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Continuum Mechanics Rotation Matrix Problem

  • Thread starter lanew
  • Start date
  • #1
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1. Homework Statement
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. Homework 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.
 

Answers and Replies

  • #2
13
0
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:

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