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