Dynamic Plane-strain Shear Rupture with a Slip-weakening Friction Law Calculated by a Boundary Integral Method

A numerical boundary integral method, relating slip and traction on a plane in an elastic medium by convolution with a discretized Green function, can be linked to a slip-dependent friction law on the fault plane. Such a method is developed here in two-dimensional plane-strain geometry. The method is more efficient for a planar source than a finite difference method, and it does not suffer from dispersion of short wavelength components. The solution for a crack growing at constant velocity agrees closely with the analytic solution, and the energy absorbed at the smeared-out crack tip in the numerical calculation agrees with energy absorbed at the analytic singularity. Spontaneous plane-strain shear ruptures can make a transition from sub-Rayleigh to near-P propagation velocity. Results from the boundary integral method agree with earlier results from a finite difference method on the location of this transition in parameter space. The methods differ in their prediction of rupture velocity following the transition. The trailing edge of the cohesive zone propagates at the P-wave velocity after the transition in the boundary integral calculations. INTRODUCTION Idealized models of earthquake sources, considered to be cracks expanding in a plane in an elastic medium, have been calculated by various methods over the past decade. A boundary integral method, in which past values of traction on the fault plane are convolved numerically with a Green function, was developed by Hamano (1974) and by Das and Aki (1977) for two-dimensional problems. The method has been extended to three dimensions by Das (1980) using an efficient Green function formulation from Richards (1979). The boundary integral method does not suffer from dispersion of wavelengths on the order of the element size, as do finite difference and finite element methods, and therefore should be more accurate. Until now, the boundary integral method has been applied only to a brittle crack model, in which there is an abrupt transition from no slip to sliding at a constant prescribed traction. In this paper, working in two-dimensional plane-strain geometry, I will: (1) show how the boundary integral method can be adapted to a slip-dependent friction law on the crack plane; (2) examine the accuracy of the method; and (3) recalculate some problems concerning the transition to rupture velocity greater than S velocity of plane-strain (mode 2) shear cracks, which were done previously by a finite difference method (Andrews, 1976b). 2 D . J . ANDREWS THE BOUNDARY INTEGRAL METHOD Let the fault be the plane x3 = 0 in an infinite homogeneous isotropic elastic medium, which is at rest in static equilibrium at time t = 0. A subsequent shearing displacement discontinuity, termed slip, has components u~, u2 in the x, and x2 directions on the fault plane. The shear traction vector is T, + 7'1 °, T2 + T2 °, where T1 °, T2° is the traction vector in the initial equilibrium state. Das and Aki (1977) and Aki and Richards [1980, p. 29, their equation (2.43)] show that the slip vector ul, u2 is related to the traction change vector T1, 7'2 on the plane x3 = 0 by the convolution