Continuum mechanics/Clausius-Duhem inequality for thermoelasticity

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} \leq -{\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}}}\leq -{\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} \leq -{\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} \leq -{\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}}}\leq -{\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}}}\leq -{\cfrac {\mathbf {q} \cdot {\boldsymbol {\nabla }}T}{T}}~.

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