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

Problem 1: Part 4: Consistency condition

The consistency condition during plastic flow requires that

$\dot{f}(\mathit{\sigma },\alpha ,T)=0.$

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

The time derivative is

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

In shorter form

$\dot{f}=f_{\mathit{\sigma }}:\dot{\mathit{\sigma }}+f_{\alpha }\dot{\alpha }+f_T\dot{T}.$