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

Problem 1: Part 3: Rate form constitutive relation

Write down the rate form of the elastic stress-strain law. Assume that deformations are small so that objectivity of the rates is not a concern.

The rate form of the elastic constitutive equation is

$\dot{\boldsymbol{\sigma}} = \boldsymbol{\mathsf{C}} : \dot{\boldsymbol{\varepsilon}}^e = \boldsymbol{\mathsf{C}} : (\dot{\boldsymbol{\varepsilon}} - \dot{\boldsymbol {\varepsilon}}^p)$

Plugging in the expression for $\dot{\boldsymbol{\varepsilon}}^p$ we get

$\dot{\boldsymbol{\sigma}} = \boldsymbol{\mathsf{C}} : \left(\dot{\boldsymbol{\varepsilon}} - \dot{\gamma}\frac{\partial f}{\partial \boldsymbol {\sigma}}\right)$