= Clausius-Duhem inequality
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{wiki=Clausius–Duhem_inequality}
The <Clausius-Duhem inequality> is a local expression of the <Second law of thermodynamics> for a simple continuum whose <thermodynamic entropy flux> is <heat flux> divided by positive absolute <temperature>. With <specific entropy> $s$, outward <heat flux> $\mathbf q$ and heat supply $r$ per unit <mass>, it reads
$$
\rho\dot s+\nabla\cdot(\mathbf q/T)-\rho r/T\geq0.
$$
Combining this with the local <internal energy> balance and writing the specific <Helmholtz free energy> as $\psi=e-Ts$ gives
$$
\sigma:D-\rho(\dot\psi+s\dot T)-\frac{\mathbf q\cdot\nabla T}{T}\geq0.
$$
Here $D$ is the <rate-of-strain tensor> and the dot denotes a <material derivative>.<Applied Mechanics of Solids by A. F. Bower>[§2.6, local entropy inequality and combined free-energy imbalance]{ref}
The inequality constrains <constitutive equations>, including both mechanical dissipation and <thermal conduction>. For <Fourier's law> $\mathbf q=-k\nabla T$ with $k\geq0$, the thermal contribution is $k|\nabla T|^2/T\geq0$. This direct sign check explains why passive <thermal conduction> carries <heat> down a <temperature gradient>. More general continua can require a different <thermodynamic entropy flux>, so the stated assumption matters.
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