Balance of energy for thermoelastic materials
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Show that, for thermoelastic materials, the balance of energy
can be expressed as
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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
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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
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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.
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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,