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Normal Stress/Shear Stress from stress tensor

  1. Jan 26, 2015 #1
    1. The problem statement, all variables and given/known data
    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?

    2. Relevant equations


    3. 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: Jan 26, 2015
  2. jcsd
  3. Jan 27, 2015 #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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