Nonlinear finite elements/Homework11/Solutions/Problem 1/Part 9

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Problem 1: Part 9: Elastic-plastic tangent modulus[edit]

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

Therefore,

Some of the results used in the above derivation are shown below.

Recall (from previous homework):

and

(we have used the symmetry of the stress tensor above.)

Also,

Now,

Therefore,

Hence,

and

Plugging in expression for we get

Now,

Therefore, the elastic-plastic tangent modulus can be written as