= 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 $\operatorname{tr}H=h'(\theta_0)\vartheta$ and
$$
\sigma_{ji}=C_{jilk}H_{kl}+\beta_{ji}\vartheta+q\delta_{ji},\qquad\rho_0(\eta-\eta_0)=C_e\vartheta-\beta_{ji}H_{ij}+qh'(\theta_0),
$$
where $C_{jilk}=\rho_0\psi_{,F_{ij}F_{kl}}$, $\beta_{ji}=\rho_0\psi_{,F_{ij}\theta}$ and $C_e=-\rho_0\psi_{,\theta\theta}$ at the reference state. The temperature factor converts this entropy coefficient to a volumetric <heat capacity>: $\theta_0C_e$. Strictly constant volume is the case $h'=0$.
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