Continuum mechanics/Maxwell relations for thermoelasticity

Maxwell relations between thermodynamic quantitiesEdit

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 relationsEdit

For thermoelastic materials, show that the following relations hold:

and

Proof:

Recall,

and

Therefore,

and

Also, recall that

and

Hence,

and