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

 

and

 

Therefore,

 

This gives us

 

or,

 

Hence

 

The continuum elastic-plastic tangent modulus is therefore