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).
    Then check your result using this page: http://www.continuummechanics.org/cm/techforms/RUDecomposition.html


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


  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.