= 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 satisfies
$$
\rho\dot e=\sigma:D-\nabla\cdot\mathbf q+\rho r.
$$
Here $r$ 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.
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