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

Problem 1: Part 9: Elastic-plastic tangent modulusEdit

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