On the patterns of wing cracks along an outcrop scale flaw: A numerical modeling approach using complementarity

摘要

The 2D Displacement Discontinuity Method (DDM) is combined with a complementarity algorithm to model the quasi-static formation and patterns of wing cracks that emanate from regions of stress concentration along a sliding frictional flaw in an otherwise homogeneous and isotropic elastic material. Because stress states and geometry change with sliding on the flaw and wing crack propagation, one cannot specify the boundary conditions a priori. Under these circumstances complementarity is superior to other well-known contact algorithms. We focus on meter scale phenomena where mineralogical heterogeneity (common to centimeter-scale laboratory samples) and 3D geometry (common to kilometer-scale crustal structures) reasonably can be ignored. Analytical solutions to the elastic boundary value problem of the closed sliding flaw include those that assume no friction, uniform friction, and a cohesive end zone (CEZ), and those that assume infinitesimal or straight wing cracks. Here we generalize the problem to consider linearly varying friction in the CEZ and curved wing cracks, and we allow the sliding flaw to open when mechanical interaction with the wing crack dictates that it should. Trace lengths of 135 strike-slip faults in sandstone are linearly related to wing crack lengths ranging from 0.16 to 72 m and correspond to a range of remote principal stress ratios: 0.06 ≤ σ 2/σ 1 ≤ 0.2. Opening displacement profiles of wing cracks from the numerical model can be significantly different from analytical solutions. These solutions may produce significant errors in stable crack length for curved propagation paths. The smeared out stress concentration in the CEZ and the heterogeneity in strength of rock suggest that multiple wing cracks may form in one slip event. The mechanical interactions of these cracks leads to kink angles that increase with distance from the flaw tip, a relationship commonly observed in nature.