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Decomposing Uniaxial Stresses

  1. Mar 1, 2016 #1
    1. The problem statement, all variables and given/known data
    I am having trouble decomposing a uniaxial compressive stress into hydrostatic and pure shear components.

    2. Relevant equations


    3. The attempt at a solution
    I am starting with

    ##
    \begin{pmatrix}
    -\sigma & 0 & 0 \\
    0 & 0 & 0\\
    0 & 0 & 0
    \end{pmatrix}
    ##

    I then do
    ##
    \begin{pmatrix}
    -\sigma & 0 & 0 \\
    0 & 0 & 0\\
    0 & 0 & 0
    \end{pmatrix} =

    \begin{pmatrix}
    -p & 0 & 0 \\
    0 & -p & 0\\
    0 & 0 & -p
    \end{pmatrix}

    +

    \begin{pmatrix}
    -2p & 0 & 0 \\
    0 & p & 0\\
    0 & 0 & p
    \end{pmatrix}


    ##

    where ## p ## is the hydrostatic pressure. I don't think that this looks correct. Any thoughts?
     
  2. jcsd
  3. Mar 2, 2016 #2
    What is 1/3 of the trace of the stress tensor?
     
  4. Mar 2, 2016 #3
    Chestermiller,

    Do you mean the hydrostatic pressure, ## p ## ?

    ## p = -\sigma/3 ##
     
  5. Mar 2, 2016 #4
    Chestermiller,

    Do you mean the hydrostatic pressure, ## p ## ?

    ## p = −\sigma/3 ##

    I know that the deviatoric tensor is indeed pure shear, because the sum of the diagonal = 0, but my assignment says that there are two states of pure shear. I am either misunderstanding the wording of the problem or I am misunderstanding how to decompose a stress tensor.
     
  6. Mar 2, 2016 #5
    Just replace the p's in your post #1 by ##\sigma/3##, and you'll have the right answer.
     
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