# Shear stress and the deviator

1. Mar 2, 2013

### Dustinsfl

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].