Continuum mechanics/Specific heats of thermoelastic materials

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Relation between specific heats - 1
For thermoelastic materials, show that the specific heats are related by the relation

C_p - C_v = \cfrac{1}{\rho_0}\left(\boldsymbol{S}-T~\frac{\partial \hat{\boldsymbol{S}}}{\partial T}\right): \frac{\partial \tilde{\boldsymbol{E}}}{\partial T} ~. $$
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Proof:

Recall that

C_v := \frac{\partial \hat{e}(\boldsymbol{E},T)}{\partial T} = T~\frac{\partial \hat{\eta}}{\partial T} $$ and

C_p := \frac{\partial \tilde{e}(\boldsymbol{S},T)}{\partial T} = T~\frac{\partial \tilde{\eta}}{\partial T} + \cfrac{1}{\rho_0}~\boldsymbol{S}:\frac{\partial \tilde{\boldsymbol{E}}}{\partial T} ~. $$ Therefore,

C_p - C_v = T~\frac{\partial \tilde{\eta}}{\partial T}       + \cfrac{1}{\rho_0}~\boldsymbol{S}:\frac{\partial \tilde{\boldsymbol{E}}}{\partial T}         - T~\frac{\partial \hat{\eta}}{\partial T} ~. $$ Also recall that

\eta = \hat{\eta}(\boldsymbol{E}, T) = \tilde{\eta}(\boldsymbol{S}, T) ~. $$ Therefore, keeping $$\boldsymbol{S}$$ constant while differentiating, we have

\frac{\partial \tilde{\eta}}{\partial T} = \frac{\partial \hat{\eta}}{\partial \boldsymbol{E}}:\frac{\partial \boldsymbol{E}}{\partial T} + \frac{\partial \hat{\eta}}{\partial T} ~. $$ Noting that $$\boldsymbol{E} = \tilde{\boldsymbol{E}}(\boldsymbol{S},T)$$, and plugging back into the equation for the difference between the two specific heats, we have

C_p - C_v = T~\frac{\partial \hat{\eta}}{\partial \boldsymbol{E}}:\frac{\partial \tilde{\boldsymbol{E}}}{\partial T}       + \cfrac{1}{\rho_0}~\boldsymbol{S}:\frac{\partial \tilde{\boldsymbol{E}}}{\partial T}  ~. $$ Recalling that

\frac{\partial \hat{\eta}}{\partial \boldsymbol{E}} = - \cfrac{1}{\rho_0}~\frac{\partial \hat{\boldsymbol{S}}}{\partial T} $$ we get

{   C_p - C_v = \cfrac{1}{\rho_0}\left(\boldsymbol{S} -T~\frac{\partial \hat{\boldsymbol{S}}}{\partial T}\right): \frac{\partial \tilde{\boldsymbol{E}}}{\partial T} ~. } $$

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Relation between specific heats - 2
For thermoelastic materials, show that the specific heats can also be related by the equations

C_p - C_v = \cfrac{1}{\rho_0}~\boldsymbol{S}:\frac{\partial \boldsymbol{E}}{\partial T} + \frac{\partial \boldsymbol{E}}{\partial T}:\left(\frac{\partial^2 \psi}{\partial \boldsymbol{E}\partial\boldsymbol{E}}:        \frac{\partial \boldsymbol{E}}{\partial T}\right) = \cfrac{1}{\rho_0}~\boldsymbol{S}:\frac{\partial \boldsymbol{E}}{\partial T} + \cfrac{T}{\rho_0}~ \frac{\partial \boldsymbol{E}}{\partial T}:\left(\frac{\partial \boldsymbol{S}}{\partial \boldsymbol{E}}:\frac{\partial \boldsymbol{E}}{\partial T}\right) ~. $$ We can also write the above as

C_p - C_v = \cfrac{1}{\rho_0}~\boldsymbol{S}:\boldsymbol{\alpha}_E + \cfrac{T}{\rho_0}~\boldsymbol{\alpha}_E:\boldsymbol{\mathsf{C}}:\boldsymbol{\alpha}_E $$ where $$\boldsymbol{\alpha}_E := \frac{\partial \boldsymbol{E}}{\partial T} $$ is the thermal expansion tensor and $$\boldsymbol{\mathsf{C}} := \frac{\partial \boldsymbol{S}}{\partial \boldsymbol{E}}$$ is the stiffness tensor.
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Proof:

Recall that

\boldsymbol{S} = \rho_0~\frac{\partial \psi}{\partial \boldsymbol{E}} = \rho_0~\boldsymbol{f}(\boldsymbol{E}(\boldsymbol{S},T),T)~. $$ Recall the chain rule which states that if

g(u,t) = f(x(u,t), y(u,t)) $$ then, if we keep $$u$$ fixed, the partial derivative of $$g$$ with respect to $$t$$ is given by

\frac{\partial g}{\partial t} = \frac{\partial f}{\partial x}~\frac{\partial x}{\partial t} + \frac{\partial f}{\partial y}~\frac{\partial y}{\partial t} ~. $$ In our case,

u = \boldsymbol{S}, t = T, g(\boldsymbol{S}, T) = \boldsymbol{S}, x(\boldsymbol{S},T) = \boldsymbol{E}(\boldsymbol{S},T), y(\boldsymbol{S},T) = T, \text{and} f = \rho_0~\boldsymbol{f}. $$ Hence, we have

\boldsymbol{S} = g(\boldsymbol{S}, T) = f(\boldsymbol{E}(\boldsymbol{S},T), T) = \rho_0~\boldsymbol{f}(\boldsymbol{E}(\boldsymbol{S},T),T)~. $$ Taking the derivative with respect to $$T$$ keeping $$\boldsymbol{S}$$ constant, we have

\frac{\partial g}{\partial T} = \frac{\partial \boldsymbol{S}}{\partial T} = \rho_0~\left[\frac{\partial \boldsymbol{f}}{\partial \boldsymbol{E}}: \frac{\partial \boldsymbol{E}}{\partial T} +\frac{\partial \boldsymbol{f}}{\partial T}~\frac{\partial T}{\partial T}\right] $$ or,

\mathbf{0} = \frac{\partial \boldsymbol{f}}{\partial \boldsymbol{E}}:\frac{\partial \boldsymbol{E}}{\partial T} + \frac{\partial \boldsymbol{f}}{\partial T}~. $$ Now,

\boldsymbol{f} = \frac{\partial \psi}{\partial \boldsymbol{E}} \qquad \implies \qquad \frac{\partial \boldsymbol{f}}{\partial \boldsymbol{E}} = \frac{\partial^2 \psi}{\partial \boldsymbol{E}\partial\boldsymbol{E}} \quad \text{and} \quad \frac{\partial \boldsymbol{f}}{\partial T} = \frac{\partial^2 \psi}{\partial T\partial\boldsymbol{E}} ~. $$ Therefore,

\mathbf{0} = \frac{\partial^2 \psi}{\partial \boldsymbol{E}\partial\boldsymbol{E}}:\frac{\partial \boldsymbol{E}}{\partial T} + \frac{\partial^2 \psi}{\partial T\partial\boldsymbol{E}} = \frac{\partial }{\partial \boldsymbol{E}}\left(\frac{\partial \psi}{\partial \boldsymbol{E}}\right): \frac{\partial \boldsymbol{E}}{\partial T} + \frac{\partial }{\partial T}\left(\frac{\partial \psi}{\partial \boldsymbol{E}}\right) ~. $$ Again recall that,

\frac{\partial \psi}{\partial \boldsymbol{E}} = \cfrac{1}{\rho_0}~\boldsymbol{S} ~. $$ Plugging into the above, we get

\mathbf{0} = \frac{\partial^2 \psi}{\partial \boldsymbol{E}\partial\boldsymbol{E}}:\frac{\partial \boldsymbol{E}}{\partial T} + \cfrac{1}{\rho_0}~\frac{\partial \boldsymbol{S}}{\partial T} = \cfrac{1}{\rho_0}~\frac{\partial \boldsymbol{S}}{\partial \boldsymbol{E}}:\frac{\partial \boldsymbol{E}}{\partial T} + \cfrac{1}{\rho_0}~\frac{\partial \boldsymbol{S}}{\partial T} ~. $$ Therefore, we get the following relation for $$\partial \boldsymbol{S}/\partial T$$:

\frac{\partial \boldsymbol{S}}{\partial T}      = - \rho_0~\frac{\partial^2 \psi}{\partial \boldsymbol{E}\partial\boldsymbol{E}}:\frac{\partial \boldsymbol{E}}{\partial T}       = - \frac{\partial \boldsymbol{S}}{\partial \boldsymbol{E}}:\frac{\partial \boldsymbol{E}}{\partial T} ~. $$ Recall that

C_p - C_v = \cfrac{1}{\rho_0}\left(\boldsymbol{S}-T~\frac{\partial \boldsymbol{S}}{\partial T}\right): \frac{\partial \boldsymbol{E}}{\partial T} ~. $$ Plugging in the expressions for $$\partial \boldsymbol{S}/\partial T$$ we get:

C_p - C_v = \cfrac{1}{\rho_0} \left(\boldsymbol{S}+T~\rho_0~\frac{\partial^2 \psi}{\partial \boldsymbol{E}\partial\boldsymbol{E}}:\frac{\partial \boldsymbol{E}}{\partial T}       \right): \frac{\partial \boldsymbol{E}}{\partial T}      = \cfrac{1}{\rho_0} \left(\boldsymbol{S}+T~\frac{\partial \boldsymbol{S}}{\partial \boldsymbol{E}}:\frac{\partial \boldsymbol{E}}{\partial T}\right) :\frac{\partial \boldsymbol{E}}{\partial T} ~. $$ Therefore,

C_p - C_v = \cfrac{1}{\rho_0}~\boldsymbol{S}:\frac{\partial \boldsymbol{E}}{\partial T} + T~\left(\frac{\partial^2 \psi}{\partial \boldsymbol{E}\partial\boldsymbol{E}}:\frac{\partial \boldsymbol{E}}{\partial T}       \right): \frac{\partial \boldsymbol{E}}{\partial T}      = \cfrac{1}{\rho_0}~\boldsymbol{S}:\frac{\partial \boldsymbol{E}}{\partial T} + \cfrac{T}{\rho_0}~\left(\frac{\partial \boldsymbol{S}}{\partial \boldsymbol{E}}:\frac{\partial \boldsymbol{E}}{\partial T}\right) :\frac{\partial \boldsymbol{E}}{\partial T} ~. $$ Using the identity $$ (\boldsymbol{\mathsf{A}}:\boldsymbol{B}):\boldsymbol{C} = \boldsymbol{C}:(\boldsymbol{\mathsf{A}}:\boldsymbol{B})$$, we have

{  C_p - C_v = \cfrac{1}{\rho_0}~\boldsymbol{S}:\frac{\partial \boldsymbol{E}}{\partial T} + T~\frac{\partial \boldsymbol{E}}{\partial T}:\left(\frac{\partial^2 \psi}{\partial \boldsymbol{E}\partial\boldsymbol{E}}:\frac{\partial \boldsymbol{E}}{\partial T}       \right) = \cfrac{1}{\rho_0}~\boldsymbol{S}:\frac{\partial \boldsymbol{E}}{\partial T} + \cfrac{T}{\rho_0}~ \frac{\partial \boldsymbol{E}}{\partial T}:\left(\frac{\partial \boldsymbol{S}}{\partial \boldsymbol{E}}:\frac{\partial \boldsymbol{E}}{\partial T}\right) ~.  } $$ {| cellspacing="0" cellpadding="0" style="margin:0em 0em 1em 0em; width:80%" 
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Specific heats of Saint-Venant–Kirchhoff material
Consider an isotropic thermoelastic material that has a constant coefficient of thermal expansion and which follows the Saint-Venant–Kirchhoff model, i.e,

\boldsymbol{\alpha}_E = \alpha~\boldsymbol{\mathit{1}} \qquad\text{and}\qquad \boldsymbol{\mathsf{C}} = \lambda~\boldsymbol{\mathit{1}}\otimes\boldsymbol{\mathit{1}} + 2\mu~\boldsymbol{\mathsf{I}} $$ where $$\alpha$$ is the coefficient of thermal expansion and $$3~\lambda = 3~K - 2~\mu$$ where $$K, \mu$$ are the bulk and shear moduli, respectively.

Show that the specific heats related by the equation

C_p - C_v = \cfrac{1}{\rho_0}\left[\alpha~\text{tr}{\boldsymbol{S}} + 9~\alpha^2~K~T\right]~. $$
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Proof:

Recall that,

C_p - C_v = \cfrac{1}{\rho_0}~\boldsymbol{S}:\boldsymbol{\alpha}_E + \cfrac{T}{\rho_0}~\boldsymbol{\alpha}_E:\boldsymbol{\mathsf{C}}:\boldsymbol{\alpha}_E ~. $$ Plugging the expressions of $$\boldsymbol{\alpha}_E$$ and $$\boldsymbol{\mathsf{C}}$$ into the above equation, we have

\begin{align} C_p - C_v & = \cfrac{1}{\rho_0}~\boldsymbol{S}:(\alpha~\boldsymbol{\mathit{1}}) + \cfrac{T}{\rho_0}~(\alpha~\boldsymbol{\mathit{1}}): (\lambda~\boldsymbol{\mathit{1}}\otimes\boldsymbol{\mathit{1}} + 2\mu~\boldsymbol{\mathsf{I}}): (\alpha~\boldsymbol{\mathit{1}}) \\ & = \cfrac{\alpha}{\rho_0}~\text{tr}{\boldsymbol{S}} + \cfrac{\alpha^2~T}{\rho_0}~\boldsymbol{\mathit{1}}: (\lambda~\boldsymbol{\mathit{1}}\otimes\boldsymbol{\mathit{1}} + 2\mu~\boldsymbol{\mathsf{I}}): \boldsymbol{\mathit{1}} \\ & = \cfrac{\alpha}{\rho_0}~\text{tr}{\boldsymbol{S}} + \cfrac{\alpha^2~T}{\rho_0}~\boldsymbol{\mathit{1}}: (\lambda~\text{tr}{\boldsymbol{\mathit{1}}}~\boldsymbol{\mathit{1}} + 2\mu~\boldsymbol{\mathit{1}})\\ & = \cfrac{\alpha}{\rho_0}~\text{tr}{\boldsymbol{S}} + \cfrac{\alpha^2~T}{\rho_0}~ (3~\lambda~\text{tr}{\boldsymbol{\mathit{1}}} + 2\mu~\text{tr}{\boldsymbol{\mathit{1}}})\\ & = \cfrac{\alpha}{\rho_0}~\text{tr}{\boldsymbol{S}} + \cfrac{3~\alpha^2~T}{\rho_0}~ (3~\lambda + 2\mu)\\ & = \cfrac{\alpha~\text{tr}{\boldsymbol{S}}}{\rho_0} + \cfrac{9~\alpha^2~K~T}{\rho_0}~. \end{align} $$ Therefore,

{  C_p - C_v = \cfrac{1}{\rho_0}\left[\alpha~\text{tr}{\boldsymbol{S}} + 9~\alpha^2~K~T\right]~. } $$