# Homework #5

Feel free to use http://www.continuummechanics.org/interactivecalcs.html when applicable.

1. Show that
$\epsilon_{ijk} \, \epsilon_{ijk} = 6$

2. 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?

$\left[ \matrix{ 4.0341 & 27.291 & 14.519 \\ 27.291 & 76.619 & 46.048 \\ 14.519 & 46.048 & 29.347 } \right] \qquad \qquad \qquad \left[ \matrix{ \;\;\;30.597 & -5.733 & -15.201 \\ \;-5.733 & \;41.305 & \;\;\;18.926 \\ -15.201 & \;18.926 & \;\;\;48.098 } \right]$

3. 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.