# 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

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

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

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

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