Balance of energy for thermoelastic materialsShow that, for thermoelastic materials, the balance of energy can be expressed as 
Proof:
Since , we have
Plug into energy equation to get
Recall,
Hence,
Now, . Therefore, using the identity , we have
Plugging into the energy equation, we have
or,
Rate of internal energy/entropy for thermoelastic materialsFor thermoelastic materials, the specific internal energy is given by where is the Green strain and η is the specific entropy. Show that where ρ_{0} is the initial density, T is the absolute temperature, is the 2nd PiolaKirchhoff stress, and a dot over a quantity indicates the material time derivative. 
Taking the material time derivative of the specific internal energy, we get
Now, for thermoelastic materials,
Therefore,
Now,
Therefore,
Also,
Hence,
Energy equation for thermoelastic materialsFor thermoelastic materials, show that the balance of energy equation can be expressed as either or where 
Proof:
If the independent variables are and T, then
On the other hand, if we consider and T to be the independent variables
Since
we have, either
or
The equation for balance of energy in terms of the specific entropy is
Using the two forms of , we get two forms of the energy equation:
and
From Fourier's law of heat conduction
Therefore,
and
Rearranging,
or,
