Homework #5
Feel free to use
http://www.continuummechanics.org/interactivecalcs.html when applicable.
-
Show that
\[
\epsilon_{ijk} \, \epsilon_{ijk} = 6
\]
-
Given the stress tensor
\[
\boldsymbol{\sigma} =
\left[ \matrix{
10 & 20 & 30 \\
20 & 40 & 50 \\
30 & 50 & 60 }
\right]
\qquad
\]
One of the two stress tensors below is equivalent to the one above,
differing only by a coordinate transformation. The other one
represents a different stress state. Which is equivalent and
which is different?
\[
\left[ \matrix{
4.0341 & 27.291 & 14.519 \\
27.291 & 76.619 & 46.048 \\
14.519 & 46.048 & 29.347 }
\right]
\qquad \qquad \qquad
\left[ \matrix{
\;\;\;30.597 & -5.733 & -15.201 \\
\;-5.733 & \;41.305 & \;\;\;18.926 \\
-15.201 & \;18.926 & \;\;\;48.098 }
\right]
\]
-
For \(E = 10 \, \text{MPa}\), \(\nu = 0.333\), and tension in the 1-direction such that
\(\epsilon_{11} = 0.1\), \(\epsilon_{22} = -0.0333\), and \(\epsilon_{33} = -0.0333\):
Calculate \(C_{1111}\), \(C_{1122}\), and \(C_{1133}\) and use them with the strains to
calculate \(\sigma_{11}\).
It should simply equal \(\sigma_{11} = 1 \, \text{MPa}\) because this satisfies
\(\sigma_{11} = E \, \epsilon_{11}\) for this case of uniaxial tension.