Clausius–Duhem inequality

The Clausius–Duhem inequality is a way of expressing the second law of thermodynamics that is used in continuum mechanics. This inequality is particularly useful in determining whether the constitutive relation of a material is thermodynamically allowable.

This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved. It was named after the German physicist Rudolf Clausius and French physicist Pierre Duhem.

The Clausius–Duhem inequality can be expressed in integral form as

In this equation ${\displaystyle t\,}$ is the time, ${\displaystyle \Omega \,}$ represents a body and the integration is over the volume of the body, ${\displaystyle \partial \Omega \,}$ represents the surface of the body, ${\displaystyle \rho \,}$ is the mass density of the body, ${\displaystyle \eta \,}$ is the specific entropy (entropy per unit mass), ${\displaystyle u_{n}\,}$ is the normal velocity of ${\displaystyle \partial \Omega \,}$, ${\displaystyle \mathbf {v} }$ is the velocity of particles inside ${\displaystyle \Omega \,}$, ${\displaystyle \mathbf {n} }$ is the unit normal to the surface, ${\displaystyle \mathbf {q} }$ is the heat flux vector, ${\displaystyle s\,}$ is an energy source per unit mass, and ${\displaystyle T\,}$ is the absolute temperature. All the variables are functions of a material point at ${\displaystyle \mathbf {x} }$ at time ${\displaystyle t\,}$.

...
Wikipedia