Nonlinear finite elements/Homework11/Solutions/Problem 1/Part 4

Problem 1: Part 4: Consistency condition

The consistency condition during plastic flow requires that

${\displaystyle {\dot {f}}({\boldsymbol {\sigma }},\alpha ,T)=0~.}$

Write down an expression for the time derivative of ${\displaystyle f({\boldsymbol {\sigma }},\alpha ,T)}$  using the chain rule.

The time derivative is

${\displaystyle {\dot {f}}({\boldsymbol {\sigma }},\alpha ,T)={\frac {\partial f}{\partial t}}={\frac {\partial f}{\partial {\boldsymbol {\sigma }}}}:{\frac {\partial {\boldsymbol {\sigma }}}{\partial t}}+{\frac {\partial f}{\partial \alpha }}{\frac {\partial \alpha }{\partial t}}+{\frac {\partial f}{\partial T}}{\frac {\partial T}{\partial t}}~.}$

In shorter form

${\displaystyle {\dot {f}}=f_{\boldsymbol {\sigma }}:{\dot {\boldsymbol {\sigma }}}+f_{\alpha }~{\dot {\alpha }}+f_{T}~{\dot {T}}~.}$