Nonlinear finite elements/Homework 11/Solutions/Problem 1/Part 6

Problem 1: Part 6: Continuum elastic-plastic tangent modulusEdit

The continuum elastic-plastic tangent modulus is defined by the following relation


Derive an expression for the elastic plastic tangent modulus using the results you have derived in the previous parts.

The stress rate is given by


From the previous part


Plug in expression for stress rate to get


We have to express the above in the form


Since the denominator is a scalar, we don't have to worry about it for this calculation. In that case we can write


In index notation, we can write


Let us manipulate the numerator of the second term above so that we get what we need. Thus


In the above






This gives us






The continuum elastic-plastic tangent modulus is therefore