Nonlinear finite elements/Homework 11/Solutions/Problem 1/Part 3

Problem 1: Part 3: Rate form constitutive relation

The rate form of the elastic constitutive equation is

$\dot{\mathit{\sigma }}=𝖢:{\dot{\mathit{\epsilon }}}^e=𝖢:(\dot{\mathit{\epsilon }}-{\dot{\mathit{\epsilon }}}^p)$

Plugging in the expression for ${\dot{\mathit{\epsilon }}}^p$ we get

$\dot{\mathit{\sigma }}=𝖢:(\dot{\mathit{\epsilon }}-\dot{\gamma }\frac{\partial f}{\partial \mathit{\sigma }})$