CONTINUUM MECHANICS for ENGINEERS

FIGURE 3.14A
Octant of small sphercal portion of body together with plane at P with normal n i referred to
principal axes Ox 1*x 2*x 3*.
FIGURE 3.14B
Mohr’s stress semicircle for octant of Figure 3.14A.
If ˆ ˆ so that its intersection point Q coincides with A, σN = σI. Likewise,
when Q coincides with B, σN = σII, and with C, σN = σIII. In all three cases, σS will be zero. In the Mohr’s circle diagram (Figure 3.14B), these stress values are
located at points a, b and c, respectively. If now θ is set equal to and φ
allowed to vary from zero to (β will concurrently go from to zero), Q
will move along the quarter-circle arc AB from A to B. In the stress space of
Figure 3.14B, the stress point q (the image point of Q) having coordinates σN and σS will simultaneously move along the semicircle of C3 from a to b. (Note
that as Q moves 90° along AB in physical space, q moves 180° along the
semicircle, joining a to b in stress space.) Similarly, when Q is located on the
quarter circle BC, or CA of Figure 3.14A, point q will occupy a corresponding
position on the semicircles of bc and ca, respectively, in Figure 3.14B.
Now let the angle φ be given some fixed value less than , say φ = φ1, and
imagine that β and θ take on all values compatible with the movement of Q
*
n= e1
π
2
π
π
2
2
π
2