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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 readsCombining this with the local internal energy balance and writing the specific Helmholtz free energy as $\psi =e-Ts$ givesHere $D$ is the rate-of-strain tensor and the dot denotes a material derivative.^[ref]

The inequality constrains constitutive equations, including both mechanical dissipation and thermal conduction. For Fourier's law $\mathbf{q}=-k\nabla T$ with $k\ge 0$, the thermal contribution is $k|\nabla T{|}^2/T\ge 0$. 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.

The local entropy balance has nonnegative production, with heat supply ${\rho }_{0}r/\theta$ and thermodynamic entropy flux $q/\theta$ at positive temperature $\theta$. Its integral version holds on every material reference subvolume.

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 isHere $b$ 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:The integral Lagrangian entropy inequality iswhere $\theta >0$ is temperature and $q/\theta$ is the thermodynamic entropy flux.

Since the subvolume is arbitrary, the divergence theorem givesExpand $\dot{U}=U_{,F_{iI}}{\dot{F}}_{iI}+U_{,\eta }\dot{\eta }+U_{,{\xi }_r}{\dot{\xi }}_r$ and eliminate the heat supply between these relations. The Clausius-Duhem inequality becomesUnder the usual local constitutive independence assumptions, the reversible mechanical and thermal variations can have either sign. The Coleman-Noll procedure therefore imposesDefine the thermodynamic force conjugate to an internal variable by $f_r=-{\rho }_0{U}_{,{\xi }_r}$. The energy balance and the remaining entropy constraint now give, respectively,