Clausius–Duhem inequality

Remarks

It is the local form of the second law of thermodynamics for continua. Its integral form concerns a part $P$ of the body. The entropy of $P$ grows at least at the rate of entropy supplied by heat sources in $P$ plus entropy carried in by heat flow across the boundary. The divergence theorem turns this integral form into the local one. Let $e$ be the specific internal energy and $\psi = e - \theta \eta$ the specific free energy. With the local energy balance $\rho \dot e = \sigma : D - \nabla \cdot q + \rho r$, the inequality takes the reduced form $$\sigma : D - \rho (\dot\psi + \eta \dot\theta) - \frac{q \cdot \nabla \theta}{\theta} \ge 0,$$ where $\sigma$ is the Cauchy stress tensor and $D$ is the rate of deformation. In the method of Coleman and Noll one requires this inequality for every possible process. If the free energy depends on the deformation gradient and the temperature, this forces the stress and the entropy to be derivatives of the free energy. For a rigid heat conductor it gives $q \cdot \nabla \theta \le 0$: heat flows from hot to cold.

Going deeper

For a student

In continuum mechanics the balance laws of mass, momentum and energy hold for every material, and the Clausius–Duhem inequality is the extra restriction that comes from the second law. Its main use is in the theory of constitutive equations: one asks which constitutive laws make the reduced inequality hold in all processes. For thermoelastic materials this gives $\sigma$ as a derivative of $\psi$ with respect to strain, and $\eta = -\partial \psi / \partial \theta$. For viscous and plastic materials it says that the dissipation, the left side of the reduced inequality, is never negative.