Clausius-Duhem inequality for thermoelasticity
For thermoelastic materials, the internal energy is a function only of the deformation gradient and the temperature, i.e., . Show that, for thermoelastic materials, the Clausius-Duhem inequality
![{\displaystyle \rho ~({\dot {e}}-T~{\dot {\eta }})-{\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \leq -{\cfrac {\mathbf {q} \cdot {\boldsymbol {\nabla }}T}{T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1b2e93f680052b122e55f72983604d2995ffe58)
can be expressed as
![{\displaystyle \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}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f96ddd7a5d3e82aee24bcd40c962b62bd22f996d)
|
Proof:
Since
, we have
![{\displaystyle {\dot {e}}={\frac {\partial e}{\partial {\boldsymbol {F}}}}:{\dot {\boldsymbol {F}}}+{\frac {\partial e}{\partial \eta }}~{\dot {\eta }}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d988ff46782cbdec0139db23d6adb0532568419)
Therefore,
![{\displaystyle \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}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a80d7587e14c9f52b97c210183e7b7742de608a6)
Now,
. Therefore, using the
identity
, we have
![{\displaystyle {\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} ={\boldsymbol {\sigma }}:({\dot {\boldsymbol {F}}}\cdot {\boldsymbol {F}}^{-1})=({\boldsymbol {\sigma }}\cdot {\boldsymbol {F}}^{-T}):{\dot {\boldsymbol {F}}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac128150043c087f6193591f54318325dfb180b)
Hence,
![{\displaystyle \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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4568aad0ef9174b25177aab7278a89c717f1b8c)
or,
![{\displaystyle \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}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f96ddd7a5d3e82aee24bcd40c962b62bd22f996d)