# 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, $${\bf a}$$.