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Clausius-Duhem inequality for thermoelasticity

For thermoelastic materials, the internal energy is a function only of the deformation gradient and the temperature, i.e., e = e(\boldsymbol{F}, T). Show that, for thermoelastic materials, the Clausius-Duhem inequality

 \rho~(\dot{e} - T~\dot{\eta}) - \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v} \le - \cfrac{\mathbf{q}\cdot\boldsymbol{\nabla} T}{T}

can be expressed as

 \rho~\left(\frac{\partial e}{\partial \eta} - T\right)~\dot{\eta} + \left(\rho~\frac{\partial e}{\partial \boldsymbol{F}} - \boldsymbol{\sigma}\cdot\boldsymbol{F}^{-T}\right):\dot{\boldsymbol{F}} \le - \cfrac{\mathbf{q}\cdot\boldsymbol{\nabla} T}{T} ~.

Proof:

Since e = e(\boldsymbol{F}, T), we have

 \dot{e} = \frac{\partial e}{\partial \boldsymbol{F}}:\dot{\boldsymbol{F}} + \frac{\partial e}{\partial \eta}~\dot{\eta} ~.

Therefore,

 \rho~\left(\frac{\partial e}{\partial \boldsymbol{F}}:\dot{\boldsymbol{F}} + \frac{\partial e}{\partial \eta}~\dot{\eta} - T~\dot{\eta}\right) - \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v} \le - \cfrac{\mathbf{q}\cdot\boldsymbol{\nabla} T}{T} \qquad\text{or}\qquad \rho\left(\frac{\partial e}{\partial \eta} - T\right)~\dot{\eta} + \rho~\frac{\partial e}{\partial \boldsymbol{F}}:\dot{\boldsymbol{F}} - \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v} \le - \cfrac{\mathbf{q}\cdot\boldsymbol{\nabla} T}{T} ~.

Now, \boldsymbol{\nabla}\mathbf{v} = \boldsymbol{l} = \dot{\boldsymbol{F}}\cdot\boldsymbol{F}^{-1}. Therefore, using the identity \boldsymbol{A}:(\boldsymbol{B}\cdot\boldsymbol{C}) = (\boldsymbol{A}\cdot\boldsymbol{C}^T):\boldsymbol{B}, we have

 \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v} = \boldsymbol{\sigma}:(\dot{\boldsymbol{F}}\cdot\boldsymbol{F}^{-1}) = (\boldsymbol{\sigma}\cdot\boldsymbol{F}^{-T}):\dot{\boldsymbol{F}} ~.

Hence,

 \rho\left(\frac{\partial e}{\partial \eta} - T\right)~\dot{\eta} + \rho~\frac{\partial e}{\partial \boldsymbol{F}}:\dot{\boldsymbol{F}} - (\boldsymbol{\sigma}\cdot\boldsymbol{F}^{-T}):\dot{\boldsymbol{F}} \le - \cfrac{\mathbf{q}\cdot\boldsymbol{\nabla} T}{T}

or,

 \rho~\left(\frac{\partial e}{\partial \eta} - T\right)~\dot{\eta} + \left(\rho~\frac{\partial e}{\partial \boldsymbol{F}} - \boldsymbol{\sigma}\cdot\boldsymbol{F}^{-T}\right):\dot{\boldsymbol{F}} \le - \cfrac{\mathbf{q}\cdot\boldsymbol{\nabla} T}{T} ~.

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