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.

1. An incompressible rubber object with an initial cross-sectional area of 100 mm2 is stretched along the x-axis to twice its original length by a force of 200 N, and rotated 20° counter clockwise in the x-y plane.

1. What is the full 3-D Cauchy stress tensor?

2. What is the full 3-D 2nd Piola-Kirchhoff stress tensor?

3. Propose a definition for the engineering stress tensor and use it to calculate a full 3-D engineering stress tensor.

2. Someone gives you a 2nd Piola-Kirchhoff stress tensor like the one below and wants to know the corresponding Cauchy stress tensor. What prevents you from doing the calculation?

$\boldsymbol{\sigma}^{\text{PK2}} = \left[ \matrix{ 10 & 20 & 30 \\ 20 & 40 & 50 \\ 30 & 50 & 60 } \right]$
3. 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.

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

$\boldsymbol{\sigma} = \left[ \matrix{ 20 & 5 & 10 \\ 5 & 10 & 15 \\ 10 & 15 & 30 } \right]$