173, 174] STRESS -POTENTIAL. 457 



principal diagonal being equal to the corresponding fifteen on the 

 other side. 



If the body besides being homogeneous is isotropic, that is, at 

 any point its properties are the same with respect to all directions, 

 there are many relations between the coefficients, so that the number 

 of independent constants is much reduced. In an anisotropic or eolo- 

 tropic body there are generally certain directions (the same for all 

 parts of the body) with reference to which there is a certain symmetry, 

 so that there are various relations involving a reduction in the number 

 of constants. Such bodies are known as crystals. We shall deal 

 here only with isotropic bodies. 



174. Energy Function for Isotropic Bodies. In isotropic 

 bodies the stresses developed depend only on the magnitude of the 

 strains, not on their absolute directions with respect to the body. 

 Accordingly if we change the axes of coordinates the expression for 

 the energy must remain unchanged, or the energy function is an 

 invariant for a change of axes. The cubic for the axes of the 

 elongation quadric 58) belonging to the shift -equations 50) is the 

 determinant 



A, g, , g y 



g, , s y - I, g x 



9y > 9* , S* 



or expanding the determinant, 



128) 4 3 - (s x + s y + s.) tf 4- (s y s z + s z s x + s x s y gl g} gl) I 



+ s x gl + s y gl + s,gl - s x s y s, - 2g x g 1J g z = 0. 

 If the roots are A 17 Z 2? A 3 , the equation is 



129) tf - (^ + A 2 + * 3 ) tf + (M 2 + Vs + Mi) * - M 2 *3 = 0. 



If we transform to another set of axes X'Y'Z' with the same origin, 

 so that the strain components are s x ', s y ', s z ; g x r , g y ; g z ', since the 

 elongation quadric is a definite surface, the equation for its axes 

 must have the same roots as before. Accordingly its coefficients are 

 invariants. The roots A 1? A 2 , A 3 are the stretches for the directions 

 of the principal axes of the strain. Therefore we have the three 

 strain invariants, symmetrical functions of the roots, 



/! = ^ + ^ 2 + Ag = S x + S y + 8,, 



130) I 2 = A^ 2 + M 3 + Vi = s v s z + B*SX + s x s y -gl-gl gl, 



J 3 = ^A^g = 2g x g y g, -f s x s y s, - s x gl - s v g* y s,g*,. 



The invariant J x represents the cubical dilatation (?, which by its 

 geometrical definition is evidently independent of the choice of axes. 



127) 



0, 



