# Homework #13

1. 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.

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

2. Start with this equation for strain energy from the Mooney-Rivlin page

$W = C_{10} \left( I_1 - 3 \right)$
and show how to get to the following equation for uniaxial tension.

$\sigma^\text{Eng} = 2 \, C_{10} \left( \lambda - {1 \over \lambda^2 } \right)$

3. 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 $$C_{01} = 0.04$$ ).