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.