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}}~. } $$