Homework #6
Feel free to use
http://www.continuummechanics.org/interactivecalcs.html when applicable.

For \(E = 10 \, \text{MPa}\), \(\nu = 0.333\), and tension in the 1direction such that
\(\epsilon_{11} = 0.1\), \(\epsilon_{22} = 0.0333\), and \(\epsilon_{33} = 0.0333\):
Calculate \(C_{1111}\), \(C_{1122}\), and \(C_{1133}\) and use them with the strains to
calculate \(\sigma_{11}\).
It should simply equal \(\sigma_{11} = 1 \, \text{MPa}\) because this satisfies
\(\sigma_{11} = E \, \epsilon_{11}\) for this case of uniaxial tension.
Use the following deformation gradient for Problems 24.
\[
{\bf F} =
\left[ \matrix{
\;\;\; 1.5 & \;\;\; 0.3 & 0.2 \\
0.1 & \;\;\;1.2 & \;\;\; 0.1 \\
\;\;\; 0.3 & 0.2 & \;\;\; 1.1
} \right]
\]

Determine \({\bf R}\) and \({\bf U}\) in \({\bf F} = {\bf R} \cdot {\bf U}\).
Do it the hard way by multiplying all the matrices out yourself (but not by hand! use software).
Then check your result using this page:
http://www.continuummechanics.org/cm/techforms/RUDecomposition.html

How many degrees of rigid body rotation are in this deformation gradient, and
what is the axis of rotation?

Now that you have \({\bf R}\) and \({\bf U}\) from #2, calculate \({\bf V}\) and use
http://www.continuummechanics.org/cm/techforms/VRDecomposition.html
to check that you got it right.