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

Problem 1: Part 4: Consistency condition

The consistency condition during plastic flow requires that

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

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

The time derivative is

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

In shorter form

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