APPENDIX TO CHAPTER VII. EQUATIONS OF EQUILIBRIUM OF AN ELASTIC SOLID DEDUCED Appendix C. (a.) Let a solid composed of matter fulfilling no condition of isotropy in any part, and not homogeneous from part to part, be given of any shape, unstrained, and let every point of its surface be altered in position to a given distance in a given direction. It is required to find the displacement of every point of its substance, in equilibrium. Let x, y, z be the co-ordinates of any particle, P, of the substance in its undisturbed position, and x+a, y+ß, z+y its co-ordinates when displaced in the manner Strain, not specified that is to say, let a, ß, y be the components of the small, specirequired displacement. Then, if for brevity we put necessarily fied by six elements. +1) + (+1)+ dx dy dx dy these six quantities A, B, C, a, b, c are proved [§ 190 (e) and § 181 (5)] to thoroughly determine the strain experienced by the substance infinitely near the particle P (irrespectively of any rotation it may experience), in the following manner : (b.) Let έ, n, be the undisturbed co-ordinates of a particle infinitely near P, relatively to axes through P parallel to those of x, y, z respectively; and let , n, be the co-ordinates relative still to axes through P, when the solid is in its strained condition. Then §‚2+n‚a+¿,2=A§2+Bn2+CS+2an$+2b§§+2c§n (2); and therefore all particles which in the strained state lie on spherical surface Appendix C. Anticipatory application of the Carnot and Clausius thermodynamic law: its combination with Joule's law expressed analytically for elastic solid. Potential energy of deforination; a minimum for stable equilibrium. are in the unstrained state, on the ellipsoidal surface, A§2+Bn2+C§2+2an$+2b§§+2c&n=r,2. This (§§ 155-165) completely defines the homogeneous strain of the matter in the neighbourhood of P. (c.) Hence, the thermodynamic principles by which, in a paper on the Thermo-elastic Properties of Matter, in the first number of the Quarterly Mathematical Journal (April 1855), of which an account will be given in Vol. III., Green's dynamical theory of elastic solids was demonstrated as part of the modern dynamical theory of heat, show that if wdxdydz denote the work required to alter an infinitely small undisturbed volume, dxdydz, of the solid, into its disturbed condition, when its temperature is kept constant, we must have where ƒ denotes a positive function of the six elements, which vanishes when A-1, B−1, C-1, a, b, c each vanish. And if I denote the whole work required to produce the change actually experienced by the whole solid, we have where the triple integral is extended through the space occupied by the undisturbed solid. (d.) The position assumed by every particle in the interior of the solid will be such as to make this a minimum subject to the condition that every particle of the surface takes the position given to it; this being the elementary condition of stable equilibrium. Hence, by the method of variations SW=fff8wdxdydz=0 But, exhibiting only terms depending on da, we have (5). Hence, integrating by parts, and observing that da, &ß, dy vanish dP (65) where for brevity P, Q, R denote the multipliers of doa doa da dx' dy' d respectively, in the preceding expression. In order that SW may Appendix C. vanish, the multipliers of da, 8B, dy in the expression now found for it must each vanish, and hence we have, as the equations of equilibrium of which the second and third, not exhibited, may be written down merely by attending to the symmetry. unique when (e.) From the property of w that it is necessarily positive when there is any strain, it follows that there must be some distribution of strain through the interior which shall make Лffwdxdydz the least possible subject to the prescribed surface condition; and therefore that the solution of equations (7) subject to this con- Their solution proved dition, is possible. If, whatever be the nature of the solid as possible and to difference of elasticity in different directions, in any part, and surface disas to heterogeneousness from part to part, and whatever be the placement is extent of the change of form and dimensions to which it is there can be subjected, there cannot be any internal configuration of unstable equilibrium : equilibrium, nor consequently any but one of stable equilibrium, with the prescribed surface displacement, and no disturbing force on the interior; then, besides being always positive, w must be such a function of A, B, etc., that there can be only one solution of the equations. This is obviously the case when the unstrained hence neces solid is homogeneous. given, unless unstable sarily unique for a homogeneous (f.) It is easy to include, in a general investigation similar to solid. the preceding, the effects of any force on the interior substance, such as we have considered particularly for a spherical shell, of homogeneous isotropic matter, in §§ 730...737 above. It is also easy to adapt the general investigation to superficial data of force, Extension instead of displacement. lysis to include bodily data of sur (g.) Whatever be the general form of the function f for any force, aud part of the substance, since it is always positive it cannot change face force, in sign when A-1, B-1, C-1, a, b, c, have their signs changed; ensy. and therefore for infinitely small values of these quantities it must be a homogeneous quadratic function of them with constant coefficients. (And it may be useful to observe that for all values of Appendix C. Case of in finitely small strains :Green's theory: the variables A, B, etc., it must therefore be expressible in the same form, with varying coefficients, each of which is always finite, for all values of the variables.) Thus, for infinitely small strains we have Green's theory of elastic solids, founded on a homogeneous quadratic function of the components of strain, expressing the work required to produce it. Thus, putting and denoting by (e, e), (f, f),...(e, f),...(e, a),... the coeffi- w=} {(e,e)e2+(ƒ, ƒ)ƒ2+(g, g)g2+(a,a) a2+(b,b)b2 +(c,c) c2} +(f,g)fg+(f,a) fa+(f,b)ƒb+(f,c)fe +(g,a)ga+(g,b)gb+(g,c)ge +(a,b)ab+(a,c)ac +(b, c)bc du da (h.) When the strains are infinitely small the products dw da db dz dA dr etc., are each infinitely small, of the second order. We therefore omit them; and then attending to (8), we reduce (7) to dynamic and relative kinematic equations. which are the equations of interior equilibrium. Attending to dw dw are linear functions of e, f, g, a, (9) we see that dw de =(e, e)e+(e,f)ƒ+(e,g)g+(e, a)a+(e,b)b+(e,c)c (11). And, a, ẞ, y denoting, as before, the component displacements of any interior particle, P, from its undisturbed position (x, y, z) we have, by (8) and (1) It is to be observed that the coefficients (e, e) (e, f), etc., will be Appendix C. in general functions of (x, y, z), but will be each constant when Case of inthe unstrained solid is homogeneous. finitely small strains: unique (.) It is now easy to prove directly, for the case of infinitely small strains, that the solution of the equations of interior equilibrium, whether for a heterogeneous or a homogeneous solid, solution subject to the prescribed surface condition, is unique. For, let proved a, B, y be components of displacement fulfilling the equations, and let a', B', y' denote any other functions of x, y, z, having is homothe same surface values as a, ß, y, and let e',. w' denote heterogenefunctions depending on them in the same way as e, f, ..., w de- surface dispend on a, ẞ, y. Thus by Taylor's theorem, dw de dw dro dg du da dw dw whether the body geneous or ous, when placement is given: (g'—g) + '" (a'--a) + ' 1" (b' —b) + 110 (c' —c) + H, where II denotes the same homogeneous quadratic function of e'—e, etc., that w is of e, etc. If for e'-e, etc., we substitute their values by (12), this becomes Multiplying by dxdydz, integrating by parts, observing that a'—a, ẞ'— ẞ, y'-y vanish at the bounding surface, and taking account of (10), we find simply But H is essentially positive. Therefore every other interior condition than that specified by a, B, y, provided only it has the same bounding surface, requires a greater amount of work than w to produce it: and the excess is equal to the work that would be required to produce, from a state of no displacement, such a displacement as superimposed on a, B, y, would produce the other. And inasmuch as a, ß, 7, fulfil only the conditions of satisfying (11) and having the given surface values, it follows that no other than one solution can fulfil these conditions. necessarily So when the are of force. (j.) But (as has been pointed out to us by Stokes) when the but not surface data are of force, not of displacement, or when force acts from without, on the interior substance of the body, the solution surface data is not in general unique, and there may be configurations of unstable equilibrium even with infinitely small displacement. For instance, let part of the body be composed of a steel-bar magnet; and let a magnet be held outside in the same line, and with a pole of the same name in its end nearest to one end of the inner magnet. The equilibrium will be unstable, and there will |