Expressing Constitutive Equation for an Elastic Solid

[sara_87]

Homework Statement

Show that the constitutive equation for an elastic solid can be expressed in the form:

T_ij=$$\frac{1}{2}$$$$\frac{\rho}{\rho₀}$$$$\frac{\partial(x_i)}{\partial(X_R)}$$$$\frac{\partial(x_j)}{\partial(X_S)}$$($$\frac{\partial(W)}{\partial(\gamma_RS)}$$+$$\frac{\partial(W)}{\partial(\gamma_SR)}$$)

Homework Equations

A constitutive equation for finite elastic solid is:
T_ij=$$\frac{\rho}{\rho₀}$$$$\frac{\partial(x_i)}{\partial(X_R)}$$$$\frac{\partial(x_j)}{\partial(X_S)}$$($$\frac{\partial(W)}{\partial(C_RS)}$$+$$\frac{\partial(W)}{\partial(C_SR)}$$)

where $$\gamma$$=$$\frac{1}{2}$$(C-I) (I is the identity matrix)

The Attempt at a Solution

so therefore i have to show that $$\frac{\partial(W)}{\partial(C_RS)}$$+$$\frac{\partial(W)}{\partial(C_SR)}$$=$$\frac{1}{2}$$($$\frac{\partial(W)}{\partial(\gamma_RS)}$$+$$\frac{\partial(W)}{\partial(\gamma_SR)}$$)
using the fact that $$\gamma$$=$$\frac{1}{2}$$(C-I),
$$\frac{\partial(W)}{\partial(C_RS)}$$+$$\frac{\partial(W)}{\partial(C_SR)}$$=$$\frac{1}{2}$$($$\frac{\partial(W)}{\partial(\gamma_RS)+(1/2)I}$$+$$\frac{\partial(W)}{\partial(\gamma_SR)+(1/2)I}$$)

so what do i do with the (1/2)I, did i make a mistake?

 

[sara_87]

oh no, the latex didnt come out right...and it took me ages! i hope it looks understandable.