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. |undefined

The relevant equations are

\begin{align} 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{align} $$ A Forward Euler time discretization gives us

\begin{align} \boldsymbol{\varepsilon}^p_{n+1} & = \boldsymbol{\varepsilon}^p_n + \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}^p_n:\mathbf{n}_n}

{\lVert\boldsymbol{\varepsilon}^p_n\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{align} $$ or

{ \begin{align} \boldsymbol{\varepsilon}^p_{n+1} & = \boldsymbol{\varepsilon}^p_n + \sqrt{\cfrac{3}{2}}

~\Delta\gamma~\mathbf{n}_n \\ \alpha_{n+1}& = \alpha_n + \Delta\gamma~\cfrac{\boldsymbol{\varepsilon}^p_n:\mathbf{n}_n}{\lVert\boldsymbol

{\varepsilon}^p_n\rVert_{}} \\ T_{n+1} & = T_n + \sqrt{\cfrac{3}{2}}~\cfrac{\chi~\Delta\gamma}{\rho_n~C_p} ~\lVert\mathbf{s}_n\rVert_{} \end{align} } $$