# Homework #12

Reminder - you're going to need these webpages: http://www.continuummechanics.org/techforms/index.html to do this homework.... unless you prefer to use Matlab, Mathematica, etc. Don't bother doing anything by hand anymore. Take advantage of software programs to do all the matrix multiplication and other procedures when you have a chance.

Given the following stress tensor

$\boldsymbol{\sigma} = \left[ \matrix{ 10 & 20 & 30 \\ 20 & 40 & 50 \\ 30 & 50 & 60 } \right]$
1. What is the value of the von Mises stress?

2. Propose 2 other stress tensors that will have the same von Mises stress?

3. Do all stress tensors having the same von Mises stress also have the same principal stresses?

4. Do all stress tensors having the same principal stresses also have the same von Mises stress?

5. Start with the relationship between Caucy stress and 2nd Piola-Kirchhoff stress and take the time derivative to figure out the relationship between $$\dot{\boldsymbol{\sigma}}$$ and $$\dot{\boldsymbol{\sigma}}^\text{PK2}$$.
Hint - You will get a Lie derivative along the way.

$\boldsymbol{\sigma} = {1 \over J} \, {\bf F} \cdot \boldsymbol{\sigma}^\text{PK2} \cdot {\bf F}^T$