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
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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