Advanced elasticity/Maxwell relations for thermoelasticity

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Maxwell relations between thermodynamic quantities[edit]

For thermoelastic materials, show that the following relations hold:

where is the Helmholtz free energy and is the Gibbs free energy.

Also show that

Proof:

Recall that

and

(Note that we can choose any functional dependence that we like, because the quantities , , are the actual quantities and not any particular functional relations).

The derivatives are

and

Hence,

From the above, we have

and

Hence,

More Maxwell relations[edit]

For thermoelastic materials, show that the following relations hold:

and

Proof:

Recall,

and

Therefore,

and

Also, recall that

and

Hence,

and