Relation between Cauchy stress and Green strain Show that, for thermoelastic materials, the Cauchy stress can be expressed in terms of the Green strain as 
Proof:
Recall that the Cauchy stress is given by
The Green strain and . Hence, using the chain rule,
Now,
Taking the derivative with respect to , we get
Therefore,
Recall,
Therefore,
or,
or,
From the symmetry of the Cauchy stress, we have
Therefore,
and we get
