Nonlinear finite elements/Homework11/Solutions/Problem 1/Part 9
Problem 1: Part 9: Elastic-plastic tangent modulus
Assume that the elastic response of the material is linear, i.e.,
Derive the expression for the elastic-plastic tangent modulus for a von Mises yield condition with Johnson-Cook flow stress for a linear elastic material using the expressions that you have derived in the previous parts.
The elastic-plastic tangent modulus is given by
From the previous parts
Some of the results used in the above derivation are shown below.
Recall (from previous homework):
(we have used the symmetry of the stress tensor above.)
Plugging in expression for we get
Therefore, the elastic-plastic tangent modulus can be written as