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.

An incompressible rubber object with an initial crosssectional area
of 100 mm^{2} is stretched along the xaxis to twice
its original length by a force of 200 N, and rotated 20°
counter clockwise in the xy plane.
 What is the full 3D Cauchy stress tensor?
 What is the full 3D 2nd PiolaKirchhoff stress tensor?
 Propose a definition for the engineering stress tensor
and use it to calculate a full 3D engineering stress tensor.

Someone gives you a 2nd PiolaKirchhoff stress tensor like the one
below and wants to know the corresponding Cauchy stress tensor.
What prevents you from doing the calculation?
\[
\boldsymbol{\sigma}^{\text{PK2}} =
\left[ \matrix{
10 & 20 & 30 \\
20 & 40 & 50 \\
30 & 50 & 60 }
\right]
\]

For the stress tensor here, find at least one unit normal vector
to a cut plane on which the normal
stress (yes normal, not shear) is zero.
\[
\boldsymbol{\sigma} =
\left[ \matrix{
20 & 5 & 10 \\
5 & 10 & 15 \\
10 & 15 & 5 }
\right]
\]

Given the stress state below, you could search forever and never
find an orientation with zero normal stress, because there isn't one. Why?
\[
\boldsymbol{\sigma} =
\left[ \matrix{
20 & 5 & 10 \\
5 & 10 & 15 \\
10 & 15 & 30 }
\right]
\]