COORDINATES. [252. 14. If the f and tj surfaces are conjugate surfaces of revolu- tion, as in § 249, show that (V 2 ^ + (V*7)" = (**?■ 15. Show from equations (16) that the >? surfaces of § 251 (ii.) and of § 252, are of anticlastic curvature. 16. In the case of § 251 (i.), what locus is represented by f = , = ^» 17. In the case of § 251 (ii.), what locus is represented by 18. In the case of § 252, what loci are represented by and tj = f = 5 s , respectively ? 19. Deduce from equations (57) the conditions that a Line of Stress may transmit a constant traction or pressure in the direction of its length, under no Applied Forces. 20. Deduce from equations (56) the conditions that a Tube of Stress may transmit a constant tension or thrust in the direction of its length, under no Applied Forces. 21. Show from equations (57) that an isotropic medium may be held in equilibrium, under no Applied Forces, by the system of stresses w ,--^--£ra -r\dsj where K is a constant, and SF a function of £ only, satisfying Laplace's equation [According to the theory of Faraday and Clerk Maxwell, this represents the condition of a dielectric medium in the neighbourhood of charged con- ductors. A' is the specific inductive capacity of the medium, and * is the electrostatic potential, so that the £ surfaces are the equipotentials.] 22. Assuming equations (47), (46), (22), (24), (25), (42), deduce (41) and (45) by the method of § 219. 23. Lame" obtains, in his Coardonne'es Curvilignes, many groups of equations involving Ii^, h 2 , h 3 and the curvatures of the coordinate surfaces. The following examples may all be deduced from the formulae of § 230-232 ; each is, of course, the type of a group of similar equations which can easily be deduced from it by the principle of symmetry. 252.] CURVILINEAE COORDINATES. 261 (i) l/Vt.-C-'W- (***.) (*»•) —J = X„ . CT. + A, . J3 f A 2 |(AA) + ^ ^PZ&'iHFrP^F&PrM- (m.) (vii.) h x 3i?3£ rf-rn" BT rJ BT { = 0. 3s. W'ft^-^ CHAPTER VI. GENERAL SOLUTIONS AND EXAMPLES. The General Problem: — Preliminary Theorems. 253.] Recapitulation of the General Problem. Let a homogeneous body of natural density p be subjected to a small strain ; and let u, v, w be the component displacements, parallel to rectangular axes fixed in space, of that point of the body which in the natural state occupies the position (x, y, z). Then, if e, f, g be the component elongations of the element described about that point, a, b, c the component shears, A the cubical dilatation, and 6 V 2 , 6 3 the component rotations, -we have by §123 3i* -. ox ou ~bw V ^=r> c ~dv ou 3-c oy •(1) . ou 3« 3w A= + + — ox oy dz u , fdw ov\ a i Cou "dw\ a , /ov ~du\ " i= % - ^)' 2= H^~^} 3= H^ - ^ Also, if P, Q, R; S, T, U be the normal and tangential com- ponents of- the stress at the point, we have by §212 for an isotropic body P = (m + n)e + (m - n)(J+ g) Q = (m + n]f+ (to - n)(ff + e) R=(m + n)g + (to - n)(e +/) S = na T=vh U=nc (2) 253.] GENERAL SOLUTIONS AND EXAMPLES. 263 (3) and by §214 e = P/q- 1 = I) 1 + D i = 0. 254. J GENERAL SOLUTIONS AND EXAMPLES. 267 Thus and D^D a = D a = 0, u = A l -C$ + C- i e\ = A 2 -C s z + CpV (13) W = A S -GjX+ Cgl/J The only distribution of displacement which can be maintained without Applied Forces or Surface Tractions is therefore com- pounded of the bodily translation u = A 1 \ w = A s and the bodily rotation (§§ 85, 86) which is simply such as can be suffered by any perfectly rigid body (§ 48), and does not constitute a strain at all. 255.] THEOREM II. The distribution of Strain through- out a perfectly elastic solid in equilibrium under any given system of Applied Forces and Surface Tractions is perfectly determinate. Consequently, the distribution of Displacement under the same conditions is also determinate, with the exception of an arbitrary displacement of the hind that can be suffered by a perfectly rigid body. For if X, Y, Z be the components of the given system of Applied Forces, and F, G, II of the given system of Surface Tractions : and if {e, f, g, a, b, c} be a distribution of strain con- sistent with the given conditions, it must satisfy the equations (m + nj—Y (m ox (m + n)J- + ( m da , 3c\ , 'dx i "dx ~Oy Wx=o ) 7 ■(14) throughout the body, and A[(m + n)e + (m- n)(f+ g)] + fine + vnb = F\ Arac + ii[(m + n)f+(m-n)(ff + e)] + vna = G J- (14 Xnb + (ina + v[(m + n)g + (m - n)(e +/)] = H) 26b GENERAL SOLUTIONS AND EXAMPLES. [255. at the bounding surface. Similarly, if {e',f, g', a', V, c'} by any other distribution of strain consistent with the given conditions, we must have (m + n)— + (m ox — £♦<-" ■»&*!)"(y*s)-"- and A.[(m + n)e + (m - n)(/' + g')] + fine' + vnb' = F \ Xnc' + /*[(wi + »)/' + (m - «)(#' + e')] + *W = G I. Xn6' +jj.na' + v[(m + n)g' + (m - n)(«' +/')] = fl" J Let e' = e + e", /'=/+/*, 9' =9+9", a' = a+a', b'=b + b", c' = c + c". Then by subtraction of the two systems of linear equations we find that 3/" (hi + n) *<»-«>(¥ + l>"(I + ^)=° <—>¥♦<— »(s*¥) + -(s + W)- 9y .(16) throughout the body, and that X[(m + n)e" + (m - «.)(/" + #")] + /anc" + ra&" = CA Xnc" + p[(m + n)f" + (m - n)(#'' + «")] t vna" = 01 (17). Xnb" + fma' + v[(m + n)g" + (m - ?i)(e" +/")] = I at the boundary surface. Comparing (16) and (17) with the standard forms of the equations [(47) and (49) of § 217], we see at once that {e",f", g", «.", b", c"} is the specification of a strain such as could be main- tained unaltered without Applied Forces or Surface Tractions. Thus, by Theorem I, e - f" = g" = a" = b" = c" = ; and consequently e' = e,/'=f, g' = g, a' = a, b' = b, c' = c. Thus only one distribution of Strain can satisfy the given conditions, and the solution is completely determinate as to the strain. Consequently, the distribution of displacement is also deter- minate, in so far as it constitutes a strain : that is to say, with 255.] GENERAL SOLUTIONS AND EXAMPLES. 269 the sole exception of an arbitrary translation and rotation of the body as a whole. Of course, as we expressly excluded such displacements from consideration (§§ 48-50), we cannot expect our equations to give us any information on the subject. It should be observed that when the surface displacements (10) are given, the distribution of displacement is absolutely determinate. 256.] THEOREM III. The most general distribution of motion possible in a perfectly elastic body, free from Applied, Forces and Surface Tractions, consists of a series of superposed small harmonic vibrations of the points of the body about their natural positions ; translations and rotations of the body as a whole being excluded. The equations of motion (7) become, when the Applied Forces are zero, ox\px m M^i' ::. : \+»v^ -p ?>n ~dv . "dw\ . , "dhi "dy ~dz) 2-?! ■dy\dx dy ?>zj v r 3« 2 3 2 M» 3 Cdu T)v 3«A , .(18) These equations are linear, and consequently the most general solutions for u, v, 10, as functions of x, y, z, and t, may be con- sidered as built up by adding together simpler values of u, v, w, which will simultaneously satisfy (18). But any function of x, y, z, t may be expanded in a series of terms, each of which is of one of the three following forms : — (i.) Function of t only. (ii.) Function of (x, y, z) only. (Hi.) Product of a function of i into a function of (x, y, z). Now any solution which gives u, v, w, or any of them in the form (i.)— that is, independent of x, y, z — represents a transla- tion of the body as a whole. Also any solution which gives u, v, w in form (ii.) — that is, independent of t — must also be a solution of equations (16) and (17), and therefore represents a superposed translation and rotation of the body as a whole (Theorem I). Both these solutions have been expressly excluded, and we must, therefore, assume that the most general solution for strain- ing motion is of the form U = MjTj + U 2 T 2 + MjT 4 + j v=v 1 t 1 ' + v 2 t 2 + V t T- + ,- (19) w — v)f{ + w 2 r 2 " + w.Tj" + J 270 GENERAL SOLUTIONS AND EXAMPLES. [256. where u v { , w, are functions of x, y, z only, and t« t/, t," of t only ; and v = v^l W = WjTj" are simultaneous simple solutions of equations (18). Substituting in these equations, we get m \ Ti W + T ' 3^% + T ' 3.3* } + nT ^ Ui = pu < d*T t and two similar equations. Now it is obviously impossible, in general, that these equa- tions can be satisfied by values of u„ v { , w t which are independent of t, unless all functions of t can be cleared from the left-hand side. The necessary and sufficient conditions for this are T i=T ( =T t for all values of i. Making this assumption, the equations may now be written m 3 | dUi ~dv t "dWi \ u, 3e \ cte 3y 3z I n. t. m 3 [ "dUj ~dv t 'dw i \ ■ v t ~dy \ "dx 3y 3: 1 9 P m 3 ( "du t ~dv t "duo, I n „ p — •tT \ ^- + ^-+-^ ( +— V«°« = - to ( oz I O03 oy oz ) w t Tj ' ~dP dhj dfi Here we have three expressions which are known to be inde- pendent of t equated to an expression which is known to be independent of x, y, z. In order that this may be possible, each of the four expressions must be equal to an absolute constant. . p i Let this constant be denoted by i ; then we shall have .(20) _ 3 ( ?)»-, , 'dv i dw, ) „ . . "dv, i r' + — 1 + ny 2 v t + piv, = dz 3z | f 3m, 3» ( 3io ) . .(21) 256.] GENERAL SOLUTIONS AND EXAMPLES. 271 and the general solution will be of the form u = 2(m 1 t 4 )'| ^ = 2(^)1 (22) w = 2(10^,) J The solution of (20) depends upon the form of i. If i be real and positive Tj = A sin *Ji. t + B cos Ji.t; if i be real and negative T ( =Ae vzr - t + Be-* /::i - t , where e denotes the base of Napier's logarithms : and lastly, if i be partly real and partly imaginary, the solution is of a mixed form. We now proceed to shew that every possible value of i which permits of solutions of (21) consistent with the boundary condi- tions is both real and positive. Let e i r i,fj t , A^ be the components of the strain corresponding to the partial solution u == upA »=*(T 1 l (23) and let P^i, QiTi, Un be the components of the corresponding stress. Then by (1) *=*' 1 'duo, "dv. f' and by (2) P i = mA ( + 2ne | S { = na„ 272 GENERAL SOLUTIONS AND EXAMPLES. [256. .(24) Thus (21) may be written dP t ?>U, dT t ,, ox ay oz ■dUi 3& 3 ^ +i v =0 "dx ~dy ~bz dT, ZS ( VRj ■ "dx ~dy 3z while the conditions to be satisfied at the bounding surface are kP ( + fiU, + vT i = 0\ AZ7 4 + / i©, + i/aS' i = oL (25) X-Ti + pSt + vR^O) Let any other particular solution of equations (18) be .(26) and let a similar notation be adopted with regard to j and the functions depending upon it. Consider the integral I y -/Yy[' u i u ) + v i°j + w t v)j\dxdyde, taken throughout the entire volume of the body. Substituting for v it v t , w t from (24) «-M<&< J +Wi '\_ "dx 3y 3z _| + wA D5T VSi ?>R-\ ) , , , Integrating by parts, as in §§ 146, 194, 219, we have - i/tfy ~ff{ Vj[XP t + pU t + vT ( ] + Vj[XU, + pQi + v# ( ] + wJ[kT t + pS i + vR l ]}dS -Iff lp%i+<&t + R?»> \ 'dx *dy 'dz -«(£♦!?) ♦'<&♦£) *<£♦%) }*** 256.] GENERAL SOLUTIONS AND EXAMPLES. 273 By (25) the surface integral vanishes, and thus V 1 " =fff{ e f* + £& + 9 A + «A + bjT t + Cj U ( }dxdydz. Now, looking hack to the original form of I ^ it is obvious that it is symmetrical with regard to i sad-j. That is and, interchanging i and j and i and j in the formula just obtained, jpl* =///{**, +M + 9 A + /^T v s = v + /?v' J - 1 r Vj = v - )8v' J - 1 tCj = W + /Jw' J- 1) Wj= w-)8w' J— Ij where u, v, w, u', v', w' are all real. s 274 GENERAL SOLUTIONS AND EXAMPLES. [256. Thus \ ij =fff{xfi + v 2 + w 2 + £ 2 (u' 2 + v' 2 + w' 2 )}dxdydz, which is the sum of six essentially positive quantities ; and since I,j is identically zero, each of these six quantities must vanish separately. Thus at every point of the body u= v=w=0 1 and the solution is therefore null. Thus it is conclusively shown that any value of i which admits of solutions of (21) or (24), consistent with the boundary condi- tions (25), must be real. Again, consider the second integral Ii — ffj\ u i + v ? + v>i)dxdydz. Since i is necessarily real, I f is necessarily positive : but by (24) \ ox ay oz / Integrating by parts, as before, ipli ^fffieft +/ i Q t + gfit + a t S, + l t T, + cjl^dxdydz = =lff/V i dxdydz l>y (19) and (20) of £ 199 ; V lTi being the potential energy per unit volume, and W iTi the total potential energy of the body, due to the partial solution (23) above. Or, which amounts to the same thing, W t is the potential energy due to the strain {«.-, A, g t , «» b„ c,{. Thus il, is essentially positive, as well as I,; and consequently i is also essentially positive. Finally then we see that every value of i which, admits of a solution of (21) consistent ivith the boundary conditions (25) is essentially real and positive. Thus we may obviously write i = i 2 ; m .(28) 256.] GENERAL SOLUTIONS AND EXAMPLES. 275 and we shall then have, for the most general equations of strain- ing motion, under no Applied Forces or Surface Tractions, u = 2(mj sin it + Mj'cos it)\ v = 2( v t sin it + v^cosit)\ (27) w = 2(iOj sin it + mj/cos it)\ where u it v„ w t and «/, v t ', wl are any two sets of values of u, v, w which satisfy throughout the body, and the conditions (25) over the bounding surface. The most general form of small straining motion, under no Applied Forces or Surface Tractions, is therefore to be obtained by superposing all such possible systems of small simple harmonic vibrations of points in the body about their natural positions. Equations (27), (28), and (25) thus present to us the analytical statement of the Problem of Free Vibrations. In general there are an infinite number of partial solutions of (28) for each value of i. In any particular problem, the boundary conditions, combined with considerations of symmetry, .restric- tions on the mode of propagation, etc., will enable us to select appropriate solutions. 257.] THEOREM IV. The most general distribution of motion possible in a perfectly elastic body, under any system whatever of Applied Forces and Surface Tractions capable of maintaining equilibrium, consists of a series of superposed small harmonic vibrdtwns — governed by the same laws as in the preceding case, and consequently dependent only on the properties of the body itself — of points in the body about mean positions which are identical with those to which the same points are displaced, when the body is in equilibrium under the same system of Applied Forces and' Surf ace Tractions. For if u, v, w be the component displacements at time t of the point which in the natural state occupies the position (x, y, z), we have 3 fdu dv ?)w\ 9 , , ,- •■ , „ dx\ax oy oz] etc., 276 GENERAL SOLUTIONS AND EXAMPLES. [257. throughout the body, and etc., at every point of its surface. But if u, v', w' be the displacements which the same point would experience, if the body were in equilibrium under the same system of Applied Forces and Surface Tractions, then CIX dx\ox oy oz J etc., throughout the body, and etc., over the surface. Thus, if we assume u = u' + u", v = v' + v', w = w' +w", the displacements u", if, w" satisfy 3 fdw" 3»" 3w>"\ , 2 „ ..„. oy\ ox ay oz J 3 fdu" 3u" ~dw"\ o „ ../, throughout the body, and -I - / N 3tt" / \/cV 3te"\~l /3u" 3i/'\ (3m'' 3m>"\ „ 3^ + 3^) = °' etc., over the bounding surface. The distribution of motion represented by u", v", vf is there- fore such as might take place if the body were in motion under no Applied Forces or Surface Tractions, and by the last Theorem we know that this consists of a series of small superposed har- monic vibrations about the positions given by 257.] GENERAL SOLUTIONS AND EXAMPLES. 277 by and this system of vibrations is of course absolutely independent of the external forces. The assumption in the enunciation of this Theorem that the Applied Forces and Surface Tractions are such as are capable of •maintaining a state of equilibrium, expressly excludes all such systems as vary with the time. These latter affect the mode of vibration, but not the mean configuration : the most important case is that in which all the external forces are harmonic func- tions of the time, of the same period throughout. The problem then becomes that of Forced Vibrations, to be considered in the next Theorem. 258.] THEOREM V. A system of Applied Forces and Surface Tractions which varies as any simple harmonic function of the time, gives rise to a distribution of Forced Vibrations, of the same period as themselves, about the natural configuration ; and the most general form of straining motion possible under such a system consists of this (perfectly definite) mode of forced vibration, with any (perfectly arbitrary) modes of free vibration superposed on it. For if the Applied Forces and Surface Tractions be given by X = Xsini« , | ^=FsiniA F = Ysint«l ff = Gsint*L Z= 2, sin it) # = H sin it) the general equations of motion will be "dx 'dy\'dx 3 3m "dv dx dy (du : | 3a i 3m 3» 3w ~dy dz I ~du "dv 3ie l3z + - + + ny 2 M — pu + pS. sin it = ' pv + pY sini*=0 4 rv^fv ~dz I 3a; 3i/ 3« and the boundary conditions } > + nyhn - pw + pZ sin it = J \\ (m + n)— + (ra dx + ra Jdv 3w>\"l /3w 3m\\ 3* + 3*J = EW ' .(29) .(30) 278 GENERAL SOLUTIONS AND EXAMPLES. [23S. In the tirst place it is obvious that if (u, v, w) represent any distribution whatever of displacement, satisfying the general equations (18) of free vibration, and the corresponding boundary conditions, viz., F=G = H=0, the distribution of displacement (w+u, r + v, -w + w) formed by superposing this system on the particular solution of (29) and (30) which depend upon X, Y, Z, F, G, B, will also satisfy (29) and (30). That is to say, the most general solution of (29) and (30) consists of the particular solu- tion, with any arbitrary modes of free vibration superposed upon it. From the form of (29) and (50) it is obvious that the parti- cular solution must be of the form u= Vi&init\ v= vsinijj- (31) w — w siaitj where U, V, W are determined by the general equations «| \ g + |? + 5? I +Vu + /<*i +X) = 01 ox ( ox oy oz ) .(29a) . (30a) iii^ -! -^-+ ^-r+^r \ + »V 2 W + p(i 2 W + Z) = oz I ox oy oz ) J and the boundary conditions ^'-"^'•'•-••'(t^KS+t)] etc., It therefore consists of a system of simple harmonic vibrations about the natural configuration, having the same period as the external forces, while the mode of vibration — or distribution of amplitudes as functions of x, y, z — depends on the form of these forces. 259.] Subdivision of the General Problem. Availing ourselves of these Theorems, we may now greatly simplify the General Problem by subdividing it into the five following : — (i) The problem of Free Vibrations, under no Applied Forces or Surface Tractions. (ii.) The problem of Forced Vibrations under any given 'periodic system of Surface Tractions only. (Hi.) The problem of Forced Vibrations under any given periodic system of Applied Forces and Surface Tractions. 259.] GENERAL SOLUTIONS AND EXAMPLES. 279 (iv.) The problem of Equilibrium under no Applied Forces, with a given distribution of equilibrating Surface Tractions, or of Surface Displacements. (v.) The problem of Equilibrium under any given equili- brating systems of Applied Forces and Surface Tractions. The Problem of Free Vibrations. 260.] General Statement of the Problem. The general equations to be satisfied throughout the body are 3A , , m— - + rfn-u ox 3A m.— + ur^ P dt* p w =0 .(32) and the boundary conditions are \U+fiQ + vS=ol \.T+nS + vlt = o\ By § 250, the general solution is of the form u - 2( u t sin it + u- cos itj\ v = 2( v t sin it + v[ cos it) l w = 2(tOj sin it + u>[ cos it) J the strain components being given by s = 2( e t sin it + e- cos it) etc., « = £(rtj sin it + a- cos it) etc., (33) .(34) .(35) where 3«i , e, = —~, etc., cte ', etc. .(36) ' "by ~oz and the stress components by P = 2(P S sin it + PI cos it) \ eta j .(37) 280 where GENERAL SOLUTIONS AND EXAMPLES. [260. .(38) P i =(m + n)e t + (m - n)(/( + g',)} v etc., Si = na t etc. Similarly, if the motion be irrotational, the displacement potential must be of the form (ft = 2( t sin it + 'i cos it) (39) where px = 3& 3y .3& " 3z 3y .(40) Selecting the partial solution of order i from (34) and sub- stituting in (32) and (33), we see that each of the systems of displacement {u\, v it w t ) and (u' t , v\, w' t ) satisfies the general equations m -— ' + nrfiut + oi 2 u ( = ox m-zr-t + ny*v t + pPv { =01 (41) m-Jp + nrftOi + pihiOi — oz ' and the boundary conditions kPl + fiUt + vT—O} KUi + t^Qt+vS^Oi (42) kTt + n$ t +vB t = 0) 261.] How does i enter into the solutions for u t , v t , w, ? Writing in (41) and (42) ** = x « W = Ja iz = z» they become 3 /3w 4 , dv t , 3wA / 3 2 . 3 2 , 32 \ , _ , ^fe + 3y; + 3^) +ji te + 3^ + 3^h + ^= ' etc - . I~, n 3m( , v/Sw , 'dw l \~~\ lov, . 3wA foil. 3ioA „ ^ (m+TO) ^ +(w _ w) ^ + _JJ + ^ + _^ + ^_. + _^ = o etc. 261.] GENERAL SOLUTIONS AND EXAMPLES. 281 and it is obvious that these equations retain precisely the same form when any other suffix is substituted for i. Thus we must have u—F^ix, iy,iz)\ •» 4 =F 2 (m!, iy, iz)\ w,= Y 3 {ix, iy, iz)) where the forms of the functions F x , F 2 , F 3 are independent of i. 262.] Distribution of Kinetic and Potential Energy among the partial components. The potential energy due to any distribution of strain may [§ 199 (20)] be put into the form TF= \///[l'e + Qf+ Jig + &'a +Tb + Uc]dxdydz. Substituting from (1) for e, ... c, and integrating by parts, W= \f/\u(XP + nU+vT) + «(A U + iiQ + vS) + w{\T + pS + vR)]d£i. + w[ ~ + — t- ^- j |<2ax2y<& \dx ?>y ~dz j Thus in a state of free vibration [putting X= Y=Z=0 in (6), and F=G = H =0 in (9)] we have W= - ^JTflyM + ^ + ww\dxdydz (43) Also, if C 3E be the kinetic energy, <$ = £ fff\u 2 + v 2 + w 2 ]dxdydz (44) Thus, if W t and ^ be the potential and kinetic energies due to the partial component of order i, Wi = ^ {sin 2 tifff\u? ■+ •»? + wfldxdydz + cos 2 it/YV'[u' i ' 2 + v[ 2 + w'i 2 ~\dxdydz + 2 sin it cos it JTf]y>{u-i + v&i + w t wf\dxdydz} (45) % = tE{~ + l '/" + u\ 2 ]dxdydz. . .. (47) which is independent of the time, as of course it ought to be— no work being done on the body. Now, substituting from (34) in (43), we obtain for the resul- tant potential energy H r = ^ /77"{2t 2 ( u : sin it + u[ cos it) . 2(% sin jt + u' } cosjt) + 2i-( »j sin it + v- cos it) . 2( y, sinji + v- cos jt) + 2i 2 (i«j sin it + w- cos &) . 2(te y sin;* + w/ cosjt)}dxdydz ; where both i and j are to receive in succession all the values included in the series (34). This again may be written W= ^{2i 2 sin 2 itfff\uc + «," + Vf]dxdydz + X 2 cos 2 i< fff\_ u i' 2 + '";' ' + w^yixdydz + 22i 2 sin i< cos ii f[[\y>{u-l + *<"/ + wjw'^dxdydz + 2i:i 2 sin t< sinj* /T/ [«-,«,' + *i"j + lOfio^dxdydz + i.'i.'i 2 cos i< cosji /// [ M ;' w ; + '"/'j + w'lWj^dxdydz + 2Si 2 sin ii cosjt fjj[u t u- -f »,-!?,' + iv^Vj \dxdydz} ; where the single summations are to be taken for all the values of i included in the series (34), and the double summations for all different values of i and j. But it is obvious that each term included in either of the double summations is of the same form as the integral I*,- of § 256, and is therefore identically zero. Thus we finally have 11'= § 2i 2 {sin% j/ ^/'(V + ^ 2 + iof\dxdydz + cos% /77"[it/ 2 + v- 2 + w- 2 ]dxdydz + 2 sin it cos it Jjf \uiu[ + vpl + w{w[\dxdydz\ , and, comparing this with (45), we see that W=-2{W t ) (48) In a precisely similar manner we may show that % = ?(%) (49) ■262.] GKNERAL SOLUTIONS AND EXAMPLES. 283 This is a very remarkable result, the significance of which must not be overlooked. It extends the principle of superposition — in the case of free vibrations only — from the components of strain and stress, which are linear functions of u, v, w, to W and "xE which are volume-integrals* of homogeneous quadratic func- tions of the same quantities. The theorem is originally due to Barre de St. Venant (Comptes Rendus; 1865, p. 3 and 1866, p. 195). If (B be the total energy of the body, due to the resultant strain, we have <£=W + c «i; = 2(W i + %) = 2(® i ) \ = ?'2i i /yy[u i 2 + v? + W; 2 + u[' z + vl' 1 + w-'^dxdydz j " ■■'•\ I Let Wi and % t denote the mean values of W t and ^Lt, through- out any interval of time which is an exact multiple of the period (2-7r/i) of the corresponding displacements. Then since ~ — / sinHldt = ~ — / cosHtdt = i i) u and / sin it cos itdt = u J for all integral values of p, it is easy to show from (45), (46) and (47) that ir,=l«=i&; (51) and consequently, if 17=2(1^), f = 2(l), we have from (50) and (51) W=<1S, = ^<& (52) Equations (51) and (52) are the analytical expressions of a statement which the student is very likely to meet with : — that, in any state of free vibratory motion, the energy is cm the whole equally distributed into kinetic and potential energies. * The student must be careful to avoid the error of applying this principle to the quadratic f uuctions themselves — viz., the potential and kinetic energies per unit volume. It is only on integration throughout the whole volume of the body that the terms involving products of j-functioiis and ./-functions disappear. Thus, although it is true that JJJ Vdxdijdz = tfff V dvdydz, it is not true that as will be at once obvious on substituting from (5;. 284 GENERAL SOLUTIONS AND EXAMPLES. [263. 263.] Future investigation may be confined to one partial solution. It is sufficiently obvious from the last three Articles that, when the properties of the partial solution of order i are completely known, those of the most general solution can be at once deduced by simple summation as to i. We shall therefore confine ourselves to the discussion of Equations (41 ) and (42). In the former, we may conveniently drop the suffix, of which the presence of i 2 will be a sufficient reminder, and write 3A .(53) m — v uivi 2 u + pi'-u = ox m~ + tvd 1 v + pih> = oy m^— + nvi'w + pi*w = (J oz , ,3A „ /36L 30,\ ., -1 or (m + n) 2n| —* - —A 1 + px z u = ox \oy oz f {m + n) lTz- 2n (**~W + pl ~ W The boundary conditions, however, it will be better to retain in the form (42). .(54) The Problem of Forced Vibrations, under Periodic Surface Tractions only. 264.] General Equations. Let the body be free from all Applied Forces, as before, but subject to any distribution of Surface Tractions that is strictly periodic as to the time. The traction components will then be of the general form F = 2(F P sin^ji! + F p cospt) .(55) G = ^(Gpsinpt + G p cos pi) H=*Z(B p sn\pt + HpCospt) Equations (32) will still represent the conditions to be satisfied throughout the body, and these may be decomposed, as before into systems of the form (41) for all values of i. The boundary conditions (9) may be written 2[(A.Z > i + p.U t + vT t ) sin it] + 2[(XP/ + pU t ' + vT t ') cos it] - 2[F P sin pt] + ~2{F p ' cos pi] and so on. 264.] GENERAL SOLUTIONS AND EXAMPLES. 285 Thus to every value of i in the series (34) which coincides with a value of p occurring in the series (55), corresponds a solution of (41) satisfying the boundary conditions XU p + l iQ p + vS p = G p ; \.T p +pS p + vE p = ff p while the solutions corresponding to all values of i which are absent from the series (55) satisfy the boundary conditions (42), as before. In other words, the general solution consists of a system of forced vibrations of the same periods as, and depending upon the form of the Surface Tractions, with any arbitrarily chosen system of free vibrations, satisfying (41) and (42), superposed upon it. It is thus convenient to consider the two problems together, taking (53) or (54) for the general equations of vibration, and \U t + itQ t + vS t = oA (56) XTt+pSt+vB^ffJ for the general boundary conditions ; F t , G t , H being each zero, except when i represents one of the values of p in the series (55). The Two Problems Combined. Sir William Thomson's Method of Solution. 265.] Resolution of the Strain. By the principle of superposition and its converse (§ 88), any system of small dis- placements, or the system of small strains produced by it, may be resolved in an arbitrary manner into any number of systems, subject only to the condition that the algebraic sums of the components of the latter shall be identically equal to the corresponding components of the original system. Since the three component displacements at each point of the body must in general be supposed independent functions of position, the number of independent functions involved in any resolution of the strain must, be exactly three in the most general case. Now the most general form of strain consists of dilatation, distortion and rotation ; and it is characteristic of vibrations under no applied forces that the strain must involve either dilata- tion or rotation at every point. [This is at once obvious from equations (58) below.] This strain then may be resolved into two, of which the first may be supposed to give rise to cubical 286 GENERAL SOLUTIONS AND EXAMPLES. [265. dilatation, but to be irrotational, and therefore to involve only one independent function of position — its displacement potential. Thus, if we suppose the second (rotational) strain to be indepen- dent of the first, its component displacements must be connected with one another by some one arbitrary relation: — such, for instance, as that they shall contribute nothing to the cubical dilatation at any point of the body. We shall then have resolved the most general form of small strain into two independent small strains, one of which con- tributes dilatation and distortion without rotation, and the other distortion and rotation without dilatation. 266. Decomposition of the General Equations. If we write in equations (54) fl 2 = (m + n)lp : n' 2 = n/p (57) they become dx \ ~dy ~dz ) a* ^ - 2fl*/22i - ™a\ + ?v - o ~dy \dz ~dx ) fi2 3A ~2tt dz \3z ~dx "(i-5) +i ""° .(56) (i.) Let us suppose the mode of vibration to be irrotational, with a displacement potential . We have then OX .(59) or by (59) of § 123 _3 c)x 3 dz (fl- v 2 4> + * 2 . Any solu- tion of it will obviously satisfy (60), and therefore also (59). (ii.) Let us suppose the strain to be rotational, and such that the cubical dilatation is everywhere zero. If u, v, w be the component displacements in this case, the requisite condition is [by (1)] M^- <«> while equations (58) now reduce to \ dz Oy) 2 W^8_^i\ + i2 v=0 . \dx Zz) 2fi,,/30 1 _3^W w = o \dy dx} j 288 GENERAL SOLUTIONS AND EXAMPLES. [266. .(65) fl'V« + * 2 w = °1 fl'Vv+i 2 v=oi. J2 ,2 y 2 w + »*w = oj Equations (65) and (64) are therefore the general equation? to be satisfied by u, v, w in this case. In virtue of (6t) only two of these quantities are independent. (iii.) If we write in equations (58) «• ox o i> = _r + v y °y OX they become | ( «V* ♦ *> ♦ «v ♦ ft ♦ «£(* ♦ 1 ♦ £) - o »,» V* ♦ *> ♦ OV ♦ «* ♦ «££ ♦ g ♦ £) . o . _(fi v + i2< M + fi V w + i2w + fi2 JYl^ + ^ + — ^ = o .(66) h These equations are obviously satisfied identically, if we take in (66), for any solution of (63), and for u, v, w any solutions of (65) which satisfy (64). Thus, so far as the general equations go, u, v, w may be supposed perfectly independent of solution. 267.] General Equations. If the mode of vibration be wholly irrotational, the potential

is a function of a; only, and (63) reduces to da? QT ' every solution of which is of the form (/>, = Af sin _ + B, cos — . The corresponding partial solution for

= 2C,sini (x -ilt- ft) + 2Cysinl(a: + Qt - ft') (71) which includes waves of all periods and wave-lengths. In the case of a finite body, we find by substituting in (67) that the maintenance of this state of vibration requires the con- tinual exertion of the system of periodic surface tractions F= - vXf^sin l(x - I2u - ft) \ + W/ S in^(.r + fi 1 !-A'f| G = - /v^LnvrW, sin lix -Ot- /3,.) + i*c; sin ^(x+ at -&')! h= - vfSlzHy r^csm it* -at- ft) + i 2 C t 'sm^(x + at-P i ')~\ It is therefore impossible for it to exist alone as a state of free vibration in a body which is bounded on all sides. 270.] Transmission of free sound vibrations through an infinite plate of any thickness. If however we suppose the body to be indefinitely extended in all directions perpendicular to Ox, and bounded only by two planes perpendicular to that axis, .(72) .(73) 292 GENERAL SOLUTIONS AND EXAMPLES. [270. we shall only have to deal with the surface conditions over these two faces, the direction-cosines at every point of which are A= ±1, /a = 0. v = 0. Thus if the faces are given by x = d, x = — d\ the only boundary conditions to be satisfied are 2is[C ( sisi( id n ft ft ^ . i . id D , id ,, A • sin — + B { cos _- = .., . id r> id' ~, A) sin — - B, cos — - = . , ■ id' „ , id' . A t sin _ - B t cos _ = This system of equations admits of three solutions. (i.) Let I = d + d' be the thickness of the plate. Then equa- tions (73) are satisfied by ._i7rft B t B! , i ff rf — - = _- . = — tan — a, a; i where i is any integer. This gives for the general solution ^ = yf^sm 1 J^) s i n l!K f2 i^) (74) 1=0 ' i * t where the summation includes all positive integral values of i, and 2£, and a t are arbitrary constants. (ii.) Again, if the ratio d : d' be reduced to its lowest terms, and then take the form r : s, so that r and s are integers of which one at least must be odd, a second solution of (73) is ? ._ "r(r + s)ft 1 B t = B; = P 270.] GENERAL SOLUTIONS AND EXAMPLES. 293 This gives for the general solution «^=V^ S in 1 >±^5 s i n ^^y-A) (75) 1=0 ' ' to be summed for all positive integral values of i, as before. (Hi.) Finally, a third solution of (73) is given by ._ (2Ul)ir(r + «)m 2P+ 1 . I { a,-a; = o J' where 2 M is the highest power of 2 in the product rs. This gives for the general solution ^^^^iM^^miii^^L^ (76) The sum of the three series (74), (75) and (76) represents all the plane waves of normal vibration that can maintain themselves unchanged in such a plate, without the application of periodic tractions over its faces. 271 .] As a simple example, let the plane of yz be so chosen as to bisect the thickness of the plate ; so that d = d' = \l, r = s=\, p = 0. The series (74) will then split up into two, which will respectively include (75) and (76). The most general solution in this case may thus be written _x -£?"«* ■ 2im; ■ 2iir(fl«-j8,) = V SO; sin — - sm ±— — t-'/ iio ' l '=» (2i + lWa; . (2i + l)ir(fl«-v i ) .__, + V G>iCOs>- — .-J— sin^ '—>. L" (77) i=U Since this is a case of free motion, we may apply the formulae of § 262 to determine the energy possessed by each prismatic portion of the plate having its generators parallel to Ox, and its transverse section of unit area. "Writing first of all in (45) i = 2iirQ.ll 2\ir m %xx 2iirB i Mi = _SS« cos— — cos— -£.* 2iir, tt 2ijra; . 2im3 4 | «•/ = =- SDi cos —j— sin — j-' ' I Co 294 GENERAL SOLUTIONS AND EXAMPLES. [271. we get, for the first part of the potential energy, _l 2 Treating the second series in the same way, we iind for the total potential energy of the prism w = 47r4flL Wj«B,2 sin 2 2 M^-A ) + ^2(2i + l)^ s i^ 2i+1 H m -^. And similarly we may deduce from (46) for the kinetic energy of the prism P I + !^2(2i + i>w C0B « ( 3i+i y-y') . Thus the total energy of the prism will be e =^$i( 2i > 4a5 ' 2+ ( 2i+i ) 4e ' 2 j ( ?8 > and in order that this series may be convergent, it is necessary that 93i 2 should vary inversely as some power of i higher* than the 5th ; and similarly for Q>?. Let us take, for example, (2iyil (2i+l) 3 i2 where B, 0, U are constants independent of i. Then „. *-yu 2 ( if 2 / 1 1 1 I \ n J 1 1 1 1 \ I The two infinite series within the brackets are convergent, and their sumsf are known to be -71^/6 and 7^/8 respectively. Thus and if B and G be so related that * Todhuqter's Algebra, Art. 562. t Todhunter's P?an« Trigonometry, Ch. xxiii., Ex. 1, 3. 271.] GENERAL SOLUTIONS AND EXAMPLES. 295 the total energy possessed by the prism — and consequently also that of the whole plate — will be precisely the same if it were moving bodily with a velocity of translation U. Substituting in (77) we get for the potential due to this equi- valent state of vibration (see -Chapter IX., below) , UP ITS orv*£° 1 • 2im; . 2i7r(fl«.-&) + Igcg_l_ cos Pi Y>™sin < 2i + W* ~ *> (79) where C is completely arbitrary. If we wish to impose the further restriction that the origin (and with it the whole median plane of the plate) shall remain at rest, we have only to make -=? = 0, when x = 0. ax This requires that 0=4/7r, and we then have , 4UZ 2 ^" 1 (2i+lWa; . (2i+ 1)tt{12*- y t ) /finN 272.] Solution in terms of Spherical Harmonics. The general equation (63), when transformed to spherical polars by means of formula (65) of § 243, becomes Let us assume that ^ = 2 t (*i,H.) = 2,^ where H, is a surface harmonic of order s, and <1> M a function of r only. Then tf> t , must satisfy (81) for all values of i and s, and since H, satisfies identically vVH.Ho, (81) reduces to This equation may also be written in the form and the solution which gives finite values for $ it and dQJdr at the origin is 290 GENERAL SOLUTIONS AND EXAMPLES. [272. where J, + , denotes Bessel's function of the first kind, and of order (« + J)- Thus finally we have a solution of the form + ^eo si ^.[H;.J. +i (|)]} .(82) where H„ H,' represent two surface harmonics of order s. This solution is adapted to a solid of spherical form, having the origin at its centre. Stokes' solution, suitable for infinite space outside the sphere, will be found in Lord Rayleigh's Theory of Sound, § 323. The choice of harmonics is not unrestricted, because the pre- servation of the continuity of the body demands that -i = 0, when sin = ; (see § 287, below). Hence all the harmonics included in the solution must satisfy the condition 3H. 361 = 0, when sin = 0. At the surface of the sphere we have, by (72) of § 243, S' = P, H'=U, Z'=T; and on substitution from (73) in (68) of that Article, and thence in (46) of § 239, we find, after availing ourselves of (63) above, „, „ 3 2 f The admissible values of s are therefore the positive integral roots of the equation s + 1 (g -!)(«+ 2) * + 3 (£+ 1)2(8+2) " * 2s+3' 1! (2* + 3)(2s + 5)' 2! s + 5 (s-l)3 (s+ 2)3 _ (2V+'3)(2« + 5)(2s + 7) ' 3! "'" ' I am not aware that the roots of this equation have ever been investigated, but it may be observed that it has at least one positive integral root, namely s=l, and that the value of i corresponding to this value of s is zero. Thus no surface harmonic of the first order can enter into a form of free vibration. and i = g N /2( s -l)( S +2)(- 298 GENERAL SOLUTIONS AND KXAMPLES. [273. 273.] Spherical Sound Waves. In one particular case- namely, that in which the only harmonics present in the solution are of order zero — the two latter of the conditions (83) are satisfied identically, H being a constant, and the admissible values of i are given by . fii * = — i r a 7 ~ 4IF 2 r 1 ^ or the equivalent* equation * icoti = 1 -jJ i2 ( 84 ) The solution (82) now takes the form ^SC^.J^sin^-*), or, as it may also be written,* , /2 V/1 r . ir . i£2/. * d> = . / -ZVi— sin — sin — (t - yX \ ir ir r r The corresponding value for the radial displacement u may be written in either* of the forms , ( =-2cJIjiir\ S i,i^V 7i)) V rr i\ r ) r /2 v C,r »»• r . ir~l . ifi,, , or «* = a; -^— cos — sin — sin — U - yX V 7r r l_ r \r r_| r This solution evidently represents a series of free spherical waves of radial vibration, propagated inwards and outwards with the same radial velocity £2. 274.] Sound Waves in general. Possible Forms. In order that the family of surfaces represented by the general equation x( x , y> z ) = £. where £ is a variable parameter, may represent a possible form of sound waves, sustainable without the aid of Applied Forces, the parameter £ must satisfy two conditions. For let

= <>; or, by (14) of § 231, ^ + V^-| + g = (85) and Thus, being a function of £ only, it follows that a* 2 3y 2 a«2 [ (2NINi must both be capable of expression as functions of £ only, or as constants. The rotational, or u, v, w solution. 275.] General Equations. If the strain be rotational, but unaccompanied by alterations of density, the component displace- ments must satisfy the general equations fi'V u +* 2 u=(n fi'V +* 2v =ol (65) G'V W + * 2w = °J and s*g + s"« <"> all such solutions being excluded as make uete + vdy + wcfe a perfect differential. The boundary conditions (42) may, in virtue of (64), be thrown into the form *§♦•&'♦£)♦»(£♦£)-*•■ w 300 GENERAL SOLUTIONS AND EXAMPLES. [276. 276.] Plane Waves of Transverse, Tangential, or Dis- tortional Vibrations. Let us suppose that v is a function of x only, while u and w are both zero : (64) is then satisfied identi- cally, while (65) gives — H v = 0. dx- Q,'* Thus Vj = A ; sin -=- + B t cos — and the full solution is of the form v = 2C, sin 1(* - £27 - ft) + ZC! sin L( x + 0,'t - ft'). This represents a series of plane waves, of vibrations which are transverse to the direction of propagation, or in the wave fronts, propagated with the same velocity Q' independent of their periods, in the positive and negative directions of Ox. These vibrations are of the same character as those by which light is propagated through the luminiferous ether. Thus if the ether were composed of homogeneous and isotropic "continuous" matter, the velocity of light would be the same whatever its colour. Moreover, it is easy to shew that the same result would hold for light of all colours, propagated in any given direction, if the ether were crystalline, but still " continuous." Now the dispersion of white light into its coloured constituents, by ordinary refraction at the bounding surface of any two trans- parent media of different densities, is proved to be due to the different velocities with which light of various colours is propa- gated in either medium. This familiar phaenomenon is con 1 -: sequently sufficient in itself to prove that the luminiferous ether — at least, as it exists in the interior of solid and liquid bodies — ' cannot possess the properties of " continuous " matter. The fascinating problem of the structure and properties of the ether is too wide and too difficult to be more than alluded to in this place. The student who wishes to follow up the subject should consult Sir William Thomson's Lectures on Molecular Dynamics, delivered at the John Hopkins University, Baltimore, U.S.A., in 1884. These lectures contain a, most interesting summary of the various hypotheses which have been framed to account for the phsenomena of dispersion, polarisation, double refraction, etc., with the grounds on which each has failed, together with a fuller development of Sir William Thomson's own remarkable conception. 277.] The General Solution. The problem, as stated in § 275, appears rather complicated, but it is easy to present it in a form which is of the utmost admissible generality, and yet satisfies all the conditions identically. 277.] GENERAL SOLUTIONS AND EXAMPLES. Thus let us assume ~dy ?lz ~dz ~dx 'dx ~by 301 (87) where \fr v \fs v \Js 3 are any three solutions whatever of the equation Q'Yty + *V = (88) It is obvious that equations (64) and (65) are satisfied identically by these values of u, v, w ; and it is also easy to shew that they cannot possibly make ndx + vdy + vrdz a perfect differential. For in that case we should have or the equivalent relations between \js } , i/r 2 , \Js 3 dx\dx dy dz J VYl fi' 2n which make **♦**+**- -$■$+£♦£) a perfeet differential ; and this would require that u = 0, v = 0, w = 0. Thus (87) and (88) constitute a solution which satisfies identically all the conditions imposed, while, since it involves three arbitrary solutions of (88) which is of the same form as (65), it is of the utmost possible generality. The problem is now reduced to the solution of the funda- mental equation (88), which is similar to (63) and does not therefore need further illustration. 302 GENERAL SOLUTIONS AND EXAMPLES. [278. Poisson's Integrals. 278.] Having given any one partial solution of (63) or (8ft), to express the complete solution as the sum of two definite integrals. Equation (63), which should properly be written fiV of order i, and results [compare the general equations (32) and (41) of § 260] from the decomposition of the perfectly general equation OV4>-$ = (90) satisfied by the resultant potential, as a whole. Writing this latter in the form (| 2 -av)* = o, and remembering that the operator v. being independent of t, behaves as a constant in combination with functions of t or the operator 3/3£, we obtain the symbolical solution tfiy where t = *J — 1 , and <£>, = &, <£ = <£. Now, with the notation of (39), § 260, - 2( f sin it + t ' cos it) \ = ~2i( cos it - { sin it) J and consequently, when I = 0, <£ = 2(/, f are at present parallel to Ox, Oy, Oz, but the operators d/dx, d/dy, d/dz behaving as constants within the integral, and the surface being symmetrical as to g, q, £, we may transform to new axes of f, r{, £' through the centre, such that V&C 2 V & s The integral thus becomes -±- i JJ'e !V dS. X (x,y,z) —r 1 / t -»"v\ / \ = 2^(e -e )x(«,y,«) Thus, writing r = Q£, the symbolical expression tSziy 304 GENERAL SOLUTIONS AND EXAMPLES. [278. represents the mean value of the function x, taken over a sphere of radius Qt with its centre at (x, y, z). Consequently we may write % ^T x{x ' y ' z) = iiffa + nt sin 6 cos "■ II y + ilt sin 6 sin w, z + Slt cos 0)sin ddddu; where 6, w are the spherical angles of § 243. Differentiating both sides as to t, 1 3 f r f iT cos(tl2«y)x(;c, y, z) = — t I /x( x + Qt s i n $ cos <°> y + Q.t sin sin o>, 2 + i2* cos 0)sin OdOdu. Substituting these integrals in (91) we have for the full solution of (90) = — 2 { *' I /&(* + ^' si 11 @ cos w, 3/ + fi< sin 6 sin + ^-t J /i(x + Sit sin cos i(x, y, z) . sin it + $>!(x, y, 2) . cos it, we can at once deduce the complete solution, as the sum of two definite integrals. These integrals may also be regarded as giving the value of

[ = 2(0/)] and [ = 1.{im 0d0du> I , 278.] GENERAL SOLUTIONS AND EXAMPLES. 305 giving, when t = 0, <£ = 2.4 4 sin ^Ax — a ( ) The solution for i/r (§ 277) is of precisely the same form, the sole distinction being the substitution of Q' for Q. Tee Problem of Forced Vibrations under Periodic Surface Tractions, and Periodic Applied Forces deriv- able from a Potential. 279.] General Equations. The only Applied Forces with which we have to deal under natural conditions are forces of attraction and repulsion, whose components are always deriv- able by differentiation from a Force Potential. We shall therefore always assume, except when the contrary is expressly stated, that a function "$r exists such that X J& yJW^ z J3* ~dx' 'dy' ~dz at every point of the body. It may easily be verified that the components of the same force, referred to any system of curvi- linear coordinates, are, with the notation of Chapter V., s i, 3lp vr i, 31p 7 j, 31p IQA\ *=K W H = A 2 _, Z = A 3 _ (94) If the Applied Forces be strictly periodic, their potential must be of the form , *' = 2( 1 i r ,sins< + -* , ;cos««) (95) and on substitution in the general equations of motion (29) and (29a) of § 258, we see that the partial displacement-amplitudes of order i must satisfy dfdu^dvi 3w\ , 2 , /.„ M^A ' ~dx / 3/3«i , 3«j , 3wA , o , /-o c^A n m -dz 3 /dui , 'dVi dwi\ , , / «, 3*P A -, .(96) where ^ is to be supposed zero, unless the value of i coincides with any one of the values of s in the series (95). The boundary conditions will still be expressed by equations (56) of § 264. u 300 GENERAL SOLUTIONS AND EXAMPLES. [280. 280.] The forced vibrations constitute a pure strain. Omitting the suffix from equations (96), and writing them in the form h&& + ¥1 - 2fl' 2 R- 3 - ^2~| + i*u = tc L J ]_?>y 3zJ [02A + ¥] - 2I2' 2 r^i - ^s~| + i 2 t> = 5-[fl*A + •*■]- 20'2f^2 _ ^i~j + i* w = o 3 3. 3 3y |_ 3a; 3y_j we may eliminate their first terms by cross-differentiation, and we thus obtain n'V*. + ™, = oJ Now these are precisely the same equations that would be obtained by cross-differentiation of (65) in § 266 (ii). Hence we conclude that the rotational part of the vibrations is of the same form as if there were no Applied Forces. Or, in other words, the forced vibrations due to a system of Applied Forces having a potential are such as to produce dilatation and shear, and any distribution of rotations which may exist is due to superposed free vibrations independent of "¥. 281.] Dilatation and Shear. Expressing the strain com- ponents in terms of the displacement potential , equations (96) become d 3a 1 3 3 [QV^ + t^ + T,]^ ^[flV*« + *** + *J = o whence we deduce* GV^ + ^ + ^O- •07) * Since we have to deal only with the derivatives of our potentials (dis- placements, forces, etc.), they are always indeterminate to the extent of an additive constant. 281.] GENERAL SOLUTIONS AND EXAMPLES. 307 Thus, by (61) of § 124, oyoz ozox acoy and consequently : (i.) If ¥ satisfies V 2 *' = 0, (y«) which is equivalent to cte ~dy "dz the dilatation is independent of the form of -$r, and the forced vibrations are purely distortional. (ii.) If ¥ satisfies ^IS^SH <*» c^rc (Sot cacty which are equivalent to 3x_?x_3r_3r_?.zr_agr 3y cU; 3z 3d Bit 3y the shears are independent of the form of "$", and the forced vibrations are purely dilatational. 282.] Example. Radial Force. If the force at every point be in the direction of the radius from the origin to the point, we deduce from (94) that (with the notation of § 243) ^ is a function of r only, and dr' Thus a is symmetrical about the origin, and the forced vibra- tions will clearly be radial, so that also will be independent of 6,w. Thus (97) may be written or 30S GENERAL SOLUTIONS AND EXAMPLES. [282. We need only concern ourselves with the Particular Integral, as the Complementary Function gives free vibrations. Thus the symbolical solution of which gives <£. = ._icos— lr sin — ^Pyfr - sin-=- /rcos— ^ t dr \ . vr [ 11^/ 12 S2_y il J Hence corresponding to the force potential ■*• = 2(-*-, sin st + %' cos st) we have the displacement potential of forced vibrations 4> = 2— < cos — lr sin — {¥. sin st + 1 F,' cos st)dr sr \ Q.J ST Rt* f^ fit* I - sin _ lr cos— (^ sin st + "$?,' cos st)dr > . 283.] General Solution. Equation (97), satisfied by the partial component which represents the relation existing at each point of the body between the resultant displacement potential and the resultant force potential at the point. The most general solution of this equation, consistent with the assumed form (95) of ty, may be found as follows : — The function ¥ is finite and continuous in value (§§ 223-228) throughout the body, though not necessarily continuous in form. Let (x', y', z') represent the coordinates of any point within the body, and let X(x -x, y'-y,z'-z, t) represent any continuous function which never becomes infinite, except when x' - X = y — y = z — z = 0. Then if we assume + =///*&' V'> «0 • X • dx'dy'dz', 283.] GENERAL SOLUTIONS AND EXAMPLES. 309 the integral being taken throughout the volume of the body, we may apply Boussinesq's Theorem (see § 310, below) to the differ- entiation of 0, and write r 2v r K + ^{ x > y, z)l I {x( - n/k 2 - V 2 > V cos *»> V si" w > - x( v k2 - v\ V c °s *>, V s i n <"> t)}i}dt)dta ; where the double integral is ultimately to receive the limiting value which it assumes when k = 0. Now assume that x is of the form r where r = J(x - xf + (y' - yf + (z' - z) 2 . The double integral then becomes f f\{^, t)-F(K, t)}^^ = ttk{F(k, «)-F(k, t)} T Q; and consequently Applying the same theorem to the second differentiation of , we have ^rfff^ y> -0 • £[>, ■>] • <*** 2tt k The double integral is in this case 310 GENERAL SOLUTIONS AND EXAMPLES. [283. Thus, if we also differentiate (f> twice as to y, and as to z, and add the symmetrical results, we have ultimately V s * = /j7*(*'. ?/'. s')V 2 P; F ( r ' t)~\tedy'dz + 4*¥(x, y, *)[*£-*(*, 0-F<«, «)]■ Now let F <^ = 4^ sini H)' so that Then, by formula (65) of § 243, = ^-. . - sin %[ t - — 1. 4irQ 4 r V n / Also and *<«•*>- s^™ < ( - i> Thus, proceeding to the limit in which /c = 0, we have finally But and therefore O V<£ ~4> + *( a; . 2/. 2 ) sin *< = 0. Similarly, if we assume we find il\ 2 - + ^'(x, y, z) cos it = 0. 283. J GENERAL SOLUTIONS AND EXAMPLES. oil The complete solution of (100), corresponding to the force potential •*" = 2(*- j . sin it + ■>?/ . cos it), is therefore + ¥,'« ,/,z>).co S i(t-0}*!^; (101) and the partial components of , of order i, are hence easily shown to be « ^JIf{ ™ y '> z>) ■ C0S S- w * *> ■ ■*£ 1^, .(102) The triple integrals are in every case to be taken within the limits of the body. 284.] Return to the Preceding Problem. When the point (x, y, z) lies altogether outside the limits of integration : that is,, when x'-x, y — y, z' -z can never vanish, and conse- quently x can never become infinite : Boussinesq's formula reduces to _9 'UP -¥(a>, y, a ) . || . dx'dydz. Hence we easily deduce that if the triple integrals in formulae (101) and (102) be taken throughout any regions of space wholly external to the body, "$r, ¥' being, as before, finite and continuous functions of position, these formulae will repre- sent a general solution of the equations (90), (89) of irrotational free vibrations The student will find no difficulty in proving, by direct differentiation, that the integrals (101) and (102) do satisfy these equations! 312 GENERAL SOLUTIONS AND EXAMPLES. [285. The Problem of Equilibrium under Surface Tractions only. 285.] General Equations. When the body is in equi- librium in a state of strain maintained by surface tractions only, equations (6), (7), (8) take the simple forms 3x oy "dz ox "dy 3z 3T 'dx B5 + affi = "dy 'dz ?A , 2 ox 3A 2 n rn. 1- nrr'v = dy 3A „ m—- + nyno -- az j .).. (m + n) — ox \ &y (m + n) - - 2n[ —J — —-» I oy \ oz Ox f (m + n) 2n| — 2 J I = C V '-dz \dx -by) oa: .(103) (104) (105) The conditions to be satisfied over the bounding surface will take the form u- v ■- .(106) or the form \P + pU+vT = F\ kU+nQ + vS = G I (107) \T+pS + vR = B j according as the values of the surface displacements or of the surface tractions are given. 286] The Solution Determinate. We know from §255 that the problem of finding a solution of (103), (104), or (105), which will satisfy (107) over the whole surface, is quite deter- minate as regards the strain, and therefore also as regards the 286.] GENERAL SOLUTIONS AND EXAMPLES. 313 stress ; while the solution in terms of the displacements is only- indeterminate to the extent of an arbitrary translation and rotation of the body as a whole. The solution of (103), (104), or (105), which satisfies (106) at all points of the surface — or, indeed, which assigns given dis- placements to any three points in the body, or on its surface, which are not in the same straight line — is consequently abso- lutely unique. Thus, in seeking the solution of any given problem, we may avail ourselves with perfect confidence of • considerations of symmetry, and all other devices which may simplify the forms of the equations, knowing that from any solution which satisfies all the conditions of the problem all other possible solutions can be deduced — even in the most general and unrestricted case — by superposition of an arbitrary displacement of the body as a whole. 287.] Preservation of Continuity. Finally, we may observe that the necessity of preserving the continuity of the substance of the body imposes certain restrictions upon our choice of a solution — even when continuous* in form — by which it may gain in definiteness. For example, the radial displacement u of § 243 must vanish with r, if the origin be contained within the substance of the body ; while the displacement v of § 243 must vanish with sin0,-f- and the displacement u of § 244 with r, if any portion of Oz lies within the substance of the body. It is obvious that these precautions are necessary to guard against spherical, conical, and cylindrical ruptures, respectively. Example I. 288.] Circular Cylindrical Tube under uniform in- ternal and external normal pressures. A shell bounded by infinitely long coaxial circular cylinders, of radii A (internal) and B (external), is subjected to a uniform normal pressure II over the whole of its inner surface, and a uniform normal pressure IT over the whole of its outer surface. Eequired the distribution of strain. The symmetry of the conditions leads us to expect that the displacement of every point in the shell will be wholly radial : that is, in the direction of the straight line drawn from the point perpendicular to the axis ; and also that the magnitude and sign * See §§ 223-228 for the restrictions imposed upon discontinuous solutions, t See § 272 for an example. 314 GENERAL SOLUTIONS AND EXAMPLES. [288. of this displacement will be the same for all points situated on any circular cylindrical surface coaxial with the bounding surfaces of the tube. Taking the axis of the tube for the axis of z, and choosing arbitrarily the origin and axes of x and y, we will then assume, with the notation of § 244, that v=.w=0, and that u is inde- pendent of 6 and z. On this assumption we have A = - -,-(w) / r dr > i i Q^ = 2 = 3 = ' and, on substitution in equations (88) of § 244, Integrating this equation twice, c r where C, C are arbitrary constants. In this case both terms are admissible (§ 287), because Oz is not withinVthe substance of the body. ■ v s - At the inner surface we have r = A,^= -l,S' = n ; dr and at the outer surface r = £,f = i,s'=-_ir. dr Hence by equations (89) of § 244, P= -II, when r = A | But P= -LT', when ■ du „ C\ dr r 2 f= u - = C+%\ and therefore P=(m + n)(c-^ + { m -n){c + V) = 2mC-M.'. 288.] GENERAL SOLUTIONS AND EXAMPLES. 315 Thus the boundary conditions become A* ' and consequently c = Am-Bm' ^ 2m(B 2 -A 2 ) C ,_ A 2 B 2 (U-W) 2n{B 2 -A 2 ) ; Substituting for C and C, we have finally „_ (^n-ffn> i^ ( n-n-) If II — IT and A 2 H — B 2 H' be of opposite signs : that is, if -B 2 /4 2 >n/ir>i: u will be zero when n Bm'-Am (W ' In order however that u may vanish at any points within the substance of the tube, we must impose the further restriction that this value of r shall be between A and B. The necessary and sufficient conditions are mA 2 + nB 2 n ^ (m + n)^ . (tn + n)A 2 IT mB 2 + nA 2 ' and if these be fulfilled, the cylindrical surface described in the body with the above radius will retain its form and dimensions unaltered ; the inner and outer shells into which it divides the tube being compressed upon it from either side. If n_ (m + r^B 2 U' mW + nA 2 the inner surface of the tube retains its natural dimensions, and if II _ mA 2 + nB 2 Tl~ (m + n)A 2 the outer surface does so, 316 GENEEAL SOLUTIONS AND EXAMPLES. [289. 289.] Principal Stresses. Lines of Stress. It is obvious that equations (91) of § 244 are satisfied identically; so that r, 9, z are the principal coordinates of the strain. The principal stresses are by (55) of § 241 .(110) N = _ mV(r* - A*) + Am(W - r 2 ) 1_ (B"*-Aty „ (sm 1 - Am)r* - A*B 2 (n - n') 2_ (B*~A*)r* „ _ (m-y,)(Bm'-Am) 3 m(B*-A 2 ) The corresponding Lines of Stress (§ 216) are respectively — (1) those portions of the radii drawn perpendicular to the axis which are intercepted within the substance of the tube ; (2) circles in planes perpendicular to the axis, and having their centres in the axis ; (3) straight lines parallel to the axis. These three systems we shall refer to as the radial, circular, and longitudinal systems respectively. Since B>r>A, it is evident that iVj is always negative, and consequently all the radial stress lines are Struts (§ 216) throughout their length. The pressure transmitted by these lines increases or decreases continuously from the limit II at the inner surface to the limit II' at the outer surface. The stress JV S , transmitted along the longitudinal stress lines, is constant, and its sign depends only on that of B 2 U'—A 2 n. Thus these lines are Struts or Ties according as n <; B2 IT >A* In the limiting case, in which U_B 2 if" 15' these are lines of zero stress, and the stress, as well as the strain, is in two dimensions. Since dNJdr is negative, the third principal stress regarded as a pressure increases continuously with r. Thus, if M 2 is a pressure at the inner surface, it will be a pressure everywhere ; while, if it is a traction at the outer surface, it will be a traction everywhere. Hence we deduce that, if II 2B 2 If At + B? 289.] GENERAL SOLUTIONS AND EXAMPLES. 317 the circular lines of stress are Struts throughout, transmitting a pressure which increases with their radius. But, if n A* + W n (> 2A 2 ' these lines are Ties throughout the body, transmitting a traction which diminishes as their radius increases. Finally, if the pressure-ratio falls within these limits: that is, if A* + B* n 2&> 2A* W A 2 + £* the stress transmitted along the circular lines of stress will be a traction at the inner surface, and a pressure at the outer surface, vanishing and changing sign when / » ( n---n') . (in) -4 so that all the circular stress lines with this radius will be lines of zero stress. It may be observed that the limits for the existence of a cylinder of zero circular stress, fall within the limits (108), for the existence of a cylinder of zero radial displacement; the two cylinders do not however coincide, as will appear on comparing their radii, given by (111) and (109). 290.] Strength of the Tube. It would be interesting to discuss the various ways in which the perfect elasticity of the tube may be endangered, by approach of one or other of the principal stresses, at the point where it is greatest, to the elastic strength of the material under tension or compression. We must confine ourselves here however to a single example. Let n/IT>-B 2 /J. 2 Then the radial pressure has its maxi- mum value II when r = A, the circular traction has its maximum value (A* + ^n - iBm' B 2 -A 2 when r = A, and the longitudinal traction has the uniform value ( m - n )(Am-Bm') m(B>-A*) The second of these is the greatest, so that if the elastic strenoih of the material be about the same for tension and com- pression [Table (bis), p. 202], the first yielding of the tube will take the form of transverse stretching, or increase of its diameter beyond its power of elastic recovery. 318 GENERAL SOLUTIONS AND EXAMPLES. [290. If T be the elastic strength of the material under tension, the condition for elastic safety is B*-A* ' so that we have, as guides for the proper dimensions of the tube, when the pressures to which it is to be subjected are known, n > & > n + T ir a* 2n'-n + T 291.] Application to cylindrical boilers. The case which we have just considered — when the ratio II/II' is considerable, especially in comparison with WjA % — may be taken fairly to represent the strain suffered by a long cylindrical boiler (except in the neighbourhood of its ends). Thus, if T represents the working strength of the material [Table (D), p. 203] which allows for a large " factor of safety," the proper thickness t for a boiler of internal radius A, to be worked at steam pressure IT under atmospheric pressure 11' will, with due regard to economy of material, be given by (t + Af = n + T A* 2IT + T-ri' Example. It is required to determine the proper thickness for a cylindrical wrought iron boiler, 4 feet in diameter, to be worked at a maximum pressure of 120 pounds to the square inch in the open air. The working strength of wrought iron is given in Table (D) at 4 5 tons to the square inch, and the atmospheric pressure may be taken at about 15 pounds per square inch. Thus, reducing lengths to inches, and stresses to pounds per square inch, we have A = 24 n - 120 IT = 15 T = 10080, and consequently the thickness in inches is given by 120+10080 (« + 24) 2 =(24) 2 , 30 + 10080 - 120 •509. 291.] GENERAL SOLUTIONS AND EXAMPLES. 319 The employment of " half -inch plate " for the construction of such a hoiler will therefore allow an ample factor of safety, to guard against the danger of accidental rise of pressure. In fact, a boiler so constructed would not begin to give until the steam pressure had risen to over 500 pounds per square inch : always supposing that the portions near the ends were able to sustain as great a stress as the middle portion. Example II. 292.] Circular Cylindrical Shear. A body, bounded by two coaxial circular cylinders of infinite length, has its inner surface (radius A) rigidly attached to an immoveable cylinder of the same radius ; while its external surface (radius B) is subjected to a uniform tangential traction F, everywhere per- pendicular to the axis of the cylinder. Required the nature of the strain produced. In this example, as in the last, the conditions present com- plete symmetry about the axis, and complete uniformity in the direction of the axis. It is therefore natural to assume that the resultant displacement of each point is in the plane, per- pendicular to the axis, which contains the point, and that the amount of this displacement depends only on the distance of the point from the axis. Thus, with the notation of § 244, we shall assume that w = 0, and that u and v (and therefore also /3) are independent of 6 and z. We have then r dr e x =e 2 = o 2G S = !.£(«■) ■* r dr and, on substitution in (88) of § 244, (2A __ dQ 3 _ f. dr dr Integrating, we get u — Cr v = Dr + D'V The given boundary conditions are partly of the one type, and partly of the other [§ 285, (106), (107)] ; for when r=A, we are to have u = v = 0: and when r = B, P = 0, U=F. 320 GENERAL SOLUTIONS AND EXAMPLES. [292 The first two conditions give c J)' A A and on substitution from (85) of § 244 in (46) of § 239, the latter conditions become nT> 2nC n 2nD' ■& Thus C=C' = A*D= -D' = B 2 F/2n and, finally, u = and 2n\A 2 rV Each cylindrical surface in the body coaxial with the bounding surfaces is therefore simply rotated about the axis through an angle P~u\T* vy F ' where r is its radius, without any change in its form or dimen- sions. The amount of rotation increases from within outwards, and the strain amounts to a circular shearing motion (in planes perpendicular to the axis) of cylindrical layers of the body, without any changes of density. Each line in the body parallel to the axis is shifted as a whole, parallel to itself, while each radial line is distorted into a hyperbolic form. For instance, the radius of the shell which initially coincides with the axis of x assumes the curve B*F(x* A which is a hyperbola, having for its asymptotes the lines x = 0, y = B 2 Fx/2n. Since the strain is supposed small, P will be very small com- pared with n, and the hyperbolas will be nearly rectangular, as well as of very small curvature in the portion intercepted by the shell. In Figure 34, the dotted lines represent the above hyperbola and its asymptotes, the portion distinguished by an unbroken line being the strained form of the radius of the shell initially coinciding with Ox. This figure is drawn for an exaggerated case, in which P = "00523 n. 293.] GENERAL SOLUTIONS AND EXAMPLES. 321 293.] Lines of Stress. Since P = q = r = S=T = 0, and U = nr-f = — ^-, dr r l all lines in the body parallel to the axis are Lines of zero stress, and the two principal stresses in any plane perpendicular to the axis are the remaining roots of the discriminating cubic (21) of § 163, which here reduces to Thus 2^= -N 2 =U=WFh*. Fig\34 In the system of coordinates which we are now employing, the directions of the " axes of reference " of § 163, at each point of the body, are those of the elementary lines dr, rdB, dz : thus, if ds be an element of a Line of Stress, and X, fi, v the cosines of the angles which it makes with the coordinate elements, we have Xds = dr, fids-rd6, vds = dz, X 322 GENERAL SOLUTIONS AND EXAMPLES. [293. and the differential equations of the Line of Stress corresponding to the principal stress iV (see note on § 241, at end of the volume) are , and the Struts the similar spirals r = Ce' e , each system cutting all radii at the constant angle 7r/4, while the traction or pressure transmitted along each diminishes, as the inverse square of the distance from the pole, from B 2 F/A 2 at the inner surface to P at the outer surface of the shell. In Figure 35, the whole lines represent the Ties, and the dotted lines the Struts ; if these are studied in connection with the direction of the Surface Tractions (indicated by the arrows), the simultaneous dragging and squeezing effects of the latter will readily be understood. The traction exerted on the inner surface by the fixed cylin- drical core is equal to the value of U when r = A ; it is therefore A 2 This is otherwise obvious ; for, in order that equilibrium may be possible, the external couples on the body must balance one another, precisely as if it were rigid (§ 146). Thus, considering a unit length of the shell, we must have F'.A.2irA=F.B.2TrB, or F.4* = F..B 2 . Example III. 294.] Spherical Shell under internal and external normal pressures whose intensities vary directly as the 294.] GENERAL SOLUTIONS AND EXAMPLES. 323 distance of the point of application from a given dia- metral plane. A spherical shell suffers a normal pressure II cos over its inner surface (radius A), and a normal pressure IT cos over its outer surface (radius B); 6 being the angle which the radius vector of any point makes with the given diameter Oz, and II, II' being constants. Required the conditions of equilibrium, and the nature of the strain produced. Adopting the notation of § 243, it is evident that the condi- tions are symmetrical about Oz, so that the displacement of every point will take place in the plane which contains Oz and the point, and the strain will be altogether independent of to. Pig.35 The conditions of equilibrium (§ 146) are the same as for a rigid body : that is to say, the forces on the shell due to the two systems of surface traction must balance one another. From symmetry the resultant force due to the pressure on each surface Of the shell is parallel to Oz, and by resolving in that direction 324 GENERAL SOLUTIONS AND EXAMPLES. [294. the force on each element of surface we find the condition of equilibrium to be fi cos 6 . cos 6 . 27r.4 2 sin Odd -ft. cos 6 . cos 6 . 2jr.# 2 sin 6d6 or TLA* = n'B* (112) Assuming that this necessary condition is satisfied, we have by (68) and (69) of § 243 A = - 2 hn*) + -4-j, |> sin 6) r 2 or r sin (too r 3r while the general equations of equilibrium (71) become 1 3w r 3(9 , >3A 2n 3 , n . m n (to + «)— — . (0„8in O) = v ; 3r rsin0 30 V 3 ' (» + «)^ + 2nl(9,r) = j The boundary conditions (72) reduce to P= -Ilcos0, tf = 0, when r = A, P= -ITcostf, Z7 = 0, when r = B ; and on substitution from (68) these become (m - n) A + 2ra?^ + II cos 6 = o) or \.... when r = A \ (m - ra)A + 2n?^ + n' cos 6 = o\ or i and when r = 5 3 /«\ 1 3m , , i ' when r = A or B Finally, by § 287, we must have .(113) .(114) .(115) .(116) .(117) v = 0, when sin 9 = 0. .(118) 294.] GENERAL SOLUTIONS AND EXAMPLES. 325 Now equations (114) may be written (to + «Wn 6^ - 2nhe 3 r sin 6) = 0} or oO (m + »)sin 6^ + 2»^(e 3 r sin d) = 0) and on elimination of 9, we have 3r|_ ory 30[_ W J ' or r*dr\ -dr) r*sm.6o6\ 3d) Comparing this equation with (65) of § 243 we see that V 2 A = 0* (119) It also appears from the boundary conditions (115) and (116) that at either surface A must be equal to cos 6, multiplied by a constant factor. But cos 6 is a surface harmonic of order 1, and thus the solution of (119) must be the sum of two solid har- monics of orders + 1 and — 2. Let us assume A = (cr + ^\coa0; (120) then, on substitution in (114) we have and the solution of these equations is obviously „ a m + n/Cr D\ ■ a /■,m\ 2e ^-n-(^-^) Sm6 - (121) Again, substituting from (120) and (121) in (113), I - (ur*) + l „ J?Jv sin 6) = (Cr + ^cos 0\ r 2 dr K ' rsin0 30 V ' \ r^) \ 1 B r "dr h- K ' rW M \ 2 r?/ * This equation is satisfied by A in all cases in which there are no applied forces, as may be deduced directly from equations (104) above. See Article 295 below. 320 GENERAL SOLUTIONS AND EXAMPLES. [294. and the form of these equations, in connection with the boundary conditions (115), (116), (117) suggests that we should assume for the form of u and v u = u cos a, v -■ v sin 0. where u and v are functions of r only. The general equations to be satisfied by u and v then become 1 d, „. 2v „ B - _(ur 2 ) + _ = Cr + Id, v u m + n - ^-( vr ) + -= r dr r n These are easily put into the form (Cr_£>\ \ 2 r*) 2vr = CrU D- ?-(ur 2 ) ) dr dr n \ r I J and on elimination of 2vr, we have d 2 , n n 2n - m~ „ 2(m + n) D -}-.("»-) " 2u = Cr ' + — l ~ ' ar J n n r .(122) the complete integral of which is 10» 'CV 2 m + n D + c- D where (7 and D' are arbitrary constants. The first of equations (122) then gives at once 2m + n„ , m + 2n D .-,, D' v = Cr' + C . 10re 2n r 2^ Thus, finally, the radial and transverse displacements are —— —Cr* + C - — Icos 6 lOre n r r 3 _J __ — Cr 2 + — C - — -; sin 6 lOn 2n r 2r*_\ .(123) The four arbitrary constants are to be evaluated by means of the four boundary conditions (115), (116), (117); it being obvious that (118) is satisfied identically. 294.] GENERAL SOLUTIONS AND EXAMPLES. 327 Taking first (117) we have, when r = A, and when r—B, lOra r 3 r 5 Therefore 3kA=C - 10n(4 2 2> - 3D') = 1 MB* C - lOn^D - 3D') = J ' Next from (115) 3kA*C + 5(3m + n)AW + 30nD' + 5 A*U = ; and similarly from (116) 3kB*C + 5(3™ + n)B*D + ZOnlT + 5.8*11' = 0, where by (112) Am=nm: Thus c = iM?iz^Vi n 1 9A(«n-n)(5*-il*) Am .(124) 3(m + ?i) jy_ ' (B 3 -A 3 )A i B i n 9(m + ra)(.8 5 -^ s ) It will be observed that C does not appear in the boundary equations, and is consequently indeterminate. The reason is that the displacement whose components are u = C cos 0, v — —C sin amounts merely to a bodily translation of the shell through a distance C in the positive direction of Oz. This term conse- quently contributes nothing to the strain, and we may put C = 0. Substituting from (124) in (123), we have finally r^tt-mX-ff-^yilr 2 Am_ ( B s -A i )A 4 £m "I „ M "|_ 9i(m + »)(#- A") 3nr 9(m + n)(V-A , yJ CM r(2m + n)(B , -A*)A*Ilr' (m + 2n)A*n _ (B>-A*)A*mi ~] . fl H 125 ) V ~\_" 9k(m + n)(B !> -A*) 6n(m + n)r 18{m + n)(B* - A*yJ J To investigate the deformation suffered by the body: we bave m = u cos 6, v = v sin 6, where u, v are functions of r, of the order H/n, so that u 2 and v' 2 328 GENERAL SOLUTIONS AND EXAMPLES. [294. are negligible in comparison with x 2 , etc. If {x\ y', z') be the strained position of the point initially at (x, y, z) Jx"' + y 12 = J a? + if + u sin 6 + v cos 8 = Jo? + y* + (n + v) sin cos 6 z' = z + u cos 6 — v sin 6 = z + ucos 2 6— v sin a 5 = a + n - (u + v) sin 2 6. Now to a first approximation (see § 68) we may regard the coefficients of u and v as functions of the initial or final coordinates, indifferently. Thus we may write z Jx' 2 + y' 2 J.c- + y 2 = v/a' 2 + 2/' 2 -(ii + v) r- .'2 j_ ,,/2 z = z - u + (u + v)- — 1L and any spherical surface .2 i ,,,2 i „2 .- . a;-' + y i + z- in the unstrained body, concentric with the bounding surfaces, is strained into the surface (^ + ^){l-(u + v)i;} 2 + {/-u + (u + v)^±^!r 2 .r- or (approximately) into the sphere a/ 2 + 2/' 2 + (3'-u) 2 =r 2 . Every such sphere is therefore shifted, without change oiform, through a distance u in the positive direction of Oz. This dis- tance depends upon the radius of the sphere, being given by |- (2n -«.)(#■ -^»)r» 1 (B*- A*)A*B» ~] Am \_9k(m + n)(B b - A b ) Snr 9(m + n)(B 5 - A 5 )r 3 J It is easily shewn that u will or will not vanish for some value of r between A and B, according as the equation, 9(wi 2 + mn - n?)a* + 5ra 2 a, 3 - 3A(3m + 4w) = has or has not a real root between A/B and B/A. If there be an odd number of such roots, the bounding surfaces will be displaced in the same direction ; if an even number in opposite directions. Although, however, these spherical surfaces retain their form unaltered, yet their surfaces suffer areal dilatation or contraction (page 61), which varies from point to point so that the cones 6 = constant in the unstrained body become surfaces of revolution of the sixth degree. .(104) 295.] GENEBAL SOLUTIONS AND EXAMPLES. 329 Solution in terms of Spherical Harmonics* 295.] The Cubical Dilatation. The general equations are 3A „ . , m— + nvfiu = ox 3A « A nt— + reyfy =0 By )«— - + »iy a io = oz Differentiating these as to x, y, z respectively and adding, we obtain V 2 A = 0; (126) an equation which is satisfied in all cases of equilibrium under Surface Tractions only. Thus A is in this case always capable of being expanded in a series of Solid Spherical Harmonics. 296.] Application to Spherical Shell Let us suppose the values of A to be given at every point of the concentric surfaces of a spherical shell, of which the internal and external radii are A and B. These values must then be capable of expansion in series of Surface Harmonics, so that we shall have, in general : .([27) when r = A, A = J£(H<)'; j when r = B, A-2(HT). «— o Here Hi and H s ' denote any surface harmonics of order i ; and, A being always supposed small, these two series are necessarily convergent. At any point within the substance of the shell, at a distance r from the centre, the value of A will be given by A ^ (/^'H, - A^my - UBy»(A 3o>2 I I vW) = (H2)(H3)U, + .L. a 3 /ska t ' ^. ' ) sin 8 30\ W ) sm 2 3w 2 J 297.^ Thus GENERAL SOLUTIONS AND EXAMPLES. 331 V 2 (r 2 U.) = [(» + 2)(» + 3) - s(s + 1)]U. - 2(2* + 3)U„ and consequently VVU.-O = 2(2i+l)U J . 1 1 vV 2 U_ i _ 2 )=-2(2i+l)U_ i -J The solution of (129), and the corresponding equations for v and w. are consequently 2n ae2/ (2i+l)( J B 24+1 -^ ss+1 ) 2?i dy^- o (2* + 1)(£* +1 - A** 1 ) mr* d_ ■ ^(-g iM H i - 4 i+1 H i ')r i + (iitfy+^'H* - -ffH/)*- 1(130) (2i+l)(5 2i+1 -4 !!itl ) where the complementary functions u, v, w, are solutions of the equations v 2u =°. V 2v =0, v 2w =° ; which must be so adjusted that the expressions (130) for u, v, w may satisfy identically the equation 3w 3t> "dw _ » .(131) T 3x 3y 3z Now, if , be any homogeneous function of (x, y, z) of degree s, = rV) oo v =2( v i +v '-'-i) we must have (J"-'H ( -^ w H,y = _n(2i + 1) /3«, + , 3^ 3w i+1 \ #*+i _ 4*+i m i + TO (2i + 1)\ 3a; 3 ^ w, + w _,_, - — g- |ro(i_i) +ra (2i_i) ~ ^(i + 2) + m(2t+3) __ J ) where u, w'_,_, may be any solid spherical harmonics of their respective orders which satisfy the conditions of § 287, viz. : — ux + vy - = = 0, when x — y = ; ^jx' + y* ' y vy+wz - 7 _ = 0,whenj, = 2 =0; wz + ux -- = 0, when z = x = 0. Jz 2 + X 1 This complete solution is of course only adapted to a solid of finite extent which does not include the origin — such as the spherical shell with which we started. For a solid sphere with its centre at the origin we must retain only the harmonics of positive orders, and for an infinitely ex- tended body with such a sphere hollowed out in its substance only the harmonics of negative orders. 297.] GENERAL SOLUTIONS AND EXAMPLES. 333 Finally, we may add to the solution (132) for u, v, w comple- mentary terms of the form a 3<£ 3$ Ox 3v/ 3z respectively, where is any solution whatever of the equation v 2 4>=o that satisfies the conditions of § 287; for ohviously any such function will disappear on substitution in equations (104). The most general and complete forms of the solution, for a spherical shell with either the surface displacements or the surface tractions given over both its surfaces, will be found in Thomson and Tait's Natural Philosophy, §§ 736, 737. We shall confine ourselves here to the simpler case of the solid sphere. 298.] Complete Solution for a Solid Sphere with surface displacements given. Let A be the radius of the sphere, and let the component displacements at each point of its surface be given by «o = 2(H l ), "o = 2(H;), w = 2(H i "). Selecting from the general solution (132) the positive harmonic terms, and adding the arbitrary complementary functions, we have ox [_ ox _■ % L where as ] .(133) m 2[m(i-\) + n(2i-l)]\ Yt ~ l dx dy dz > (134) and satisfies vV = 0- Thus we may legitimately assume and we shall then have at the surface of the sphere (r=A) 334 GENERAL SOLUTIONS AND EXAMPLES. [298. Thus all the conditions of the problem are satisfied by making -=($)'* - ,-6)k-) and consequently and on substituting these values in the general formulae we finally obtain the complete solution ^|(,)W + wi(.4 2 - r 2 )— > t ^v^i], m(A 2 — r 2 ) — m(»-l) + «,(2*-l)]4 < v /rV n „ ^ 'dz ;m(t-l) + ro(2»-l)].4' (135) m(»-l) + n(2i-l)]^' 299.] Complete solution for a Solid Sphere with surface tractions given. The components F, G, H, parallel to the coordinate axes, of the stress across any concentric spherical surface of radius r are by (107) F= x -P + Uu + -T, etc. Thus Fr =*[<"-"> 4+2 "s>»i be any solid harmonic of order i, and Si the corresponding surface harmonic. Then & = ^S„ and the twin solid harmonic is Thus or ox 3 *-'-' = - (t + nar-'-'S, + r- j -'P P OS! OX ^ or dx ox \ _,w?t!=.i + (i + i) XP r-r-»^ I' 336 GENERAL SOLUTIONS AND EXAMPLES. [299. and consequently ^ = 2^i[^-^t] < U0 > Applying this result to (139), and bearing in mind (134), £ +1 = ^{[1 - (i - 1 )(2i + l)*/,>V<-i - &+>}, Ji + i where *-'"'{l,(^) + IW + l(^)} < U1 > Differentiating, and again making use of (140), ■ l SxV^-yJ dx ' "" 2»- 1 L ^ 2i + (H2) 1 3*+, 2t+l Sc ' Substituting from (138) and (142) in 136, and once more availing ourselves of (140) we obtain r/ , 1 _ I =«<*-i)ru,-jr^M 2*-l L St 'dx\r a >- 1 /J „[1 - (i - 1)(M + l)y,] r .3^,-, _ 2r*+' ^M-Al 2i-l L 3x 2t + 1 BxVr 2 '- 1 / J 2i + 1 3.K and finally ^, 1 = .{ (i-l)v ( -2 (i -2 W ,^-^|(^)-^ ^ } -/,,_,-«{ (i -lK-2 ( i- W^-M^y*^-^- %} where /■ ^ m(t + 2)-w(2i-l) , . . '' (2i+l)[m(i-l) + rc(2i-l)] v ' Now let the radius of the sphere be A, and let the components of the surface traction be given at every point of the surface by ^ = 2(H<), £ = 2(H0, #=2(H/') (145) (14 3) •299. GENERAL SOLUTIONS AND EXAMPLES. The expressions in (143) consist of solid harmonics depending on surface harmonics of the orders i and i - 2 respectively. Thus, picking out the terms which involve surface harmonics of order i only, we must have % i (i - 1)«, - 2ur t ^&ba - B^lftizi) _ ] **>+> 1 A I V ' ' + " ftr. ' arV*" 1 / 2; + 1 3.r I etc., when i* = .4 . Thus (;-i 3.r: ' dx\r"~ 1 / 'li + 1 3.r n..4''~' 2^1 1 i _ £,.«+! 3 /f*-A _ i ? 1/ 4 s'Tif - F r *'+ l — I il~A _ l ?^«+" = !^M f ' ' ,+= " fy *' di/\r'-'J 2i+l 3y «4'-' (t-i)w,-2f.y <+ .^^ , + | . 3z Differentiating these as to a;, ?/, 2, respectively, and adding the results, + £(< J H,")] (147) Also substituting for u„ v„ w, in (141) their values as given by (146), 2t* I+1 + 2t(» + l)(2i+ l)XW i+! ^, = ^ri/HA + 3/H/\ + 3/HA-| (U8) Thus we obtain i//- from (3 47), and then from (148), and lastly u, v, w from (146); and we have only to substitute the values so obtained in the general solution --{"-"■tr] :i33) If we write X,_, = | (r'H,) + J^H/) + 1 (rBD ) 3a; oy o* ' (149) 338 GENERAL SOLUTIONS ANT) EXAMPLES. [299. the final form of the solution is u = l 2 L_ f,. H+ -_ 1 — 3 *iH « (i-l)/l'-'{ ' 2i(2i+l) 3.c ^•-i)TO(^ 2 -r 2 ) ax^, + 2[(ir 2 + 1 )«» - (2i - 1 )n] 3.« [(f + 2)w-(2t-l)n]^' 3AVAI , (2i+l)[(2i*+l)m-(2t-l)«] aMS=ff,fd!S ~ff?dS = %kA\ equations (152) will be completely satisfied if only M" = N', N=L", L'=M; that is if r ox x r dy ■ r 6z where U 2 is any solid harmonic of degree 2. 339 .(152a) General Solutions. 301.J Application of Sir William Thomson's Method to express the component displacements in the form of potentials. We have already seen (§ 295) that, in all cases of the present problem, the cubical dilatation satisfies the equation V*A = 0. .(126) Again, by successive differentiations of equations (105), we deduce V 1 ~dx\dx + dy + -dz) V 2 d,/\dx dy dz) ' ia 3/30, 30., 30A but it follows at once from equations (1) of § 253 that 3* 1 + 3fl ! + 30, = o (153) ?.C 01/ o.~ so that we also have V 20 1 = V 20 2 = V !!0 3 = O (126 a) 340 GENERAL SOLUTIONS AND EXAMPLES. [301. If now we resolve the strain, after the method of §§ 265, 266, 275, 277, into its dilatational and rotational elements, writing dx dy dz dy dz dx 'dz dr. dy and assuming that \/<-,, t/<\,, i^ 3 satisfy the condition .(154) ^1 + ^2 + ^ = 0, (155 ) ox dy dz we have, on differentiation, A = v ^ 2*i = 2e,= ■vVJ ,(156) Hence it follows from (126) and (126 a) that V^ = vVi = V 4 ^ = vV., = (1ST) Equations (156) are satisfied by the assumptions 4>= — — /// — dx'dii'dz ^JJJ r ^kjffl 6 r dx ' dy ' ,h ' where the notation is similar to that of § 266 ; and since by equation (216) of § 311, below .(156 C.r 3 '/'2 + 3f 3 = l ffffi e < + ™2 ^ V0 3 yxdy'y 3S 3s = = 3* 3y 3i? 3s~ = .(159) are satisfied identically by the assumptions /' = ^*i + 32 X» ciy- 3z 2 3s 2 3z 2 3x 2 3y 2 ~dydz T= - ^ 3z3a; 3 2 v ' 3x3// J The functions Xi> X2- X3 mav ^ e continuous or discontinuous in form, but their second derivatives must of course satisfy the conditions (63) of § 226, as well as (126) of § 295. This method of solution is particularly useful in cases of stress in one plane (§§ 175-184), as everything is then made to depend upon a single arbitrary function. Taking the plane of xy so as to coincide with the plane of the stress, the solution is in this case dy- Q-= ._c?X Stc 2 .(ICO) 3x3y Illustrations will be found in §§ 307-309 below. The method is originally due to Sir G. B. Airy* who arrived at it in a much less direct way, by an application of the Calculus of Variations. * Report of live British Association; Cambridge, 1862: p. 82. 342 GENERAL SOLUTIONS AND EXAMPLES. [303. The Problem of Equilibrium under a Conservative System of Applied Forces, with or without Surface Tra ctio.ys. 303.] General Equations. For reasons stated in § 279, we shall confine ourselves to the consideration of bodies influ- enced by such systems of Applied Forces as are derivable by differentiation from a Potential. Let M' denote the Force Potential, as in § 279, so that the component forces per unit mass on any element of the body are given as before by equations (93) and (94). The equations (6), (7), (8) of equilibrium then take the forms ?.■■ 3// ds 3£ r + a[fr + ,A»'i + a.v_ ~d& dj Hz 'djc 'bij "dz -pi ~-[»»A + fP¥] + ny 2 M = tf -[m.A + pV] + >iy°-v=0 ■ .(161) ■p. ■jAmL + pV] + ny'% = .(162) ■dx 1 (163) -[(m + n)A + pV] - 2n/ 3 _3 - 3 1A =, G\ k - \3.y 3z/ |-[(m + n)A + /,*] - 2n^i - ^A = oj/ \3.s 3a;/ 3 > + »>* + ^-3«(S-f)-0, and from the latter set we easily deduce V 2 L(m + n)A + / ^] = (164) Similarly, the general equations (48) of § 239 become £<" "> 4 ♦<*] " -[*• 4© ^ "■ I,©] - «• £<" + " ,a+ '' ! ']- 3 "["'l,(!, , )-".|,(S)]=» -<■«> 3 3». 303.] GENERAL SOLUTIONS AND EXAMPLES. 343 The boundary conditions will be given, as before, by (10G) or (107) ; and the considerations of §§ 286, 287 apply equally to this problem. The simplest method of proceeding to solve this problem is, in general, to obtain the particular integrals of the above equa- tions, depending upon the form of ¥, and then to add comple- mentary functions satisfying the conditions of the last problem, and so chosen that the complete solutions may satisfy the boundary conditions (106) or (107). Case in which ¥ can be expressed in a series of Solid Spherical Harmonics. 304.] Solution of the General Equations. Let the Force Potential be expanded in the series of solid harmonics ¥ = 2^);. .(16b) then y 2 '"J r = 0, and we at once deduce from (164) that V 2 A = 0. Thus A must also be capable of expansion in a series of solid harmonics : let this series be represented by A = 2(A,) (167) These series may be supposed in general to include all values of i, positive and negative. Substituting in (162), they become v s «=--Sl 1 (»A + P*.) , | n ox V 2 a ,= _i2|-(»A + P^ i ) n oz (168) and, on solving these equations as we did (129) of § 297, we obtain U= U - —2, 2n '2i + 1 ~dx 3 ( TO A i + P * i )' 2n ii+l 3y r* v 1 w = w - — - ~ In 2i + 1 cte (mAi + p^) .(169) }44 GENERAL SOLUTIONS AND EXAMPLES. [304. where u, v, w are solutions of y-u = 0, y-v = 0, y^v - 0. hi order that (131) may be satisfied we must have ox oy Oz n Ji+l and on picking out all the terms corresponding to the harmonics of order i in the solution (109) _ ^- I )"(§-; + ^ + ^-)-^- 1 )^--' '■' )»t(t-i)+>i(it-i) Substituting in (169) and adopting the abridged notation of (134) § 298, we have finally (170) The complementary functions in (170) are of course the complete solutions (133)* of the problem of § 297. The par- ticular integrals given by this method of solution are , = - '"- m in \ "by ) m , \ Oz J .(171) We may however adopt another mode of solution.-f and obtain particular integrals of a different form, as follow : — Assuming that bx . °H , w t = _0 t - bz * The arbitrary complements of (133) appear as the complementary solutions of the equations obtained below for , by the second method. t See Thomson and Tait's Natural Philoiophy, Articles 733, 834. 304.] GENE11AL SOLUTIONS AND EXAMPLES. 34£ and substituting in the general equations (168), we obtain ■v-[(m + J»)y- i+1 + />*,_,] = and the particular integral of these equations is obviously r ' +1 2(m + n) 2t+l" giving for the corresponding particular integrals of (168) „=_ /» y?^Y i(«t + n)^ac\2t'+l/ v =- -±- .v^^-iV ■2(m + n)—'dz\2i+l) .(172) Thus it is clear that both of the particular integrals, (171) and (172), are partial ; and in fact, if we assume the particular integrals to be of the form ox ox (173) etc., etc. J and substitute in (168), we obtain the single relation mC i /M i +2(2i + l){m + n)C i ' = P (174) between d and C'/, so that one of these constants is altogether arbitrary so long as we are concerned only with the general equations The former solution (171) satisfies (174) by making C^pMJm, C/ = 0; and the solution (172) also satisfies (174) by making 00, C7 = p/2<2i + l )(#» + «). 346 GENERAL SOLUTIONS AND EXAMPLES. [305 305] Complete Solution for a Solid Sphere with surface displacements given. The complete harmonic solu- tion in its most general form is " 2.P'' + %' - '■% W;-. + t' t %.,)-C:~(r-%-S] - i -'\_ OX OX OX _J — 1_ diy % djj J w = ^ W; + d J^±l - r'-'l(/)/^,_, + CIV,) - being any solid harmonics which allow u, v, w to satisfy the conditions of § 287 (see § 297, page 332). This form of the solution corresponds to (133) of § 298. In applying it to the case of a solid sphere we must of course assume that the series (175) include only positive values of i. Let A be the radius of the sphere, and let the components of the surface displacements be given by m, = 2(H,), r„ = 2(H.), itf = 2(H,"). Then we must have, when r = A, u, + _*+■ - r£(i. ,*,_, + C,*,.,) - dhw^) = H„ etc. ; ox ox ox ur, differentiating the term involving C, and making use of (140) above, u, + ?*±' - M/&t±} = (2i-l)g. + (2*+l)g. rg 3y.., 3x ~dx 2i - 1 cte . 20,' __, +1 3 /¥,_,> 2i — 1 3aA The solution is obviously (compare that of § 298) determined by and yjs in (175), and perform- ing the differentiations indicated in the last terms of those expressions, we have finally « = 2[©'H, + (— <(%<-)] Thus the arbitrary constant C" disappears from the complete solution for the displacement : or, in other words, it is indifferent whether we take for the particular integrals of our general equations the form (171), or the form (172), or any combination of these which satisfies the condition (174). This is an excellent illustration of the statement made in §§ 255 and 286 — that the solution is absolutely determinate when the surface displacements 306.] Complete Solution for a Solid Sphere with surface tractions given. Let A be the radius of the sphere, and let the components of the surface traction parallel to the coordinate axes Ox, Oy, Oz be given by ^^(H,), -i)py ,.,^,3/y.A (1M) 2»-l M^ -1 / with symmetrical expressions for S,' and S,". Substituting from (183) in (181), Y , v , 2i(i- l)C2i + l)p M i A<-^ X ,., = X,._, + 2^- 1 v,_, where * and X are given by (149). ...(184) :r>0 GENERAL SOLUTIONS AND EXAMPLES. [306 Compounding (179) and (182), and substituting from (183) and (184), we have finally for the complete solution* ! = ] y- 1 $ r'H, + , n^(i-\)A'- 1 i"'" l '' r 2t(2i+l) dc (i-l )m(A* - r 2 ) 3X,_, ^ [(i + 2)7»-(2t-l)ra|r" +1 2[(2i 2 + l)/«-(2i-l)n] dx (2i + 1 )[(2i 2 + 1 )m - (2i - 1 )n] dx i— I f (i + l)m - 7i j 2 ^Vi (f^)l '*y m-(2i- \)n { 2(i-2) i(2i + l)m 2 3.c 2(2/- l)[(i-l)m + (2i- 3^^_ w ,- 8p„\l (185 . l)n] 3.r 2/-1 ar^r 2 '- 1 / ) V -4/tjf',s' Method. 307.] Airy's Solution for the components of Strain and Stress. It has been shown in § 302 that the general equations (103) admit of a very simple and general solution (159) for the stress components. On comparing (161) with (103) it is at once evident that the same form of solution is applicable, the only changes required being the substitution of P + pty, Q + pty, R + p^ior P, Q, R. Thus, corresponding to the solution (159) we have the more general form 3,y 2 a-; 2 -fl^ dx 1 3,t 2 -t& 3a; 2 3/y 2 -P* dijilz ~dzd,r. 3.u3// / .(180) The principal application of this method is, as already stated in S 302, to cases of Plane Stress. For example, take the case of a body in the form of a rectangular parallelepiped of any pro- portions, placed with its three pairs of opposite faces parallel * For the conditions of equilibrium in this problem, see Example 20, at the end of this Chapter. 307.] GENERAL SOLUTIONS AND EXAMPLES. 351 respectively to the three coordinate planes. Let this body be free from surface tractions over the pair of faces perpendicular to Oz, and let it be acted upon by impressed forces and by surface tractions over the remaining two pairs of faces, everywhere per- pendicular to Oz, and in magnitude independent of z. Then the stress-components R, S, T will be independent of z, and zero over every face of the body: and consequently they must be zero throughout. The force-potential '\t r will also be independent of z, and so therefore will the remaining stress component* P,Q, U. Thus the stress at every point of the body is wholly in the plane perpendicular to Oz (§§ 175-184), and the solution (186) reduces to the simple form 3y2 .(187) 'dxdy where x is a function of x and y, to be so chosen as to give P, Q, and IT their proper values over the bounding surfaces. Two of the examples considered by Sir G. B. Airy in his original paper* will be investigated in the following articles : the remainder will be found amongst the Examples at the end of this Chapter. 308.] Case of a heavy rectangular beam, with one end clamped to a vertical wall and the other end free ; the faces of the beam being horizontal and vertical. Let L be the length of the beam (horizontal), B its breadth (horizontal), and D its depth (vertical). Take the origin at the centre of the fixed end, Ox in the direction of the length (axis of the beam), and Oy vertically downwards. Then ty = gy, and equations (187) become P = 3 1* gPV a^- gpy -' 'dxdy * Report of the British Association ; Cambridge, 1862 : p. 82. 352 GEXEKAL SOLUTIONS AND EXAMPLES. If we make these equations take the more manageable form ox- [30S. .(188) U= - &+ 3.c3// .(189) Ujt [/■ T^r Fig.36 Since the end x = is the only portion of the surface in contact with solid matter, the surface tractions must vanish over all the other faces. Thus and 7>=0, U=0, wbena: = Z; Q = 0, U=0, when y= ±\D. Also, since the whole weight of the beam is supported by the integral tangential stress over the fixed end, B I Udy = gpBDL, when a: = 0. -in Substituting for P, Q and U their values in terms of yfr, the surface conditions become P^lo.g-t.^o.l for all valnea of y J'" (190) 308.] GENERAL SOLUTIONS AND EXAMPLES. 353 :*/>]= ±i»»0,|^[.y=±i0]-O,) for all values of x J .(191) §£[* = 0,j,= -JZ>]-g[ K = 0,y= + ±D]=g P DL (192) From (190) it appeal's that both \}s and dyfr/dx vanish when x = L; and from (191) that 3 2 i/r/3a^ is independent of x and changes sign with y, while dxf^/dy vanishes when y = ± JZ>. Hence we infer that i/r only involves x in the form of the factor (L— x) 2 , that it contains only odd powers of y, and that (iD 2 — y 2 ) is a factor of d\p-/dy. From these data we deduce without difficulty that i/r is necessarily of the form ^ V L 4.1 4.3 4.5 ...] where C v (7 S , C' s , ... are constants, to be determined by the remaining conditions. These are (192), and the first condition of (191), and it will be found on substitution that both give rise to the same equation : namely — Cy + MM* - 4C i) + A<-ZW» - 4C 3 ) + .. . = SgpiD*. Since only one of the arbitrary constants is determinate, we will adopt the simplest hypothesis and assume We shall then have = 0. *-#*-*tf?-9 and giving on substitution in (189) .(193) 2)2- 2gP. K?- y2 ) U^{L x) (?"') (194) and it will be found on trial that these values satisfy all the imposed conditions. 354 GENERAL SOLUTIONS AND EXAMPLES. 1308. Figure 37 is reproduced from Airy's sketch of the Lines of Stress. Their equations are not integrable in finite terms, but this approximation was arrived at by determining the directions [by means of § 129 (H4)] of the lines of stress passing through each of a great number of points, and joining up the elementary ares so obtained. The obliquity of many of the intersections proves that the Lines of Stress are not very accurately represented, but a good idea is doubtless given of their general tendency. The same remarks apply to Figures 39, 40 and 41, which illustrate examples 29, 30 and 31 at the end of the Chapter. In all cases the whole curves represent the Ties and the dotted curves the Struts. 309.] A rectangular beam (placed as in the last ex- ample) is supported at both ends, but not clamped— so that no couple acts upon it at either end : while a given load is uniformly distributed over a certain portion of its length. Take the axes of reference as in the last example ; Fig.38. let the total load be W, and let it be distributed uniformly over the upper face of the beam from x = A to x=C. It is obvious that the normal component Q of surface traction over the upper face of the beam will be a discontinuous function of x, and that the discontinuities of value will occur at the lines x = A a.ndx=C. [Q = from x = to x = A ; Q=-W/B(C-A) s eg 309.] GENERAL SOLUTIONS AND EXAMPLES. 355 from x=A to x=C; Q = Q from x=G to x = L\. The principles of § 228 therefore lead us to assume that \fr will be a discontinuous function of x, the discontinuities of form and value occurring at the planes x = A and x=G, subject to the conditions of stress continuity (63) of § 226. Let us then assume that if = ip v from x = to x = A ; \p = i/'j, from ,r = A to x = C ; ^ = \f/ s , from x = C to x = L. If Pj, Qj, 1^ ; P 2 , Q 2 , f7 2 ; P 3 , Q 3 , U s be the stress components in the three regions into which we thus divide the beam, as deduced from \jr v \^- 2 , \fr 3 respectively by means of equations (189), the conditions of stress continuity require that P 2 -P 1 = U i -U 1 = 0, when x = A;) P 2 -P s =U 2 -U s = 0, when x=C. I The conditions that there may be no couples on the ends of the beam are yrhD yPfdy - 0, when x = A -i-B >- /i/P s dy = 0, when x = L > -ID Since there is no tangential stress on any portion of the beam's surface except its ends, Cj = t T g=ET 8 = 0, when2/=+i». Since there is no normal stress on the sides of the beam within the first and third regions, #, = #3 = 0, when y=±iD; and similarly for the lower surface of the loaded region (? 2 =0, wl!Pll ? /= +2-Z). The integral normal pressure on the surface of the loaded reoion is of course equal to the weight of the load, and since this is uniformly distributed B(C-A)Q 2 =-W. Finally, the integral tangential stresses, reckoned upwards, over the two ends support between them the total weight of the beam and of the load ; so that B/u x dy[x = 0] - bJI'Mx = L] = gpBDL + W. -4.0 -i" 356 GENERAL SOLUTIONS AND EXAMPLES. [309. Substituting for the stress components their values in terms of y]s v i/r 2 , i/'j, these various conditions become ? 2 (+2 - *l) = ^7.(^1 - W = °. Whe ° X = A dy ~dxdy ^-(f.-W-O.wl.en.-C. (195) (196) ' + ^[* = 0,y=-J2>] = (197) |Z){|^[.r = X, y =12)] + ^ = Z, y= -|Z>] } -^[x = £, y = *2>] + ^ 3 [x = X, y = -$Z>]=0 (198) (199) ^ii = ^«=^*=0,wh e iiy=±J2) 3a;3y dxdy 'dxdy | ¥ i=S=±4gPA^eny=±^ 2-^-JgpA wheny = Jfl ^i?= -Agp.D- ^ , wben y= -42) dx* jBP B(C-A)' J f 3 ^[.r = i, y = *Z>] - ^[x = L,y=- h D]-^[x = 0,y = |Z>] ■d^ w .(200) .(201) .(202) .(203) ^[x=0,y= -lD] = g P DL+ lt From (199) it appears that dyjsjdy, d^r^jdy, and d\/f 3 /dy must all contain the factor (\D 2 — y 2 ), and from (200) that t£ x and \fs s cannot involve even powers of y. From (200), (201) and (202) we deduce that d^-yfrjdx 2 , dPyfsJdx 2 , atyjdx 2 must all be independent of x. We shall satisfy (qualitatively) all these conditions, and at the same time (197) and (198), if we assume n /j2 t x-L)<* + 8)^-^ where a, /3, y, <5, X, yu are constants, to be determined from the remaining conditions. 309.] GENERAL SOLUTIONS AND EXAMPLES. 357 If we write for brevity = W/gpBD(C-A), (204) we must, in order to satisfy (201) and (202) quantitatively, make fi=l-U\fD s = 12X/DU 261+1. Hence we find \= -0Z> 3 /12, fi^l + 6; and consequently + t = %l* + fr + y)[0 + V(&-fy-!fQ (205) Substitution in (203) gives us 8-a = L + 26(C-A), (20G) while (195) and (196) become , +fj ^ A(A+a) _ 2A+a _ (0-L)(C + 8) _2C-L + 8 A^ + Afl + y 2A+/3 C 3 + C/3 + y 2C + /3 -^'l The latter group is solved without difficulty, and gives a= -L-6(C -A)(2L -C- A)i'L P= - [Z 2 + 6(2CL - C a + A2)]/(l + 6)L y = 6U*/(1 + 61) 8=6(0*- A*)/L and it will be found on trial that these values satisfy (206) identically.* All the constants are determined, and the complete solution is expressed by *-^*-*.- I v*-v- fl *T '»'»(g , -fl-%] ■■<*»> where is defined by (204). The student should calculate the values of the stress com- ponents in the three regions, by means of equations (189), and convince himself that all the boundary conditions are satisfied — especially the condition of continuity of P and U throughout the body. *This identity simply expresses the fact that the beam passes on the weight of the load, unaltered, to its supports. 358 GENERAL SOLUTIONS AND EXAMPLES. [307 bis. 307 bis.] Important Addition and Correction. The solutions of the problems suggested in the last two Articles were given — as has already been stated — on the authority of a paper by the late Astronomer Royal, published in a Report of the British Association. I now observe, however — when the printing of the Articles and engraving of the Figures is already completed — that they cannot be accepted as true solutions, inasmuch as they do not satisfy the general equation (164) of § 303. It is perhaps as well that they should be preserved as a warning to the student against the insidious and comparatively rare error of choosiDg a solution which satisfies completely all the boundary conditions, without satisfying the fundamental conditions of strain, and which is therefore of course not a solution at all. The indeter- miuatcness of the C constants in Article 308 should have served as a timely warning, by its inconsistency with the general Theorem of Article 255. As for the diagrams of the Lines of Stress, they are only given as approximations, and a little consideration will convince the student— especially when he has mastered Chapter VII. — that they do represent the general character of the distribution of Tension and Thrust. The remainder of this Article is to be considered as a continuation of S 307. The functions Xi> Xz< Xs> * u terms of which we have expressed the six components of strain, are not wholly arbitrary, nor in general wholly independent. The six strain components being obtained by differentiation from three independent displacements, certain relations must exist between their derivatives in order to ensure the possibility of re -integration. From equations (1) of § 253, combined with (153) of § 301, we easily deduce 2 30 1 = 3& dc a*' 3y "da dy 2^J = 2^-?- a - (A) ■dz ■dz' with similar formulae for the derivatives of 9 2 and 6 S : these may be verified by substitution, and are identities. On eliminating 8 V 6. 2 , 8 3 by cross differentiation, in all possible ways, between these nine differential equations of the first order, we obtain the following six of the second order 3--V ay _ 3 2 a ay* a* 2 d-ydz 3-e afy m dz 2 7)x l ~dzc>x ay av 3 2 c a.t 2 ~by' 1 chcdy 2/3% + avv = Xd.i/'dz ?)x i ) _ 3/3a 34 k 3.c\3a; 3y ~bz) 2 /ay + w\ \dzdx 3y 2 / 3 /3a 36 + 3c\ 3y\3cc !>y dz) \dxdy 3zV " _ 3/3a + 36 3c\ a«\3a: "dy 'dz) •(B) 307 bis.] GENERAL SOLUTIONS AND EXAMPLES. 359 These six relations must be satisfied identically by every system of values that can legitimately be assumed for the six component strains, and every system of functions which satisfies these and equations (161) will also satisfy (164) and all other equations deducible from (161) by differentiation. Substituting from (186) in (B), we have for the six fundamental relations between Airy's functions "by- ^at* - (i + »>v*xj + §[*-(! + ")v 2 x 2 j = o g 2 [* - (l + c) v Vl + |k* - (1 + "VxJ = ° g[* - (1 i v) V %] + |J[* - + + y*)]+£, (K) where the complementary function £ is arcy solution of i*?-' <«> 360 GENERAL SOLUTIONS AND EXAMPLES. [310. Boussinjssq' s Potential Solutions 310.] Boussinesq's Theorem on the Differentiation of Potential Functions. The ordinary "gravitation potential" of any distribution M of matter, at a given point P {x, y, z) is dM f- where r is the distance from P of the element dM of the Potentiating Matter.* The same nomenclature may be con- veniently extended to the purely analytical function MP where r = J{x - x) 2 + ( V> Z ')X(*' ~ x > V' -y,z'~ z)dx'dy'dz', (210) integrated throughout space, for all values of x, y, z : — (i.) in the case where the continuous function x can on 'y become infinite at one point in space, namely, where x' = x, y' = y, z'^z; (ii.) in the case where the continuous function x becomes infinite for all those points where x' = x, y' = y, whatever be the value of z'. First Case (applicable to I, D and their derivatives). Bous- sinesq's investigation rests on the following principle : — The values of the integral, and of all its derivatives, wiU be unaltered, if we suppose the limits of integration to include all space except that enclosed by a small sphere with its centre at P, and an elementary radius k; the value of k being reduced to zero after evaluation. If we accept this principle and remember that 34> , denotes simply the increase in the value of 3?, produced by the translation of the point P through an elementary distance dx in the positive direction of Ox, it is clear that (P being supposed always to carry with it its little enveloping sphere) this increase will consist of two distinct portions : — 362 GENEKAL SOLUTIONS AND EXAMPLES. [310. First, the gain due to the displacement of the sphere, which has left behind it, on the negative side of P, a small volume now to be included within the limits of integration, and has taken from these limits an equal volume on the positive side of P. (See Fig. 42.) Secondly, the increase, due to the displacement of P, in the value of that portion of the integral which is taken throughout all space not included within the sphere in either of its positions. Fig.42. Figure 42 represents the translation of the sphere, and shows clearly the loss and gain of equal volumes by outside space. To find the net gain of the integral * ] • xE - "A* -(y -yf- (z' - zf, y - y, 2' - z] + -Jx"-ly-yf-(z'-z) 2 , y', 2'l-x[+ J^-iy'-yf-iz'-z) 2 , y'-y, z'-z]}dardx due tu this cylinder will be very nearly \f[.c- Jk* -f[. The whole gain due to the shifting of the sphere is found by integrating this expression as to a, over the whole area of the circle in which the two spherical surfaces intersect. If we write y = y + r\ cos w z = z + 1) sin U + V cos , .: + i) sin ui] . x[ + Vk 2 — i? 2 , '/ cos o', >/ sin »/>(), and k is ultimately to be made zero, this may- be written $(x, y, z)dvj /{x[- six 1 -if, ij cos to, >j sin w] II ~ x[ +• v K " ~ v 2 > v cos ""i 'j s '" <"] } 'iJydv- Again, the gain in that portion of the integral which is taken throughout all space excluded by the sphere in both positions is of course ^g- ///'P( x '> y'< z ')x( x ' - x > y -y, * - z)dz'dy\h' or d.c I J I \j/ ILdxdyds the limits of integration excluding both spheres. Adding together these two terms, and proceeding to the limit, we have finally* =//H{x, y', z ')^Xk-»> ~ *> y' - y> *' - z)dx'dy'ilz + #*-■» V' *)/ /{x[ - o/* 3 - '/ 2 . v cos <°, i sin <"] - x[ + n/k 2 - 'J 2 . */cos (-^12) where the triple integral is taken throughout all space, and the double integral is to lie given the limiting value it assumes when /c = 0. If P is in free space -^(x, y, z) = 0, andf ^JJjf>P{x',y\z').^.d X 'dy'dz (213) The formulae for differentiation as to y and z may be deduced by symmetry from (212) and (213). It may be observed, as a useful general rule, that if x is a homogeneous function of order n, the residual (double) integral is of the dimensions -2. * Quoted in Article 283. t Quoted iu Article 284. 364 GENERAL SOLUTIONS AND EXAMPLES. [310. Second Case (applicable to L p L 2 , and their derivatives). When the function x becomes infinite if x'=x, y' = y, for all values of z', we must replace the sphere by a circular cylinder of infinite length and radius k (ultimately zero, as before), with the line x' = x, y' = y for axis Taking y' - y - »; j X' - X = n/k* - rf ) and remembering that a shifting of the cylinder parallel to Oz does not affect the limits of integration, we find as before ^T = JJJ '/'(•«> y> z ')~£dx'dydz +J xj,(x, y, z)J { X [ - vV - rf, v , z - z] -oo — K -X[+ J*=?,il,z' -*]}*&, (214) with a symmetrical formula for cfa/dy ; while a» 'jS/'^' y '' ^i dx ' d ' i >' d ^ < 2 ! 5 ) whatever be the position of P. If \}r(x', i/', z') is zero for all values of z' ', when x' = x, y' = y, the formula (214) becomes symmetrical with (215), as the residual integral then vanishes identically. 311.] Alternative Formulae. If we shift the origin a distance dx in the negative direction of Ox, we change x, x' into x + dx, x' + dx, without altering y, y', z, z', or the infinite limits of integration. Thus, since (x'+dx) — (x+dx) = x' — x, <&(x + dx, y, z) =fffo(x + dx, y', z')x(x - x, y' - y, z -z)dx'dy'dz', and therefore * S^g.x.-H (2i6) etc., etc. I This formula is independent of the position of P and of the form of the function x- 312.] Application of the formulae of differentiation to the Potential Functions. It is left as an exercise for the student to show, by successive applications of the formulae (212), (214) and (215) to the Potential Functions (209), that & ^ a*» -dz -1 ' (217) * Quoted in Articles 266 (?) and 301. 312.] GENERAL SOLUTIONS AND EXAMPLES. 365 (218) y 2 D = 2I, v 2 I=-W; y 2 L 1 = 47r/i/'(a;, y, z')dz : S V 2 L 2 = 4y be any function whatever that satisfies Laplace's equation y 2 ^> = 0, we have i(~A\ = 0.^ V 2 (^) Oz .(221) identically, and consequently _3 Ox If therefore we assume a ( 2 SHi^>>-- Ox CjJ we shall have w = — ^-(zd>) + w oz C/2 and on substitution in (104), we find, as the necessary and sufficient condition that the general equations of equilibrium may be satisfied, 2(m + n) , m Let us then assume for our general solution 3<£ .(222) 0 m + 2n , w = — z_T +

?3' 3' being any root of Laplace's equation which for great values of r is of zero or negative dimensions in r, the general equations and the conditions required at infinite distances will both be satisfied. We deduce from (225) A = (226) T=1nl ~dzdx S = 2n ^i \ (227) S = 2n- dz* 316.] Special forms of the Potential functions, when reduced to Surface Integrals. The imaginary distribution of potentiating matter, to which the functions (209) of § 310 may be supposed due, may of course be confined to a surface layer, of surface density ip- per unit area, over the whole or portions of the plane of xy. In this case the first three integrals reduce to the forms* i-J/w.rf^ .(228) D =ff4>{x', y')rdx'dy' L 1 = /^ty(K\ y')\og{z + r)Jx'dy\ r= ^Hx-riy + iy -!/)* + #; (229) and, since in this case \^- = throughout the interior of the body, we deduce from the equations of § 312 where CIZ ' I, y 2 D = 2I, V =I = 0,y2L 1 = 0. .(230) Also, yfr being finite or zero at all points of the surface, I is of the dimensions required in § 314, and L, of those required in § 315. * The student will observe the applicability of these surface integrals to the method suggested in Article 284 for the solution of the Problem of Free Vibrations. 368 GENERAL SOLUTIONS AND EXAMPLES. [317. 317. Special form of the first type of solution. Putting = I in equations (222), (223), (224), they become __ 3 l~r i\idxdy 3 rr^clx'dy mi In/ Tfdx'dy' 'dzJJ r m JJ r (231) A = — ?- f f^ (,x ' d '!l ' (232) 7'_9 ^V n ff 4 ,dx '< '//'_„ ^ /T'_H x jhl~\ ) ?Sx\jnJ/ r ^zJJ >■ J „ _ 3 Trt rr^/dxdy _ __~b_ /7 ~j'd.r'dy' ~\ I ~^%L™JJ r ^JJ r J | D .. r"»« + w 3 /~/~ \l/dx'dy' 3 2 /7" '\pdx'di/~\ l , (233) Performing the differentiations as to z, by means of formula (213) of § 310, we obtain #z\j/dx'dy m+2n frxj/dx'dy r' m JJ r- A = 2n /V'zxf'dx'dy' w y'l r=2n— T- fr t dx ' d v' + z rr^dxdy'~[ ,9=2w^r- r r^dx'dy + z /Tzj'dx'dy-\ ii= - inV n - f f z ^ dx ' d y ' + 2,fr^ ,dxd!l '~\ The integral rr^/dx'dy and its derivatives as to a; and y are certainly finite for all values of 2, and. in order to determine fully the surface conditions, it only remains to evaluate the integrals ff^K.ff z^xf/dx'dy 317.] GENERAL SOLUTIONS AND EXAMPLES. 369 To do this, transform the independent variables by the substi- tutions x = x + zr) cos o) ) so that y = y + z v sln *° r '- s s/l + rf, dx'dy' = zhjdrfdio. Since \}r = 0, except over certain finite areas of the surface, we may extend the limits of integration so as to include the entire plane of xy ; thus the limits of i\ are and oo, and those of and may be written yfs(x, y) in the above integrals. The surface values are therefore •(23+) .// »' -/ (1 + t)^ Substituting these values in (231), (232), (233), and putting z = 0, we obtain for the surface values of u, v, w, A, T, S, It n n m + 2n frrpdx'dy •. \= - — #«> y) m ~<->xJJ t r ,_2n i d r f^/dx'dy' ~ ~™ ^yJJ r H=- ^< m + n) ^x,y) to 2a . (235) 370 GENERAL SOLUTIONS AND EXAMPLES. [318. 318.] Special Form of the second type of solution. Writing L t for in the equations of § 31 5, they give us u = — /7

3 2 f r^dx'dy' r "bzdxjj r a_z 3 2 rftydx'dy' ~2ir TiydzJJ r R ■_L Yz^- ff ^M. 3 / T+dx'dy' -] 2*- L 3*lZ/ r 3*^// " r J .(244) throughout the body : while at the surface u 1 1. 47rm 3a; ^y ^ log rdx'dy' ™ Wyjfi /l0 * rd * dy ' _m + n PC Rdx'dy 1 4-irm.njf/ r torn A„ = \Kg. y). J=0, = 0, H=+(x,y) ,.(245) .(246) .(247) We have therefore obtained a complete and perfectly general solution of the problem proposed in § 313. i?a:arapZes. 320.] Regarding the earth as an infinite isotropic solid, with one plane face, to determine the strain produoed by a very small but heavy mass lying on the surface. The surface of contact being very small, the integrals will each reduce to a single element, which we will suppose to be at the origin. We shall then have x' = Q,y' = 0, and Rdx'dy" = Hdx'dy = - W, where W is the incumbent weight. Substituting in equations (242 — 245), and performing the indicated differentiations on r -1 and log(r+z), we obtain 372 GENERAL SOLUTIONS AND EXAMPLES. [320. — v~ , 1 i i i \ '■• \ Q • N '\ : ,■ j / CO 1; i il i 1 = * wr z i ~i r ' 47T j_7W 2 'M? + z ) J y wr Z i r Air |_w s m(r + z) z wr » , "i r 4ir \_nr z mnz 5 ] A = - r= - ,v = i2 = zW~ ] a; 3^W r ' 2nr 4 2/ 3z 2 W 2wr* z 3z 2 W 2jtt* throughout the hody ; and x W \ "0" Wo r 4irmr _ y W r 47rmr (m + n)W 4jthmw over the surface. W must of course be supposed very small in comparison with the weight moduli of the ground ; it must also be remembered that, while W/r represents fairly at dis- tant points the potential of the last Article, yet these formulae cannot be considered to hold right up to the origin. The above values of u , v , w represent the strained sur- face as formed by the revolution about Oz of the hyperbola h + ^ n A = (^\ *u + 4jrm but the central depression will in fact be rounded instead of conical (see Figure 43), in accordance with the statement of § 55 that discontinuous curvature cannot be produced by small strain. 321.] GENERAL SOLUTIONS AND EXAMPLES. 373 321.] A right circular cylinder, formed of homo- geneous and perfectly rigid material, stands on end upon the ground; required the deformation produced by its weight. Let W be the weight of the cylinder, and A the radius of its base : let the centre of its base be placed at the origin. Since the cylinder is rigid, its base will remain plane, and consequently we must have w constant all over the area of contact. Also the conditions will be symmetrical about the axis of z, and therefore we may write i/Kf) i' or V r ( J5 '» V') an( * •27T /~A J J ^rfjrjdio im- 1 1 . Thus the problem reduces itself to the following : — Required a function \}Arj), such that J J4>{v)nd-0du>= -W, (248) while \f/(r])rjdrjd(o If >J(x — y cos o)) 2 + (y — i] sin u) 2 .(249) is constant for all values of x and y that make x i + y i Thus if we make m—mZ'-* (251) *The formula here quoted are easily deduced from those of the ellipsoidal shell, by making (with the notation of Article 252) B=A, C=0. 374 GENERAL SOLUTIONS AND EXAMPLES. [321. from >? = to 17 = A, (248) will be satisfied identically, while the value of the integral (249) will be — ttW/2A all over the area of contact, and for all the rest of space r 2 "r A MnWndu _ ^tan-'-i*. (252) J J >J(x - 1\ cos w) 2 + (y — rj sin to) 2 + z 2 A j£ £ being given by (250). Finally we may note that ~r I I log[z + *J(x — t] cos - f ** f- i(t])r}d-r)du> j j J(x - ij cos a)) 2 + (y - 7] sin o o - f° r iT f- ^(rj)rjdrjdtodz J J J J(x — ij cos a)) 2 + (y — rj sin oj) 2 + z 2 z ^f^'Tt (253) Substituting in the equations of § 319, we have at the surface of the body ^v' r iT r A ( x — cos nt'yqdnidut 8Tr 2 mA J J \_(x — rj cos to) 2 + (y - ij sin to) 2 ] -J A 2 — rf f* 2T C A (y — V s i n J J [(x— 17 cos = - i- — , li x i + y i A 2 omnA |_ 7t .4 _J and throughout its substance « = -- _ — tan 1 — F .-t tan 1 — ^tfe 4ttto4 dx[_ ?i ,/£ y n/£ J W arm, .,i /"*>, _^ ,-1 «= j :=- — tan 1 — - _/ tan l —-dz AirmA ?>yl_ n Jf: J Jg J W r~mz d _m + n~] _ x A iirmA \_n 'dz n _] ^/Z ..(253) ,.(254) 321.] GENERAL SOLUTIONS AND EXAMPLES. 375 We deduce without difficulty that over the area of contact iirm(x Wy Airmfx while over the free surface Wo; 4irro(a; 2 + y 2 ) | Wy 4irm(x 2 + J/ 2 )-' Thus the horizontal component of the surface displacement is directed towards the axis of z, and its magnitude is W or 4irm Jx 2 + y 2 5? n_ /r^y-i 47rm^a; 2 + y 2 |_ A/ ^ 2 J' according as the displaced point is on the free surface or within the area of contact. The student must be referred to Boussinesq's original memoir Sur V application des Potentials a I'etude de I'equilibre et du mowvement des Solides Elastiques (Gauthier-Villars, Paris ; 1885) for more extended applications of this theory, together with some interesting examples. EXAMPLES. [Unless the contrary is expressly stated, it is to be assumed that the body under consideration is free from Applied Forces.] Vibrations. ' 1. The following forms of $ all satisfy equation (63), and consequently represent possible forms of free irrotational vibration : {Note. exp[0] is equivalent to e"). :J7(J GENERAL SOLUTIONS AND EXAMPLES. "2. The following is the general solution for symmetrical waves of longitudinal displacement, radiating from or converging to a single centre : * = ha, «., <(«+£-«,)! ~dx dy 3~' ' Special cases are considered in § 273, and in the following Example. 3. A spherical shell whose internal and external radii are A and B, vibrates radially, the motion being symmetrical about the centre. Prove that the admissible values of i are given by I = iO, where tH,fu - tan-'( n f™ X] = iJTiB - tan-Y J™£ Y] On making .4=0, £ = r, this reduces to the formula (84) of § 273. (NJ2. — The i of the present Example corresponds to i/r in the former case.) 4. A circular cylinder of radius A and infinite length, per- forms symmetrical radial vibrations. Prove that (with the notation of § 244) u = 2C, Jj/^ \ain(it + a 4 ), the admissible values of i being given by i = iQ/A, where i is any root of iJ 1 '(i) + (l-5 2 )j 1 (i) = 0. ■' 5. The following is the general solution for waves of trans- verse vibrations (§§ 275-277) in a given plane radiating sym- metrically from a single centre : Investigate the form assumed by this solution when r is very great in comparison with the wave length (2x£2'/*)- 6. A solution for waves of transverse vibrations may be constructed by making Q 1 dx 2 c)xdy . ~dxdz GENERAL SOLUTIONS AND EXAMPLES. 377 \jr having the same form as in the last Example, or being any solution of equation (88) of § 277. Investigate the form of the motion when r is very great in comparison with the wave length. o 7. Prove that the equations of free periodic vibrations, per- formed symmetrically in planes through Oz, may be written, with the notation of § 243, in the form 3^__1_ . ?£, r _l o i+ 1 3ft „. 3r r 2 sin0 30' ' r 30 rsm0~3>' where t and i/r, are independent functions of r and 6 satisfying the equations ^ + s ^3/j_3ft\ i 2 ^ f 3r 2 30^sin0 30/ fl' 2 U J r* - 8. Equations (53) of § 263 may be put into the form giving (v , + gsV-°- It is easy to show that fi 2 3A 12 2 3A fl 2 d& i J ox i* oy i 2 3s are particular integrals of these equations; the complementary functions being of course solutions of ( v 2 + i 2 /fi' 2 )u = 0, etc. Lord Rayleigh* has obtained a solution specially adapted to the case of free vibrations propagated parallel to the plane surface of an infinite solid, which the student will have no difficulty in con- structing for himself. Taking s = for the plane surface, assuming that A,, u„ etc., vary as the sine or cosine of i(\x+/uiy), and determining the complementary functions so as to satisfy (131) : and then adjusting the arbitrary constants so that R, S, and T may vanish when 3 = : the solution finally reduces to the form 3 o = 8*2^' cos i[P r cos (° -«.) + «- A] {exp( - iz s/^i/fl 2 ) - (1 - l/2fi ,2 p 2 ) . exp( - is Jp*-1 /&'*)}, W = -^ T^-l^ T cos * r COs{e - a J + i ~ ft) • exp( - iz Jp^Tjmj : p being a root of the bicubic 1 6i2' c (fi 2 - n' 2 ) p s - 8fl' 4 (30 2 - 2Q'> 4 + 8i2 2 i2 V - fl 2 = 0, and A,a,f3 being arbitrary constants. The corresponding cubical dilatation is A = - 24 , cos i[pr cos(0 - a,) + 1 - j3,] . exp( - iz Jp 2 - 1/J2 2 ). The symbols r, 6 here denote the cylindrical polars of § 244 ; the student will find it a good exercise to prove by actual substi- tution that the displacements or r op 02 satisfy equations (88) and (89) of that Article, when £? = H = Z = S' = H' = Z' = 0, and = *. 9. Plane sound waves travelling through an isotropic elastic medium (m v n v p v QJ impinge obliquely on the plane surface separating this from a second medium (m 2 , n 2 , p 2 , 2 ; ). Prove that the disturbance is partly " reflected " into the first medium, and partly " refracted " into the second ; and that if the directions of displacement in the incident, reflected and refracted waves make angles \[r, \Js', yjs", respectively with the normal to the dividing surface, then f = 7T - ^ ; f = sin- 1 [(fi 2 sin ^)/J2J. Investigate also the distribution of energy between the reflected and refracted waves. The surface conditions in this problem reduce to those necessary for per- manent contact between the two media ; and these are that the normal com- ponents of displacement and of stress in the two media be equal at every point of the surface. Take the dividing surface for plane of xy, Oz being directed into the second medium, and assume 0j - A sin-i (acos ■ji + y sin f + Qj) + B sin-l(a; cos f + y sin \j/ + Qj), "i "i 2 =Csin— (x cob if/' +y nia \l/' + il 2 t). "a The potential 2ir + T, it will be impossible, by any expenditure of material, to make the tube strong enough to resist the stress produced. Expose the inconsistency of this reasoning. 11. Assuming that the rivetted seams of a boiler are its weakest parts, compare the strengths of two cylindrical boilers (§§ 289-291) which are alike in all respects except that in one the seams are parallel and perpendicular to the axis, while in the other they are everywhere inclined to it at angles of 45°. 12. A solid sphere is subjected to a normal pressure Ccosfl over its whole surface; required the strain produced. A very elegant solution has been obtained for this problem, but it is unfor- tunately disqualified by an inherent impossibility. What is this ? 13. A solid sphere is subjected to a normal surface traction C cos 6 over the hemisphere from 6 = to 6 = \ir, and to a normal traction — Ccosfl over the other hemisphere. Prove that equi- librium will be maintained, and determine the strained form of the sphere (i.) when C is positive, (ii.) when C is negative. 14. A spherical shell (internal and external radii A and B) is subjected to uniform normal pressures LT, II' over its surfaces : show that the radial displacement is given by _ (Am - Bm')r A^B%U - IT) U 3k(B3-A*) ^(B 3 - A s )r 2 Adopt the notation of Article 243, and assume the strain to be symmetrical about the centre. • 15. If 11 = in the case of the last Example, determine the value of IT at which the limit of stable elastic resistance will be reached. 16. A sphere of " incompressible " material (i.e., an imaginary substance for which the ratio Jc/n is infinitely great) is subjected to a surface traction whose components are the harmonics F= H ; , G s = Hi / , H=~H." ; prove that the radial displacement is i f (i - im 2 ^ (i - i)(2i + syxt.j _ # <+1 ) ^Z^t 2(2i 2 +l) 2(2t+l)(2i 2 + l) 2i(2i+l)$' where A is the radius of the sphere, and X, $ are solid harmonics, such that («H. + ySU + zOCY = r2Xi 2i~i' +1 - 380 GKNERAL SOLUTIONS AND EXAMPLES. 17. An isotropic cylinder of elliptic section is slightly deformed in such a way that the section of its bounding surface (which remains cylindrical) becomes a confocal ellipse. Deter- mine the displacement throughout the solid. Use the elliptic cylindrics of Article 246, or a system analogous to the spheroidal s of Article 251. The surface condition is that a shall be inde- pendent of t) all over the surface. 18. A solid sphere is subjected to tangential surface traction, everywhere parallel to the plane of xy and of magnitude SC,Pi/ sin 6, where 6 has the same meaning as in § 243, and P< is Legendre's coefficient of order i. Show that the system is in equilibrium, and that the point (r, 6) will be displaced parallel to the plane of xy through an arc ,7ib27'(2) < P <- P -> where A is the radius of the sphere, and £ i =C i +C i _,+ C i _ i + If the surface traction be G(P 2 — P 4 )/sin 6, discuss its distri- bution over the surface, and draw the curve into which any superficial meridian is deformed. [P 2 = £(3 cos J - 1,) P 4 = $(35 cos 4 - 30 cos 2 + 3)]. 19. Investigate the system of forces and , tractions required to produce in a solid sphere the distribution of displacement u = tx + yy - /3z+ \(x 2 - y 2 - z 2 ) + 2\ixy + 2vzx\ v = ty + az-yx + 2\xy + /j.(y 2 - z 2 - x 2 ) + 2vyz K w = a + (Sx-ay + 2\zx + 2pyz + v(z 2 - x 2 - y 2 ) J where all the coefficients are constants. 20. If any body bounded only by a sphere, or by two con- centric spheres, be submitted to any conservative system of impressed forces, the action on the body as a whole reduces to a single resultant force. 21. In the case of § 306, the conditions of equilibrium are ^^-A^Ho + yHo' + ^Ho"] \ 1 3U 2 tt,_1 3U 2 tt'._1 3U r ox r oy U 2 being any solid harmonic of degree 2 rr- 1 au 2 fT'- 1 d U 2 tt»_1 3U 2 | r ox r oy r oz J GENERAL SOLUTIONS AND EXAMPLES. 381 22. A vertical cylindrical hole of circular section is cut in a rigid body, and an elastic cylinder of density p, which, if freed from the action of gravity, would exactly fit the hole, is placed in it and stands upon the bottom. Prove that the sides of the hole suffer the same hydrostatic pressure as if it were filled with a liquid of density p(m — n)/(m + ri). 23. A glacier fills a valley which is perfectly symmetrical about a vertical plane, and which narrows as it descends. Assuming that ice at temperatures below the freezing point, and under moderate stresses behaves as an isotropic elastic solid, investigate the general character of the strain produced in the superficial lamina of ice by (i.) the weight of the glacier tending down the valley, (ii.) the lateral compression as the valley narrows, {Hi.) the friction against the sides. Show that there will be a tendency to form " crevasses " or cracks extending across the glacier, symmetrical about the middle line and uritfi their concavities turned down the valley. [W. Hopkins.] ^ 24. Investigate the strain produced in a solid sphere by the mutual gravitation of its parte. Show that if k represent the mutual attraction of two unit masses concentrated at points separated by the unit distance, a uniform normal surface traction ±Trmicp 2 A 2 /lD(m+n) will preserve the volume of the sphere unaltered : the cubical dilatation at the surface being in this case 4nrKp 2 A 2 /15(m+n), and the cubical compression at the centre 2TTKp 2 A 2 /5(m+n). s 25. Substituting from equations (187) in (3) and thence in (1), and making use of equations (F, G) of § 307 bis, we have in all cases of plane stress under gravity, such as those of §§ 308 and 309, ^£qu + (1 + and \fr denote arbitrary functions. 3y GENERAL SOLUTIONS AND EXAMPLES. 383 Hence show, by substituting in (A) the values of the strain components, and integrating, that q e l+ (i + «r)^=-x>i) =| {/[* - (i + -)v 2 x 8 ]^ + Uv) q0 3+ {\ + 2 (*)) 9 « =/[$ - (1 + o-) V 2 x 2 ]^ - (1 + °)^(Xi + Xi - X») + &(») + *,(*) ?M > =/[* - (l + M J In applying these general formulae to the case of Plane Stress, worked out independently in Example 25, put Xl = X2 = 0, Y 3 = y. We must also write as will at once appear on forming the equations analogous to (C) of § 307 bis. The and i/r functions are quite determinate, the arbitrary terms which appear on integration representing bodily translations and rotations. y 34. Obtain a solution analogous to that of § 307, when a plane stress is caused by the Applied Forces X=— F= — ~dy 3jc' SP being any function of x and y which satisfies (f tl tyjdoix)y = 0, and the surface tractions being such as to admit of Plane Stress (§ 307) : and determine, by the method of § 307 bis, the necessary limita- tions to the form of the function in terms of which the stress components are expressed. 35. A free charge E of electricity distributes itself over a plane disc bounded by the ellipse A* h"- 384 GENERAL SOLUTIONS AND EXAMPLES. with surface density E on either side of the disc : the potential produced being e r* 2 J : 2 J JMAZ+\)(W+\) at points within the disc, and E r m dX. V J\(A* + k)(W+k) at points without it ; £ being the greatest root of the cubic : + -^ — Z + -v = 1. A* + £ £* + $ £ Hence deduce, as in § 321, that a rigid right cylinder of weight W whose normal section is of the same form as the disc will, if placed upright upon the ground, descend vertically through a distance o = y — i-j- . F(e, \tt), where e is the eccentricity of the elliptic section, and F denotes the elliptic integral of the first kind; and determine the distribu- tion of displacement throughout the earth. [Take $= - WfiTrAB Jl - x 2 /A 2 - y*/W]. 36.] A cylindrical vessel is filled with liquid to a height D, in vacuo. The vessel and its contents are then weighed under an atmospheric pressure II, and at the same temperature as before. The mean density of the liquid in the vessel being thus found to be p',show that its true natural density maybe deduced by the formula , f , II gp'D ) where k is its compressibility for the given temperature. CHAPTER VII. BEAMS AND WIRES. Introductory. 322.] Definitions. The terms Beam, Wire, and Hoop, in the most extended sense in which they will be employed in the present Chapter, denote bodies which have the following charac- teristic property in common : — Each is so related to a certain straight or continuously curved line, called its Central Axis, that the centre of gravity of every section by a plane perpendicular to the Central Axis lies in that Axis. The Central Axis itself may be situated wholly or partly within or without the substance of the body. The Central Axis of a Beam is a straight line, and — unless the contrary be expressly stated — the beam is to be supposed cylindrical or prismatic in form, the generators of the lateral surface being parallel to the Axis, and the plane ends of the beam being perpendicular to it and of dimensions comparable with its length. The Central Axis of a Hoop is any closed curve of continuous curvature, and the form of the Hoop is denned by that of its Central Axis and by those of its normal sections. We shall only deal with uniform circular hoops.in which the Axis is a circle, while all sections in planes perpendicular to the Axis are equal and similar figures, similarly situated with regard to its polar line (i.e., the straight line drawn through its centre perpendicular to its plane.) A beam, or hoop of any form, the dimensions of whose trans- verse sections are all very small in comparison with the length of the central axis (but yet finite) will be called a Wire. For purely geometrical purposes, a wire may be regarded as coincident with its Axis. We shall confine ourselves to the consideration of wires of naturally uniform transverse section, but no restriction will be placed upon the natural form of the Axis. 2b 386 BEAMS AND WIRES. [323. 323.] The class of Strains to be investigated. Ex- clusion of Lateral Surface Tractions. The main object of this Chapter is to obtain reliable data for the employment of beams in structures and mechanism, where their function is to transmit from one body to another forces or couples, the straining effect of which upon themselves is in general very great in com- parison with that of their weight. The distinctive character of all the strains discussed will therefore be the absence of all stress across the lateral surfaces of the beams, wires, or hoops. The straining of beams will be considered as due to forces and couples applied by means of surface tractions acting over their ends alone. These may be supplemented, in the case of terminated wires, by impressed forces. Since closed hoops have no ends, they will be regarded as under the influence of impressed forces only. St. Venant's Problem: Straining of a naturally Cylindrical Beam, free from Impressed Forces, by Surface Tractions applied to its ends alone. 324.] Anticipation of the General Character of the Strain. Geometrical conditions imposed. It is sufficiently obvious, from a superficial view of the conditions of equilibrium of the beam as a whole (§ 146), that the most general form of small strain which external action of the supposed kind will tend to produce must be compounded of the three comparatively simple types — (i.) Longitudinal Extension of the beam, accompanied by lateral contraction : due to equilibrating forces parallel to the Axis. (ii.) Torsion, or twisting of the beam about some straight line parallel to its Central Axis, with or without warping of the transverse sections and distortion of the lateral surfaces ; due to equilibrating couples in planes perpendicular to the Axis. (Hi.) Flexion of the beam, of such a kind that the Central Axis assumes the form of a plane curve: due to equilibrating couples in planes parallel to the Axis. We shall find, on analysing the general equations of strain obtained in § 327 below, that this anticipation is fully borne out. To simplify the geometrical conditions of the problem, we shall suppose the centre of gravity of the area of one end of the beam (henceforth referred to as the Base) to be an absolutely fixed point, which we shall take for origin. The Central Axis of the beam will be our axis of z, and the principal Axes of Inertia of the area of the base our axes of x and y. ■(3) 324.] BEAMS AND WIRES. 387 Thus we shall have, Jjxdxdy =JJydxdy =Jjxydxdy = (1) identically, where the integrals are taken over the whole area of any normal section of the beam. Also, if JC denote the area of the transverse section, £v J2 its moments of inertia about the principal axes through its centre of gravity parallel to Ox, Oy, and JQ its moment of inertia about the Central Axis of the beam, ffdrdy = & ffy*dxdy = $ 1 Jfi?dxdy = g 2 //(<# + y*)dmdy=M 1+ § 2 = $ 3 These quantities are of course constants which depend only on the natural form and dimensions of the beam, and not at all on its material. We shall further suppose that the element of the base immedi- ately surrounding the origin always retains its initial plane, and that an elementary line in that plane — for simplicity, say the initial element of Ox — retains its natural direction.* The geo- metrical conditions to be satisfied at the origin are therefore A 3io ~dw ~dv A /0 > „_*-„ = 0, _ = -=_ = 0. (3) 325.] Conditions of Equilibrium. Besides the general equations of equilibrium (103) or (104) of § 285, the stress com- ponents must satisfy identically six relations imposed upon them by the peculiar circumstances of the strain. Since the sides of the beam are free from stress, it follows that any length of it, bounded by normal sections, is held in equilibrium solely by the total stresses across its ends. Hence the component forces and couples across those ends must be equal, and it follows from equations (6) and (7) of § 146 that the six integrals ffMxdy, f/Sdxdy, f/Tdxdy, } ff(yR-zS)dxdy, ff{zT-xR)dxdy, ff(xS - yT)dxdy] taken over any transverse section, must be absolutely independent of z. * It will appear later that if any line in the fixed plane element of the base be constrained to retain its initial direction, every line in that element will do the same. 388 BEAMS AND WIRES. [326. 326.] Statement of St. Venant's Problem. Since the lateral surfaces, over which the stress components are everywhere zero, are parallel to the Axis, the boundary conditions reduce to XP + f iU=X.U+ii.Q = \T + tJi.S = 0. The first two of these will be satisfied identically if we assume * that P=Q = U=0 (5) throughout the body ; and in this case the only condition to be satisfied by the special values of the stress components at the lateral bounding surface is X.T+pS = Q (6) To obtain a solution of the general equations which will satisfy (4) and (5) throughout the body (6) all over the lateral surface, and (3) at the origin, is the problem justly named by Clebsch " St. Venant's Problem." The peculiarity of the solution is that c = 0, and e =/= — a-g throughout the body, so that each longitudinal "fibre," or ele- mentary prism parallel to the Axis is extended longitudinally and contracted laterally just as if it were solitary (§ 213), while its transverse sections do not suffer shear. 327.] Solution of the Problem equations (40) of § 214, we have ?>z q. ~dv _ 'dw ~dy 3z 3d 3ii_ fl ~dx dy Substituting from (5) in (7) (8) (9) and the general equations (104) of § 285 reduce to 3 2 « 3 2 i» _ „~. 3« 2 ~dzdx 3z 2 ~dydz ?Pw 3 2 M> -,C) 2 U) __ + — _ + 2 — = ~dx l "dy 2 3z 2 .(10) .(11) .(12) * This will probably appear to the student a very sweeping assumption to make at such an early stage of the investigation. The solution of the general problem is, however, of a " semi-inverse " character, the conditions of each of the simpler component strains of Article 324 having been fully analysed by St. Venant in his two splendid memoirs — " Sur la Torsion des Prismes," Mem. des Sav. Etr.: t. xiv. (1855), "Sur la Flexion des Prismes," Liouville: 2° ser. t. i. (1856). It must be remembered that any solution which satisfies all the conditions is the solution. 327.] BEAMS AND WIRES. 389 while the boundary condition (6) may be written ^SM^sH < i3 > Differentiating (10), (11), (12) as to x, y, z respectively, and subtracting the first two results from the third, 3 2 /^dw _ "du _ "dv\ _ n and therefore by (8) £-» <»> -'V5.(S*SH Again, differentiating (10) as to y and (11) as to x, and adding the results, ?>xdydz and therefore by (9) ^ w =0. "dxdydz Lastly, differentiating (12) as to z, and taking account of (14), 2Pw cfiw _ ~. _ = 0, ~dx^dz 'dy^dz but if we differentiate (10) as to y, and (11) as to a;, and subtract the results, 3 3 ie _ %) dg 392 BEAMS AND WIRES. [327. where ds is an elementary arc of the periphery of the section. Now, we have by (2) &=ffdxdy = yi^ + H/) d "> so that the boundary condition satisfied by w' requires that which is obviovisly inconsistent with 3 2 w' 3 2 w' _ _ 3a: 2 Sy 2 Hence it follows that /3 must be zero. It will be found, on substituting from (19) in (4) and taking account of (1), that the first, second, third and sixth conditions of equilibrium are satisfied identically. In order that the fourth and fifth may be satisfied we must have jTf^dy = J&K2 - a) J 2 - (4 + StJJJ I Jf^dxdy = i&[(2 - «r) Jf j - (4 + 32/ a throughout the body, and the boundary conditions ,3w 3w . A lte + ^S = (2 + °" )Aa:y + ^ + i "^ 2 - «*)] over its lateral surface : and also the conditions of equilibrium .(24) .(25) .(26) Jf^ lxdy \//'% dxdy = JTf^dxdy =JJ"^dxdy - tf(2 - «r) J 2 - (4 + 8«r) J J = ...(27) ...(28) ...(29) where the double integrals are taken all over any transverse section. It should be observed that the stress components S and T are independent of z, and therefore constant along each longitudinal "fibre." 328.] Determinateness of the Solution. We already know from general principles (§ 255) that the, solution is perfectly determinate when the distribution of stress over the ends of the beam is given. We may however show that the solution (21), subject to the boundary conditions (24-26), is perfectly deter- 394 BEAMS AND WIBES. [328. minate in itself, so that the distribution of stress over the ends, as deduced by means of (22), is not at all arbitrary, but is governed by fixed laws depending only on the form and dimen- sions of the beam. To prove this, it will be sufficient to show, by a method equally applicable to all * that any one of the w functions is completely determined by (23) and the appropriate boundary condition (24), (25) or (26). If possible let these conditions be both satisfied by two different values of w (for example): let £ be the difference of these two values. Then £ must satisfy 3a; 2 3// 2 throughout the body, and ox oy all over the lateral surface. Now if we integrate the expression by parts it becomes and each of these terms is identically zero. But (compare §§ 254, 256) the original integral is the sum of a number of essentially positive quantities, each of which must therefore vanish separ- ately. Consequently ~dx 'dy throughout the body, and since w is supposed to vanish at the origin, and thus cannot involve a constant term, f must be zero throughout. Hence the two values of w are identical, and it is obvious that the same may be proved, in precisely the same words, of w x and w 2 . First Component. — Simple Extension. 329.] Complete Solution. Making all the arbitrary con- stants zero, with the exception of e, we have the simple strain u = - (rex, v = — aey, w = iz;\ giving [ (30) B = qe, S=T = 0. ) * This is, in effect, a special proof of Green's general theorem, adapted to the case in which the solution of Laplace's equation is independent of z, while the surfaces bounding the region within which that equation is satisfied are either parallel or perpendicular to Oz. 329.] BEAMS AND WIRES. 395 This is the case already fully discussed in § 213. A longi- tudinal tension E is applied to the beam by means of a uniform traction E/Jl over each end, and produces a uniform extension e = 1H/Q,q throughout the beam, accompanied by a uniform con- traction a-e in every transverse direction. The ratio t = Qq of the tension to the consequent elongation is called the Coefficient of Longitudinal Extension of the beam, and sometimes Hooke's modulus ; but it must be remembered that it depends upon the dimensions of the body, as well as on the properties of the material, so that it is not a true specific modulus, in the sense in which we have always employed the term. For a beam of given material it is proportional to the sectional area, and for a beam of given section to the Young's modulus of the material. If L be the length of the beam, equation (41) of § 214 gives for the total potential energy due to extension W=R*L&l2q = LWl2t = lLi over the lateral surface, and the conditions of equilibrium Jjf^i xd y = Jj^^ dxdy= Q (36) 331. J Geometrical character of the Strain. The sim- plest way of ascertaining this is to investigate (i.) the curve assumed by any longitudinal fibre of the beam, and (ii.) the surface into which any (initially plane) normal section is warped. Let the point initially at (x, y, z) be displaced by the strain to {x ', y', z'), so that x' = x + u, y' = y + v, z' = z+w. (i.) Along any longitudinal fibre of the beam the initial coordinates x, y are constants. Thus (see § 68) the form assumed by any such fibre is represented by the equations x' = x — TS? y =y + T2' x' — X y "(s) y -y ■♦(*).' Thus, when the strain is very small, each fibre remains a straight line, and the foot of each fibre (the point in which it cuts the plane of xy) retains its initial position. In general each fibre is inclined to the Axis of the beam at a small angle 331.] BEAMS AND WIRES. 397 but the particular fibre for which is altogether unaffected by the strain. This fibre is called the Axis of Torsion, and the strain is said to be a Torsion about this axis. Again, each strained fibre lies in a plane *'-<"S)J^-'C'-©J- which is perpendicular to the straight line joining its foot to that of the Axis of Torsion. Thus the generators of any circular cylinder [■♦®J-['-©J-* described about the Axis of Torsion in the unstrained beam, become one set of generators of the one-sheet hyperboloid of revolution (x' - xf + (y' - yf = CW 2 . This surface is represented, on an exaggerated scale of torsion, in Figure 44. The strained fibres may however, to the same degree of approximation, be regarded as helices of pitch -WRIJ^FDJ described on circular cylinders about the Axis of Torsion, and this is the form they actually take under torsion of finite amount. (ii.) Over any naturally plane normal section of the beam, the initial coordinate z is constant, and it appears by applying § 68 to the third of equations (32) that every such section is warped into the (gener- ally) curved surface --Z + T where w' denotes the same function of x' and y that w does of x and y. FIG.44. 398 BEAMS AND WIRES. [332. Torsion about the Central Axis of the Beam. St. Venant's Solution for a certain class of Beams. 332.] Equations of the Strain. "When the Axis of Torsion coincides with the Central Axis of the beam, we have e);Q= w '=° (37) In this case, equations (32) reduce to u=-ryz, v = txz, w = tw; (38) the other equations of § 330 being unaffected. The strain now obviously consists of the bodily rotation of each normal section through an angle rz about the axis, together with a general warping of these sections by longitudinal dis- placement. The quantity t is called the Twist per unit length of beam, or the Amount of Torsion. 333.] St. Venant's Solution. The problem can now be readily solved for a large and important class of beams, as follows. Let us suppose that the equation of the cylindrical surface (or of the closed curve bounding the base) can be put into the form 4> + i(* 2 + 2/ 2 ) = C, (39) where is any solution of S + p=°' <«» and C is a constant. Then ox oy and the boundary condition (35) becomes 3w ?> 3w ?> fdw 3<£\ /3w_3<£\_„ 'ox "dx oy 'by \dx by) \by bx) ' thus (34) and (35) will in all such cases be satisfied if we suppose 3^_3^ = 3w + 3^, = 0) by cte cte ~by that is if we choose w so that and w may be Conjugate Func- tions * of x and y. This is St. Venant's celebrated solution which is developed so skilfully and with such beautiful results in his Memoir on the Torsion of Prisms, already referred to. * See Article 245, and Examples 1-4 on Chapter V. 333.] BEAMS AND WIRES. 399 The conditions of equilibrium (36) will also be satisfied identi- cally upon this assumption, for they may now be written or ydx =/<\>dy = 0, the single integrals being taken round the perimeter of the section. But by (1) JYxdxdy = fjydxdy = 0, and therefore must be such that fx 2 dy =Jy 2 dx = ; also, since (39) must be supposed to represent a closed curve, jy 2 dy = /x 2 dx — 0. and ydy=Jdx = 0. Since + $(a?+y s ) = C all round the periphery, these last equations give us /[* + W + y 2 )¥* =/[* + i(<«* + y 2 )¥v = °> and consequently Jdx =J$dy = identically. In order that (37) may be satisfied w must not contain a linear function of x and y : nor therefore must . But this con- dition is necessarily satisfied ; for if did involve any such terms, the integrals fipdx -and ffy&y would involve terms of the form Jxdy and Jydx, which are proportional to the area Jl of the transverse section, and consequently cannot vanish. Hence all the conditions of the problem are satisfied by any value of which satisfies (40) and makes the boundary (39) of the base a closed curve, provided that the included area has the origin for its centre of gravity and Ox and Oy for its principal axes of inertia. 334.] The Torsion-Couple, Coefficient of Torsion and Potential Energy of the Strain. It follows from (1) and (36) that the distribution of stress (33) over any transverse section of the beam (which is the same for all such sections) reduces to a couple in the plane of the section. 400 BEAMS AND WIRES. [334. If T be the magnitude of the couple applied to either end of the beam T=ff(xS-yT)dxdy by (2) and (41). Thus if K^( a l +y l)H (42) the torsion-couple required to produce an amount of torsion t is given by T = ir, (43) and t may be called the Coefficient of Torsion * of the beam. For a beam of given material it depends only on the form and dimensions of the transverse section, and for a beam of given dimensions it is proportional to the rigidity n of the material. Again, if L be the length of the beam, the total potential energy of the strain is by (41) of § 214. L - /7{S 2 + T 2 )dxdy W = ^ ,(" + w)'*( , '~W}™" x 2 + y 2 + x-^f- + ty-i J dx *d-y + - m 2 L Now = C f^x dX ^% dy \ bj (39) a,ld (40) ' = 0, because the periphery is a closed curve. * Also known as the Torsional Rigidity. 334.] BEAMS AND WIBES. 401 Thus finally TT=JZlT 2 =ZT 2 /2t; (44) this formula should be compared with (31) of § 329. 335.] Circular Cylinder. If the base of the beam be a circle its centre must be at the origin, and we must therefore put = in (39) and take $A 2 for the constant term. A will then be the radius of the base. It is evident that w vanishes with is zero for both surfaces, and w is zero throughout the body, as before. The coefficient of torsion will be in this case f = nJ,' = J«4«(l-«*); and if we compare this with the coefficient of the solid circular beam, as given by (46), we find i' : t : : 1 - ** : 1. But the masses of the two beams, if their lengths are equal, are in the ratio M' : M : : 1 - k 2 : 1, 2c 402 BEAMS AND WIRES. [336. so that ¥/M':t/M::l+K*:l; whence we deduce that the resistance to torsion, proportionally to its mass, of a circular cylindrical beam of given length and external radius is increased by making it hollow. This principle is of great importance in the economy of struc- tural materials, and will be referred to again later. 337.] Elliptic Cylinder. If in equation (39) we make <£ = h a (v 2 - a;2 )' it becomes (l-a)x2 + (l+o)2/2 = 2C, (48) and if G is positive, and a is positive and less than unity, this represents an ellipse having its major and minor axes along Ox and Oy. y Fig.45. If A and B be the semi-axes of this ellipse C and we have thus A*-£2 AW A 2 + B 2 ' A*-W A* + B* u-— ryz, v = rxz, w = ' A 2 + B? Txy. .(49) 337.] BEAMS AND WIRES. 403 Also t = 4Jf s + a(J 2 -J 1 )] = lirnABlAZ + W + a(A* - B*)] = xn AB(A* + B* ) 2 A* + B* The transverse sections are in this case warped into hyperbolic paraboloids, any one of which is cut by planes perpendicular to the Central Axis in a series of hyperbolas having their asymptotes coincident with the principal axes of the unstrained elliptic section. It is evident from the form and sign of w that these surfaces are concave towards the positive direc- tion of Oz in the ( + x, + y) and ( — x, — y) quad- rants, and convex in the remaining quadrants. Figure 45 represents the "contour lines" (coupes topographiques) in which the warped section is cut by a series of planes perpen- dicular to the axis. The principal axes A. A', BR of the unstrained section are unaltered by the strain, being merely twisted bodily through an angle tz about the axis of torsion ; they are therefore the contour lines for the original level of the section. The dotted hyperbolas in the quadrants AB, A'R are below the original level (as looked at from the free end of the beam), and those in the remaining quadrants are above it. Figure 46 shows very clearly the warping of the sections, as it may be realised in practice on a greatly exaggerated scale, by twisting an incharubber band of elliptic section. .(50) FIG.46. 338.] Hollow Beam, bounded by cylindrical surfaces of similar Elliptic sections. If the beam is hollow and bounded internally by the similar and coaxial elliptic cylinder A* & ' is of the same form for both surfaces, and w and w have the same form as before. If t' be the coefficient of torsion of such a beam t'-*LI, +««.-«,)] _™ AB(A* * B *) n ,, ~T' A*+B*— {1 ~'- 404 BEAMS AND WIRES. [338. Thus the resistance to torsion of an elliptic cylindrical beam of given material and given length, per unit mass of the beam, is increased in the ratio l+/c 2 :l by hollowing it. [Compare the corresponding result for beams of circular section in § 336.] It is easy to show from equation (42) that this result applies to beams of any section, provided that the internal surface is similar to and similarly situated with the external surface. 339.] Beam of Equilateral Triangular Section. If in equation (39) we write for the constant term C 2 /!^, and put <£ = (3a^ 2 -ar')/C-v/3, it becomes a^-3a y 2 a: 2 + y 8 _C 2 Q Cv/3 2 18" ' or 6 Jl(a? - 3a*/ 2 ) - 9C(a: 2 + i •' / Fig.48. referring to Example 4 (vii.), page 258, it is at once apparent that the appropriate form of solution is 406 BEAMS AND WIRES. [340. The first of the required conditions will be satisfied identically if we suppose all the values of p included in this series to be of the form (2i+l)ir/G, where i is any integer, or zero : and the solution will then be fully determined by the remaining conditions ^C 2 - y 2 = 2" r^e ,2i+1,,r '- + ,B i e- , * +1|,r/!! "lcos(2; + l)Try/C = 2T-4 i e- ,;ii+1| ' r/ - + B i e ,li+1>lrni ~\cos(2i + \)irylC, from which it is at once evident that A { = B t . By Fourier's Theorem * and consequently '-*- .. 7£nnwf fo-*»&^ 2i (2i+l) 8 ir 8 C0Bh(H*±i^ it (2t + Thus finally and + ' * '=»(2i + ])3coshl^±i^ C The contour lines are represented as before in Figure 48, and Figure 49 gives a view of the warped sections for comparison with those of the Elliptic Beam. 341 .] Character of the Stress. By equations (5) and (33) the only existing stress components are R and S ; thus equations (21) and (22) of § 163 reduce to W(N 2 -S S -T 2 ) = () \ Tv_Sv_ Tk + S l A, _ :N .{ (51) X fj. v ; * Todhuiiter's Integral Calculus, Article 326, formula 5. t Example 4 (iv.), page 258. 341. J BEAMS AND WIRES. 407 One of the principal stresses therefore vanishes at every point, and since the directions of the lines of zero stress are given by v = 0, TX. + S/i = 0, they are plane curves in planes perpendicular to the Central Axis and cutting the lateral surface at right angles. The remaining principal stresses are equal in magnitude and of opposite sign, so that the stress at every point is a simple shearing stress, of magnitude in a plane parallel to Oz the direction cosines of which are given by A The system of Principal Surfaces enveloping these planes has for its differential equation Sdx-Tdy = 0, the integral of which is, by (33) and (41) (f> + £(re 2 + y 2 ) = constant. This system therefore includes the lateral surface of the beam. It may also be deduced from (51) that the directions of the principal traction and pressure at every point are inclined at angles of 45° to the Central Axis. The magnitude of the resultant shearing stress is FIG.49. S = i 'Ve^-K)- Let * = $ + i(as« + y*), so that the surfaces $ = constant are those Principal Surfaces across which there is no stress, but which envelope the planes of shearing stress ; then S -W(SHI) ! <» Since w, dw/dx, dw/dy all vanish at the origin, so also do , d

/dy, and $, d$/dx and d^/dy. $ therefore increases 408 BEAMS AND WIRES. [341. continuously in numerical value from (along the Central Axis) to C (over the lateral surface). Similarly, S is zero along the Central Axis, and for corresponding points * on the different $ surfaces S increases continuously with the numerical value of # until we reach the surface. To determine therefore the points of maximum stress (" points dangereux ") we have only to determine those points on the lateral surface of the body at which the expression (52) for S becomes a maximum. It follows at once from (52) that S is zero at any projecting angle (such as the edges of the square and triangular beams) and infinite at any reentrant angle. Angular grooves f are therefore fatal to beams intended to sustain torsion, and the slightest crack in the surface will tend to spread indefinitely until the beam is destroyed. On the other hand, angular ridges add nothing to the torsional strength of the beam. St. Venant has, however, proved a more general, and perhaps more striking property of torsion-shear. This is that the stress at the surface is always a maximum, at those points nearest to the axis, and a minimum, at those points farthest from it. We can prove this property without difficulty for the cases which we have solved. (i.) Circular Beam: Here * = %{x 2 + y 2 ), S — rn Va: 2 + y 2 = nrA. Thus S is constant all over the surface. (ii.) Elliptic Beam : Here $ = (4y + W)/(l 2 +^), and S - 2jit JlYTWx 2 /(A? + B 2 ) = 2nrB[A i - (A* - B 2 )x 2 ]*/(A 2 + B 2 ). Thus S has its maximum value 2n T A 2 B/(A 2 + B 2 ) when x = 0, i.e., at the extremities of the minor axis, and its minimum * Corresponding points on any family of curves, involving one variable parameter, are those points in which the family are cut by any one of the orthogonal system. With the notation of Chapter V., any function taken along the curve ri — const., t;=consl., can only vary with {. t It seems possible that the curious twisting of old poplar trees, growing in situations where they are exposed to prevalent winds in a fairly definite direction, may be due in part to the presence of deep and sharply cut longi- tudinal grooves in the trunk. The unsynimetrical growth of the boughs affords a leverage to the wind, which thus exerts a powerful torsion couple. This tendency is of course greatly increased when the trees form an avenue, for they are then much more exposed on one side than on any other. The Cambridge student will find excellent examples of the action here referred to in the old poplar avenue at Newnham Croft, near the University Swimming Club's sheds. 341.] BEAMS AND WIRES. 409 2nrAJP/(A i + B i ) when x= ±A, i.e., at the extremities of the major axis. There are consequently two lines of minimum stress (A A, A' A' in Figure 46), and two lines of maximum stress (BB, BE) along the whole length of the beam. (Hi.) Equilateral Triangular Beam. Here * = (3xy* - a*)/C J 3 + \(a? + y>), and S = ™V[(2/ 2 -a 2 ) j3 + Cxf + y*-(2x J3 + C)*/C. The sides of the beam are represented by 2xj3 + C = 0, xj3 + 3y-C = 0, xj3-3y-C = 0. Thus over the first side S=nrj3(lC 2 -y 2 )/C, and this expression vanishes when y=±\G (i.e., along the edges which bound the side), and has its maximum value £titCV3 when y = (i.e., along the straight line drawn parallel to the Axis to bisect the face). Similarly for the other two sides. (iv.) Square Beam. Here *-„ 2 + ^ jT ( 1)C ° Sh — P— ^ (W+lta . the component stresses can easily be deduced by differentiation, and calculated numerically, and the resultant deduced. St. Venant gives tables of the results (Memoire sur la Torsion des Prismes, pp. 393, 394), which show conclusively that S is a maximum when x = 0, y = ±\0, and when x= ±JC, y = 0, the four corresponding values being equal, and that S vanishes at the angles. This latter property may very easily be proved directly : for when ±x=±y — G, 3/3a; = 0, (Todhunter's Plane Trigonometry, Ch. xxiii., Ex. 3.) Thus it appears that very little strength is gained by making beams, intended to resist torsion only, with projecting longitudinal ridges, or flanges. We shall however presently see that such flanges are of the greatest possible value in resisting fiexion, if properly disposed. 410 BEAMS AND WIRES. [342. 342.] Erroneous Extension of Coulomb's Formula. It was assumed by engineers,* before St. Venant had obtained the complete solution of the problem, that all beams — of what- ever section — behaved under torsion like circular cylinders : i.e., that their normal sections rotated without distortion in their own planes. Thus the formula T = ii$ 3 t was supposed to be univer- sally applicable, whereas we know from formula (42) that it is a unique property of the circular cylinder. The true value of t as calculated from (42) for a beam of any other form, is found always to be less than that given by the application of Coulomb's formula, and also (as we might have expected from the last Article) less than that of a circular cylinder of the same sectional area. Figure 50 shows the results of St. Venant's comparison : the first line of numbers giving the ratios of the values of t for beams of the sections represented to (A) (B) (C) (D) (E) 8435 •8833 •8186 ■8666 •7783 ■8276 Fig.50. ■5374 •6745 •6000 •7255 those deduced from the fallacious theory just referred to, and the second line their ratios to the value of t for a circular cylinder of the same sectional area. The waste of material in forming projecting ridges is very conspicuous in case (D). Third Component. — Flexion. 343.] Equations of Strain. Retainiug only the terms in the second lines of equations (21) — i.e., annulling all the arbitrary constants but s^ and /^ — the displacements take the form v = ^[My 2 - * 2 ) + & 2 ] - &*[V(2/ 2 - <* 2 ) + ¥ 2 - fe) 1 .(53) * This statement is made by St. Venant, and quoted by Thomson and Tait. Neither authority gives any references, and I have Dot been able to verify it personally. 343.] BEAMS AND WIEES. 411 while the stress components are *--A"[* + W-*>-^]| (54) The function w x must satisfy the conditions throughout the body, cte 2 'dy 2 X ^ 1 + ^ = (2 + so that all fibres initially in the plane of zx — and in particular the Central Axis — retain their natural lengths unaltered. We have thus a Plane of Zero Extension dividing the beam longi- tudinally into two portions. If ^V/Sj be numerically greater than the length L of the beam, all fibres on one side of this plane will be elongated, and all fibres on the other side of it contracted, throughout their whole length. If however ^//Sj be numerically less than L, the elongation of every fibre not in the plane of zero extension changes sign at a point initially distant ^J^ from its foot. The curvature of the Central Axis, and indeed of every fibre, changes sign at the same distance from the base, so that in this case each strained fibre has a point of inflexion. The plane transverse sections are deformed into the surfaces *' — ^(^) -^-*^ + ^)J + Aw » (59) where, as in § 331, w/ bears the same relation to x', y' as w x to x, y. The tangent plane to any such surface at the point where it is cut by the Central Axis is found on expanding w x ' by MacLaurin's Theorem to be it is therefore parallel to Ox. 345.] The Second Flexion Component. The terms in- volving cr 2 , /3 2 and w 2 in equations (21), (22), (23), (26), (29) may be deduced from those discussed in the last two Articles by inter- changing the suffixes 1 and 2, and the coordinates x and y. The strain represented by them is therefore a flexion of precisely the same character, only in converse relation to the principal planes of the beam. Plane Circular Flexion in a Principal Plane. 346.] Reduction of the Strain. If we now annul fi v and with it all the terms involving Wj (see § 327), and retain only those which have cjj for coefficient, the character of the strain is greatly simplified. 346.] BEAMS AND WIRES. 413 The displacements become « = £OTjV(^ 2 -a: 2 )+« 2 ]l (60) w=-7S 1 yz J and the stress components R= -VStfy, P=Q = S=T=U=0 (61) 347.] Geometrical Character of the Strain. The easiest way to realise the effects of this strain is to resolve the displace- ments (60) into the three simple component systems u = \ u = TSjO-xy \ u-0 1 r = JCT 1 « 2 V(i.) v = — JCTjO-a; 2 Wit.) V - \T3^ry i l( Hi. ) w = — tttfz) w = w = (i.) First, we have y'-y=%1Z 1 z 2 , z" -z= -VS^yz; thus (y' -y)(2-2^! 1 y) = 7S l zz, and to our order of approximation (y'- 2 /)(2-CT 1 y-CTy)=W 1 «' 2 , or &- (67) and the potential energy W=lLfjBf = LP,*/2p a (68) Circular Flexion in any Plane. 351.] Equations of Displacement. Let the beam be flexed in such a way that the Central Axis takes the form of a circular arc of curvature or in a plane inclined at an angle a to the principal plane of zx. Then, by a simple application of Meunier's Theorem the com- ponent curvatures of the Axis in the two principal planes are CTj = CTcosa, CX, = GXsin a (69) Making these substitutions in equations (21) and (22), we have for the displacements. u = 7S{ (75) the sign being taken so that it tends to bend the Axis towards the plane of yz, in which the coefficient of flexion is p r In other words, this couple is necessary to prevent the beam from acquiring the given amount of flexion in the easiest possible way, i.e., in that principal plane in which the coefficient of flexion is least. If the plane of the resultant couple make an angle \fr with zx, the component couple perpendicular to this plane vanishes, so that sin \j/JJRydocdy - cos xj/JYRxdxdy = 0, or tan^ = the ellipses become circles, and V = Vi = Vi- The beam is then said to be equally flexible in all directions, and flexion takes place accurately in the plane of the applied couple. 354.] The Potential Energy of Flexion. By equation (20) of § 199 W=\Lf/Rgdkdy = \LqXtfyj\y cos a + x sin a) 2 dxdy = lLr^\ J lC os 2 a + J 2 sin 2 o) = iZ4H3»-ZP»/2p, (77) as before. 355.] The Stress. Economy of Material in Beams designed to resist Flexion only. The I beam. It follows at once from (71) that — whatever be the form of the transverse section, the longitudinal traction is zero throughout the Neutral Plane, drawn through the Central Axis perpendicular to the Plane of Flexion. The traction also has its maximum positive and negative values along those generators of the beam which are farthest from the Neutral Plane, on either side. Since the coeflicients of flexion, like that of torsion, depend upon the moments of inertia of the cross section, a precisely similar economy of material, or increase of strength per unit mass, is effected by hollowing out that portion of the beam which surrounds the Central Axis. * Ibid, Article 26. 420 BEAMS AND WIRES. [355. Flange suhje'it to Thrust When the plane of the straining couple is determinate — in actual structures this is usually the vertical plane through the Central Axis — a still greater economy of material is possible, because our only object is then to make the coefficient of flexion in one given plane as great as possible, while that in the perpen- dicular plane may theoretically be reduced to any extent. We shall therefore gain by concentrating the substance of the beam as near as possible to the plane of flexion, and as far as possible from the neutral plane. In practice we have to take into account possible small flexions in other planes, as well as accidental torsions, so that the reduction of material in the central portion of the beam must not be carried too far. The best practical compromise is found in the "I beam," in which the section consists of two flanges con- nected by a web, the whole being symmetrical about the plane of flexion. In the case of wrought iron, and the other more perfectly elastic materials in which the working strengths under tension and compres- sion are approximately equal [see Table (D), page 203], the Neutral Plane should be equidistant from the two extremes of the section In cast iron, however, the work- ing strength under compres- sion is nearly three times that under tension, so that the greatest economy of strength will be gained by making the distances of the Neutral Plane from the extreme sur- faces of the flanges in the same ratio. Since the centre of gravity of the entire section A'lutral /■'lung i- subject to Tension Fig.54. Plane must always lie in the neutral plane, this consideration of course requires that the sectional area of the stretched flange should be considerably greater than that of the compressed flange. The Coefficient of Flexion for an I beam of given dimensions is easily calculated.* Let "depth" denote dimensions parallel * In practice the inward angles are rounded off, to guard against acciden- tal torsion, and other shearing actions (Art. 341). 355.] BEAMS AND WIRES. 421 to the plane of flexion (yz), and " breadth " dimensions perpen- dicular to this plane. Let B v F x be the breadth and depth of the flange on the side towards which flexion takes place, and which is therefore subject to longitudinal thrust (§ 34b), and let B 2 , F 2 be the dimensions of the flange subject to tension. Let B s be the breadth of the web, and Y v Y 2 the distances from the neutral plane (xz) of the extreme surfaces of the contracted and extended flanges. Then (i.) Since the neutral plane contains the centre of gravity of the section hS,{ }\ -F 1+ Y 2 - F 2 )( Y 1 -F 1 -Y 2 + Fj = 5 2 ^(r 2 -p' 2 )- J B lJ f 1 (r i -^ 1 ) (78) (ii.) Since the extreme stresses, and therefore also by (61) the extreme ordinates, are to be in the ratio of the working strength under thrust (0) to that under tension (T), Y X :Y 2 :: 0:T (79) {Hi.) By (63) of § 349 V/9 = A^tW + < *i - W] + V.[tW + ( V, ~ W] + £ 3 (Y 1 -F 1 +Y 2 -F 2 )[ T \(Y 1 -F 1+ Y 2 -F 2 )* + l(Y 1 -F 1 -Y 2 + F 2 )>]...(80) By means of (78) and (79) p may easily be found in terms of the dimensions of the flanges, and of the total depth (F a + T 2 ) of the beam. The dimensions adopted in practice are such that the strength of the beam is about six times that of a simple rect- angular beam of the same sectional area (Cotterill). The Small Strain compounded of Uniform Extension, Uniform Torsion about the Central Axis, and Plane Circular Flexion. 356.] The Displacements and Stress Components. Compounding the equations of §§ 329, 332 and 350, we obtain for the resultant displacements u = - 422 BEAMS AND WIRES. [357. 357.] The External Forces and Couples to which this Strain is due. Independence of their effects. Over either end of the heam T=F, S=G, R = H, and on substituting from formulae (82) in the surface integrals (6) and (7) of § 146, and integrating over the area of the transverse section, we find for the system of external forces and couples which must be applied to the ends of the beam to produce the strain represented by (81) (i.) A force, parallel to the Axis, of amount E = ?2U"W, (83) (ii.) A couple, in the principal plane of yz, of amount * -Pj= -q$jpcosa= -pjGTcosa. (84) (in.) A couple, in the principal plane of zx, of amount P 2 = q^bjp sin a = p 2 CT sin a (85) {iv.) A couple, in the plane of xy perpendicular to the Central Axis, of amount T=wT {^ + #(%-^H} =tT (86) Thus each of the component distortions e, £^( = 57 cos a), & 2 ( = gj sin a), t, is related to the corresponding external action as if it existed alone. Consequently, if the external action on the ends of the beam is distributed according to the laws of equation (82), the effects of the longitudinal force and the component couples will be entirely independent. Let E be the longitudinal tension, and C the resultant couple applied in the plane having X, yu, v for its direction cosines : then by equations (83-86) : = E/e .(87) o-CMA/PiP+Wft)* a = tan- 1 (- /t p 1 /Ap 2 ) T = vC/t It should also be observed that, even in the most general form of strain, the force and couple across any transverse section of the beam are transmitted, unaltered in magnitude, from one end to the other. 358.] The Total Potential Energy. By equation (20) of § 199, we have V ' W= \Lff{Rg + Sa+ Tb}dxdy, * The couples are here taken in the standard directions of Appendix I. If the plane of flexion lies between the positive directions of Ox and Oy, the effective couple in the plane of yz must be negative. 358.] BEAMS AND WIBES. 423 and it is easily deduced, on integrating and using where necessary the results of the previous Articles, that W= $£[««» + w + v p* + ir *\ , = lL[zf&-&da v etc., or A + pf^iSi ~ ^i')^-i = ^o j Again, the direction cosines of the tangent to the Central Axis (which is the axis of the torsion couple) are dx'/ds, dy'/ds, dz'/ds, sc that the components of this couple about the axes of reference are . 3a:' , ~dy , 32* 3s ' 3s ' 3s ' Similarly, the direction cosines of the binormal to the Central Axis (which is the axis of the flexion couple) being CT\3s 3s 2 3s 3s 2 /' the components of this couple are v yds 3s 2 3* 3s 2 / Thus, taking moments about Ox, we have . cte" /3y' 3V_3*' ^y'\_\' tT dx'~\ _ VJW 3V_3£ ay\-| 3s P \37 os 2 3s W) L ^Jo L \a» &* & 3s 2 /Jo + y'C - z'B + P /^(y 1 'B 1 - *& + ^ 1 )ds 1 = pfMv& - «, *i' + li)rf«, i^<¥s-%'soi <-> Now, if we multiply the second of equations (92) by z, and the third' by y', and subtract one from the other, we obtain y'C - z'B = qy - Bf! - pfMv'&x - *i') - 2 '(li - yi)¥\ > or + / 42S BEAMS AND WIRES. [360. and on substituting* this result in (93) it becomes * s + Kl' '% -IS)* <* - "-• * '/*» - «* « p/"*,(W - »' x& - =.') - <«.' - »'xs, - ill*, r. ax' /ay' a 2 z 3* ay\~| , Q , and two similar equations may be obtained by taking moments in the remaining coordinate planes. For practical purposes equations (92) and (94) are more readily available after being differentiated as to s; so that finally we write ^ + P&(2-2/')=0. (95) ^4-pJl(E-i') = OS and [on elimination of A , B , by means of (92) after differen- tiation of (94)] 3»L 3s ^3s 3s 2 3s 3s 2 /J 3s 3s r. 3v' /3a' L tT 3! + H^ 3z' 'She' ~dx' 'd-z 3s 2 3* )] as|_ a* \a» 3s2 3s 3s2 /J a * + pJKI£-1) = os 3s + />Jl(ja-m) = A* 1 ' V .(96) /3r' S 2 ^' 3y' "SPx^ + P < |t(it-n) = / We also have, as the analytical expression of assumption (ii.) above, 7^»*' —Tin' . nz .(97) .3e' i}3')=0. If a be the length of either arm, we have per = C a when x' = a, so the constant of integration is zero, and or the curvature at every point is numerically proportional to the distance from the line of tension. Transforming the inde- pendent variable from s to z', this equation may be written and, on multiplying by Idx'fdz' and integrating, C n (D 2 -x' 2 )dx +/ \ Jip 2 - C*{D 2 - x"Y where D is an arbitrary parameter. This is then the equation of the curve into which the wire is strained; it is known as the Linea Elastica. When dx'/dz' = 0, x'= ± JD 2 ± 2p/C , so that if C be taken to represent the numerical magnitude * of the tension, the maximum distance of the curve from the line of tension is s/ u'^+'Zp/Cg, and the minimum distance (if any true minimum exists) is s/D 2 — 2\r/C . In the cases of Figures 55-58 D 2 must be therefore taken less than 2p/C , and in the case of Figure 60 greater. Figure 59 represents the intermediate case, in which D 2 = 2p/C , where the equation of the curve reduces to the integrable from s - = /T _ Jp :'* /p~. 2y/p + 74p - 6' .r'- "-V^-VS*"'^ Figures 55-60 are taken from Thomson and Tait's Natural Philosophy : they are copied from the actual forms assumed by flat springs of such small breadth that no appreciable tortuosity (and consequent torsion) was introduced by the crossing of the different branches. * This is positive if the ends are pulled apart as in Figures 58, 59, 60, and negative if they are pulled towards one another as in Figures 55, 56, 57. 361.] BEAMS AND WIRES. 431 432 BEAMS AND WIRES. [362. 362.] The Helix of Equilibrium of a Uniform Wire under no Impressed Force or Couple. Writing in the general equations of equilibrium ts for the resultant curvature, (X, p., v) for the direction cosines of the tangent, and (X', fx, v) for those of the binormal to the curve assumed by the Central Axis, under no impressed force or couple, they become dA dB _ ds ds dC_ ds \ as rX + JJCTX'] + C/i -Bv-- = —[tr/* - pSJ/i'] f Av- -CX-- = fr rv + pO>'] + .BX- -Ap- = XA+pB + vC = = ee Thus A=A , B = B , C=C throughout the wire, or the tension is constant in magnitude and direction. Let E be its magnitude, and let us choose the arbitrary axes of reference so that Oz may be parallel to its direction. Then (7=E, A=£ = 0, and the equations of equilibrium will be satisfied by the assumptions that e, r, cr are also constant, and that v=«e/E v tr— - + »EJ— - + E/* = as as ax ds .(98) dv'_ ds = Thus v, v are constant and \p! — XV = 0, or the tangent and binormal are inclined at constant angles to Oz, and the principal normal is everywhere perpendicular to Oz. The curvature being constant, it is obvious that the Central Axis of the wire assumes the form of a regular Helix, described upon a right circular cylinder having Oz for a generator. If r be the radius of this cylinder, and a the pitch of the helix, r = cos 2 a/CT, e = Esina/«. If we now transform the origin to a point in the axis of the cylinder, and choose Ox so that it shall pass through the end z' = of the wire, and if (f> denote the angle through which the arc 8 of the wire turns about Oz, we shall have x' = r cos , y' = r sin , «' = r<£ tan a, s = r<£ sec a, and the second and third of equations (98) both reduce to pCT sin a-tTCOSa + rE = (99) 362.] BEAMS AND WIBES. 433 When the magnitude of the couple applied to either end and the inclination of its axis to Oz are given, cr and t can be deter- mined, and (99) will then serve to determine a ; whence finally r and e can be found. 363.] Equilibrium under Terminal Couples only. Writing E = 0, we have e=0, and if P, T be the flexion and torsion couples (99) reduces to Psina — Tcosa = 0, while r = p cos 2 a/P. Let the magnitude of the couple be 0, and let its axis in any possible position of equilibrium make an angle \Js with Oz, on the same side of it as the tangent to the curve. Then P = O cos(a + ty, T = O sin(a + ^), and by (99) sini/r = 0. Conse- quently the axis of the helix is perpendicular to the planes of the terminal couples. Thus if equal opposing couples in parallel planes be applied to the ends of a fine wire, it will take the form of a uniform helix described upon a cylinder with its axis perpendicular to the planes of the couples, the length 1 and radius r of the cylinder, the pitch a of the helix (or angle at which it cuts the planes of the couples) and the angle through which it turns about the axis, being connected by the equations r = pcosa/C, l=Lsina, =LC/p (100) If only the magnitude of the couples be given, there are an infinite number of possible positions of equilibrium, but if in addition one of the geometrical quantities 1, r, or a be known, the solution is completely determinate. The curvature of the wire will be C cos a/p and its torsion C sin a/p throughout. 364.] Simplified form of the equations, when the maximum curvature of the Central Axis is very small. In this case the transverse displacements of all points of the Axis are small, and if Oz be taken to coincide with its unstrained direction, x' or u and y' or v may be regarded as small quantities, as well as z — z or v; (§ 359). To the first order of small quan- tities, s is equal to z\ and the operator d/ds is equivalent to d/dz. Thus equations (93) may be written \ .(101) g + P&(B-fi)-o The elongation of the wire at any point will now of course be given by -g, (102) 2e 434 BEAMS AND WIRES. [364. and the approximate values of t, 1, in, n may be found as follows. The curvature being very small, the displacement of any point in the transverse section Jt, relative to its centre of gravity, will be very approximately perpendicular to Oz, and due entirely to twist about the Central Axis. If 6 be the angular rotation of this section about the Central Axis due to torsion (§ 332), which we must in general suppose to vary from one section to another (§ 359) under the influence of impressed couple, and which is not necessarily very small, the amount of torsion, or rate of twist per unit length of wire, is evidently T = | fl (103) oz Let (£, tj, z) be the initial coordinates of that point of the section Jt whose strained coordinates we have denoted in § 360 by (x' + i', y' + y' , z '+f) i then we shall have very approximately Substituting these values in (91), and remembering that every diameter of the section through its centre of gravity (§ 353) is a principal axis of inertia, we find (on the assumption that the angular velocity Q of rotation of transverse sections about the Central Axis is small) l-ttt = 0; ^n = J 3 6> (104) Equations ( where i is any integer, and M t , a t are arbitrary constants. The velocity of transmission of sound along a wire is therefore y/qjp. This result should be compared with the velocity »J(m+ri)/p, obtained in § 268, for transmission through an infinite medium. We shall write Q 1 for s/q/p. 366.] Lateral Vibrations in a Fixed Plane. The wire being equally flexible in all directions, we may take any plane 436 BEAMS AND WIRES. [366. through the Central Axis, e.g. the plane of zx, as plane of vibra- tion. Annulling therefore v, tu, and 8, we have B = 0, 1£ = 0, A^+C = 0\ dz oz Eliminating C between the third and fifth of these equations, and neglecting the square of du/dz, we obtain and, on differentiating as to z and eliminating A, the equation of motion reduces to The tension is, to our order of approximation, entirely transverse (i.e. due to shearing stress only), and its value is A =~% < 108 > the flexion couple being P CT = +V^n (109) OZ" The equation satis6ed by the amplitude u, (§ 260) is or, if we write then p£'-^<=°' ■ = v_ I v 1 vi P & d*u { i* M-IP ( 110 > The general solution of this equation is « i = ^/ i sin^ + J//cos 1 * + i\r i 6inh^ + jr/coshi5 (Ill) L h L L where the four coefficients are arbitrary constants, to be deter- mined by means of the terminal conditions at the two ends. These are as follows : — 366.] BEAMS AND WIRES. 437 (i.) at an end that is absolutely free, there can be neither force nor flexion couple, so that at such an end • S=°> *-° < 112 > (ii.) at an end that is fixed in position, but so that the ter- minal portion of the wire is free to change its direction (e.g. a hinge), the displacement is zero, and so is the flexion couple, and at such an end « = °. gh° < 113 > (Hi.) at an end that is clamped so that the terminal portion of the wire is fixed in direction, the tangent to the Axis at its termination must coincide with its initial direction : at such an end therefore « = 0, g = (114) (iv.) at an end carrying a rigid mass M, but otherwise free, the couple vanishes, while the transverse force must obviously be equal to the effective force on M. Thus at such an end 3ii = °> ^--M. ( 115 > For the applications of these conditions, and the different forms of the general solution to which they give rise, the student is referred to Lord Rayleigh's "Theory of Sound," Chapter VIII., and to Donkin's " Acoustics," Chapter IX Deflection of Uniform Mods from the Horizontal, under Gravity. 367.] General Equation of Equilibrium. Let a thin rod or wire rest upon any number of rigid supports in one horizontal straight line. It is required to determine the small deflections of the rod from the horizontal, between. the supports, caused by its own weight. Take any point in the line of supports for origin, and that line for axis of z, and let Ox be directed vertically downwards. It is evident that the deflection will be entirely in the vertical plane of zx, and we have X = g, Y= Z— 0, so that JE = g, D = E = 0. IP = Jft = £L = 0. Thus the equations of equilibrium reduce to 438 BEAMS AND WIRES. [367. and, on elimination of A, ?-& = *&• Integrating this equation four times, we see that the curve assumed by each portion of the rod terminated by consecutive supports, or by a free end and a support, is represented by an equation of the form u = k + « x z + K 2 Z 2 + Kg? + /ogJU*/24p, (116) where k , k v k 2 , k s are constants, different in general for each such portion. To show that the solution is completely determinate, we will take the general case in which there are p supports at given distances apart, and the ends are either free or clamped. The rod will be divided into p + 1 curves, each represented by an equation of the form (116), and there will consequently be 4p + 4 constants to be determined. Now, (i.) the line of supports being the axis of z, the values of u, as deduced from the equation of any portion of the rod, must vanish at each of the supports which bound that portion (2p equations), (ii.) the curvature of the rod being necessarily continuous, the values of du/dz and d 2 u/dz 2 at each support, as deduced from the equations of the curves on either side of it, are necessarily equal (2jo equations). (Hi) at either end, whether free or clamped (§ 366) two conditions must be satisfied (4 equations). Thus, on the whole, we have exactly 4p + 4 equations of con- dition to determine the 4p+4 constants involved. The thrust on any support is equal to the difference in the values of A, immediately on either side of it. 368.] Rod supported by one end only, that end being clamped in a horizontal position. In this case, u and dujdz must vanish at the clamped end (z = 0), while d 2 u/dz 2 and d^/dz 3 vanish at the free end (z = L) (§ 366). Thus «„ = Kl = 2gJl// 4 /o84p. The ends of a rod supported by its middle point suffer there- fore only 4 of the depression experienced by the middle point of a rod supported by its ends. Greenhill's Problems on Stability. 371.] Method of Investigation. In the three following- problems we have to determine the conditions under which the natural straight form of a beam, in given circumstances, becomes unstable, In each case it is obvious that this instability will be caused by the undue increase of a geometrical quantity involved (in §§ 372 and 374 the length of the beam, and in § 373 the angular velocity of rotation). Professor Greenhill's method con- sists in supposing the limit to be passed, and a small deflection of the Central Axis from the straight, line to have taken place, and in determining the least value of the critical quantity for which this deflection can be maintained. This, being the limiting value which divides the two states, is obviously also the greatest value consistent with stability in the original form. 372.] The maximum height of a vertical pole, con- sistent with stability under gravity.* Let a pole of length L, in the form of any solid of revolution about a vertical axis, * Mathematical Tripos, 1879, and Proceedings of Canib. Phil. Society, vol. iv., page 65. 440 BEAMS AND WIRES. [372. with its base rigidly fixed in a vertical direction, be supposed slightly deflected from its naturally straight form. Taking the highest point of the axis for origin, we have X= 7=0, Z=g, and consequently J = 1 = 0, Z = g, W = M = & = °- K the plane of xz coincide with the plane of flexion, equations (101) and (lOo) reduce to dzVdz 2 ) dz J and, when C has been eliminated, the equation of equilibrium becomes d ( d 2 u\ du C z .*. , A If r denote the radius of the section ,31 at distance z from the summit, 3>=7rr 2 , and y=\-wqr" (§§ 349, 353), so that d/ 4 dhA + ipg du /" iadg = (120 ) dz\ dz 2 / q dzj o When r varies as any given power of z, the solution may be obtained in terms of Bessel's functions, for on putting r = z"/D p ~ 1 our equation reduces to ^(du\ ±d(du\ ip gD>»-> du = ~ dz\dz) F 'dz\dz) (2p + l^z 2 *- 3 dz ' the solution of which — satisfying the terminal condition that the curvature d 2 u/dz 2 vanishes at the free end 2 = — is du = j.^j |~ 42)"-' / _pg -| dz ' ^*L(2p-3)^-*V(2p+l)s-J where « is an arbitrary constant. Since the base of the pole is rigidly fixed, we must have du/dz = 0, when z = L, and consequently , r 4Z>- / Pg -i Thus, if i be the £eas£ positive root of the equation Jto-i(i) = 0, (121) 3-2p and if x = r i6pg^2 -1*=-. ... {122) L(2^ + l)(2^-3)V 2 J ' V ' 372.] BEAMS AND WIRES. 441 then if L 2 u, v= —a>\ and the equations of motion become d 3 u , d s v t, . dv lt du n First of all, by eliminating A and B between the last three equations, we obtain da d 3 v _ dv d 3 u _ „ dz dz 3 dz dz 3 du d 2 v dv d 2 u !«w*^ =constaat: and, since du/dz and dv/dz both vanish at the bearings, this con- stant is zero. Thus the component curvature of the Axis in the plane per- pendicular to Oz is everywhere zero, or the form assumed by the strained Axis is a plane curve. Writing then r=(u 2 + tf)t for the radial displacement, we have the solution of which (compare § 366) is r = M cosh '- + M' sinh i? + Ncos^r + N' sin l * L L h L where i* .= pJUM* * Mathematical Tripos, 1878. 442 BEAMS AND W1KES. [373. Since r and drfdz vanish at both bearings (z = and z = L), the coefficients must satisfy the relations M + N= M' + N' = -¥cosh i + M' sinh i + iVcos i + N' sin i = M sinh i + M' cosh i — N sin i + JY' cos i = 0, and consequently on elimination of M, M', N, N', cos i . coshi= 1 (123) If therefore i be the least positive root of (123), the critical angular velocity is given by i 2 IV . ^- (124) for if (o< * Mathematical Tripos, 1881, and Proceedings of the Institution of Mechanical Engineers, April, 1883. .(126) 374.] BEAMS AND WIRES. 443 throughout the shaft, if *efe* ds? da 2 ^ J A possible solution is m = KjCos iz + k 2 cos/z + k 3 cos az cosh /Ja + K 4 sin az sinh /8« 1 i; = — KjSin ia - R,siiy's — « 3 sin az cosh /?« — k 4 cos az sinh /3z j where i, j are the real roots, and a ± t/3 are the imaginary roots of the quartic equation pA.4 - TA.3 + EA.2 - />JU 2 = (127) 375.] The most general case is complicated to work out, but two simple cases may be solved. If we suppose that the ends of the shaft are absolutely fixed in position, but able to change their directions by exerting couple on the bearings -the only terminal condition necessarily satisfied is that u = v = at each end. Similarly, if the ends be clamped rigidly in their initial direction, but able to force the bearings aside from their initial line the only necessary terminal condition is that du/dz = dv/dz = at each end. In either of these cases, we need only retain the completely periodic terms in (126), the solution being in the first case of the form u = k(cos iz — cos jz) ) v = — /<(sin iz- sin/2) J and in the second case of the form (cos iz cos jz\ i -r—r) I (sin iz sin jz\ j — »~ ~rV with the condition in each case that (»' ~j)L = 2/>Jr, p being any positive integer. The critical length in these cases is therefore given by L = 2,r/(t~j) (128) 376.] In the case where the angular velocity of rotation is moderate, and the thrust and torsion couple very great, so that the effects of inertia are negligible in comparison, the quartic (127) reduces to the quadratic jjA.2 - TA + E = 0, 444 BEAMS AND WIRES. [376. the roots of which, E being essentially negative, are real. Thus i~j= JT 2 -ipE/\>, and the critical length L is given by ^ = T!- E (129) V 4p 2 V ' This solution is very approximately applicable to the screw shafts of large steamers, and accurately true in the case of equi- librium under thrust and twisting couple. Naturally Curved Wires. Circular Hoops. 377.] Equations of Motion and Terminal Conditions. If a wire of infinitely small section have for its Central Axis a curve of any form, but everywhere of finite curvature, an elementary length of the wire can always be measured from any transverse section, such that its length is of at least the same order of dimensions as its greatest diameter, and yet so small that the portion is practically straight. The conditions of strain and stress in such an element may be taken to be the same as in a naturally straight beam, and by superposing one such element upon another until the curvature of their aggregate becomes sensible, it will appear that the conditions of strain in a wire of naturally finite curvature may be deduced from those of a naturally straight wire simply by substituting for "curvature due to strain " " change of curvature due to strain," and for •' direction of the unstrained Central Axis " " direction of the tangent to the unstrained Central Axis at any point!' With these changes the considerations (i— v) on page 425 are applicable to naturally curved wires of equal flexibility in all directions, as are the terminal conditions on page 429, with the exception of (iv), vs denoting the change of curvature due to strain. Equations (95) and (97) also retain the same form, but equations (96) and the terminal condition (iv) become more com- plicated, owing to the form of the flexion couples. If k = Ws d* ~ Ws **> etc -' x = i a? ~ Ts 3?' etc "' th en the natu ral curvature at the point (x, y, z) is approximately vA 2 + yu 2 +j/ 2 , and the altered curvature at the corresponding point (x, y', z') is s trictly J\' 2 + fJL '2 + v '2 : thu s the resultant flexion couple is p(s/X' 2 + ju' 2 + v' 1 - J X 1 + ^ + v 1 ), and the direc- tion cosines of the osculating plane of the strained Axis, in which this couple acts, are as before X'/s/X^+fi'^+v'i The components 377.] BEAMS AND WIRES. 445 of the flexion couples, parallel to the arbitrary coordinate planes, are now therefore p\'( Jk'2 + // 2 + v' 2 - JJ? + (1? + v 2 ) . etc., instead of simply pX', etc. 378.] The Energy Method. Owing to this complication of the ordinary equations of motion and equilibrium, problems on curved wires and hoops are generally attacked by means of the Energy Method. Let the natural form of a uniform wire of unequal flexibilities be such that the curvature at any point of the Central Axis is vs, the osculating plane at that point making an angle a with that principal plane of inertia in which the coefficient of flexion is p, ; and let the component curvatures in the principal planes be zs v and ct 2 , so that S x = ex cos a, cr 2 = st sin a. Then, if the effect of the strain be to change t5, vs v 5 2 , a to vs, vs v vs 2 , , and to produce at the same time extension e and torsion t, the potential energy per unit length of wire at the point in question will be, by § 358, M = H^ + Fi(^i-5 1 ) 2 + p 2 (CT 2 -5 2 )2 + tr2} (130) where ts 1 = zs cos <£, nr 2 = gt sin . The tension, torsion couple and flexion couples will be with our previous notation .(131) oe T = tr or If the wire be of equal flexibility in all directions, the poten- tial energy per unit length is ffit = J{w 2 + p(cj-CT) 2 + tT 2 } (132) and the resultant flexion couple in the final osculating plane at any point is P = p(C7-5) (133) If £ be any one of a system of coordinates defining the con- figuration of the wire in any state of strain, the resistance per unit length offered by an elementary portion of the wire to the 44(i BEAMS AND WIRES. [378. increase of £ is of course 3«&t/3£ In other words, there is a force 9cS/3£ if g is linear, or a couple 9SB/3£ if £ is angular, per unit length, on each element of wire, tending to diminish £. From (130) and (131) we easily deduce that this action may be ex- pressed in the form ^-- E l +T |* p .f* p -1*- < 1M > 379.] Rotation of a wire of equal flexibility about its Central Axis. The formula (133) shows that the energy of such a wire, whatever its natural form, depends only upon extension, torsion, and change of resultant curvature. If there- fore the wire be set in motion in such a way that each point describes a circle about the centre of gravity of the normal transverse section in which it lies, no resistance will be offered to the motion except that due to inertia. We have thus an ideal means of transferring rotatory motion without loss of energy from one rigid axis to another in any other direction, by con- necting them to the terminals of a perfectly elastic wire of uniform flexibility, so placed that in its natural form the tan- gents to the Central Axis at either end coincide exactly with the axes of rotation. This result does not apply to curved wires of unequal flexi- bilities, because, even if the resultant curvature be maintained constant, the component curvatures in the principal planes of inertia at each point must change periodically during each rotation. (See § 382.) Applications of the Energy Method. 380.] Stretching of a uniform circular hoop of equal flexibilities in all directions. If the Central Axis of a uniform wire be in its natural state a circle of radius r, and if the wire be stretched, without change of the circular form and without torsion, until this radius is increased to r, the length of the wire will be increased from 2ttt to 2irr, and its curvature diminished from 1/r to l/r. Thus we shall have e—(r — r)/r, and ™=i{<^^-m _ (r - r) 2 (tr 2 + y) 2rV The tension will be t(r — r)/r, and the flexion couple in the plane of the wire exerted across each transverse section will be p(r — r)/rr. 380.] BEAMS AND WIRES. 447 Since the strain is expressed entirely in terms of the linear coordinate r, the resultant action on each element of the wire is a force towards the centre, of magnitude dM_ (r-r)(t* a + yr) dr vV per unit length. 381. J Small radial vibrations in the plane of the hoop. Let us now suppose that the wire performs small vibra- tions about its natural configuration, the displacement of each point being wholly radial, and the form of the Axis always circular. The velocity of each point will be r, and the kinetic energy per unit length will be faQr 2 . The principle of conser- vation of energy therefore gives us 2*r { ip&r* + (r-'y+P) I = constant . Differentiating as to t, and writing r = r + u, where u is a small quantity of the first order compared with r, the equation of motion reduces to er 2 + v n U + M^=— P = 0, so that the periodic time of small vibrations is 27rV P JU7(tr !! + p). 382.] Wire hoop of unequal flexibilities. A hoop has naturally the form of a circle of radius r, the plane of greatest flexibility (i.e., that in which the coefficient of flexion is least) at each point malting an angle a with the plane of the circle. It is stretched into a circle of radius r and held so that the plane of greatest flexibility makes everywhere an angle (j> with the plane of the circle. Here, if ft be the least and ft the greatest coefficient of flexion, we have with the notation of § 378, ST, = cos a/r, G7 2 = sin a/r, ETj = cos /r, E7 2 = sin /r, and € = (■? — r)/r. Thus __. , f (r - r\ 2 /cos cos a\ 2 /sin <£ sin a\2 ) « = *{*(—) +*(—?-— ) + P<-^-— ) }. and the action on each element of the wire consists of (*.) a radial force tjr-r) - ft /cos a _ cos /cos a _ cos \ ,. „.. r\r r ) r \ r r / ^ per unit length, in the normal plane of the wire at each point, tending to turn it about its Central Axis, and restore to the angle <£ its initial value a. This couple vanishes when r _r (p i! -y 1 )sin2^ (137) 2 ' p„cos sin a — pjSin cos a so that, for every value of less than tan _1 (p 2 tan a/pj), it is possible to stretch the hoop so that its elements may experience only radial force. For every assigned value of r it is possible to determine $ so that the turning couple may vanish. If the external action be confined to keeping the hoop stretched, it will assume a form in which

2pj. The first value of

= sin _1 (r sin a/r) ; or, in other words, the change of the component curvature in the plane of small flexibility is negligible in comparison with that of the other component. The flexion couple in the plane of greatest flexibility will be (cos a cos \ — ~-r)> and, since from symmetry the resultant flexion couple must be in the plane of the hoop, this latter will be y t /cos a cos \ cos \ r r )' BEAMS AND WIRES. 449 EXAMPLES. [In the following Examples all wires — unless the contrary is expressly stated — are to be supposed uniform, naturally straight, of equal flexibility in all directions, and free from, the action of any impressed force or couple, except at their end,s or " points of support."] 1. Opposing couples of 10° centimetre-grammes are applied to the two ends of a round bar of iron (Young's modulus about 1000 million grammes weight per square centimetre) of 2 J centi- metres diameter. Find the curvature produced. Also the greatest curvature possible within the elastic limits of the material, and the couple required to produce it, assuming three million grammes weight per square centimetre to be the tenacity of the iron. 2. Show that, if a wire be subjected to tension and flexion couple of such magnitude that the product me is sensible, the coefficient of flexion will be p(l + e). 3. If a uniform bar, both of whose ends are fixed, be so dis- placed longitudinally that initially one half is*uniformly extended and the other half uniformly contracted, prove that the displace- ment at time t will be given by 2e£J=r' 1 (2i'+1)ttz (2i+l)7rJ2.« W = V t^t- — ttv; cos 5^ — cos r ' *" ~o(" 2i+1 ) L L where e is the initial extension, and Q. x the velocity of sound (§ 365) along the bar. 4. The extremities of a bar are attached to springs of equal strength. Show that, if w, = M- t sin (}z : L) + JS T { cor ()z/'L) be an amplitude for longitudinal vibrations, then i is a root of (iV - £V) *<" i + ~^ L l l = °> where /x is the " strength " of either spring {i.e., the ratio of the tension applied to the increase of length produced). 5. A steel cylinder of small elliptic section, and L centi- metres in length, is clamped at one end. The gravest note when vibrating transversely in one principal plane of flexion, and the note above the gravest when vibrating in the other principal plane, both have a frequency of 256 per second. Show that the 2f 45() BEAMS AND WIRES. semiaxes of the section are about 000179L 2 and = 0, and the terminal conditions are (taking the origin at the lowest point) 6 = a sin it when z = L, and. tdd/dz — 10 = when z = 0. In the case of free vibrations, i must be such that the total energy of the system remains constant, so that i/Vls^ 2 + t(d^) 2 ]dz + J/0*,_ is independent of t. Evaluate this expression and equate co- efficients of cos 2 it and sin 2 i<.] 8. A wire is held bent by suitable forces between two points A and B so that, the area between the wire and A B being given, the work expended in bending the wire may be the least possible. Show that the curvature at any point varies as r 2 — D 2 , where AB = 2D, and r is the distance of the point from the middle point of AB. Show also that, if the wire be bent completely round to satisfy the same conditions, the curve assumed will be of the form I s = C 3 cos 3d. BEAMS AND WIRES. 451 9. A square ABGD is formed of four rods, each of length L, clamped together at the corners, the rod AB being elastic, and the other three rigid. If the system revolve about CD with uniform angular velocity &>, such that \L is a small quantity where X is given by X 4 = pJU) 2 /?. the displacement of any point of AB at a distance z from its centre will be j-sinh £XL(cos Xz - cos JA.Z) + sin JAL(cosh Xz - cosh \XL) sinh \XL cos %XL + sin %XL cosh ^XL 10. If, in the case of § 367, the rod be clamped at any one support in a position other than that which it would naturally assume when freely supported, an impossible identity will apparently be introduced. Explain this. 11. Calculate the pressure on each support of a heavy bar which rests upon four equidistant supports, two of which are at its ends. 12. A heavy beam of length L is cut in half, and one of the halves is again divided into four equal portions, which are placed upright in a row at equal distances \L from one another. If the remaining half be placed upon these, and if P v P 2 be the thrusts on the intermediate and extreme vertical beams, show that P 1 :P 2 : \p&L:: H«i 2 + 486p : 4e£ 2 + 486p : 10e£ 2 + 64 8p. v^ 13. A heavy uniform wire is supported at a series of points in the same horizontal straight line. If P 4 be the flexion couple at the ith point of support, and D, the distance between the (i — l)th and ith supports, prove that P M z>, + 2PXA + A +1 ) + Pm A +1 = \pM(W + A +1 3 ). 14. A weightless wire is supported at its two ends, and at its middle point 0, and a small weight W is suspended from it at a point Q. Show that the downward vertical displacement u of any point P is given by - *v(* - 1 £)(~*i - m ± *h\w{z* + **) + &*]}, where z, z v D are the numerical values of the distances OP, OQ, PQ, and the upper or lower sign is to be taken in the ambiguity according as P lies on the same side of as Q, or on the opposite side. 15. A rod OABC is constrained to pass through three fixed points A, B, G in one straight line. Show that the deflection of the part OA, due to forces acting on that part only, is the same as if the rod had been constrained to pass through two points A and X only, where ' . „ .JSAB + iBC 452 BEAMS AND WIRES. 16. If the rod be constrained to pass through an infinite number of points, at intervals each equal to AB, the constraint — as regards the part OA — will be the same as if the rod had been constrained to pass through A and Fonly, where A F= £v 3 . AB. j 17. The maximum height for stability under gravity of a conical pole of semi-vertical angle a is T _ 3gi 2 tan 2 a where i is the least positive root of the equation J 3 (i) = 0. * 18. The maximum height for a paraboloid of revolution, of latus rectum D, planted with its vertex upwards, is where i is the least positive root of the equation J,(i) = 0. 19. If a rod revolve about its Central Axis under a given tension, prove that the straight form will be unstable when the number of revolutions per second exceeds the number of lateral vibrations executed in a second by the same rod under the same tension. ^ 20. A rod of given length, securely clamped at one end and with the other end free, rotates about its Central Axis. Show that the greatest angular velocity consistent with the stability of the straight form is given by the least positive root of cos i . cosh i = — 1, where i has the same value as in § 373. 21. If the clamp in the last example be replaced by a universal joint, and the angular velocity be such that i is a root (other than the least) of the equation tani = tanhi, the Central Axis of the wire will describe the surface of revolution r = Ml sin i sinli t + sinh i sin j \. 22. If the hoop of § 380 be cut, and the ends twisted through p complete turns and then joined again, and if the hoop be con- fined between two perfectly smooth parallel planes, so that the Central Axis must remain always a plane curve, the radius of the hoop in equilibrium will be a root of the quartic xr 4 — .err 3 + prr - (p +p*{)r* = 0. If this radius be denoted by i v the time of a small vibration about the position of equilibrium will be 2 T j_ziLM r v^ .. <• • > £- , ; .*> .. ,. \ > e>V + m(3r - 2r,) + 3pH* BEAMS AND WIRES. 453 23. A wire is twisted, and strained into the form of a helix, and its ends are then joined, so that it forms an endless spiral curve round a tuhular core. Find the direction and magnitude of the resultant stress across any transverse section. V 24. £ and r\ are conjugate functions of x and y, the f curves being closed. If a wire have for its Central Axis a curve of the £ family, and if it execute small vibrations in its own plane, so that each point moves along the principal normal to the Central Axis, and this latter remains always a curve of the same family, the time of a small oscillation will be n ]' where the integrals are to be taken all round the curve, and f is to be given its initial value after differentiation. 25. Apply this result to obtain in terms of elliptic integrals the time of vibration of an elliptic wire which always retains the form of a confocal ellipse. 26. A perfectly rough bar has in its natural state the form of a circular arc of length L and curvature ST. A ring is formed by joining the ends of the bar, and is laid upon a smooth hori- zontal table, with its centre over that of a circular hole in the table. A perfectly rough cone, of very obtuse vertical angle 2/3, and of weight 2ir%q, is placed gently on the ring, with its axis vertical and vertex downwards, and sinks under gravity. Prove that if the ring never approaches the edges of the hole, if its inertia be neglected, and if its section JV be supposed always to remain circular, the time of a small oscillation about a position of equilibrium is to the first approximation V g(8ir + Lis cosec 2 /? cos y)' where 7 = ---7=- cosec 2 /? cos /3. j> BO t/27. Show that if ajod be set in vibration with initial trans- verse displacements and velocities given by u = (z), u = yjs(z) : the displacement at any subsequent time t may be represented by u= X f f \ <£(£) cos tj'^t + ripain if-wt {. cos r)(z - g)d^h h — 00 —x where co 2 = vlp'&. (Fourier's solution.) /£w£<^ L < ^^> . 454 BEAMS AND WIRES. 28. Obtain directly St. Venant's solution for the torsion of beams (§ 333) by adopting the conjugate cylindrical coordinates £ rj, z of Example 4 (i.), page 258, (making p unity), and assum- ing u = 0, j8 = 2t. [It wall be found that the general equations of equilibrium are satisfied by making div/dz = and d 2 w/d£ 2 + chv/dT? = 0, while the boundary conditions reduce to 3f aje \dq /dr) The differential equation of the bounding surface must there- fore be (| + CVe=<), S -|^0; or, if and w be any conjugate functions of £ and >?, and therefore also of x and y (Example 2, page 257), \3f / oil This is satisfied by all surfaces of the form + %C 2 Te^ = constant, or + \r(x 2 + y 2 ) = constant. [See also Boussinesq's Application des Poientiels cb Ve'tude de I'e'quilibre et du mouvement des Solides Elastiques, pp. 435-4G3.] APPENDIX V. Strength of Materials under Torsion and Flexion. Strain- Reversal (Nachwirkung). Strength under Torsion. A cylindrical bar, subjected only to torsion couple applied to its ends, experiences only shearing stress (§ 330). The elastic strain produced depends therefore entirely on the rigidity of the material, and the only elastic limit involved is its hardness (page 181). To exhibit clearly the form of yielding to torsion stresses exceeding the elastic limit we will consider the simple case of a right circular cylinder (§§ 335, 341), following the account of Prof. James Thomson.* * Cambridge and Dvblin Mathematical Journal, Nov. 1848, sections 10-20 ; reproduced in Sir W. Thomson's article on Elasticity in Encyclopaedia Britannica, section 9. BEAMS AND WIBES. 455 The shearing stress under torsion t is [§ 341 (i.)] S = nrr, where r is the distance from the Central Axis: the torsion couple (§ 335) within the elastic limits is T = Jxw^V, where A is the extreme radius of the cylinder. Let us first suppose the material to be perfectly plastic (p. 170) and of solidity S. If the torsion couple be gradually increased until r = S/nA and T = Jtt^ 3 S, the limit of elasticity will be reached for the extreme bounding portion of the cylinder, which will then begin to flow. When still greater torsion has been produced, so that r s S/n T is less than A, all that portion of the cylinder between the surfaces r and A will be in a state of flow with a uniform stress S throughout, while the portion comprised within the surface r will still be in a state of elastic strain. The torsion couple will then obviously be T = r r -^-.t\. 2th-,*-! + f A 8 . *i . %rr x dv x r = ^r 3 S + H4 3 -r 3 )S. As the torsion steadily increases, the radius r of the surface limiting the flow diminishes, until we reach a limit at which all but that portion of the cylinder in the immediate neighbourhood of the Central Axis has suffered flow. At this limit the maximum value T = §7r-4 3 S of the torsion couple is reached, which is £ of its value at the elastic limit. This therefore represents the inaximuyn resistance to torsion of a plastic cylinder of radius A. If now the torsion couple be gradually removed, there will be an elastic torsional recoil or untwisting of the wire, which will diminish the stress at each point by an amount proportional to its distance from the Central Axis. Suppose that, when the couple is entirely removed, the stress at distance r from the Axis is S — Gr : then /' (S-CV).r.27rnfr = 0, or C=^S/A. Thus the permanent stress due to set is a shearing stress S(l — ir/A). All that portion of the bar contained within the surface r = \A is permanently strained in the direction in which torsion took place, while all the portion without that surface is permanently strained in the opposite direction. The bar is, in fact, left in a very marked state of constraint (see page 182). The permanent stress being everywhere within the elastic limits, it follows from the principle of superposition that the stress produced by the application of a fresh torsion couple, in either direction, will be of the same form as if the bar were in a state of ease (see page 185), i.e., proportional to r. Thus if Tj be 450 BEAMS AND WIRES. the couple sufficient to produce flow, when applied in the original direction, we shall have f (V"V + S(1 - lr/A)]r. 2imfr = T 1 , where G X A — £S = S ; and consequently T 1 = ^ttA^S. Similarly, if T 2 he the Couple required to produce flow in the opposite direction, f [G.f - S(l - PI A )]r . iv-rdr = T,, where C' 2 ^1 + JS = S; and therefore T 2 = ^7r^. 8 S. Thus the strength of the bar under torsion is twice us great in ike direction in which it was originally twisted, as in the other. These results are only true numerically of bars of perfectly plastic material, but the principle is obviously applicable also to ductile materials, the hardness of which increases during flow. Thus it is evident that the apparent strength of a bar under torsion may depend very largely upon its previous elastic history. Strength under Flexion. In this case the strain is a longitudinal traction or pressure, proportional to the distance from the neutral plane. Taking the case of a rectangular bar of plastic material, of depth 2D and breadth B (see page 420), and of equal strength C under tension and thrust, it is easy to show that (('.) The elastic limit is reached when the flexion couple amounts to §D 2 BG. (ii.) The maximum strength to resist flexion is D 2 BC. {Hi.) On removal of the couple the tension of fibres distant y from the original neutral plane is R= —G(l — Sy/2D), the axis of y being taken as in §§ 343-349. Thus the stress vanishes at distances ± §D from the original neutral plane. (iv.) If the beam be bent again in the same direction, its strength is given by (ii.). (v.) If it be bent in the opposite direction, its strength will be |D 2 i?C, or one third of the former. We see therefore that in the case of flexion, as in that of torsion, the apparent strength of a bar, even of the most regular kind of material that we can imagine, depends chiefly upon the relation of the method of testing to the processes to which the bar may have been previously subjected. Strain-Reversal (Nachwirkung). This phaenomenon is described in this place because it appears most prominently, and was first observed, in connection with torsional strains. It is, BEAMS AND WIRES. 457 however, in all probability an invariable accompaniment of all strains which approach or surpass the elastic limits of the body. The following account is extracted from Prof.Tait's "Properties of Matter," §§ 251, 252. The phenomenon is of purely physical interest, and no satisfactory explanation of it has yet been advanced. '* All this part of our subject is still very imperfectly worked out. . . . There is no doubt that all elastic recovery in solids is gradual, so that, for instance; in . . . torsion vibra- tions . . . , even when there is no sensible viscous resistance, the middle point of the range does not coincide with the original untwisted position of the wire. It is always shifted towards the side to which torsion was applied, and to a greater extent the longer the wire has been kept twisted before being allowed to vibrate. With every vibration, however, it creeps slowly back towards the original undisturbed position, but usually comes to rest before reaching it. . . . These phenomena are seen in a more striking form when we dispense with oscillation. Thus, f or example, suppose the wire to be kept twisted through 90° to the right for six hours, then for half an hour 90° to the left, and be then so gradually let go that there is no oscillation. When it is left to itself it turns slowly towards the right, gradually undoing- part of the effect of the more recent twist, then stops, and twists still more slowly towards the left, thus undoing part of the quasi- permanent effect of the earlier twist. Thus the behaviour of such a wire, strictly speaking, is an excessively complex one, depending as it were upon its whole previous history ; though of course the trace left by each stage of its treatment is less marked as the date of that stage is more remote. This subject has of late attracted great attention in Germany, and, under the name Elastisclie A ' achwirkung , has been the object of numerous researches by Wiedemann, Kohlrausch, Boltzmann, etc." A sketch of Clerk Maxwell's theory of this peculiar action is given in § 253 of the same work. 458 PLATES AND SHELLS. [384. CHAPTER VIII. PLATES AND SHELLS. Introductory. 384.] Definitions. The term Plate will be used in this Chapter to denote a body cut from a right cylinder or right prism of any form by two (necessarily parallel) normal sections. These sections form the Faces of the Plate, while the intercepted portion of the original bounding surface of the prism forms the Edge or Edges of the Plate. The Thickness of the Plate is the normal distance between its faces. A Thin Plate is one whose thickness is a small quantity of the first order compared with its least transverse dimension. A plane drawn parallel to either face, and equidistant from both, will be called the Median Plane, and the section of the plate by this plane its Median Surface. The centre of gravity of this section is the Centre of the Plate. The section of the plate by any plane perpendicular to its faces is called a Normal Surface. The straight line drawn through the Centre, perpen- dicular to the faces, is the Normal Axis. The form of the plate is determined by that of the prism from which it is cut : — thus a Circular Plate is derived from a right circular cylinder, a Square Plate from a right prism of square section, and so on. A Closed Shell is a body contained by two surfaces belong- ing to one of a set of three orthogonal families; the surfaces being each of one sheet, and one of them entirely enclosing the other : e.g. — a Closed Spherical Shell is contained between two complete spherical surfaces, one of which entirely encloses the other, but which may or may not be concentric. An Open Shell has for its faces two surfaces of one family, while its edges are formed by surfaces of one or both of the remaining families of an orthogonal system. 384.] PLATES AND SHELLS. 459 In a Thin Shell the thickness — here measured by the length of arc of the orthogonal curve intercepted between the faces of the shell — is a small quantity of the first order compared with its least superficial dimension. 385.] The Class of Strains to be investigated. Exclu- sion of Surface Tractions on the Faces of the Plate or Shell. Throughout the present Chapter, plates and shells will be supposed subjected to the action of Surface Tractions on their edges only, with or without the accompaniment of Impressed Forces. No stress will in any case be supposed to act across the faces of the plate or shell. Closed shells will be considered as performing vibrations under normal impressed forces only. Glebsch' Problem: Straining of a Plath of Finite Thickness, free from Impressed Forces, by Surface Tractions applied to its Edges alone, in directions everywhere parallel to its faces. 386.] Statement of the Problem. Taking the origin at the Centre, and the Normal Axis for axis of z, the boundary con- ditions M = S=T=0 must be satisfied over the entire area of either face, with the further condition \T+/nS = over the whole of the edges. Since the edges are everywhere parallel to Oz, the three stress components R, S,.T are not involved in the actual surface tractions, which will therefore be none the less entirely arbitrary if we at once make the assumption * that S = S=T.= (1) throughout the substance of the plate. The boundary conditions will then be satisfied identically, as also will the third of the general equations of equilibrium [(1 03) of § 285], and it only remains to determine the most general values of u, v, and w that will satisfy the first and second of these, as well as (1) above, throughout the plate. 387.] Solution of the Problem. Substituting from (1) in equations (40) of § 214, we have 'dm civ _ 3m ~dw _ .., .„. 3g/ ~dz "dz "dx I] «-'>5"(£ + §H <3> * Compare the assumption made in (5) of Article 326, page 388. 460 PLATES AND SHELLS. [387 to be satisfied concurrently with the first two of the general equations, viz. : — 3 (~du ~do 3wA dAfix dy 3 /du 3u . 3«) 3y\3a: dy + (l-2V M^V 3*VSV ( whence 3^ = C :Cc + (8) whei-e 6' is a constant, and w a function of x and i/, which by (G) must satisfy W 3' 2 (u 3 2 ■ + IVigey), where x is a second function of x and y, and nr„ zs 2 , cr s are con- stants. The last term is introduced on purpose to simplify the equation of condition satisfied by x : on substituting in the first of equations (5), we see that, if we make C= -(ct 1 + ct 2 )/(1-o-), we shall have simply 3a:' 2 ~dy 2 .(10) 387.] Thus PLATES AND SHELLS. 461 ^1 -a-) •while equations (2) give 3w 3y 3<= or on integration -BT^-^-g-.g ■ 7 V --,"' -V. V .— SJ By .(12) .(14) _ _ 3v z 2 3(0 . .. 3y 2 3y ^ where and ^ are functions of x and 2/ only. Substitution from (11) and (12) in (3) gives (1-^ + ^ + 2^ = 0, (13) and the equatious of condition to be satisfied by and \fr are then found from (4) to be <•-'«)*<• ">|(g4JH It will be found, on differentiating the first of equations (14) as to x, and the second as to y, and adding the results, that the value of io as given by (13) satisfies (9) identically. Thus we may eliminate a> from (11) and (12) by means of (13), and the complete and most general solution of the proposed problem will finally be represented by 3v , o-z 2 3 /3 3iA where , y, ^ are any functions of x and y which satisfy (10) and (14) identically. Also, since JK = 0, it follows from equations (3) of § 253 that P = q {e + of)i{\-ar% (? = ? (/+ and yp- are due to tensions and thrusts in direc- tions parallel to the faces, or to couples in planes perpendicular to the Normal Axis. Flexion by Couples only. 388.] Case of Uniform Flexion of the Median Surface. If in equations (1 5) we annul all the terms but those involving the constant coefficients & v sr 2 , cr 3 , they reduce to u — — VS^yz — VljZX w = K^x^ + V,y + 2v 3 xy) + ^i + yj ' (17) •ST* /--or* g ^±^t\ 1 — IT \ a = 0, 6 = 0, c= - 2CT„2 and consequently P- Q = U-- 9(CTj + * !- (22) ,-_n- f--n* a - "^i + U J Z ) W r ' S l-T I (23) « = h = c --= I so that the shear disappears, and the new axes of x and y are the principal axes of the strain. It is obvious that the Median Surface is a Neutral Plane (S 347), i.e., it simply suffers warping without strain of any kind. The analysis of § 3+7 will sufficiently explain the nature of the strain. 391.] The Flexion Couples, f Returning to the arbitrarily directed axes Ox, Oy of § 388, the components of the stress, at any point (x, y, z), across a Normal Surface of the plate the perpendicular on which from the Centre makes an angle 6 with Ox, are E, _ qz\ (CTj + 12(1_ the expression (27) for the flexion couple proper may be written s . %{V5 X + CT 2 ) + a . [£(5^ - gt 2 )cos 20 + CT 3 sin 20] (31) By § 389 the coefficient of s is the curvature of " fibres " of the Median Surface due to synclastic flexion of the plate, and the coefficient of a is the curvature of fibres of the Median Surface perpendicular to the Normal Surface across which the flexion couple acts (or parallel to the plane of the couple) due to anti- 2G 466 PLATES AND SHELLS. [391. clastic flexion of the plate. Thus, with a notation analogous to that of § 349, e and a may be termed Coefficients of Synclastic and Anticlastic Flexion respectively.* If we write the components (26) of the couple per unit length across any Normal Surface perpendicular to Ox will be P 3 and — Pj, while those of the couple per unit length across Normal Surfaces perpendicular to Oy will be P 2 and — P 3 : all being reckoned in the standard directions. The flexion couple proper (31), across any Normal Surface, is !(P 1 + P 2 ) + £(P 1 -P 2 )cos20 + P 3 sin20 (33) per unit length. Equation (29) may be written in the equivalent form tan 26= ?\ (34) 392.] The Potential Energy. By equation (20) of § 199 we have Tr= kfffiPe + Qf+ Uc)dxdydz = 2ifr-fe) [CTi2 + *** + 2 J where LTj and II 2 are the principal curvatures. If SW be the increase of energy due to a small increment of each of the curvatures, SW= 12 »"j^v [( g i + o^ 2 )to x + (cr 2 + rtr,)te, + 2(1 - v = ~ z ~r> W = X (39) cte dy where x satisfies (10). If we analyse this strain by the method of § 389, we find that if J^X+y^K. and «g^f-+2/jp be both infinitely small within the limits of the plate, the curvature of any normal section of the strained Median Surface of the plate is given by *(3-!-H 9+ s|r 2 * which consists of two systems of anticlastic curvature. Neither of these can vanish for any position of the axes of x and y, unless X is a quadratic function of these coordinates, in which case the flexion is uniform, and the strain is included in the type just considered. Stravning without Flexion Couple. 394.] The Remaining Terms. The terms depending upon (j> and \fr in the general equations (15) represent a strain which is due solely to tensions and thrusts applied round the Edges of the plate in directions parallel to the Faces, and couples in planes perpendicular to the Normal Axis. It leaves the Median Surface absolutely unchanged, and pro- duces anticlastic curvature in all parallel surfaces, proportional to their distance from that surface. We shall not further concern ourselves with this strain. 46« PLATES AND SHELLS. [395. Equilibrium of a Thin Plate undicr Impressed Forces throughout its mass, and surface tractions applied to its edges, such that the component forces parallel to the faces on any portion of the plate bounded by A'ormal Surfaces are either evanescent or reducible to COUPLES. 395.] Preliminary. The results obtained in §§ 388-392 for the case of uniform flexion of a plate of any thickness by surface tractions applied to its edges in directions parallel to its faces, and everywhere reducible to couples or evanescent, are extended to the case of a thin plate subject to impressed forces and surface tractions the components of which parallel to the faces satisfy this condition, by a procedure very much like that of § 360. We assume in fact that the stress due to the applied couples will be everywhere of the form of that just discussed, only that the curvatures will vary from point to point of the Median Surface, and that the applied forces (necessarily perpendicular to the faces) on any portion of the plate bounded by normal surfaces will intro- duce shearing stresses in the same direction across those surfaces. The plate may be considered geometrically as coincident with its Median Surface. 396.] Equations of Equilibrium* If X, Y, Z be the components of the impressed force per unit mass at (x, y, Zj, the restriction imposed upon the form of the resultant force acting on any part of the plate bounded by Normal Surfaces requires that Xdz= I Ydz = 0, (40) Let J Zdz = rZ, (41) -St and let the components of the impressed couple on a rectangular element rdxdy of the plate, about axes through its centre(x, y, 0) parallel to Ox and Oy, be prl&dxdy, pT^&dxaly. The sole im- pressed force on the element is of course prZdxdy, acting through its centre parallel to Oz. ' Let A, B be the shearing forces per unit length at (x, y, 0) across Normal Surfaces drawn through that point perpendic- ular to Ox, Oy respectively. The components of the flexion * This and the two following articles are taken, with merely a change of notation, from Thomson and Tait's Natural Philosophy, Articles 644-648. 398.] PLATES AND SHELLS. 469 couple at (x, y, 0) across these surfaces, per unit length, are by § 391 P 3 and —P v P 2 and — P 3 respectively. Hence the equations of equilibrium are easily seen to be prZdxdy + U + l*bg£idy -I A- \d^\dy + (/? + \dy |? \dx -(b- ^dy^jdx = o pr^dxdy + Bdxdy + (P s + idx^s\dy ~ fes - i^^W + (p 2 + v^y* - (p 2 - biv^y* = o P TgS.dxd, U - Adxdy - fe + hdx^\dy+h 1 - \dx^\dy or, on simplification, .(42) ■dA -dB „ „ ox oy ^J + Sjj + A-prJR-O ox oy It may be shown, as in § 389, that if xdPw/dx^+yd^w/dxdy and x&vj/dxdy + y&w/dy 2 are infinitely small within the limits of the plate, 3 2 w? 3 2 3 2 m; CT] -i?' ^'w *'"*&!/' .(43) Substituting from (43) in (32), Pi I + o\3a: 2 2)y "-w\ tj s /3 2 w> , B^X tj „ 'BPw .... whence, since (l + o-)a = (l — «r)«, equations (42) may be written dA 3.8 „ n e 3 /3 2 to 3 2 «A „ T = 1+T 3^\3aj J+ 3P/ *""* _s_ 3/3^ 3%A A _ iH = Q 1 t- o- 3.rV3a: 2 3y 2 / ^ ^ .(45) 470 PLATES AND SHELLS. [396. On elimination of A and B between these three equations, we obtain (S^H-^^-f) «•> a linear partial differential equation of the fourth order, to be satisfied by w in all cases of equilibrium under strain of the kind supposed. 397.] The Boundary Conditions. Poisson's three bound- ary conditions are easily obtained by considering the equilibrium of a triangular element of the plate, bounded by planes of length dx, dy parallel to zx, yz and an element of the edge of length ds. Let the outward normal to ds make an angle 6 with Ox ; let H be the surface traction on the edge parallel to Oz, and let Hdz = rlt -ir so that rhds is the shearing force on the element rds of edge. Let Vds and qds be the couples on the element in the plane perpendicular to it and in its own plane. Then, on the assump- tion that the force and couples acting across the edge must be of the same form as on any Normal Surface within the substance of the plate, we have first rhds = Ady + Bdx or tK = j4 cos 0+ B sin 6 ; (47) and further by (33), (28) and (32) P = KPi + P 2 ) + *(Pi-P 2 ) cos2 + P3 sin 2^ (48) q = J(P 2 -P l )sin2fl + P g oos2^ j (49) These are Poisson's three conditions. Kirchhoff, however, has shown that the assumption involved in them (expressed in italics above) is not necessarily fulfilled, so that they express too much. The proof of this statement depends upon the fact (to be proved in the next Article) that if we apply, all round the edge of the plate, a shearing force parallel to Oz of amount t(W ~ h), P er unit length and couple round axes everywhere parallel to the Median Surface and perpendicular to the edge, of amount (Q - q) per unit length, such that T(»-h) = i?(Q-q) (50) no modification of the strain vjhatever will be produced, except at points infinitely near the edge. 397.1 PLATES AND SHELLS. 471 Thus we may suppose rW to be the shearing force per unit length and Q the couple per unit length in the tangent plane to the edge actually applied at each point, where 1§ and Q may be any quantities connected by the relation (50), h and q being given by (47) and (49). Eliminating k and q by means of (47) and (49), and A and B by means of (42) from (50), we obtain AW + P(W sin 6 - J« cos 6)} - ^ + | g [J(P 2 - Pi) sin 20 + P 3 cos 20] ^fH+ei+fH=° <•» Equations (48) and (51) are KirchhofPs two boundary conditions. If they be regarded as determining the values of w at the edge, P, Q and |$j may be treated as entirely arbitrary couples and force applied to the edge. 398.] Proof of Kirchhoffs Boundary Theorem. " The proposition stated " in the last Article " is equivalent to this : — that a certain distribution of normal* shearing force on the bounding edge of a finite plate may be determined which shall produce the same effect as any given distribution of couples round axes everywhere perpendicular to the Normal Surface supposed to constitute the edge. To prove this let equal forces act in opposite directions in lines EF, E'F' on each side of the middle line "f - and parallel to it, constituting the supposed distribution of couple. It must be understood that the forces are actually dis- tributed along their lines of action, and not, as in the abstract dynamics of ideal rigid bodies, applied indifferently at any points of these lines ; but the amount of the force per unit length, though equal in the neighbouring parts of the two lines, must differ from point to point along the edge, to constitute any other than a uniform distribution of couple. Lastly, we may suppose the forces in the opposite directions to be not confined to two lines, as shown in the diagram, but to be diffused over the two halves of the edge on the two sides of its middle line ; and further, the amount of them in equal infinitely small breadths at different distances from the middle line must be proportional to these dis- tances" [see formulae (25) of § 391] "if the given distribution of couple is to be thoroughly such as " q of § 397. "Let now the whole edge be divided into infinitely small rectangles, such as ARGD in Figure 61, by lines drawn per- * I.e., parallel to the Normal Axis of the plate. + The lioe in which the edge is cut by the Median Surface of the plate. 472 PLATES AND SHELLS. [398. pendicularly across it.* In one of these rectangles apply a balancing system of couples consisting of a diffused couple equal and opposite to the part of the given distribution of couple belonging to the area of the rectangle, and a couple of single forces in the lines AD, GB, of equal and opposite moment. This balancing system obviously cannot cause any sensible disturb- ance (stress or strain) in the plate, except within a distance comparable with the sides of the rectangle ; and, therefore, when the same thing is done in all the rectangles into which the edge is divided, the plate is only disturbed to an infinitely small distance from the edge inwards all round. But the given dis- tribution of couple is thus removed (being directly balanced by a system of diffused force equal and opposite everywhere to that constituting it), and there remains only the set of forces applied in the cross lines. Of these there are two in each cross line, derived from the operations performed in the two rectangles of which it is a common side, and their difference alone remains Pig61 effective. Thus we see that if the given distribution of couple be uniform along the edge, it may be removed without disturbing the condition of the plate except infinitely near the edge." Otherwise, "a. distribution of couple on the edge of a plate, round axes everywhere in the plane of the plate (i.e., in the plane of the unstrained Median Surface), of any given amount per unit of length of the edge, may be removed, and, instead, a dis- tribution of force perpendicular to the plate, equal in amount per unit length of the edge, to the rate of variation per unit length of the amount of the couple, without altering the flexion of the plate as a whole, or producing any disturbance in its *To the unstrained Median Surface. 398.] PLATES AND SHELLS. 473 stress or strain except infinitely near the edge." For, in Figure 61, let AB = ds, the arc s being measured from A towards B. Then, q— Q being the amount of the given couple per unit length, the amount of it on the rectangle ABCD is (q — Q)ds. Thus the forces introduced along AD, GB to form the balancing system must be of amount q— Q. Similarly, the amount of the forces introduced along BC and the next transverse line is q— Q+ffe-r-(q— Q), and finally we are left with a force of amount ds — -j along BO, and a similar force in the negative direction of the Normal Axis along every other such transverse line. And obviously we may substitute for forces of amount , ■ ds at infinitesimal intervals ds a continuous distribution as of force of amount j, — - per unit length round the whole edge, without causing disturbance in the plate except at infinitely small distances from the edge. Hence, finally, we have the result stated in equation (50) of § 397. Equilibrium and Normal Vibrations of Thin Plates subject to any distribution of normal impressed force, but free from impressed couple except at the edges; treated by the energy method. 399.] The Total Energy. The second of the expressions (35) for the potential energy of a plate of area Ji subjected to uniform flexion, may be written by means of (30) and (43) in the form The flexion of an element rdxdy may be considered uniform under any circumstances, so that we deduce for the entire potential energy of a plate subject to non-uniform flexion where t? =V/d* + V/ty. 474 PLATES AND SHELLS. [399. If the plate be executing normal vibrations, the entire kinetic energy will be % = ^ P Tffw' i dxdy, (54) and the total energy of the plate will of course be W-\-%. 400.] The Variational Equation of Motion, and the Boundary Conditions. Let us suppose the plate to be either in equilibrium, or executing normal vibrations freely or under normal forces only, and that its edges are either free or " sup- ported" or "clamped" (§ 366) all round. In the most general case, the small amount of work done in producing the incre- ment Sw of normal displacement will be, with the notation of §§ 396-397, JTprZ&wdxdy + /T(tS> - §V«> + P^~K where rds is the element of edge, and dv the element of outward drawn normal to it. [The work done by the couple Q, as couple, that is in producing flexion about axes perpendicular to the edge, is — /Q-=— rfs, which, since s is necessarily a closed curve, vanishes identically. This is the analytical justification of Kirch- hoff's principle.] Thus the variational equation of motion is ' T $_^W + p?gfU = (55) Taking the first term separately, and making use of the general theorem Jjf{W* - W4¥*dy = /Y*|£ - 4tH ( 56 ) where the double integral is taken over the entire area of the plate, and the single integral round the whole of its boundary edge, we have /TS(^! 2 w) 2 dxdy = 2// y 2 i/;Sy 2 u;c7a-c(7/ = ^ V \ i w.Sw.dxd,j + 2fL 2 w.^-'^..Sw\d li ....(r ) 7) Again, the second term of (55), on being integrated twice by parts, gives "/[( 400.J PLATES AND SHELLS. 475 JJ |_3y 2 ox 2 3x 2 cty 2 ~dxdy ~dxdy_\ = / J ( cos 6—— - sin 1 ■J l\ °y oxoyj ox V {(""V aS»X V-"™*ai) ♦ [-*£■♦««$-*.•-.£!]£}* where has the same meaning as in § 397. Integrating the first term again by parts, and neglecting the integrated portion (a being necessarily a closed curve, and the function to be integrated necessarily single-valued), we have + [«g + -«^-i*.- 2 lr +2 «aL ,nBfl0 " fl l^i-a?) +/"{(» + *> v 2 " - *[*«S + cos2 ^ -2sin0co S ^~|-2Pl.^°.rf S =O (59) OXOi/_] ) Ov Thus the general equation of vibration, to be satisfied at every point of the plate, is (s + a)y'V«' +2 / 5T («'-B) = .- (60) 47 C PLATES AND SHELLS. [400. while the two boundary conditions are ( , ^'^X7 2 W „ 3 f* . n n/d-W 3'mA + (co.^- S in^] + 2 (rl-§)}^-0 (61) and {( s + a)v*»-2a[rin«flg + ooB^ -2 8m«oosfl^-"|-2P^ = (62) oxoy_i ) av If the edge is clamped all round we have (Siy = 0, Sdw/dr = everywhere, and (61) and (62) are necessarily satisfied. If the edge is only supported Siv = Q, and we must have (6 + a)v 2 *> - 2 a r S in26l^ + cos 2 ^ - 2 sin 6 cos 6»|^-~| = 2P, |_ Ox 2 3y 2 ac9i/_J or (s - n)y 2 w + 2 a r c os 2 ^ + ain»^ + 2 sin cos ^J^l = 2P . . . (63) If the edge is free, we must have, in addition to (63), / be conjugate functions of x and y, and let the form of the edge be such that it can be represented by the equation g= constant. It is obvious that the equations of the last Article will be much more readily applicable if they can be transformed from x and y to g and q. It is an excellent example of the methods of Chapter V. to effect this transformation ah initio. The principal curvatures LTj, LT 2 of any surface (,c, y, z) = are the roots of the quadratic * h 4 n 2 ± hn[A 2 a + B% + C 2 c + 2BCa + 2CAb' + 2ABc' - K 2 v 2 *] + A*(bc - a' 2 ) + W(ca - b' 2 ) + C 2 (ab - c' 2 ) + ■2BC(b'c'-aa') + 2CA(c'a' - bb') + 2AB(a'b' - cc') = (65) * Frost's Solid Geometry, Article 60S. 401.] PLATES AND SHELLS. 477 where A=d$/dx , a = d 2 $/dx 2 , a=t<*$jdydz , and " = A 2 +B 2 +(J 2 . Putting $ = z + w, and transforming (65) from (x, y, z) to (£ ,, z) by the formulae of § 230, 231, 245, we obtain G7 1 + CT 2 =n 1+ n 2 = ■Er,»- 11,11, = * l_3P V VW J Vd| 3f 3^ - (68) (e + ^^Rg^ 3/t 3io 3A 3u; 3ry 3g 3£ 3»j )M3- T ?}<»> Of these (68) must be satisfied round a supported edge, and both round a free edge. For examples on Normal Vibrations of Plates the student is referred to Lord Rayleigh's " Theory of Sound," Chapter X., and for examples on equilibrium to Thomson and Tait's "Natural Philosophy," §§ 649-657, 719-729. We shall here confine our- selves to a single example of the latter class, to exhibit the convenience of curvilinear coordinates in cases of symmetrical strain. * There is apparently a residual double integral from the second term of W, but this vanishes with v 2 logA which is identically zero for all conjugate functions. See Note at end of volume. 478 PLATES AND SHELLS. [402. Circular Plate Symmetrically loaded and supported. 402.] The General Expression for the Displacement. A circular plate of radius C is placed so that its unstrained plane is horizontal, and loaded and supported in a perfectly symmetri- cal manner about its centre : required the general expression for the vertical downward displacement of any point. If Oz he directed vertically downwards through the centre it is evident that the load, and the boundary forces and couples (if any) must be functions of r only (§ 244). Thus taking the conjugate coordinates of Example 4 (i), page 258, £ = log(r/<7), r,= 6, (70) all the quantities involved will be independent of r\. The general equation (67) of equilibrium thus reduces to j 2 d 2 , vd?w _ 2/dtB 1 d£'' ''d^'JTs.' or, since h = ljr and d/dg= r . d/dr, 1 d d 1 d dw 2otZ /-,s - -j-t— - r— = -C — (i\) r dr dr r dr dr + a Integrating four times we have w = "'" / — / r dr I — / r%dr S + -&J rj J rj + \C'r\\ogr-\) + \C"r i + C" r iogr+C"", (72) where C, C", C", G"" are arbitrary constants. Since however it is clear from symmetry that the tangent plane to the strained plate at the centre will be horizontal, we must put C"" = (73) 403.] The Boundary Conditions. Equations (70) and (71) reduce in this case to oo ooo + |C'(0logC+|a) + £gC" (74) s^-sT**-^ < 75 > 404.] Plate under Gravity, supported by its Centre only. In this case % = g, and w = when r = ; also, since the edge is free, P=0, 1 = 0. Thus (7"' = 0, C= -gpr&fo + n), C" = gprC 2 [log C-1 + 4a/ 8 ]/(0 + a), and finally w -#££"- 0, K-Hr)] < 76 > 405.] PLATES AND SHELLS. 479 Normal Vibrations of Thin Shells under Normal Forces. 405.] Formation of the Variational Equation of Motion. We now advance from the consideration of thin plates to that of thin shells, subject only to normal impressed forces and (if open shells) to surface tractions applied to the edges only, and such that the component tensions in the tangent plane to the shell at each point of its edge reduce to couples. A portion of a shell, as denned in § 384, may be taken of such small superficial dimensions that in its natural state it is practi- cally plane, while the change of curvature produced in it by the strain is practically uniform. Thus, if the principal curvatures at any point of the shell be increased from tl v II 2 , to U v IL,, we are led, as in § 377, to the assumptions (i.) that the couples per unit length across the principal normal surfaces at any point of a thin shell are V ^ ; [ (77) p * - 12(1 -t^ - n 2 + -£ w u * = ^k 3 # < 81 > 480 PLATES AXD SHELLS. [405. where we again are to make £= C after differentiation, and (78) may now be written -p( n i-f°i»)( n .-Fr) ( 82 ) The vibrations being supposed normal, t) and £ will remain constant for each point, and the only effect of the strain will be to change the value of g from C to C+a, where a is a small quantity of the first order, and in general a function of r\ and f. The normal velocity at time t will be (§ 237) a/h v and the kinetic energy per unit of unstrained superficial area will be p/ca 2 /2A 1 3 . Thus, the element of surface (§ 230) being dridg/h 2 h 3 , if we write for the normal impressed force per unit area {J ««£ = «£/*!, (83) the variational equation of motion will be Jf {*(<• + a)8(n, + II 2 - f o, - f o f )» - 88(1^ - { o,)(n, - f or f ) where kW/K is the shearing force per unit length applied to the edge in a direction perpendicular to the faces, P is the couple per unit length in the plane parallel to this force and perpen- dicular to the edge (flexion couple), and Q is the couple per unit length in the tangent plane to the edge. In an edge formed by a portion of an jj surface ds = d£/h z> dv = dq[h i , and in an edge formed by a portion of a f surface ds = d^/h i> dv = dg/h 3 . In default of general formulae, analogous to (66), giving the sum and product of the increments of the principal curvatures in terms of a and its derivatives as to >? and £, the equation of motion and boundary conditions cannot be obtained in general terms, but each case must be solved separately from this point. 406.] Case in which the surfaces of the shell remain always members of the family to which they initially belong.* If we suppose the vibration to be of this kind, a will * Examples :— (i.) a shell bounded by coucentric spheres performing nor- mal vibrations symmetrical about the centre, (u.) an ellipsoidal shell with confocal surfaces, vibrating normally so that the surfaces remain confocal with their initial forms, etc. 406.1 PLATES AND SHELLS. 481 of course be independent of r\ and f (§ 242), and we shall have simply dJ3 033 \ 1 ' v 3£ [ n 2 = ^ f + a-|J- so that equation (84) reduces to (85) oJS ' P K JJ W, + a/jTi !±if -^ + *-^T - 'a-^ !& I ^ JJ l a L " af" "-I " 3£ ?£ j v*A - p #wr° <8C) The boundary condition to be satisfied, in all cases in which the assumed mode of vibration does not require the edge to remain fixed, is ,c$-A*i-2P|*l = (86«) rf* ov The periods of possible vibrations of this kind are independent of the impressed forces (unless these are periodic), and can be ascertained when the form and dimensions of the shell are given. Example. 407.] A spherical shell of radius C and small uniform thickness t, performs free radial vibrations symmetrical about a diameter, the amplitude of the displacement being proportional to a zonal harmonic. Required the periodic time of the vibration.* Let Oz be the axis of symmetry, and u the radial displacement. Then, with the notation of § 243, u is independent of w, and equation (84) of § 405 reduces to r {J(s + a)8(n i + 11,-2 C)- - aSfll, - 1/0(11. - 1 C) YpTuhi}sm6M = Q (87) But if P be the point (C+u, 6) on the strained shell, and PG the normal at P, meeting Oz in G, and making an angle \fr with OP, we have r=0P=C+u, tt I <** /I * Professor C. Niven, Mathematical Tripos, 1878. 2 H 482 PLATES AND SHELLS. [407. ""ft-?)' 1 dhi C- S* n, = L ?"(.*+*> = l(i + ^ cot g) r sin 6/ 9' = A'" ^rA c) c^r re- Thus, if we write we shall have a du , cos = v, p~— — u = <;>, a;; tt 1 1 (d 2 u \ 1 P/i ^d'^u du ~] CV?> 2 and consequently ( n .-$ D '-B)~tfte-)*-*S-*] so that equation (87) may be written -1 + pr J ic8udj} = (). Since 1 — \r = at both limits, the first line reduces, after integration by parts, to £/{*[<■ -'■£>•}•■»■*• -1 while the second line is equal to — o u 1 _ . . 2t .-4 | "-' •*07.] PLATES AND SHELLS. 483 ^/' 8 {(i-^(j) 2 -^}^ Thus the equation of motion is (,+a >{4[< i -^] +3 }" If we now assume that the displacement is of the form it = A sin i< . P,(cos 0) where P denotes a Legendre's coefficient, we have u= —i 2 u and { 40 1 -*£>*}«- -o--i)(i + *>«. so that the equation of motion will he satisfied if '"* = —•. 2CV^ [(S + a)(i " 1)0 ' + 2) + 2a ^ 12(1 -tr 2 )^ lU 1 >W + J ' +1 °M- whence we deduce the required periodic time 2x/i. EXAMPLES. 1. The most general solution of Clehsch' Prohlem, consistent with the absence of shear, is of the form ii = A l x- t3 1 sx + A i xy-l Bj^o-x? + y 2 - ers 2 ) - J Cj>:(frr.t s + y* - = (1 + -diSin^ 1 — ■2L for all values of z from to L. Hence wc find, by Fourier's theory, that and consequently •-^M*^"- 3 ^ < 3 > [It may be observed that, when t = and z = L, this expression reduces to = &Lt(l 1 1 \ to as of course it should.] The equation expressing the principle of conservation of energy is /'{?(!l) I+ !Cs)>^"*> ' where Jl is the transverse section ; or fm'^im^^' «> 409.] IMPACT. 489 This is easily reduced, on substitution from (3), to le^yr- i /"■ i (2 i+ i )g , (2» + i)riy ** ^ =0 (SiTipy t sin — sr- cos — 2Z — J2i h ] W» . 9 (2i + lWf2 n i! ) , T „ n ., + cos 2 !: — ' sitf! —^ — T \ dz = Zt-l/j-, or „— - > , = Le 2 ilJ, which is an identity. Whenever the time from release is an odd multiple of L/Q v or the time required for a sound vibration to travel the length of the rod, w = for all values of z and the rod passes through its natural state. Whenever the time is an even multiple of 2L/Q V iv = ez, and the rod passes through its initial state of strain. Whenever the time is an odd multiple of 2Z/'fi 1 , w= — ez and the initial state of strain is reversed. The traction on the fixed end of the bar is ■dw _WT(-1)' (M+l),^ 'a^,,, 7r^ 2i + l ' 2L This is equal to qe from t = to t = L/£l v when it suddenly changes sign (§ 408) and is equal to — gefrom t = L/Q 1 to t = 2LQ/ v this cycle being repeated indefinitely in equal periods of time. Exam/pie of Direct Collision with a Fixed Rigid Obstacle. 410.] A Rod of length L moving with velocity U in the direction of its length, comes into direct collision with a fixed rigid wall. Required the subsequent motion. During the whole time that the rod is in contact with the wall the end in contact will be absolutely fixed. Thus if we take that end for origin, and the Axis of the rod for Oz, we have w = when t = from s = to z = L, and w = 0, w = when 2 = during whole time of contact. Also, since the further end is free, dw/dz=0 when z = L for all time. The form of the displacement during contact with the wall is evidently „, . (2i+l)« . (2i+lWJ2 1 < for this satisfies all the foregoing conditions. The coastant coefficients may be determined by the consideration that at the instant of impact every point in the body is moving with a velocity U towards the wall, and consequently, for an unappreci- able interval following that instant every point in the body has 4,90 IMPACT. [410. a velocity U relative to tfic end 0. Thus we must have w= — U when t = for all values of z between and L, including the limit z = L but excluding the limit 2 = (where w = 0). Now the series 4U- 1 ■ (2t+l)7r- w X. cm ' ' ■^■2i + 1 sin- 27, = - U from i = to z = 2L, exclusive of both limits, and vanishes when z = 0. Thus we shall satisfy this condition by making (•2i + l)7r~ -«,v/,- ,n, • (2i+l)jrs 4TJ-V 1 a : n (2i + — >l(2t + l)A, sin v — f-l- — =- -Zj-t- — rSin -, 2/, 2L ir *-'2% + 1 2J , 8ZU 1 and consequently The equation of conservation of energy is which reduces to the identity 8U 2 Z,^ 1_ «(2t+l)« At the instant when t = 2L/il l iv = throughout, and . 4U^ 1 • (2i+l)ir: «; = > sm > ■ -i — 7T "^*2t + 1 2L = U throughout. At that moment therefore the rod is instantaneously in its natural state, and is moving bodily from the wall with velocity U. Contact consequently ceases after a period 2L/Q 1 from the first impact. Confining ourselves for the present to the period of contact, we deduce from (5) dw 4TJ-^ 1 ■ (2i+l)irz . (2i + \Wil.t = > sin - - si n J: L 1_ cfc~ ~ ^2^27+1 SU " 2Z 2L = _£E j v 1 Bil , (2f+i)»L/tt 1 , we get by putting t = L/i\ + i' and s = Z-.c' in (6) 'd»_ _ W | v 1 (« + 1)tt( 2 ' - fl/) cte Trfi, 1 ^2iTl ~27, + y_L- S in (ii+iM*:+_V) 1 ■"2i +1 2Z, " J ' The first of these series =— \ir from a' = to z'=Q 1 t — L and = 4"7r from £' = 0^ — L to z' = L, while the second series = ^7r from z' = to s' = Z. Thus the first series = }-7r from 3=0 to a = 2L-0^ and = — \w from a = 2Z — fi,f to z = L, while the second series = \ir from a = to z = L. Summing up results, it follows that (i.) from < = to f = £/Q, ^' = - ^ from : = 0to; = Q/, and _ - = from z = Q,i to z = L. cz ' 7i TT (ii.) from t = Z/J2, to t = 2L/Q lt ^-= -^- from a = to - = 12£- Of, and ~ = from a = 2£-Q* to a = Z. Thus a portion of the rod next the wall suffers uniform com- pression of amount U/i\, the remainder being free from strain ; and the geometrical surface separating the two portions advances from the fixed end with uniform velocity Q 1 along the rod, is reflected at the free end, and returns with the same velocity, reaching the wall again at the instant (£ = 2Z/Q 1 ) when contact ceases. The thrust on the end in contact with the wall is eU/O^ or QXJuJqp throughout the duration of contact. At the instant when the rod leaves the wall it is unstrained, and every point is moving with the initial velocity U reversed. Since no forces act on the rod, its centre of gravity will continue to move with the same velocity U, and the kinetic energy due to the motion of its centre of gravity alone will be equal to its kinetic energy before the impact. Hence the kinetic energy of 492 IMPACT. [410. motion of the parts of the rod relatively to its centre of gravity- is zero, and consequently no such motion can take place. The rod therefore retreats in its initial unstrained condition and with its initial speed U. EXAMPLES. 1. Two uniform heavy beams AB, CD, equal in every respect, are connected by a weightless inelastic string BC ; the beam AB lies unstrained on a smooth rigid horizontal table, while CD is suspended at rest under the action of gravity by the string which, being held at B, passes over a small smooth pulley at the edge of the table, and in one line with AB produced. Investigate the motion of the string when set free ; prove that its tension, after being instantaneously diminished by one half, remains constant, and that its velocity receives equal increments at equal intervals. 2. Example 3 on Chapter VII. (page 449) may be treated as a case of sudden release by the method of § 409. 3. Prove that if we make y ( = in equation (80) of § 271 it will represent the vibrations excited in an infinite plate of thick- ness I, moving with velocity U, on its median plane being instan- taneously brought to rest. 4. A uniform elastic bar is suspended vertically by one end, and to the other is attached a weight W, which is supported so that the bar is unstrained (the effect of gravity upon it being neglected). If the weight be suddenly set free, investigate the motion of the system. 5. Prove that if an elastic bar, of length L, impinges directly with velocity U on a longer bar, of length pL and the same cross section, the first bar will be reduced to rest by the impact, while the second bar will appear to move forward by successive advances of the ends with velocity U for intervals of time 2L/Q V alternating with intervals of rest of duration 2{p—l)L/Q 1 . G. If the revolution of the square described in Example 9 on Chapter VII. (page 451) be suddenly stopped by its sides striking simultaneously a smooth fixed rigid plane, prove that the dis- placement at any subsequent time t from the impact will be given by 4o^y>taii(tXZ/2h)i) J cosh(iXz/o)i) _ cos(i\z/a>i) \ .- k 2j ,-( w 2 _ fiy- \ TO 8 "h(iAX/2wi) ^s(iXZ/2^~) r° Sl ' IMPACT. 493 the summation extending to all values of i given by the equa- tion tan — — ;- + tanh — , = 0. 2d)i 2 m + n + -v— , 3 dt + p (S -it) = + p(H-v)=0 (— ♦ i-m-i-iy^m-mi + P (Z-w) = and, similarly, the boundary conditions (45) of § 238 become ■0) f(m - n)A + 2ne - 2v(^ - A~K|| + (w + vc)A. 3* 2 d v 3 (nb + vb)h,-— + (ma + va)h„—- 3£ Or) + (na + vd)h s — = hH' ■ £(m - n)A +2nff- 2v(* - ^)]a 3 -| = hZ' ...(2) where e, f, g, A, a, b, c, Q v 2 , G 3 are given in terms of u, v, w by equations (26), (27), (29), (31) of § 235, and h by equation (19) of § 233. Example. 413.] Torsional Vibrations of a viscous cylindrical rod, of circular section. With the notation of § 244, let the surface of the rod be given by r = C, the origin being at one end of the central axis, and the length of the rod being L. From the analogy of § 335 we are led to assume that the torsional motion will consist in a bodily twisting of each transverse section about the axis in its own plane ; or, analytically, that in pure torsion u = w = Q and r = r0, where is a function only of z and t. On this assumption we have « = 0,/=0, 6 = 0, r = 0, 3, 496 VISCOSITY. [413. and the equations of motion (1) reduce to the single equation / 3\3 2 <£ B 2 ,,, the conditions (2) for freedom of the lateral surface being satis- fied identically. 414.] Free Oscillations. We can now solve completely the case in which the end z = is fixed and the. end z = L, after having been held twisted through an angle tL till the rod assumes the configuration of equilibrium = tz (8 335), is set free. We have (i.) when 1 = 0, = tz, <£ = 0. (ii.) when « = 0, = 0. (Hi.) when z = L, 3/3z = 0. Thus the appropriate solution of (3) will evidently be of the form , i 2 =j i + (n-jv)f-!p; or. if { X — xA7/p as in § 266 = 2.4 „ sin ps.e' vpQ ' 1/2 " cos p& J 1 - pWa^/in- . t (4) To satisfy the remaining boundary conditions we must have f> = (2i+l)ir/2L, where i is any integer, and v . . (2i+l)irz tz = 2,A i ,bui> — — :' , I 8Xt (-1)' 77- (2i + 1)- Thus finally sir «r- (-iy 8in (2£+I)« r _(2« + i )M>iy«n * " ^"^o (2T7TT* 81n -^r - • exp L — &^1J- J r(2i+l)7rfi' /. (2*+l)Wfl'* ,~1 ,S L 2L V 16nW 'J' The effect of viscosity, in increasing the periodic times and steadily diminishing the amplitudes of the vibrations, is obvious. 4 Id.] VISCOSITY. 497 Extension to Viscous Liquids. 415.] Equations of Motion. If we regard a liquid as a limiting form of the solid state in which the elastic rigidity is absolutely zero, we can deduce the equations of motion of a viscous liquid from (41) of § 237 by simply making n = 0, and P = Q = .R=— II, where II is the hydrostatic pressure, i.e., the only elastic stress that can exist in such a body (see Appendix IV., pages 169-180). We have then in general '!{"A»-!te^Ml '3 -«m + wCKw)*KA)]} + />(S-ii)-A r -j = 0, etc. The relative displacements in a liquid may however be in- definitely great consistently with infinitely small strain (except when they are such as to produce cubical dilatation or com- pression), and it is in general impossible to identify their magnitudes or directions. All that we are concerned with practically is the relative velocity of displacement of different parts of the liquid, and this must of course be small in order that the viscous resistances may be small. If we change our notation, making u, v, w represent the velocities of displacement parallel to fixed axes, and e, /, g, a, b, c, A the rates of increase of the longitudinal extensions, shears and cubical dilatation, the equations of motion will be, when u, v, w are very small, I,,, 3/e\ 2/ 3A. ^ 1g 3A 3 „3A + Wt3 K(w) + 1(A) ] \ + k(* ' i) = W ♦***[&w)*&«jtf]K( ,, -D-W The quantities e, f, g, A, a, b, c will still be given in terms of u, v, w by the linear equations (26), (27) and (29) of § 235. 2 I ..(5) 498 VISCOSITY. [416. 416.] Liquids treated as "Incompressible" In all mobile liquids the numerical value of k is so very great in comparison with that of v, that it is usual to neglect i>A in comparison with II. In fact, if the hydrostatic pressure be supposed of the same order of small quantities as the shearing stresses due to viscosity, the rate of cubical compression will be very small in comparison with the rate of shearing. This treat- ment of liquids as " incompressible " is of course only an approxi- mation, intended solely to reduce the great analytical difficulties introduced into hydrodynamics by taking viscosity into account. On this assumption the equations of motion may be written in the form (6) -*4(&)M--S)-'.W - i 'i(is)}-''( H ~li)-' 4 *w -*^fe)}-'( z -w)-*'f; with the further condition e+f+g = (7) When referred to Cartesians these equations take the simple forms •^-S*>( x -^)= V-^*'(r-S)-o .(8) With ^tJ'+^.O; ( 9) so that, in the case of conservative impressed forces, derived from a potential ¥, we obtain by elimination V 2 (n-,,^) = (io) [Compare this with equation (164) of § 303.] 417.] VISCOSITY. 499 417.] Boundary Conditions. In practice, the bounding surfaces of a liquid must be either (i.) free, (ii.) subject to the uniform normal pressure of a gas, or (im.) in contact with a solid or another liquid; and in all but the first of these cases the contact must be maintained throughout the motion. Thus in cases (ii.) and (Hi.) we have the purely kinematic condition that the normal velocity of every point in the surface of a liquid is equal to that of the point in the surface of the other body (of whatever nature) in contact with it. The dynamical condition is when relative motion takes place between the two bodies, tangentially to the dividing surface, the shearing stress exerted on either is proportional to the relative velocity, and in a direction tending directly to retard it. Thus, let u be the velocity of any point in the surface of the liquid resolved in the tangent plane to the surface, and u' the surface velocity of the body in contact with it : then u satisfies the equation g| + rtt»_tO = (11) where n is the element of normal to the surface, measured out- wards from the liquid, and fi is a new constant — the " modulus of contact viscosity." In the more mobile liquids (ether, alcohol, etc.) the value of /x is so great that practically no relative slipping takes place at the surfaces of contact, so that the surface velocities of the liquid, in all directions, are the same as those of the body limiting it. In case (i) the boundary conditions are to be found by writing n = 0, A = 0, S' = H' = Z' = 0, P = Q = _R=-n in equations (2). Thus they become with our new notation (2ve - im-l? + w*J* + vM,— = 0^ 1 dg 2 drj "cC vch™ + (2,/- IL)h™L + vah™ = 1 _ + I/a / i2 - + (2v^-n ) A 8 _ = .(12) / Examples of the motion of Viscous Liquids will be found in Professor Lamb's " Motion of Fluids," Chapter IX. 500 VISCOSITY. APPENDIX VI. Economy of Material in Nature. A few simple examples of economy of material — i.e., the principle of producing the greatest possible elastic strength under specified types of strain, with the least expenditure of a given material — have already been discussed in Chapter VII. Numer- ous beautiful applications of this principle are to be found among organic structures, and in fact they may be looked for with con- fidence wherever great strength in proportion to the material available, or gTeat lightness in proportion to strength is an advantage. Good examples in the vegetable kingdom are to be found in the stems of the grasses and the order Umbelliferse. These plants grow thickly together, or force their way among other thickly growing plants, and often on very poor soils. They are all enormously reproductive, and bear their seeds in heavy masses. It is therefore of the utmost importance to them to use the least possible material in building up their stems, and at the same time to make them strong enough to resist considerable vertical thrust and flexion couple. They all have largely hollowed cylin- drical stems. Very young trees, which have to struggle for food with the surrounding grasses, etc., have most of their mass concentrated in an external cylindrical layer of the stem, the axial portion being occupied by a soft and light pith. As growth proceeds, however, and their leaves in the one direction, and their roots in the other, emerge from the sphere of close competition, they accumulate material beyond the strict needs of economy, and it is largely devoted to hardening of the axial portion of the stem. Conse- quently in old trees the " heart-wood " is the more durable and valuable portion of the trunk. The stem of the common rush, on the other hand, composed of a thin but very tough outer rind, requiring some exertion of strength to break it, and a pith of large relative volume but very small mass, is a good instance of the attainment of extreme light- ness without too great a sacrifice of strength. It is, however, in the complex structure of the bones of the higher animals that we find the most consistent and remarkable application of the principle of economy. It is of course advisable, in order that the muscular power may be fully utilised, that the bones, which from a mechanical point of view are simply an inert system of levers, should be as light as possible, and at the same time the exertion of that very power exposes them habitually to VISCOSITY. 501 considerable stresses. In order to explain how these varied requirements are met, we will describe the structure of the human thigh-bone as a typical example. This is a long bone, the principal office of which is to transmit half the weight of the trunk and head to the knee-joint, and thence to the ground. The principal stress to which it is subject is therefore one of longitudinal thrust. It is however also subjected, especially in walking or running, to considerable flexion couple and slight torsion couple. The bone consists of two terminal articular masses, which receive the complicated stresses from the joints and muscles, and a connecting shaft, almost the only function of which is to transmit stress from one articular mass to the other. The shaft, which for our purposes may be regarded as approxi- mately cylindrical, thus receives almost its entire stress across its end surfaces, and, in accordance with the principles of §§ 329, 336, and 355, it is extensively hollowed out throughout its length, the hard, rigid and heavy bone-substance being compactly arranged, mainly in the" form of longitudinal fibres, as a cylindrical casing, and the interior space being filled with light and semi-fluid marrow, which for practical purposes may be said to offer resist- ance only to cubical compression. The structure of the articular masses, which are subject to very varied stresses over the greater portion of their surfaces, is naturally much more complicated. Broadly speaking, it may be said that the rigid bone-substance of the shaft-cylinder divides on entering the terminal mass into a thin outer casing, and a series of thin laminae which in the main take the form of the principal surfaces of the most severe form of strain to which the mass is subject, the small orthogonal spaces enclosed by these laminae being filled with marrow. Under the specified strain the laminae are in the proper position to transmit directly the principal normal stresses, and are only subject to cubical compression, and the interspaces to change of volume, to which their contents offer a resistance comparable with that of a solid. On the other hand, the composite structure admits readily of small deformations under accidental shocks in unaccustomed directions. The advantages of this arrangement over a solid bony structure, either of the same strength or of the same weight, are obvious. Figure 62 exhibits a slightly diagrammatic view of the lines of stress* in a section of the upper portion of the thigh-bone, cut vertically from right to left and looked at from the front. It will be seen that the " head " AB has a considerable inclination inwards, like the head of a crane. The direct thrust, due to the weight of the body, falls exclusively upon the surface A, the tensions on the surfaces B, C, D, E, H and J being due to liga- * Of course they are not really plane curves. 502 VISCOSITY. ments and muscles exerting the couple necessary to maintain the upright posture. The muscles arising from F and assist in keeping the knee-joint rigid. It is evident that the main thrust will be transmitted by strut lines down the inner side of the shaft, while the orthogonal tensions required to support the "head" will act along tie lines arising from the outer side. The details of the arrangement are shown in the figure. On comparing this with Figure 63, which is from a photo- graph of an actual section of the same bone, the reader cannot fail to be struck by the extraordinary closeness with which the sections of the bony laminae correspond to the theoretical lines of stress. The small bones of the body, such as those of the spine, the wrist, and the ankle and heel, are practically in the position of PLATE IV. Fig. 63. Fig. 65. F.CONOMY OF NATUEE. (Paye 30 J.) VISCOSITY. 503 articular masses without shafts, and are constructed on the same principle. Figure 64 represents the theoretical lines of stress in Fig. 64. the bones articulating at the ankle-joint, and Figure 65 is from a photograph of a section of the heel bone.* *The student should read a charming paper, entitled "How a Bone is Built," by Dr. Donald Macalister, in the " English Illustrated Magazine " for July, 1884 (Volume I., pages 640-649). Figures 63, 64, 65 are taken from that paper, by the kind permission of the Publishers. The photographs are by Zaaijer, engraved by J. D. Cooper, and the diagram 64 is from a drawing by Professor Hermann Meyer. ADDITIONAL NOTES. I.— § 3, page 2. Sphere of Action of the Tntermolecular Forces. Some very interesting cases are known of the appreciable action between the molecules of bodies whose surfaces can be brought into really intimate contact. Professor Tait supplies the following instances (" Properties of Matter") : — (i.) Finely powdered graphite is re-solidified in the manu- facture of lead pencils by the application of powerful pressure. (ii.) Two freshly cut lead surfaces, pressed firmly together with a screwing motion, will adhere very strongly to one another. (Hi.) Sir Joseph Whitworth's steel planes are so true that, when pressed together, they offer a resistance to separation markedly greater than can be accounted for by the pressure of the atmos- phere, (in.) The surfaces of marble blocks may be so truly worked that, on being pressed together, either can be lifted suspended from the other, even in vacuo (if its weight be not too great in comparison with the area of contact), (u) All the processes of gilding, silver-plating, etc., as well as the properties of gum and glue, depend upon the cohesive forces between mole- cules brought within insensible distances of one another. II.— § 123, page 56. Expressions for the Component Strains and Rotations, to the second power of small quantities. Let the coordinates of the points P, Q, R in the natural state be (x, y, z), (x+dx, y, z), (x, y + dy, z), and let P', Q', R' be the strained positions of these points. Then, if the component displacements of P be u, v, w, the coordinates of Q' relative to P will be 1 , <^Aj 3v , 3ic, 1 + ^- Ida;, —ax, -^-dx. ~dx) ' 3a; ' 3a; ADDITIONAL NOTES. 505 But P'Q'= (1 +e)PQ= (1 +e)dx, and therefore thus to the second order of approximation Again the projections of P'R' upon the axes are and PR' = (1 +f)dy, so that sin c = cos(£jt — c) = cos Q'PR' (. chiYdu 3^/i 3iA 3w oto 'dxf'dy "dx\ ?>y/ ~dx ^by and ultimately _ 3m>/, 3 "\ , 3»/, _ 3io\ 3m 3m 3y\ 3y/ 3z\ 3s/ 3j/ 3s , _ 3m/-i _ 3w\ 3io/, _ 3w\ 3» civ dz\ 3«/ 3a; \ 3a;/ 3z 3a; _3«/, chi\ "du/, _~dv\ ^bw ?)w 3x\ 3a;/ 3y\ 3y/ ' 3a; 3y Finally, if P'Q', P'-R' make angles taDV/= ( 1+ 3|)/S' and therefore 3«/, 3»\ _ 3«/, 3tt\ 3a\ 3y/ 3y\ 3a;/ cos( + ^) = 3i 3aA 1 + 3w , 3i> iTf^Y + l^Y + 9 3 ^ 5£~1 1 3y 1W W 3a; fyj 506 ADDITIONAL NOTES. Thus ultimately ^t("S)-|(' + |) III.— § 235, page 222. Transformation of the Component Rotations. With the notation of Chapter V., _ BTu 3£ v 'drj w 3f~l 3Tm 3f v chj w 3£~| 3z 'dyXhJ dy 3»V'i/ a * MV 3 y &\V and so for 2 and # 3 . But 9 1 = X 1 1 + /x 1 2 +j/ 1 3 ; and therefore and ADDITIONAL NOTES. 507 IV.— § 239, page 229. LamPa transformation of the General Equations. Multiplying equations (52a) of § 218 by \, ^ v 1 respectively and adding, we obtain , , >/„ 3A 3A 3A\ L l \c)y -dz) ^{dz -dx) \-dx 3y/J + p[\(X - «) + ^(Y- v) + v^Z - ii)] = 0, or, with the notation of Chapter V., XT ft 9 1 3 ^ e 2 ^o. 6 ! 3f Aj oy A 2 dy A s oy 3 A x 35 A 2 3s A 3 3a' and therefore, by the results of the last Note, and so on. Consequently ^37/ 3z/ ™\a* 3a:/ ^3*" 3y/ =As 3t 2 (t;)-s! 8 (^)' and the transformed equations are etc., etc. i)8 ADDITIONAL NOTES. Y.—$ 241, page 231. Differential Equations of the Lines of Stress, referred to any Curvilinear system. It follows at once from § 163 that the principal stresses N v N 2 , N 3 , at any point are the roots of the cubic in P- U T U Q- S T S R-4> = and that the directions of the principal axes are given by Pk+Uix+Tv _ U\ + Qp + Sv _ T\ + Six + Rv _ N A fl v where A, yu, v are the cosines of the angles made by the principal axis corresponding to N with the elements ds v ds 2 , ds 3 , and the notation is throughout that of Chapter V. Now dsj\ = dsjfi. = dsjv = ds, where ds is the element of the principal axis, so that these latter equations may be written Pdsj + Uds 2 + Tds^ _ Uds { + Qds 2 + Sds^ _ Tds x + Sds 2 + Bds s _ „ ds ds, ds„ These then are the differential equations of the Lines of Stress. See § 293 for an example. VI.— § 401, page 477. A Theorem in Conjugate Functions. If £ >j be conjugate functions of x and y, and if it is required to show that v 2 log h = 0. Whatever function h may be of x and y, w^-wffi-ffi identically. But A 2 (aj + (%)' ADDITIONAL NOTES. and therefore 509 "dif ?>*R>y -dx 3a% 2 \Vxdy) ty V W7 i "V^e ac 2 3j/ 3u%/ \3aj 3a% 'dy 3t/ 2 / f r/3A 2 _ /3AH/3 2 ! _ 3!|\ + 4 M M ^l_ I v *£ — identically, since V 2 £= 0. INDEX [The Arabic numbers refer to the Articles.] Absolute moduli, 221. .Eolotropy, 201-206. AIRY, Sir G. B., general solution under surface traction only, 302 ; under applied forces also, 307-309, 307 his. Anticlastic flexion of plates, 389 ; coefficient of, 391 ; do., for thin shells, 405. Applied forces, 4; form an equilibrating system, 29 ; work done by, against stress, 21, 27, 31; measure of, 136; continuous and finite, 136; work done by, during change of strain, 193-195; vibrations under periodic, 279-283 ; equilibrium under conservative, 303-321. Areal dilatation, 129. Asymmetrical elasticity, 201. Axes of reference, choice of, 50, andApp. I.; change of, 121, 159. Axes, principal, see Principal axes. Axis of stress in one dimension, 186. Axis, central of a beam, hoop, or wire, 322. Axis of torsion, 331. Beams, defined, 322. BERNOULLI, James, the Linea Elmtica, 361. Boilers, strength of cylindrical, 291. Bone, structure of, App. VI. BOSCOVITCH, theory of intermolecular force, 37, 208. BOTTOMLET, J. T., on effects of set, 15. Boundary conditions, in Cartesians, 145, 217, 218; in curvilinears, 238; for wires, 360, 364; for thin plates, 397. 398, 400, 401 ; for viscous solids, 412 ; for viscoub liquids, 417. BOUSSINESQ, J., solution of the problem of vibrations, 283, 284 ; of equilibrium under conservative forces, 310-321. Breaking stress, App. IV. (B.) Brittle materials, 13, and App. IV. (C). CATJCHY, Aug., on the sphere of action of intermolecular forces, 38; the first to introduce the modern idea of stress, App. 111. Central axis of a beam, hoop, or wire, 322. Centre of a plate, 384. Change of direction of straight lines in homogeneous strain, 55; of axes of refer- ence, 121, 159. CLEB3CH, problem on flexion of plates, 386, 387. CLERK MAXWELL, J., on viscosity of air, App. IV. (A.) ; on Nachwirkung, App.V. ;andFaraday's theory of dielectric tension, Ex. 21, page 260. Coefficients of elasticity, 198-212 ; of longi- tudinal extension, 329 ; of torsion, 334 ; of flexion in beams, 349,352; ofsynclastic and anticlastic flexion in plates, 391 ; do. in shells, 405. Coefficient of viscosity, App. IV. (A) Collisions, 408. Components of displacement, 51, 235 ; of rotation, 86, 123, 235, and Notes II. and IU.; of strain, 89, 107, 108, 123, 234, 335 ; of stress, 142, 148-152, 237-239. Compressibility, 211. Compression, cubical, see Cubical compres- sion. Compression quadric, 74. Concurrent strains, 110 ; stresses, 156. Cone of no elongation, 77 ; of constant do., 78 ; normal, of shearing stress, 166 ; tangent, of do. do., 168. Conjugate cylindrics, 245, 246. Conservation of energy, and of momentum in impacts, 408. Conservative system, conditions for, 24. Constraint, state of, App. IV. (B.) Continuity of displacement, 47, 287; of stress, 137. Continuous elastic matter, 42. Contour lines, 337. Contraction, 53. Contrary strains, 110 ; stresses, 156. Conventional theory of elasticity adopted, 39. Coordinate surfaces in general, 230; their principal curvatures, 232. Coordinate systems ; spherical polars, 243 ; cylindrical polars, 244 ; conjugate cylin- drics, 245, 246; surfaces of revolution, 247, 248; conjugate do., 249, 250; spheroidals, 251; ellipsoidals, 252. COTTERILL, J. H., "Applied Mechanics » quoted passim. COULOMB, torsion coefficient of right circular cylinder, 335; erroneous exten- sion of do. to prisms in general 34 2. Coupes topographiques, 337. INDEX. 511 Crisis, elastic,™ ductile metals,App. IT.(B.) Cubical dilatation and compression, 102,103; uniform do., 104, 105; specification of ,112; cubical and longitudinal do., behaviour of ductile metals under, App. IT. (B.) Curvatures, principal, of coordinate sur- faces, 232. Curvilinear coordinates, 230. Cylindrical polars, 244. Cylindrics, conjugate, 245, 246. D'ALEMBEKT'S principle, 41. Deflection of uniform beams from the hori- zontal under gravity, 367-370. Density, 7. Determinateness of the solution under given boundary conditions, 255, 286; do. do. for beams, 328. Dilatation, areal, 129 ; cubical, see Cubical dilatation. Direction, change of, 55 ; directions, stand- ard, for rotors, App. I. Director quadric of stress, 167. Discontinuity, limitations of, 223-229. Discrepant measurements, of shear and shearing stress, 152; of cubical compres- sion and hydrostatic pressure, 174. Displacement, resultant, 125; lines of, 127; potential of, 124; do., in homogeneous strain, 126 ; do., in vibrations, 267. Dissipation of energy in non -uniform strain, 23. Distortion, planes of no, 93, 94, and App. II. Ductile materials, 13, and App. IT. (B.) Ductility, App. IV. (B.) Dynamics, of a particle, 40; of a rigid body, 41. Ease, state of, App. IV. (B.) Economy of material, under torsion, 336,338; under flexion, 355; in nature, App. VI. Elastic properties of actual matter, 11, and App. IV.; limits of do., 12, and App. IV; fatigue, 16; coefficients, 198-212; moduli of isotropic solids, 210-213; strength, 222, and App. IV. (B.) Elasticity, limits of, 12, 13 ; do. of shape and bulk, 14; perfect, see Perfect elas- ticity; coefficients of, 198-212; asym- metrical, 201; crystalline symmetry of, 202-206; isotropic, 207-209; of figure, 210; of volume, 211. Ellipsoid, strain, 64, 70, 123; stress, 169; position, 84. Ellipsoidal coordinates, 252. Elongation, 53, 83; quadric, 74; cones of constant, 78 ; cone of no, 77 ; principal, 83; simple, 90, 91; specification of do., 113 ; work done by stress in increasing, 189, 190. Energy, intrinsic in natural state, 20 ; potential of strain, see Potential energy ; dissipation of, in non-uniform strain, 23 ; total, of free vibrations, 262 ; conserva- tion of, in impacts, 408. Energy method, applied to solids in general, 219 ; to curved wires and hoops, 378 ; to thin plates, 399-401 ; to thin shells, 405- 406. Equations of motion and equilibrium in Cartesians ; in terms of stress, 138-143 ; in terms of strain, 217 ; in terms of dis- placement, 218 ; Lame's form, 218 ; deduced from principle of virtual work, 219 ; in curvilinears, 237 ; Lame's form, 239, and Note IV. ; of naturally straight wires, 360 ; do., when curvature small, 364 ; of thin plates, 396, 397, 400, 401 ; of thin shells, 405, 406 ; of viscous solids, 412; of viscous liquids, 415; do., treated as incompressible, 416, 417. Equilibrium, of the body as a whole, 146 ; general problem of, 285-287, 303; unstable elastic, stage of, App. IV. (B.) Equipotential surfaces of displacement,125; for homogeneous strain, 126; in curvi- linears, 240. Extension, maximum of ductile metal bars, App. IV. (B.); ultimate of do., ibid; longitudinal, behaviour of ductile metals under, ibid ; of beams, 329; coefficient of longitudinal, 329. FARAD AT, see Clerk Maxwell. Fatigue, elastic, 16. Finite shear, App. II. Flexion of beams ; plane circular in a prin- cipal plane, 346-350; in any plane, 351- 354; couple, 349, 352; coefficients of, 349-352 ; economy of material under, 355 ; strength under, App. V. ; of plates ; uni- form, 388-392; synclastic and anticlastic, 389; couples, 391; coefficients of, 391 : do. of thin shells, 405. Flow, of plastic solids and fluids, App. IV. (A.); stages of uniform and local, in ductile metal bars under tension, App. IV. (B.) Fluidity, Fluids, App. IV. (A.) Force, intermolecular, 3; applied or im- pressed, see Applied forces. Free vibrations, problem of, 260-263. General problem, 253; of vibrations, 260, 261, 264; of equilibrium under surface tractions only, 285-287; do. under con- servative appUed forces, 303. GEAY, A. and T., on effects of set, 15. GBEEN, G., his foundation of a theory of elasticity on the principle of energy, App. GREENHILL, A. G., problems on stability of beams, 371-376. HAGKNET, on testing of bars and plates, quoted in App. IT., passim. Hardness of ductile solids, App. IT. (B. ) Harmonics, spherical, see Spherical har- monics. Heat, vibrations due to, 2, 6 ; do. ignored in conventional theory of elasticity, 45. Helix of equilibrium of a naturally straight wire. 362, 3&3. Heterogeneous strain, 122 ; stress, 187 ; isotropy, 220. HODGK.INSON, on stretching of cast-iron beams, App. IT. (C.) 512 INDEX. Homogeneity of molecular structure, 7 ; of continuous matter, 43. Homogeneous strain, 59-121 ; stress, 157. HOOKE'S law, 197, App. III., App. IV. (B.) Hoops, 322; motion and equilibrium of, 377-383. HOPKINS, W., on form of crevasses in glaciers, Ex. 23, page 381. Hydrostatic pressure, 174. I-Beams, 355. Impact, denned, 408. Impressed forces, see Applied forces. Intensity of stress, 131. Intennolecular forces, 3 ; probable sphere of action of, 38 ; Boscovitch's theory of, 37; stresses, 28-33. Invariants of the strain, 111 ; of the stress, 164 ; potential energy expressed in terms of, 209. Irrotational strain, 66; conditions for, 81, 82 ; components of, 89, 107 ; displace- ment potential in, 124 ; free vibrations, 267-274. Isotropy, 207 ; heterogeneous, 220. KENNEDY, Alex. B. W., experiments on ductile metals, App. IV. (B.) KIRCHHOFF, G., boundary conditions for thin plates, 397, 398, 400, 401. KOHLRAUSCH, F., effect on rigidity of change of temperature, Table (F.), page 204. I LAME'S theory of elasticity, App. III.; form of the equations of motion in Car- tesians, 218 ; do. in curvilinears, 239, and Note IV.; analytical theorems in curvi- linears, Ex. 23, page 260. Length moduli, 221 ; of rupture, 222. Limits of elasticity, 12-15; mathematical of perfect elasticity, practical in ductile metals, of uniform flow, and of tenacity, App. IV. (B.) lAnea elastica, 361. Lines of flow, App. IV. (A.) Lines of stress, 216, etc. Lines in body, 44 ; preserve continuity of structure, and of curvature, 55 ; and per- manence of intersections, 56. Loads, maximum and terminal, of metal bars under tension, App. IV. (B. ) Local flow, stage of, App. IV. (B.) Longitudinal stress, 132, 148; extension, coefficient of, 329. MACALISTER, Donald, on economy of material in nature, App. VI. Malleable substances, 13, and App. IV. Mathematical limit of perfect elasticity in ductile solids, App. IV (B.) and Table (C), page 201. Matter, structure of , 1, 2, 3 ; solid do. , 6 ; elastic properties of, 11-16. Maximum load, extension and strength of ductile metal bars under tension, App. IV. (B.) Median surface of plate, 384. Moduli, elastic, of isotropic solids, 210 (rigidity); 211 (compression); 213 (Young's modulus); various systems of measure- ment, 221. Modulus of rupture, 222. Modulus of viscosity, App. IV. (A.) Molecular structure of matter, 1, 2. Molecules, 2 ; probable size of, 36. Momentum, how affected by collisions, 408. Motors, App. I. Nachwirkung, App. V. Natural state, 5; intrinsic energy in, 20; stability of, 21 ; of a ductile solid after manufacturing processes, App. IV. (B.) NAVIER'S theory of elasticity, App. III. Neutral plane in flexed beam, 347; do., surface in flexed plate. 390. NIVEN, C, normal vibrations of a thin spherical shell, amplitude varying as a zonal harmonic, 407. Non-rotated straight lines in homogeneous strain, 82. Normal axis of plate, 384. Normal stress, 132, 148 ; principal do., 163; cone of shearing stress, 166. Notation, for strain, 59, 71, 72, 73, 100, 103, 123, and Note II.; for stress, 142; for potential energy, 196, 198; for isotropic solids, 212. Origin of axes of reference, choice of, 50, and App. I. Parallelism of straight lines and planes, unaffected by homogeneous strain, 61, 62. Particle, dynamics of, 40. Perfect elasticity, 18; approximation of natural solids to, 19 ; mathematical limit of, in ductile solids, App. IV. (B.) Plane of stress in two dimensions, 184. Planes of no distortion, 93, 94, and App. II. Plastic substances, 13, and App. IV. (A.) Plates, 384; uniform flexion of, 388-392; thin, see Thin plates. Points in body, 44. POISSON'S integrals of the equations of free vibration, 278. Polars, spherical, 243 ; cylindrical, 244. Position ellipsoid, 84. Potential, displacement, 124; for homo- geneous strain, 126 ; for vibrations, 267. Potential energy of strain, 21, 27, 34; equal to work done by applied forces, 21, 34, 188 ; per unit volume, 196 ; relation to stresses, 32, 196 ; do. to strains, 200 ; as an invariant of the strain, 209 ; of iso- tropic solids, in terms of strain, 212 ; in terms of stress, 214. Potential energy, of beam, 358 ; of wire or hoop, 378; of plate, 392, 399, 401; of thin shell, 405. Practical elastic limits in ductile materials, App. IV. (B.), and Table (C Ms), page 202. Pressure, 131 ; surface, 133 ; hydrostatic, 174. INDEX. 513 Principal axes of Btrain, 65, 79, 80, 81, 82 ; of stress, 163 ; of strain and stress in isotropic solids, 215 ; in flexed plate. 390. Principal elongations, 83 ; normal stresses, 163; do. in terms of principal elonga- tions, 215; surfaces of strain, 216, 241, 242. Principal curvatures of coordinate surfaces, 232. Principle of superposition, 87, 88, 155, 197, 254-259. Pure strain, see Irrotational strain. Quadrics, elongation and compression, 74; strain ellipsoid, 64, 70, 123; position ellipsoid, 84; stress ellipsoid, 169; direc- tor quadric of stresB, 167 ; first quadric of stress, 162 ; fourth do. do., 172; cone of no elongation, 77; do. of constant do., 78 ; normal cone of shearing stress, 166 ; tangent do. of do. do., 168. RAYLEIGH, Lord, solution for plane waves propagated parallel to the plane face of an infinite solid, Ex. 8, page 377. Release from stress, sudden, treated as impact, 408. Resilience, 222. Resistance to stress, 135. Resolution, of small strain into arbitrarily chosen components, 116-120 ; of stresses, 144; of extension into dilatation and shear, 213. Resultant, of any number of small strains, 115; rotation, 85, 86; stress, 160; of simple extension, torsion and flexion of beams, 356-358. Revolution, surfaces of, 247, 248 ; conjugate do., 249, 250. Rigid dynamics, 41. Rigidity of an isotropic solid, 210 ; change of do. with temperature, Table (F.), page 204. Rods, see Beams and "Wires. Rotational strain, 85, 86; components of do., 108 ; free vibrations, 275-278. Rotations, component, 86, and Note II. ; in curvilinears, 235, and Note III. Rotors, App. I. St. VENANT, Barre de, adopts Cauchy's deductions from Boscovitch 's hypothesis, App. III. ; his problem on the straining of beams, 324-328 ; special solution for torsion, 333 ; on points of maximum and minimum torsion-shear, 341. Scalar quantities, App. I. Shear, simple, 92-100; axes of, 92; amount of, 94 ; notation for, 100 ; finite, 101, and App. II. ; specification of, 114 ; work done by stress in increasing, 191. Shearing motion, 95 ; finite do., App. II. Shearing stress, 132, 150-152. Shells, 384; thin, see Thin shells. Sign of stress, 131. Similar and similarly situated figures, in the same or parallel planes, remain such after homogeneous strain, 63. 2 Simple strain and stress, 33; elongation, 90,91,113; shear, 92-100, 114; dilatation, 102-105, 112. Small strain, 51-58 ; stress, 153-155. Solid matter, 6 ; body, homogeneous, 8. Solidity, App. IV. (A.) Sound, plane waves of, 268-271 ; spherical harmonic do., 272; simple spherical do., 273 ; possible forms of do., 274 ; velocity of, in infinite medium, 268; in wires, 365. Specification, in terms of standard com- ponents, 111 ; of cubical dilatation, 112 ; of simple elongation, 113 ; of simple shear, 114 ; of most general small strain, 115. Spherical harmonics, solution in terms of, for free vibrations, 272, 273 ; for equi- librium of sphere under surface tractions only, 295-300; do. under applied forces whose potential can be expanded in a series of harmonics, 304-306 ; normal vibrations of thin spherical shell, am- plitude varying as a zonal harmonic, 407, Spherical polars, 243. Spheroidal coordinates, 251. Stability of natural state, 21. Stages, of perfect elasticity according to Hooke's law, of unstable elastic equi- librium, of uniform flow, and of local flow, in ductile metals, App. IV. (B. ). Standard components, of strain, 106-110; of stress, 148-152. Standard directions for rotors, App. I. States of constraint and of ease of ductile solids, App. IV. (B.) State, natural, see Natural state. STOKES, G. G., on Boscovitch's hypo- thesis, 37, 208; on experimental proof of Hooke's law, App. III. ; on viscosity of fluids, App. IV. (A.) Strain, in molecular structure, 10 ; coordi- nates, 32, 46, 109; simple, 33; type of, 33, 110; in continuous matter, 47-50; small, 51-58; homogeneous, 59-121 ; pure or irrotational, 66 ; concurrent and con- trary types, 110; specification of, 111-115; heterogeneous, 122, 123, and Note II. ; in two dimensions, 129; geometry of, App. I. ; invariants of, 111 ; work done by stress in increasing any small, 192; in terms of stress, 200, 214 ; relation to potential energy per unit volume, 200 ; components in curvilinears, 234 ; principal axes of, 65, 79, 80, 81, 82, 215; principal surfaces of, 216, 241, 242. Strain ellipsoid, 64, 70, 123. Strain reversal (Nachwirku-ng), App. V. Strength, elastic, 222; of ductile metals under tension, and cubical and longi- tudinal compression, App. IV. (B.) ; do. under torsion and flexion, App. V. Stress, intermolecular, 28 ; an equilibrating system, 29 ; resists increase of, and vanishes with, strain, 30; type of, 33, 156 ; simple, 33 ; work done by or against, 31 ; relation to potential energy of strain, 32 ; in continuous matter, 130, 514 INDEX. 131; Bign of, 131; intensity of, 131; total, 131 ; normal or longitudinal, 132, 148 ; tangential, 132, 149 ; shearing, 132, 150-152; two aspects of, 134, 135; re- sistance to, 135 ; continuity of, 137 ; components of, 148-152 ; small, 158-155 ; homogeneous, 157 ; graphic properties of, 159-186 ; general theorems on, 159-161 ; resultant, 160; quadrics of, 162, 167, 169, 172; principal axes of, 163, 215; invariants of, 164 ; special forms of, 174 ; in two dimensions, 175-184 ; plane of do. , 184 ; in one dimension, 185, 186 ; axis of do., 186 ; heterogeneous, 187 ; work done by in, small arbitrary increase of strain, 189-192 ; relation to potential energy per unit volume, 196 ; expressed in terms of strain, 197. 212 ; principal normal, 163 ; do. in terms of principal elongations, 215 ; lines and tubes of, 216 ; breaking, maximum and terminal, of ductile metal bars under tension, App. IV. (B.) Strut lines, 216. Summary of the general problem, 253. Superposition, of small strains, 87, 88 ; of small stresses, 155 ; of strain and stress, applied to proof of Hooke's law, 197; of partial solutions of the general equa- tions, 254-259. Surfaces in body, 44; preserve continuity of structure, and of curvature, 55 ; and permanence of intersections, 56 ; of re- volution, 247, 248; conjugate do., 249, 250 ; principal, of strain, see Principal surfaces. Symmetrical elasticity, crystalline or seolo- tropic, 202-206 ; isotropic, 207 et seq. Synclastic flexion of plates, 389 ; coefficient of do. , 391 ; do. in shells, 405. Tables ; Factors for reduction from one system of units to another, (A.), page 199; Compressibility of liquids, (B.j, page 200 ; Weight moduli of solids, in O.G.S. units, (O), page 201; Practical moduli, in English measure, (C bis), page 202; Ultimate and working strengths, (D.), page 203 ; Effect on Young's modulus of change of temperature, (E.), page 204; Effect on rigidity of do., (F.), page 204 ; Velocities of plane sound waves m infinite media, page 290. TAIT, P. G., examples of sensible inter- molecular force, Note I.jf account of Nachwirkuitg, App. V. V Tangent cone of shearing stress, 168. Tangential stress components, 132, 149. Temper, 15. Temperature, 20 ; constant, 21 ; free to vary, 22. Tenacity, 222, and App. IV. Tension, 131. Terminal load of ductile metal bars under tension, App. IV. (B.) Theorems, general, on the partial solutions of the linear equations of elasticity, 254-259. Thin plates, 384 ; equations of motion and equilibrium under normal forces, 396, 397; Kirchhoff's boundary conditions, 397, 398; treatment by energy method, 399-401. Thin shells, 384; motion and equilibrium under normal forces, 405, 406. THOMSON, Sir Win., on viscosity and fatigue, 16, and App. IV. (A.); on thermoelasticity, 25 (footnote) ; on size of molecules, 36 ; on Navier and Poisson's deductions from Boscovitch's hypothesis, 37 (footnote) ; on permanent change of density, due to longitudinal extension, App. IV. (B.) ; solution for free vibra- tions, 265-267, 275 ; on theories of the luminiferous ether, 276; application of his method to obtain a general solution for equilibrium under surface tractions only in the form of potentials, 301. THOMSON and TAIT'S "Natural Philo- sophy," first combines the principles of Green and Stokea as a mathematical basis for the linear -elations between strain and stress, Ap >. III. ; spherical harmonic solutions, 295-300, 304-306 ; on equilibrium of thin plates, 396-398; also quoted passim. Thrust, 131. Tie lines, 216. Timber, App. IV. (D.) Torsion of beams, 330-342; axis of, 331; couple, 334; coefficient of, 334; economy of material under, 336, 338 ; false exten- sion of Coulomb's formula for, 342; strength under, App. V. Total stress, 131, 133. Traction, 131;- surface do., 133; resolved into dilatation and shear, 213. TBESCA, on flow of plastic Bolids, App. IV. (A.) Tubes of stress, 216. TwiBt, 332. Type of strain, 33, 110; of stress, 33, 156. Types of reference, for strain, 89-109 ; for stress, 148-152. Ultimate state of ease of a ductile solid, App. IV. (B.); do., strength of materials, Table (D.) page 203. Uniform flexion of plates, 388-392. Uniform flow, stage of, App. IV. (B. ) VectorB, App. I. Velocity of sound, in infinite media, 268; in wires, 365. Vibrations, free or under periodic surface tractions only, 260-278, 284 ; under periodic applied forces, 279-283; Bous- sinesq's solution for, 283, 284; of wires, 365, 366. Viscosity, 16, 19, 25, and App. IV. (A,B.) Viscous liquids, App. IV. (A. ) ; torsion of, 335 ; equations of motion of, 415, 416 ; boundary conditions for, 417. Viscous solids, App. IV. (A, B.); equations of motion of, and boundary conditions for, 411, 412. Weight moduli, 221. INDEX. 51 o WERTHEIM, effect on Young's modulus of change of temperature, Table (E.), page 204. Wires, 322; equilibrium and motion of naturally straight do., 359, 360; do. do., when flexion small, 364; small vibrations, 365, 366; naturally curved wires, 377- 383. Work done, by external forces during strain, 21, 27, 31; by stress in small arbitrary increase of strain, 189-192; by applied forces in do. do., 193, 195; iden- tical equality of these, 194, 195. Working strength of materials, Table D.), page 203. YOUN6,onHooke's law, App. III. ; Young's modulus, 213; change of do. with tem- perature, Table (E.), page 204. ROBERT MACLEHOSE, UNIVERSITY PRESS, OLASQOW.