View source for Continuum mechanics/Clausius-Duhem inequality for thermoelasticity

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Clausius-Duhem inequality for thermoelasticity
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<div style="padding:2em 5em 0em 3em;"> 
For thermoelastic materials, the internal energy is a function  only of the deformation gradient and the temperature, i.e.,  <math>e = e(\boldsymbol{F}, T)</math>.  Show that, for thermoelastic materials, the  Clausius-Duhem inequality
:<math>
 \rho~(\dot{e} - T~\dot{\eta}) - \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v} \le 
 - \cfrac{\mathbf{q}\cdot\boldsymbol{\nabla} T}{T} 
 </math>
can be expressed as
:<math>
 \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} ~.
 </math>
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'''Proof:'''
 
Since <math>e = e(\boldsymbol{F}, T)</math>, we have
:<math>
  \dot{e} = \frac{\partial e}{\partial \boldsymbol{F}}:\dot{\boldsymbol{F}} + \frac{\partial e}{\partial \eta}~\dot{\eta} ~.
 </math>
Therefore, 
:<math>
 \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} ~.
 </math>
Now, <math>\boldsymbol{\nabla}\mathbf{v} = \boldsymbol{l} = \dot{\boldsymbol{F}}\cdot\boldsymbol{F}^{-1}</math>.  Therefore, using the 
identity <math>\boldsymbol{A}:(\boldsymbol{B}\cdot\boldsymbol{C}) = (\boldsymbol{A}\cdot\boldsymbol{C}^T):\boldsymbol{B}</math>, we have
:<math>
  \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v} = \boldsymbol{\sigma}:(\dot{\boldsymbol{F}}\cdot\boldsymbol{F}^{-1})
               = (\boldsymbol{\sigma}\cdot\boldsymbol{F}^{-T}):\dot{\boldsymbol{F}} ~.
 </math>
Hence,
:<math>
 \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} 
 </math>
or,
:<math>
 \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} ~.
 </math>

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[[Category:Continuum mechanics]]