STRAINED ELASTIC SOLIDS 459 



The method of deaHng with the variational equation [360] is 

 essentially the same as that of dealing with the variational 

 equation [15] in the early pages of Gibbs' discussion, although 

 the presence of integral signs and merely formal differences of 

 appearance betweert [15] and [360] may mask the identity of the 

 methods. It would have been quite legitimate to write in 

 [15] f f ft'h-q'v'dx'dy'dz' for t'hri, the integration being 

 throughout the phase indicated by one accent, and so on; but it 

 was unnecessary, as the conditions were uniform throughout 

 any given phase in equilibrium. But for a solid the strain 

 may be heterogeneous, and so ■qv might well change in value 

 from point to point of the solid body with the changing values 

 of an, ai2, . . . flss. Hence the necessity for the integral. Also 

 if the strain were homogeneous we could write the second term 

 in [360] as F'ZS'Xx'San, Y' being the volume (unstrained) of 

 the solid; but in general this is not possible. Reflection on this 

 and similar considerations for the remaining terms will remove 

 any difficulty in understanding raised by pure differences of 

 form. Following this hint we see that [361], [362] and [363] 

 are the additional equations arising from constancy of total 

 entropy, from constancy of the total volume of the system 

 within the envelop, and from constancy of total mass of an 

 independent constituent of the system; they are entirely 

 analogous to equations [16], [17] and [18] respectively. Con- 

 dition [361] is straightforward. In [362] we consider any 

 element of the fluid Dv in the form of a thin disc lying between 

 an element of surface Ds of the solid and a similar element of the 

 rigid envelop. First of all the variation of the strain in the 

 solid involves displacements hx, by, 8z of the point x, y, z, the 

 center of Ds; thus Ds is displaced normally towards the envelop 

 by abx + ^by + 'ybz. This reduces the volume Dv by an 

 amount {abx -\- ^by + ybz)Ds. In addition the accretion of 

 new matter reduces it also by bNDs or vvbN'Ds' as we saw 

 above. These two causes therefore bring about a change 

 8Dv in Dv which is given by [362]. Equation [363] offers no 

 difficulty. The subsequent reasoning leading to equation 

 [369] is based on an application of Lagrange's method of 

 multipliers, referred to and used earlier in Gibbs' discussion. 



