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

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