Coleman–Noll procedure

Remarks

Example: take a thermoelastic body with reference density $\rho_0$, deformation gradient $F$, temperature $\theta > 0$ and referential temperature gradient $g$. Let its first Piola–Kirchhoff stress $P$ and referential heat flux $q_0$ be functions of $(F, \theta, g)$, like $\psi$ and $\eta$. Step 1 gives $$\rho_0 (\dot\psi + \eta \dot\theta) - P : \dot F + \frac{q_0 \cdot g}{\theta} \le 0.$$ Step 3 applies to the arbitrary rates $\dot F$, $\dot \theta$ and $\dot g$, and gives that $\psi$ does not depend on $g$, that $\eta = -\partial \psi / \partial \theta$, that $P = \rho_0\, \partial \psi / \partial F$, and that $q_0 \cdot g \le 0$. Non-example: requiring the inequality only in equilibrium states, where all rates vanish, is not this procedure, and it gives none of these restrictions. The method is named after Bernard Coleman and Walter Noll.