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

To determine the sensitivity of propagation path to the choice of r, we performed a
series of stability tests using the same model band that produced Figures 4.16 and 17. We
added a 2-cm element to one tip at a variable kink angle of α, and plotted the relationship
between α and the angle at which the next increment of propagation was predicted to
occur (β) for r = 1, 5 and 10 mm, and the bracketing remote differential stress magnitudes
(σ d = σ11 r -σ22 r ) of zero and 0.5 (Figure 4.18). The response in every case is to correct the
deviation such that β is negative for positive α along a nearly linear trend. When the
remote stress state is isotropic, |β| ≤ |α| and symmetry with σ11 r is nearly restored with the
next increment of propagation. When σ d is the maximum of 0.5, |β| ≥ |α| and the next
increment of propagation over-corrects for the original deviation. The effect of varying r
at the two stress states is also opposite: for σ d = 0, |β| varies directly with r; for σ d = 0.5,
|β| varies inversely with r. Discarding r = 1 mm as unrealistically small for the reasons
stated above, we observe that β never varies by more than 4° for a given α between r = 5
mm and r = 10 mm. Due to this limited effect, and to preserve the plausibility of the
linear elastic continuum assumption, we chose r = 5 mm for all simulations.
Finally, as the model uses linear segments to approximate CB paths that generally are
smoothly curving, we examined the stability of the propagation direction as a function of
the dislocation element length used. Naturally, the shorter the elements, the better the
model fit to the ideal (i.e. smooth) path. As a practical matter of computational efficiency,
however, longer elements are desirable. We performed a second suite of kink-angle
stability tests for tip-element lengths of 2, 6 and 10 cm at r = 5 mm (Figure 4.19). Again,
in every case β acts to oppose α along a nearly linear trend, with the response being
strongest for σ d = 0.5. The sense of the effect at the two stress states is also opposite,
with |β| varying directly with element length for σ d = 0.5, and indirectly for σ d = 0. The
spread in the directional variability is about three times greater at σ d = 0.5 and is equal to
about 85% of α. These results are entirely consistent with those reported in previous
studies of propagation path stability for segmented opening-mode cracks using the
kinked-tip approach (Broberg, 1987; Cotterell and Rice, 1980; Melin, 1983, 1987;
Thomas and Pollard, 1993), which indicate that the misorientation of successive
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