Nonlinear finite elements/Homework11/Solutions/Problem 1/Part 5

Problem 1: Part 5: Consistency condition - 1
Use the consistency condition and the expressions you have derived in the previous parts to derive an expression for $\dot{\gamma}$ in terms of $\partial f/\partial\boldsymbol{\sigma}$, $\partial f/\partial\alpha$, $\partial f/\partial T$, $\boldsymbol{\mathsf{C}}$, and $\dot {\boldsymbol{\varepsilon}}$.|undefined

From the consistency condition

\dot{f} = f_{\boldsymbol{\sigma}}:\dot{\boldsymbol{\sigma}} + f_{\alpha}~\dot{\alpha} + f_T~\dot{T} = 0 $$ From the previous parts

\dot{\boldsymbol{\sigma}} = \boldsymbol{\mathsf{C}} : \left(\dot{\boldsymbol{\varepsilon}} - \dot{\gamma}f_{\boldsymbol

{\sigma}}\right)~;\qquad \dot{\alpha} = \sqrt{\cfrac{2}{3}}~\dot{\gamma}~\cfrac{\boldsymbol{\varepsilon}^p:f_{\boldsymbol{\sigma}}} {\lVert\boldsymbol{\varepsilon}^p\rVert_{}} ~;\qquad \dot{T} = \cfrac{\chi~\dot{\gamma}}{\rho~C_p}~\boldsymbol{\sigma}:f_{\boldsymbol{\sigma}}~. $$ Plug into the consistency condition to get

f_{\boldsymbol{\sigma}}:\boldsymbol{\mathsf{C}}:\left(\dot{\boldsymbol{\varepsilon}} - \dot

{\gamma}f_{\boldsymbol{\sigma}}\right) + \sqrt{\cfrac{2}{3}}~\dot{\gamma}~f_{\alpha}~\cfrac{\boldsymbol{\varepsilon}^p:f_

{\boldsymbol{\sigma}}} {\lVert\boldsymbol{\varepsilon}^p\rVert_{}}+ \cfrac{\chi}{\rho~C_p}~\dot{\gamma}~f_T~\boldsymbol{\sigma}:f_{\boldsymbol{\sigma}} = 0 ~. $$ Collect terms containing $$\dot{\gamma}$$:

\dot{\gamma}\left[ -f_{\boldsymbol{\sigma}}:\boldsymbol{\mathsf{C}}:f_{\boldsymbol{\sigma}} + \sqrt{\cfrac{2}{3}}~f_{\alpha}~\cfrac{\boldsymbol{\varepsilon}^p:f_{\boldsymbol{\sigma}}} {\lVert\boldsymbol{\varepsilon}^p\rVert_{}}+ \cfrac{\chi}{\rho~C_p}~f_T~\boldsymbol{\sigma}:f_{\boldsymbol{\sigma}} \right] + f_{\boldsymbol{\sigma}}:\boldsymbol{\mathsf{C}}:\dot{\boldsymbol{\varepsilon}} = 0 $$ Therefore,

\dot{\gamma} = \cfrac{f_{\boldsymbol{\sigma}}:\boldsymbol{\mathsf{C}}:\dot{\boldsymbol{\varepsilon}}}{ f_{\boldsymbol{\sigma}}:\boldsymbol{\mathsf{C}}:f_{\boldsymbol{\sigma}} - \sqrt{\cfrac{2}{3}}~f_{\alpha}~\cfrac{\boldsymbol{\varepsilon}^p:f_{\boldsymbol{\sigma}}} {\lVert\boldsymbol{\varepsilon}^p\rVert_{}} - \cfrac{\chi}{\rho~C_p}~f_T~\boldsymbol{\sigma}:f_{\boldsymbol{\sigma}}} ~. $$