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Continuum Mechanics Rotation Matrix Problem

  1. Dec 14, 2011 #1
    1. The problem statement, all variables and given/known data
    The components of stress in the [itex]x_i[/itex] reference Cartesian system at a point of interested have been determined to be:

    [itex]
    \left[\begin{array}{ccc}
    500 & 0 & 300 \\
    0 & 700 & 0 \\
    300 & 0 & -100
    \end{array}\right] \mathrm{MPa}
    [/itex]

    Determine the principal values and directions of stress. Determine the rotation tensor transforming the components of stress from the principal components into components along the [itex]x_i[/itex] reference Cartesian system.

    2. Relevant equations
    [itex]\mathbf{A} = \mathbf{R}^T \mathbf{V} \mathbf{R}[/itex]

    where [itex]\mathbf{A}[/itex] is the original stress tensor, [itex]\mathbf{R}[/itex] is the rotation tensor, and [itex]\mathbf{V}[/itex] is a matrix of eigenvectors.

    3. The attempt at a solution
    I've solved for the principal values and directions, but don't know how to solve for the rotation tensor. It seems there's too many unknowns or I'm not making a necessary assumption. Does anyone have any suggestions?

    Thank You.
     
  2. jcsd
  3. Dec 14, 2011 #2
    I think I figured it out using diagonalization.

    I combined the principal eigenvectors into a matrix [itex]\mathbf{R}[/itex] and checked by multiplying it by the diagonal matrix of the principal values.

    [itex]\mathbf{A}=\mathbf{R}\mathbf{\lambda}\mathbf{R}[/itex]

    [itex]
    \left[\begin{array}{ccc}
    500 & 0 & 300 \\
    0 & 700 & 0 \\
    300 & 0 & -100
    \end{array}\right] =
    \left[\begin{array}{ccc}
    V_1 & V_2 & V_3 \\
    V_1 & V_2 & V_3 \\
    V_1 & V_2 & V_3
    \end{array}\right]
    \left[\begin{array}{ccc}
    \lambda_1 & 0 & 0 \\
    0 & \lambda_2 & 0 \\
    0 & 0 & \lambda_3
    \end{array}\right]
    \left[\begin{array}{ccc}
    V_1 & V_1 & V_1 \\
    V_2 & V_2 & V_2 \\
    V_3 & V_3 & V_3
    \end{array}\right]
    [/itex]

    When I evaluated the right side, it equated to the left.

    Does this seem correct?

    Thanks Again.
     
    Last edited: Dec 14, 2011
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