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

${\dot {T}}={\cfrac {\chi }{\rho C_{p}}}{\boldsymbol {\sigma }}:{\dot {\boldsymbol {\varepsilon }}}^{p}$

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 $\partial f/\partial {\boldsymbol {\sigma }}$. This is the evolution law for $T$.

Plugging in the expression for ${\dot {\boldsymbol {\varepsilon }}}^{p}$, we get

${{\dot {T}}={\cfrac {\chi ~{\dot {\gamma }}}{\rho ~C_{p}}}~{\boldsymbol {\sigma }}:{\frac {\partial f}{\partial {\boldsymbol {\sigma }}}}~.}$

Nonlinear finite elements/Homework11/Solutions/Problem 1