STRAINED ELASTIC SOLIDS 



441 



where th, ... T33 are nine linear functions of the stress-con- 

 stituents Xx, • ■ • Zz, involving the quantities brs in the co- 

 efficients. It will be found in fact that 



Til = bnXx + &2i^y + bziXz, 

 T12 = bnXx ~\~ 622-^ r + O32XZ, 



Tl3 = blsXx 4" 023Ay + bszX z, 



and six similar equations. Now the expression (36) represents 

 the change in the strain-energy caused by the infinitesimal 

 increase of strain in the matter occupying unit of volume in the 

 state of strain. But, as we have seen previously, this matter 

 occcupies a volume H~^ in the state of reference, and so we must 

 multiply the expression (36) by H in order to obtain the increase 

 in strain energy of the matter which occupies unit volume in 

 the state of reference. Now from the definition of brs given above 

 we see that brsH is equal to Ars- Hence we arrive finally at the 

 result that the infinitesimal increase in strain energy estimated 

 per unit of volume in the state of reference is 



Xx'^dn ~\~ Xy'Sun ~\~ Xz'Sais 

 + Fx'5a2i + Fy'5a22 + Yz'Sa^s 

 + Zx'Sasi + Zy'dasi ~\~ Zz'dazz, 



(37) 



where 



(38) 



The expression (37) occurs in Gibbs' equation [355]. It is 

 essentially his notation with the convenient simplification 

 of replacing dx/dx' by an, etc. 



It is really an important matter to realize that Gibbs' stress- 



