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.
-
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 \({\bf n} = (0.100, \; 0.700, \; 0.707)\).
-
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.
- What is the full 3-D Cauchy stress tensor?
- What is the full 3-D 2nd Piola-Kirchhoff stress tensor?
- Propose a definition for the engineering stress tensor
and use it to calculate a full 3-D engineering stress tensor.
-
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?
\[
\boldsymbol{\sigma}^{\text{PK2}} =
\left[ \matrix{
10 & 20 & 30 \\
20 & 40 & 50 \\
30 & 50 & 60 }
\right]
\]