# Continuum Mechanics Rotation Matrix Problem

1. Dec 14, 2011

### lanew

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

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

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 $x_i$ reference Cartesian system.

2. Relevant equations
$\mathbf{A} = \mathbf{R}^T \mathbf{V} \mathbf{R}$

where $\mathbf{A}$ is the original stress tensor, $\mathbf{R}$ is the rotation tensor, and $\mathbf{V}$ 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. Dec 14, 2011

### lanew

I think I figured it out using diagonalization.

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

$\mathbf{A}=\mathbf{R}\mathbf{\lambda}\mathbf{R}$

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

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

Does this seem correct?

Thanks Again.

Last edited: Dec 14, 2011