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

$\dot{𝝈}=𝖢:{\dot{𝜺}}^e=𝖢:(\dot{𝜺}-{\dot{𝜺}}^p)$

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

$\dot{𝝈}=𝖢:(\dot{𝜺}-\dot{\gamma }\frac{\partial f}{\partial 𝝈})$

Nonlinear finite elements/Homework 11/Solutions/Problem 1