Entropy field equations

The entropy flow h is related to the spatial gradient of the thermal stress (temperature) $\sigma_3$ by the thermal conductivity $\kappa_h$ giving the constitutive relation  

$${\bf h}~=~- \displaystyle \mathop{\mbox{${\bf \kappa_h}$}}_{\mbox{$\sim$}} ~{\rm \bf grad}~{\bf \sigma_3}$$ (14)

The entropy flow has to obey the conservation equation  

$${\rm \bf div}~{\bf h}~=~- {\partial \over{\partial t}} {\bf \epsilon_3}$$ (15)

which means that heat is only created by letting entropy flow in the medium. There are no other heat sources present. The linear law relating changes in elastic strain, electric and thermal displacement to changes in thermal stress is  

$${\partial \over{\partial t}} {\bf \sigma_3} ~=~ \displaystyl... ...\mbox{$\sim$}} ~{\partial \over{\partial t}} {\bf \epsilon_3 }$$ (16)

By combining Equations 14, 15 and 16 we find the coupled wave equation  

$${\bf div}~ \lbrace~ \displaystyle \mathop{\mbox{${\bf \kappa... ...)~\rbrace ~=~ {\partial^2 \over{\partial t^2}} {\bf \epsilon_3}$$ (17)