# Homework #8

Reminder - you're going to need these webpages: http://www.continuummechanics.org/techforms/index.html to do this homework.... unless you prefer to use Matlab, Mathematica, etc.

Don't bother doing anything by hand anymore. Take advantage of software programs to do all the matrix multiplication and other procedures when you have a chance.

1. We've talked about how a normal component of a Green strain tensor can never be less than -0.5, no matter how much an object is compressed. But if you calculate the principal strains of the following Green strain tensor, you will find that one of them is less than -0.5. So what's going on here?

${\bf E} = \left[ \matrix{ 0.10 & 0.60 & 0.00 \\ 0.60 & 0.00 & 0.00 \\ 0.00 & 0.00 & 0.00 } \right]$
2. A 2-D problem: Take a square with corners at (0,0) and (1,1) and apply the following coordinate mapping to it.

$\begin{eqnarray} x & = & 0.530 X - 0.530 Y + 2 \\ \\ y & = & 0.884 X + 0.884 Y + 1 \end{eqnarray}$
Do the following:

1. Sketch the undeformed square, and also its deformed shape.

2. Calculate principal strains and figure out their orientation. (It's up to you to pick a strain tensor.)

3. Sketch an inscribed undeformed square and deformed rectangle inside the deformed square, and do so at the proper orientation. (See the red square and rectangle in the figure near the top of the principal strain page for an example.)

4. Is the principal direction for your strain tensor consistent with your sketch? Why or why not?

The curve ball in this problem is that there is a lot of rotation going on. This will force you to think very carefully about reconciling the principal strain orientation you calculate with your sketch.

3. For this strain tensor

${\bf E} = \left[ \matrix{ 0.5 & \;\;\; 0.3 & \;\;\;0.2 \\ 0.3 & -0.2 & \;\;\; 0.1 \\ 0.2 & \;\;\;0.1 & -0.1 } \right]$
1. Calculate the 3 strain invariants.

2. Calculate the principal strains and use them to recalculate the invariants and confirm that you get the same answers.