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.