# Homework #6

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

1. For $$E = 10 \, \text{MPa}$$, $$\nu = 0.333$$, and tension in the 1-direction 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 2-4.

${\bf F} = \left[ \matrix{ \;\;\; 1.5 & \;\;\; 0.3 & -0.2 \\ -0.1 & \;\;\;1.2 & \;\;\; 0.1 \\ \;\;\; 0.3 & -0.2 & \;\;\; 1.1 } \right]$
1. 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).
3. 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.