Strain Energy In Terms Of Principal Stresses:

We have already derived expressions for strain energy supposing that at a time only one stress component is working. For instance, we have taken e _x as equal to s_x  /E which is true only if s_x  alone is acting. Or else, we know that,

e _x= (s _x )/E - (v s_y ) /E - (v s_y )/E

and the use of such expressions shall result in more composite expressions for strain energy and strain energy density. Therefore, in the case of a broad stress field, we might be able to obtain simpler expressions if we decrease the number of terms to be considered. A general stress field along six components of stress namely s_x, s_y, s_z, t_xy, t_yz and t_zx may be expressed in terms of equivalent principal stress components s₁, s₂ and s₃. In terms of these three components, now let us derive expressions for net strain energy.