Nonlinear finite elements/Homework11/Solutions/Problem 1/Part 7

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Problem 1: Part 7: Flow rule[edit]

The theory of plasticity also states that the material satisfies the von Mises yield condition

where is the deviatoric part of the stress . Derive an expression for in terms of the normal to the yield surface


The von Mises yield function is

We can alternatively write the yield function as

in which case the following equations take a slightly different form.

Therefore,

The deviatoric part of is

Therefore,

Now,

and

Therefore,

The derivative with respect to is

Let us use index notation to find the derivatives. In index notation,

Hence, the components of the second-order tensor are

Therefore,

Similarly,

Hence, the components of the second-order tensor are

Therefore,

Plugging the above results into the expression for the derivative of we get

Hence, we get

The required expression is