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 , outward heat flux and heat supply per unit mass, it readsCombining this with the local internal energy balance and writing the specific Helmholtz free energy as givesHere is the rate-of-strain tensor and the dot denotes a material derivative.[ref]
Applied Mechanics of Solids by A. F. Bower. §2.6, local entropy inequality and combined free-energy imbalance
The inequality constrains constitutive equations, including both mechanical dissipation and thermal conduction. For Fourier's law with , the thermal contribution is . 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.
Requiring the Clausius-Duhem inequality for independent reversible mechanical and thermal variations constrains a constitutive equation. For it gives and . Combining the Lagrangian internal energy balance with the Lagrangian entropy inequality leaves . The independence assumptions are part of the constitutive argument.
The local entropy balance has nonnegative production, with heat supply and thermodynamic entropy flux at positive temperature . Its integral version holds on every material reference subvolume.
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The Clausius-Duhem inequality is a fundamental principle in thermodynamics and continuum mechanics that expresses the second law of thermodynamics in a differential form. It serves as a mathematical statement of the irreversibility of thermodynamic processes and the concept of entropy production. In simple terms, the inequality can be stated as follows: \[ \frac{dS}{dt} \geq 0 \] where \( S \) is the entropy of a system.