structural geology, propagation mechanics and - Stanford School of ...

immediately ahead of the tip, while shear stress (σ12) is distributed in an anti-symmetric
quadrant pattern around it. Because each new increment of growth extends a small
distance (r) from the existing tip, it is natural to think in terms of a local polar coordinate
system centered at the tip (Figure 4.10), with propagation occurring in the direction for
which the circumferential normal stress (σθθ) is a maximum, provided that this exceeds
the compressive yield strength of the material. Expressed in terms of opening-mode
cracks (tension positive), this is the maximum circumferential stress criterion of Erdogan
and Sih (1963).
For the symmetric anticrack configuration described above, representing pure mode I
loading and displacements, the distribution of the polar-coordinate stresses as a function
of θ resembles those illustrated in Figure 4.12a, where σθθ max occurs at θ = 0. This
indicates that propagation would continue along a straight path. If even a small amount of
positive mode II displacement (left-lateral shear) is resolved across the tip of the
anticrack, however, the next increment of propagation will diverge at a substantial angle
to the negative (clockwise) side of the θ = 0 axis (Figure 4.12b). In effect, the anticrack
tip attempts to reorient itself to remain symmetric to the perturbed local maximum
principal compressive stress (i.e. along a path for which the local shear stress is zero).
Successive small increments of shearing along the propagation path will result in a
smoothly curving CB.
Excluding variations in effective remote loading (e.g. due to tectonic changes,
regional faulting or slip along major dune boundaries), there are two primary ways for
shear to become resolved on the model anticrack tip: either it deviates from its symmetric
path due to some grain-scale heterogeneity, or it propagates into the perturbed stress field
generated by an adjacent anticrack. The first case results in the resolution of shear on the
deviant tip from the remote loading. If σ11 r > σ22 r (i.e. a significant remote differential
stress exists), then the sense of shear resolved rotates σθθ max back toward the original
symmetric propagation path inhibiting further deviation (Figure 4.13a). As σ11 r
approaches σ22 r in compressive magnitude, however (minimal remote differential stress),
little or no path-correcting shear is resolved on the tip (Figure 4.13b). This analysis
corresponds to that of Cotterell and Rice (1980) on the effect of remote crack-parallel
stress on the propagation stability of opening-mode cracks except that, because anticracks
101