# Homework Help: Normal Stress/Shear Stress from stress tensor

1. Jan 26, 2015

### muskie25

1. The problem statement, all variables and given/known data
If $$\sigma_{ij} = \begin{pmatrix} 3 & 3 & 3 \\ 3 & 3 & 3 \\ 3 & 3 & 3 \end{pmatrix}$$ represents a stress tensor, on what plane(s) will the normal stress be a

minimum? On what plane(s) will the shear stress be a maximum?

2. Relevant equations

3. The attempt at a solution
The first two parts to the question asked for the eigenvalues and eigenvectors, which are:
$$\lambda_1 = 9 , \lambda_2 = 0 , \lambda_3 = 0$$

$$v_1 = \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix} , v_2 = \begin{bmatrix} -1\\ 1\\ 0 \end{bmatrix} , v_3 = \begin{bmatrix} -1\\ 0\\ 1 \end{bmatrix}$$

I don't understand how these relate to the normal min, shear max.

Last edited: Jan 26, 2015
2. Jan 27, 2015

### Orodruin

Staff Emeritus
What is the normal/shear stress on planes with these vectors as normal vectors? How does the stress change as you make a rotation from one to the other?

Otherwise, a good place to start is reading up on Mohr's circle: http://en.wikipedia.org/wiki/Mohr's_circle