Advanced elasticity/Clausius-Duhem inequality for thermoelasticity

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Clausius-Duhem inequality for thermoelasticity

For thermoelastic materials, the internal energy is a function only of the deformation gradient and the temperature, i.e., . Show that, for thermoelastic materials, the Clausius-Duhem inequality

can be expressed as

Proof:

Since , we have

Therefore,

Now, . Therefore, using the identity , we have

Hence,

or,