CONTINUUM MECHANICS for ENGINEERS

Solution
For this stress state, the determinant Eq 3.6-5 is given by
σ − σ σ
o o
σo σo − σ
0 0
which results in a cubic having roots (principal stress values) σ (1) = 2σ o, σ (2) =
σ (3) = 0 (as may be readily verified by Eq 3.9-4) so that, in principal axes form,
the stress matrix is
The Mohr’s circle diagram is shown in Figure E3.9-1.
Here, because of the double-zero root, one of the three Mohr’s circles
degenerates into a point (the origin) and the other two circles coincide. Also,
*
we note that physically this is simply a one-dimensional tension in the x1 direction and that the maximum shear stress values (shown by points A and
B) occur on the x1 and x2 coordinate planes which make 45° with the principal
* direction.
x 1
3.10 Deviator and Spherical Stress States
The arithmetic mean of the normal stresses,
(3.10-1)
is referred to as the mean normal stress. The state of stress having all three
principal stresses equal (and therefore equal to σ M) is called a spherical state
of stress, represented by the diagonal matrix
0
0
−σ
⎡2σ0
0⎤
* ⎢
⎥
[ σ ij]=
⎢
0 0 0
⎥
⎣
⎢ 0 0 0⎦
⎥
= 0
1
1
σ M = ( σ11 + σ22 + σ33)= σii
3
3
⎡σ
M 0 0 ⎤
⎢
⎥
[ σ ij]=
⎢
0 σ M 0
⎥
⎣
⎢ 0 0
σ M ⎦
⎥
(3.10-2)