Linear thermoelasticity with prescribed thermal volume

Linearize $\det F=h(\theta )$ at $F=I$, $\theta ={\theta }_0$, $h({\theta }_0)=1$, and zero reaction. With displacement gradient $H$ and temperature increment $\vartheta$, one has $\mathrm{tr}H=h^{\prime }({\theta }_0)\vartheta$ andwhere $C_{jilk}={\rho }_0{\psi }_{,F_{ij}{F}_{kl}}$, ${\beta }_{ji}={\rho }_0{\psi }_{,F_{ij}\theta }$ and $C_e=-{\rho }_0{\psi }_{,\theta \theta }$ at the reference state. The temperature factor converts this entropy coefficient to a volumetric heat capacity: ${\theta }_0{C}_e$. Strictly constant volume is the case $h^{\prime }=0$.