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Normal Stress/Shear Stress from stress tensor

  • Thread starter muskie25
  • Start date
  • #1
16
0

Homework Statement


If [tex] \sigma_{ij} = \begin{pmatrix}
3 & 3 & 3 \\
3 & 3 & 3 \\
3 & 3 & 3
\end{pmatrix} [/tex] 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?

Homework Equations




The Attempt at a Solution


The first two parts to the question asked for the eigenvalues and eigenvectors, which are:
[tex] \lambda_1 = 9 , \lambda_2 = 0 , \lambda_3 = 0 [/tex]

[tex] 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}
[/tex]

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

Answers and Replies

  • #2
Orodruin
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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
 

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