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.

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?
\[
{\bf E} =
\left[ \matrix{
0.10 & 0.60 & 0.00 \\
0.60 & 0.00 & 0.00 \\
0.00 & 0.00 & 0.00
} \right]
\]

A 2D 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:
 Sketch the undeformed square, and also its deformed shape.
 Calculate principal strains and figure out their orientation.
(It's up to you to pick a strain tensor.)
 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.)
 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.

For this strain tensor
\[
{\bf E} =
\left[ \matrix{
0.5 & \;\;\; 0.3 & \;\;\;0.2 \\
0.3 & 0.2 & \;\;\; 0.1 \\
0.2 & \;\;\;0.1 & 0.1
} \right]
\]
 Calculate the 3 strain invariants.
 Calculate the principal strains and use them
to recalculate the invariants and confirm that you get the same answers.