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

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Problem 1: Part 6: Continuum elastic-plastic tangent modulus[edit]

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