Static shear crack with a zone of slip-weakening

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As a prologue to our investigation of the nature of sequences of earthquake events, we investigate the problem of a finite, 2-D, antiplane, static shear crack for two different models of a slip-weakening zone near the crack tip. The crack is imbedded in an infinite elastic medium. In the limit as the size of the slip-weakening zone vanishes, the problem becomes one considered by Knopoff for a finite shear crack in a uniform field. We solve for the displacement and the stress for representative values of dimension-less stress drop for the two models. Our results indicate that these quantities are not strongly dependent on the details of the constitutive relation in the transition zone. With decreasing dimensionless stress drop, the crack length increases monotonically and the size of the slip-weakening zone decreases monotonically. These results can be interpreted as the critical conditions for the further extension of an initial earthquake fault. In this interpretation, these results imply that the initiation of an earthquake is not only controlled by the stress drop on the fault plane, but also by the size of a pre-existing fracture: an earthquake can be initiated from a small pre-existing fracture if the stress drop on the initial fault is large enough; conversely it can be initiated under the influence of a lower stress drop, if the initial fault is large enough. No solution to the static problem exists if the dimensionless stress drop is greater than a certain value; this maximum dimensionless stress drop corresponds to a minimum crack length for which the slip-weakening zone occupies the entire crack. In the small dimensionless stress drop or small-scale yielding limit, the dimensionless half-length of the crack is inversely proportional to the square of the dimensionless stress drop; the dimensionless length of the slip-weakening zone has been calculated. We generalize the formula for the effective shear fracture energy obtained by Palmer & Rice and Andrews to include the effect of the size of the slip-weakening zone. Our results indicate that because the influence of the size of of the slip-weakening zone has not been taken into account, earlier estimates of the effective shear fracture energy may have been overestimated by as much as 67 per cent.