Nonlinear finite elements/Homework11/Solutions/Problem 1/Part 10

Problem 1: Part 10: Discrete evolution equations

Discretize the equations for ${\dot {\mathbf {e} }}^{p}$ (equation 1), ${\dot {\alpha }}$ (from part 1), and ${\dot {T}}$ (from part 2) using Forward Euler.

The relevant equations are

${\begin{aligned}f_{\boldsymbol {\sigma }}&={\sqrt {\cfrac {3}{2}}}~\mathbf {n} \\{\dot {\boldsymbol {\varepsilon }}}^{p}&={\dot {\gamma }}~f_{\boldsymbol {\sigma }}={\sqrt {\cfrac {3}{2}}}~{\dot {\gamma }}~\mathbf {n} \\{\dot {\alpha }}&={\sqrt {\cfrac {2}{3}}}~{\dot {\gamma }}~{\cfrac {{\boldsymbol {\varepsilon }}^{p}:f_{\boldsymbol {\sigma }}}{\lVert {\boldsymbol {\varepsilon }}^{p}\rVert _{}}}={\dot {\gamma }}~{\cfrac {{\boldsymbol {\varepsilon }}^{p}:\mathbf {n} }{\lVert {\boldsymbol {\varepsilon }}^{p}\rVert _{}}}\\{\dot {T}}&={\cfrac {\chi ~{\dot {\gamma }}}{\rho ~C_{p}}}~{\boldsymbol {\sigma }}:{\frac {\partial f}{\partial {\boldsymbol {\sigma }}}}={\sqrt {\cfrac {3}{2}}}~{\cfrac {\chi ~{\dot {\gamma }}}{\rho ~C_{p}}}~{\boldsymbol {\sigma }}:\mathbf {n} ={\sqrt {\cfrac {3}{2}}}~{\cfrac {\chi ~{\dot {\gamma }}}{\rho ~C_{p}}}~\lVert \mathbf {s} \rVert _{}\end{aligned}}$

A Forward Euler time discretization gives us

${\begin{aligned}{\boldsymbol {\varepsilon }}_{n+1}^{p}&={\boldsymbol {\varepsilon }}_{n}^{p}+{\sqrt {\cfrac {3}{2}}}~\Delta t~{\dot {\gamma }}_{n}~\mathbf {n} _{n}\\\alpha _{n+1}&=\alpha _{n}+\Delta t~{\dot {\gamma }}_{n}~{\cfrac {{\boldsymbol {\varepsilon }}_{n}^{p}:\mathbf {n} _{n}}{\lVert {\boldsymbol {\varepsilon }}_{n}^{p}\rVert _{}}}\\T_{n+1}&=T_{n}+{\sqrt {\cfrac {3}{2}}}~{\cfrac {\chi ~\Delta t~{\dot {\gamma }}_{n}}{\rho _{n}~C_{p}}}~\lVert \mathbf {s} _{n}\rVert _{}\end{aligned}}$

or

${\begin{aligned}{\boldsymbol {\varepsilon }}_{n+1}^{p}&={\boldsymbol {\varepsilon }}_{n}^{p}+{\sqrt {\cfrac {3}{2}}}~\Delta \gamma ~\mathbf {n} _{n}\\\alpha _{n+1}&=\alpha _{n}+\Delta \gamma ~{\cfrac {{\boldsymbol {\varepsilon }}_{n}^{p}:\mathbf {n} _{n}}{\lVert {\boldsymbol {\varepsilon }}_{n}^{p}\rVert _{}}}\\T_{n+1}&=T_{n}+{\sqrt {\cfrac {3}{2}}}~{\cfrac {\chi ~\Delta \gamma }{\rho _{n}~C_{p}}}~\lVert \mathbf {s} _{n}\rVert _{}\end{aligned}}$

Nonlinear finite elements/Homework11/Solutions/Problem 1