Nonlinear finite elements/Homework11/Solutions/Problem 1/Part 2

Problem 1: Part 2: Energy equation

For an adiabatic process, the rate of change of temperature can be written as

$\dot{T}=\frac{\chi }{\rho C_p}\mathit{\sigma }:{\dot{\mathit{\epsilon }}}^p$

where $\chi$ is the Taylor-Quinney coefficient, $\rho$ is the density, and $C_p$ is the specific heat. Express $\dot{T}$ in terms of $\dot{\gamma }$ and $\partial f/\partial \mathit{\sigma }$. This is the evolution law for $T$.

Plugging in the expression for ${\dot{\mathit{\epsilon }}}^p$, we get

$\dot{T}=\frac{\chi \dot{\gamma }}{\rho C_p}\mathit{\sigma }:\frac{\partial f}{\partial \mathit{\sigma }}.$