Imperfectly Homogeneous Elastic Solid. 243 



Mathematical theories of an apparently continuous substance 

 are sometimes based upon finer molecular theories. The dis- 

 tribution, however, of the molecules and of their properties 

 presents so many irregularities and discontinuities, that it is 

 impossible to trace the changes which take place in each indi- 

 vidual, and we are thus obliged to confine ourselves to taking 

 the average of a large number of them ; and it is this average 

 with which we are concerned in explaining the sensible pro- 

 perties of matter. In this way the dynamical theory of gases 

 has been constructed ; and the earlier theories of elasticity 

 rested on a similar foundation. Green's theory of elasticity, 

 on the other hand, is based on the general consideration that, 

 whatever be the nature of the forces acting in the interior of a 

 solid, the energy required to produce the change of form of 

 an element-volume depends only on the deformations which it 

 experiences. These may be expressed in the following man- 

 ner. If u, v, iv be the displacements of a point (x, y, z), 

 u + Au, v + Au, w + Aw those of an adjacent point (x + h, y + k, 

 z + l), then 



. 7 du 1 du 7 du 



UjU/ U/if Lb/is 



provided we neglect squares and products of A, k } I. These 

 terms may be analyzed (after Helmholtz) as follows: — Putting 



K_du -p^dv ar\_dw dv £) ^_dv du 

 da? dy* " dy dz : " dx dy 



dw dv rt du dw ^ dv du 



(i> 



^ = Ty~dz> 2OT2= & - ^' 2 ^ = di~dy 



we shall have 



Au =Ah + Fk + ~El — vrjc + vr 2 l, \ 



Av^Fh + Bk + m-vrJ+^hX • • • ( 2 ) 



Aw = E h + Bk + CI — ™ 2 h + vr-Jo. J 



The terms depending on w h iz^ vt 3 represent a rotation of the 

 parts of the medium near (x, y, z) round lines parallel to the 

 axes ; and when the motions of the solid are exceedingly small 

 compared with the length of a wave in it, the remaining terms 

 express a pure strain of the substance in the neighbourhood 

 of (x, y, z). The energy contained in the element is a function 

 of A, B,C, D, E, F. 



This theory, which compares the mathematical element with 

 a homogeneously strained solid, assumes that the element may 

 be taken so small that the enclosed solid may be treated as 

 homogeneously strained, and that the whole distortion of the 



R2 



