Continuum mechanics/Balance of energy for thermoelasticity

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Balance of energy for thermoelastic materials[edit]

Show 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 materials[edit]

For thermoelastic materials, the specific internal energy is given by

where is the Green strain and is the specific entropy. Show that

where is the initial density, is the absolute temperature, is the 2nd Piola-Kirchhoff 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 materials[edit]

For thermoelastic materials, show that the balance of energy equation

can be expressed as either

or

where

For the special case where there are no sources and we can ignore heat conduction (for very fast processes), the energy equation simplifies to

where is the thermal expansion tensor which has the form for isotropic materials and is the coefficient of thermal expansion. The above equation can be used to calculate the change of temperature in thermoelasticity.

Proof:

If the independent variables are and , then

On the other hand, if we consider and 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,