Slip-weakening constitutive relation and the structure in the vicinity of a shear crack tip

Shear-crack model with a cohesive zone (or breakdown zone) is appropriate for the analysis of a fault surface in which slip distribution is strongly nonuniform. As the slipped portion advances, slip-weakening occurs over the so-called cohesive zone, a distance behind the fault tip. For a prescribed strength vs. displacement constitutive relation, however, the zone structure is difficult to determine by an analytical method except for some simple cases, thus it often requires a certain numerical procedure. This work proposes a numerical procedure to obtain approximated solutions of the problem by combining a series of elastic solutions derived bySmith (1974). The series is linearly combined and the unknown coefficients are determined by a nonlinear least square method. This method can fit a wide range of prescribed strength vs. displacement relations which may be simple algebraic relations or curves obtained by laboratory tests. By examining the residual errors and in comparison with a derived result in which linear stress is assumed within the zone, it could be concluded that the results provide good accuracy. Moreover, because the results are written in formulae, they can be easily referred to or used. By fitting constitutive curves in many different shapes, it is found that the stress distribution within the zone is more sensitive to the constitutive curve shape than the displacement. The most interesting fact is that the zone size is not sensitive to the curve shape, i.e., the zone size can be estimated by $$R = 3\mu \zeta \upsilon _c /\{ 2(1 - \upsilon )(\tau _c - \tau _f )\}$$ with ζ=1±0.11 for most cases.