# Homework #11

Reminder - you're going to need these webpages: http://www.continuummechanics.org/interactivecalcs.html to do this homework.... unless you prefer to use Matlab, Mathematica, etc. Take advantage of software programs to do all the matrix multiplication and other procedures when you have a chance.

1. If a stress tensor is

$\boldsymbol{\sigma} = \left[ \matrix{ 10 & 20 & 30 \\ 20 & 40 & 10 \\ 30 & 10 & 50 } \right]$
then calculate the traction vector on a plane with unit normal $${\bf n} = (0.100, \; 0.700, \; 0.707)$$.

2. An incompressible rubber object with an initial cross-sectional area of 100 mm2 is stretched along the x-axis to twice its original length by a force of 200 N, and rotated 20° counter clockwise in the x-y plane.

1. What is the full 3-D Cauchy stress tensor?

2. What is the full 3-D 2nd Piola-Kirchhoff stress tensor?

3. Propose a definition for the engineering stress tensor and use it to calculate a full 3-D engineering stress tensor.

3. Someone gives you a 2nd Piola-Kirchhoff stress tensor like the one below and wants to know the corresponding Cauchy stress tensor. What prevents you from doing the calculation?

$\boldsymbol{\sigma}^{\text{PK2}} = \left[ \matrix{ 10 & 20 & 30 \\ 20 & 40 & 50 \\ 30 & 50 & 60 } \right]$