Homework #10


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.


Use the following true strain tensor for the first three problems.

\[ \boldsymbol{\epsilon}_{\text{True}} = \left[ \matrix{ \;\;\; 0.758 & -0.237 & \;\;\;0.012 \\ -0.237 & -0.546 & \;\;\;0.067 \\ \;\;\; 0.012 & \;\;\; 0.067 & -0.212 \\ } \right] \]
  1. This true strain tensor is for an incompressible material. Demonstrate that this is indeed the case.


  2. Calculate \(\epsilon_{\text{Hyd}}\), \(\gamma_{\text{Max}}\), and \(\gamma_{\text{Sec}}\), and determine whether the strain tensor represents a state closer to uniaxial tension, shear, or equibiaxial tension.


  3. Figure out the corresponding Green strain tensor.



  4. 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 \({\bf n} = (0.100, \; 0.700, \; 0.707)\).