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Shear stress and the deviator

  1. Mar 2, 2013 #1
    Let the second invariant of the stress deviator be expressed in terms of its principal values, that is, by
    $$
    \text{\MakeUppercase{\romannumeral 2}}_{\text{S}} = \text{S}_{\text{\MakeUppercase{\romannumeral 1}}}\text{S}_{\text{\MakeUppercase{\romannumeral 2}}} + \text{S}_{\text{\MakeUppercase{\romannumeral 2}}}\text{S}_{\text{\MakeUppercase{\romannumeral 3}}} + \text{S}_{\text{\MakeUppercase{\romannumeral 3}}} \text{S}_{\text{\MakeUppercase{\romannumeral 1}}}.
    $$
    Show that this sum is the negative of two-thirds the sum of squares of the principal shear stresses.
    Is this really true? I used Mathematica to calculate this for an arbitrary symmetric matrix but it didn't turn out true.
    Code (Text):

    In[77]:= FullSimplify[{{a + 1/3 (-a - d - f), b, c}, {b,
        d + 1/3 (-a - d - f), e}, {c, e,
        1/3 (-a - d - f) + f}} + {{1/3*(a + d + f), 0, 0}, {0,
        1/3*(a + d + f), 0}, {0, 0, 1/3*(a + d + f)}}]

    Out[77]= {{a, b, c}, {b, d, e}, {c, e, f}}

    In[78]:= v =
      FullSimplify[
       Eigenvalues[{{a + 1/3 (-a - d - f), b, c}, {b,
          d + 1/3 (-a - d - f), e}, {c, e, 1/3 (-a - d - f) + f}}]];

    In[79]:= FullSimplify[v[[1]]*v[[2]] + v[[2]]*v[[3]] + v[[3]]*v[[1]]]

    Out[79]= 1/3 (-a^2 - d^2 - 3 (b^2 + c^2 + e^2) + d f - f^2 +
       a (d + f))

    In[74]:= w = Eigenvalues[{{a, b, c}, {b, d, e}, {c, e, f}}];

    In[75]:= FullSimplify[-2/3*(w[[1]]^2 + w[[2]]^2 + w[[3]]^2)]

    Out[75]= -(2/3) (a^2 + 2 b^2 + 2 c^2 + d^2 + 2 e^2 + f^2)
     
    Line Out[79] doesn't equal line Out[75].
     
  2. jcsd
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