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 ${\dot {\boldsymbol {\varepsilon }}}$.

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,

${\dot {\gamma }}={\cfrac {f_{\boldsymbol {\sigma }}:{\boldsymbol {\mathsf {C}}}:{\dot {\boldsymbol {\varepsilon }}}}{f_{\boldsymbol {\sigma }}:{\boldsymbol {\mathsf {C}}}:f_{\boldsymbol {\sigma }}-{\sqrt {\cfrac {2}{3}}}~f_{\alpha }~{\cfrac {{\boldsymbol {\varepsilon }}^{p}:f_{\boldsymbol {\sigma }}}{\lVert {\boldsymbol {\varepsilon }}^{p}\rVert _{}}}-{\cfrac {\chi }{\rho ~C_{p}}}~f_{T}~{\boldsymbol {\sigma }}:f_{\boldsymbol {\sigma }}}}~.$

Nonlinear finite elements/Homework11/Solutions/Problem 1