Homework #13
 Start with the relationship between Caucy stress and 2nd PiolaKirchhoff 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.
\[
\boldsymbol{\sigma} = {1 \over J} \, {\bf F} \cdot \boldsymbol{\sigma}^\text{PK2} \cdot {\bf F}^T
\]

Start with this equation for strain energy from the
MooneyRivlin page
\[
W = C_{10} \left( I_1  3 \right)
\]
and show how to get to the following equation for uniaxial tension.
\[
\sigma^\text{Eng} = 2 \, C_{10} \left( \lambda  {1 \over \lambda^2 } \right)
\]

Plot the max principal engineering stress
versus max principal engineering strain, up to \(\epsilon_\text{max} = 0.50\),
for uniaxial tension, shear, and equibiaxial tension,
for a material having (\( C_{10} = 0.4 \) and
\( C_{01} = 0.04 \) ).