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

Homework Equations

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

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

The Attempt at a Solution

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

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.