Continuum mechanics/Stress-strain relation for thermoelasticity

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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