Homework #2

  1. Identify the following tensor notation quantities. The first one is done to demonstrate what's requested.

    Tensor NotationNameVector NotationExpanded
    A.\(a_i b_i\)Vector Dot Product\({\bf a} \cdot {\bf b}\)\(a_x b_x + a_y b_y + a_z b_z\)
    B.\(\epsilon_{rst} a_r b_t\)No need to expand
    C.\(A_{rs} B_{ts}\)No need to expand
    E.\(\boldsymbol{\sigma}_{ij,j}\)No need to expand

  2. Demonstrate that starting with: \( \qquad \epsilon_{ijk} \epsilon_{lmn} = \delta_{il} (\delta_{jm} \delta_{kn} - \delta_{jn} \delta_{km}) + \delta_{im} (\delta_{jn} \delta_{kl} - \delta_{jl} \delta_{kn}) + \delta_{in} (\delta_{jl} \delta_{km} - \delta_{jm} \delta_{kl}) \)

    and multiplying both sides through by \(\delta_{il}\) produces: \( \qquad \epsilon_{ijk} \epsilon_{imn} = \delta_{jm} \delta_{kn} - \delta_{jn} \delta_{km} \)

  3. Show that \[ \nabla \times ({\bf u} \times {\bf v}) = ({\bf v} \cdot \nabla){\bf u} - {\bf v} (\nabla \cdot {\bf u}) + {\bf u} (\nabla \cdot {\bf v}) - ({\bf u} \cdot \nabla){\bf v} \]

  4. We've done the simpler version of this in class. This time, invert Hooke's Law for strain as a function of stress, to get stress as a function of strain, but with thermal expansion thrown into the mix.

    The starting point is

    \[ \epsilon_{ij} = {1 \over E} \left[ (1+\nu) \sigma_{ij} - \delta_{ij} \, \nu \, \sigma_{kk} \right] + \delta_{ij} \, \alpha (T - T_{ref}) \]
    where \(\alpha\) is the thermal expansion coefficient, and \(T\) and \(T_{ref}\) are temperatures.