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 \( C_{01} = 0.04 \) ).