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

Problem 1: Part 7: Flow ruleEdit

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