458 IX- DYNAMICS OF DEFORMABLE BODIES. 



The energy function for an isotropic body, being unchanged when 

 we change the axes, can contain the strains only in the combinations 

 I,, J 2 , I 3 , but these are of the first, second and third degrees 

 respectively, and since <& is of only the secoud degree it cannot 

 contain J 3 . 



Since it is homogeneous (except for a gas) it can contain J t 

 only through its square. We therefore have 



131) = - PIi + AI* + #/ 2 , 



where P, A, B are constants. P is zero, except for gases, and is then 

 positive, for if the gas expands it loses energy. The constant A 

 refers to a property common to all bodies, namely, resistance to 

 compression, and is positive, for work must be done to compress a 

 body. The constant B is peculiar to solids. 



All symmetrical functions of the roots may be expressed in terms 

 of the invariants, for example: 



132) &-*,) + ft -*)' + &- AJ 1 



= 2 ( v + v + V) - 2 ft A 2 + 1, ^ -t- A 3 ;g 

 A 2 + A 3 ) 2 - 6 ft;, + A 2 * 3 + A,^) 



Also 



2&J t + J,J, + J,i 1 )~(i 1 + l 1 + ^'-.(V+V + V), or 



r2 



We may accordingly write A 1^ -f BI 2 as a linear function, of J, 2 

 and of either (^ - A 2 ) 2 + (A 2 - A 3 ) 2 + (X - Aj 2 , or of V + V + A 3 2 . 

 Suppose we write the quadratic terms 



134) 



y 2 + (i s - Itf + (i, - 



which is the form given by Helmholtz. The constant H, being 

 multiplied by tf 2 , refers to changes of volume without changes of 

 form, representing in this case the whole energy, for if there is no 

 change of form the stretches of the principal axes, ^, A 2 , A 3 are equal. 

 The term in C on the other hand refers to changes of form without 

 change in volume, for it vanishes when A, = A 2 = ^ 3 , and represents 

 the whole energy if <? = 0. A perfect fluid is defined as a body in 

 which changes of form produce no stress, so that for such bodies 

 C = 0. 



We may also write 



135) A I* + -BZ, - -STCV + V + V) + K& & + I, + Itf, 



