Quasi-static crack models and the frictional stress threshold criterion for slip arrest

Two-dimensional crack problems in elastic homogeneous isotropic media are considered which describe rupture over a fault surface characterized by non-uniform stress drop. Solutions can be found in which the stress field is finite at the crack tips and the rupture surface is not assigned a priori, but is part of the solution. These crack models are found to be consistent with the frictional stress threshold criterion for slip arrest over pre-existing fault surfaces. A crack is found to stop when its contribution to the stress field is opposite to the stress drop at the crack tips. The quasi-static propagation of a crack up to the arrest configuration is studied in terms of the minimum energy principle. The crack spontaneously propagates in such a way as to make the value of the stress intensity factor at one tip equal to the value at the other tip. Furthermore a tip propagating in a region with higher friction is found to move more slowly than the other tip propagating in a region with lower friction. Simple criteria for fracture arrest are derived, in terms of a properly averaged stress drop. Piecewise constant stress drop profiles are explicitly considered yielding a variety of solutions which can be applied to modelling asperities or barriers over a fault plane. The evaluation of the amount of the energy released during the quasi-static crack propagation shows that stopping phases cannot be efficiently radiated if the crack comes to rest in a low friction region.