# Normal Stress/Shear Stress from stress tensor

## Homework Statement

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?

## 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.

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## Answers and Replies

Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
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