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

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

Nonlinear finite elements/Homework 11/Solutions/Problem 1