Homework #4
Feel free to use
http://www.continuummechanics.org/interactivecalcs.html when applicable.

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.

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

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

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