Admissible Speeds for Plane-Strain Self-Similar Shear Cracks with Friction but Lacking Cohesion 论文

摘要

Dynamic shear cracks with friction have recently been studied as models for shallow-focus earthquakes both in the laboratory and analytically. There is some evidence from laboratory experiments that zones of slip may spread with velocities greater than the S-wave speed but so far theoretical models with rupture velocities faster than the S-wave speed have not been considered. In particular the shearing version of Broberg's self-similar crack propagating at the Rayleigh speed has been suggested as the solution for cracks lacking cohesion. The present paper shows that this Rayleigh-speed crack may be valid if limiting static friction is sufficiently high, but otherwise the crack may run at the P-wave velocity. The mathematical method used is to show that a self-similar motion is likely and then to eliminate all possible crack speeds except those in a range of speeds between the Rayleigh speed vR and the S-wave speed vS, or, if the static friction is weak, the P-wave speed is found to be the only admissible speed. It is worth noting that for propagation speeds between the Rayleigh and shear speeds there is also a well-known non-physical solution analytically identical to Broberg's solution but having a stress singularity of the wrong sign. The solutions proposed in this paper do not have any stress infinities at all. Analogous solutions apply to tensile cracks but the physics in that situation is not clear. In the final section some conclusions are drawn concerning frictional sliding in a fully three-dimensional situation. Notation x1, x2 Cartesian co-ordinates. t time (usually). ui displacement relative to initial state. Ui ui specialized to x2 = 0. stress tensor. τ0ij initial stress field. P0 = -τ022 initial normal pressure across plane x2 = 0. T0 = τ022 initial shear traction across plane x2 = 0. s Laplace transform variable replacing t. ξ Fourier transform variable replacing x1. υP, υSS, υR, υC speeds of P, S, Rayleigh waves, and the crack edges. γq = l/γq, q ≡ P, S , R , C. ηq = (γq2 - ξ2)½,q ≡ P , S, R, C. R(ξ) = (½ γs2-ξ2)2+ξ2ηPηS. ζ = ξ2 = z + it A is a real constant.