CONTINUUM MECHANICS for ENGINEERS

FIGURE 3.15A
Reference angles φ and β for intersection point Q on surface of body octant.
FIGURE 3.15B
Mohr’s semicircle for stress state displayed in Figure 3.15A.
along the circle arc KD of Figure 3.15A. For this case, Eq 3.8-2a becomes
(σN – σII)(σN – σIII) + = (σI – σII)(σI – σIII) cos2φ1, which may be cast into
the standard form of a circle as
σ 2
S
2
2
⎞⎤
2
N II III S I II I III II III R ⎠ 1
⎡ 1 ⎛
⎞⎤
2 2 ⎡1
σ − ⎝σ
+ σ ⎠ + σ = ( σ −σ
) ( σ − σ ) cos φ + ⎛
1 σ −σ
⎣⎢
⎦⎥
⎣
⎢ ⎝
2
2
⎦
⎥ =
(3.8-9)
This circle is seen to have its center coincident with that of circle C1 in stress
space and to have a radius R1 indicated by Eq 3.8-9. Therefore, as Q moves
on circle arc KD in Figure 3.15A, the stress point q traces the circle arc kd
π
shown in Figure 3.15B. (Notice that if φ1 = so that cos φ1 = 0, R1 reduces
2
1
π
to ( σ σ , the radius of circle C1.) Next, let β = β1 < and then, as φ
II − III)
2 2