= Solution
Use the material-first <nominal stress tensor> convention $P_{Ii}$, with $F_{iI}$ the <deformation gradient>. For any fixed reference material subvolume $V_0$ with outward unit normal $N$, <conservation of energy> is
$$
\frac d{dt}\int_{V_0}\rho_0\left(U+\frac12|v|^2\right)dV_0
=\int_{V_0}\rho_0(b\cdot v+r)\,dV_0+\int_{\partial V_0}\left[(P^TN)\cdot v-q\cdot N\right]dA_0.
$$
Here $b$ is <body force> per mass, $r$ is heat supply per mass and $q$ is nominal <heat flux>. Using <momentum conservation> to remove the <kinetic energy> balance gives the integral <Lagrangian internal energy balance>:
$$
\frac d{dt}\int_{V_0}\rho_0U\,dV_0=\int_{V_0}(P_{Ii}\dot F_{iI}+\rho_0r)\,dV_0-\int_{\partial V_0}q_IN_I\,dA_0.
$$
The integral <Lagrangian entropy inequality> is
$$
\frac d{dt}\int_{V_0}\rho_0\eta\,dV_0\ge\int_{V_0}\frac{\rho_0r}{\theta}\,dV_0-\int_{\partial V_0}\frac{q_IN_I}{\theta}\,dA_0,
$$
where $\theta>0$ is <temperature> and $q/\theta$ is the <thermodynamic entropy flux>.
Since the subvolume is arbitrary, the <divergence theorem> gives
$$
\rho_0\dot U=P_{Ii}\dot F_{iI}+\rho_0r-q_{I,I},\qquad \rho_0\theta\dot\eta\ge\rho_0r-q_{I,I}+\frac{q_I\theta_{,I}}{\theta}.
$$
Expand $\dot U=U_{,F_{iI}}\dot F_{iI}+U_{,\eta}\dot\eta+U_{,\xi_r}\dot\xi_r$ and eliminate the heat supply between these relations. The <Clausius-Duhem inequality> becomes
$$
(P_{Ii}-\rho_0U_{,F_{iI}})\dot F_{iI}+\rho_0(\theta-U_{,\eta})\dot\eta-\rho_0U_{,\xi_r}\dot\xi_r-\frac{q_I\theta_{,I}}{\theta}\ge0.
$$
Under the usual local constitutive independence assumptions, the reversible mechanical and thermal variations can have either sign. The <Coleman-Noll procedure> therefore imposes
$$
\boxed{P_{Ii}=\rho_0\frac{\partial U}{\partial F_{iI}},\qquad\theta=\frac{\partial U}{\partial\eta}.}
$$
Define the <thermodynamic force conjugate to an internal variable> by $f_r=-\rho_0U_{,\xi_r}$. The energy balance and the remaining entropy constraint now give, respectively,
$$
\boxed{\rho_0\theta\dot\eta=\rho_0r-q_{I,I}+f_r\dot\xi_r,\qquad f_r\dot\xi_r-\frac{q_I\theta_{,I}}{\theta}\ge0.}
$$
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