# Elastic solid

1. Sep 8, 2009

### sara_87

1. The problem statement, all variables and given/known data

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

Tij=$$\frac{1}{2}$$$$\frac{\rho}{\rho0}$$$$\frac{\partial(xi)}{\partial(XR)}$$$$\frac{\partial(xj)}{\partial(XS)}$$($$\frac{\partial(W)}{\partial(\gammaRS)}$$+$$\frac{\partial(W)}{\partial(\gammaSR)}$$)

2. Relevant equations

A constitutive equation for finite elastic solid is:
Tij=$$\frac{\rho}{\rho0}$$$$\frac{\partial(xi)}{\partial(XR)}$$$$\frac{\partial(xj)}{\partial(XS)}$$($$\frac{\partial(W)}{\partial(CRS)}$$+$$\frac{\partial(W)}{\partial(CSR)}$$)

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

3. The attempt at a solution

so therefore i have to show that $$\frac{\partial(W)}{\partial(CRS)}$$+$$\frac{\partial(W)}{\partial(CSR)}$$=$$\frac{1}{2}$$($$\frac{\partial(W)}{\partial(\gammaRS)}$$+$$\frac{\partial(W)}{\partial(\gammaSR)}$$)
using the fact that $$\gamma$$=$$\frac{1}{2}$$(C-I),
$$\frac{\partial(W)}{\partial(CRS)}$$+$$\frac{\partial(W)}{\partial(CSR)}$$=$$\frac{1}{2}$$($$\frac{\partial(W)}{\partial(\gammaRS)+(1/2)I}$$+$$\frac{\partial(W)}{\partial(\gammaSR)+(1/2)I}$$)

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

2. Sep 8, 2009

### sara_87

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