Homework #4


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

  1. Show that

    \[ \int_S {\bf n} \cdot \nabla ( {\bf x} \cdot {\bf x} ) dS = 6V \]

    where \(V\) is a volume bounded by the surface, \(S\), and \({\bf n}\) is the outward unit normal vector.





  2. Use Hooke's Law to calculate strain tensors given the following stress tensor. Use E = 50 MPa and calculate two separate strain tensors: (i) one for \(\nu\) = 0.33 (metals), and (ii) the second for \(\nu\) = 0.50 (incompressibles).

    Thoughts on sensitivity of stress/strain tensors to Poisson's Ratio?

    \[ {\bf \sigma} = \left[ \matrix{ 20 & 15 & 5 \\ 15 & 30 & 0 \\ 5 & 0 & 10 } \right] \text{MPa} \]



  3. First, demonstrate that the following coordinate transformation matrix satisfies \({\bf Q} \cdot {\bf Q}^T = {\bf I}\). Yet, even though it does satisfy \({\bf Q} \cdot {\bf Q}^T = {\bf I}\), there is something fundamentally wrong with \({\bf Q}\). Can you identify what it is?

    \[ {\bf Q} = \left[ \matrix { 0 & \;\;\;0 & \;\;\;1 \\ 1 & \;\;\;0 & \;\;\;0 \\ 0 & -1 & \;\;\;0 } \right] \]




  4. If \({\bf r} = \theta \, \hat{\bf r}\) and \(\theta = \omega \, t\), then determine equations for velocity, \({\bf v}\), and acceleration, \({\bf a}\).