Continuum mechanics/Specific heats of thermoelastic materials

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Relation between specific heats - 1[edit]

For thermoelastic materials, show that the specific heats are related by the relation

Proof:

Recall that

and

Therefore,

Also recall that

Therefore, keeping constant while differentiating, we have

Noting that , and plugging back into the equation for the difference between the two specific heats, we have

Recalling that

we get

Relation between specific heats - 2[edit]

For thermoelastic materials, show that the specific heats can also be related by the equations

We can also write the above as

where is the thermal expansion tensor and is the stiffness tensor.

Proof:

Recall that

Recall the chain rule which states that if

then, if we keep fixed, the partial derivative of with respect to is given by

In our case,

Hence, we have

Taking the derivative with respect to keeping constant, we have

or,

Now,

Therefore,

Again recall that,

Plugging into the above, we get

Therefore, we get the following relation for :

Recall that

Plugging in the expressions for we get:

Therefore,

Using the identity , we have

Specific heats of Saint-Venant–Kirchhoff material[edit]

Consider an isotropic thermoelastic material that has a constant coefficient of thermal expansion and which follows the Saint-Venant–Kirchhoff model, i.e,

where is the coefficient of thermal expansion and where are the bulk and shear moduli, respectively.

Show that the specific heats related by the equation

Proof:

Recall that,

Plugging the expressions of and into the above equation, we have

Therefore,