 Continuum thermodynamics

Continuum thermodynamics applies the first law of thermodynamics and Second law of thermodynamics locally to a deforming material. A region can gain internal energy through mechanical work, heat flux through its boundary and internal heat supply. Those mechanisms must be distinguished from transport caused by the region itself moving.

With specific internal energy $e$, mean mass density $\rho$, symmetric stress tensor $\sigma$, rate-of-strain tensor $D$ and outward heat flux $\mathbf{q}$, a simple continuum satisfiesHere $r$ is heat supplied per unit mass per unit time, and the dot is a material derivative. The Clausius-Duhem inequality additionally requires nonnegative local entropy production. Together the two laws test proposed constitutive equations: an equation can satisfy mechanical balance while incorrectly predicting heat flow from cold to hot without compensating work. Elastic energy storage, viscous dissipation and thermal conduction enter the same accounting framework.