Propagation of plane strain shear cracks is calculated numerically by using finite difference equations with second-order accuracy. The rupture model, in which stress drops gradually as slip increases, combines two different rupture criteria: (1) slip begins at a finite stress level; (2) finite energy is absorbed per unit area as the crack advances. Solutions for this model are nonsingular. In some cases there may be a transition from rupture velocity less than Rayleigh velocity to rupture velocity greater than shear wave velocity. The locus of this transition is surveyed in the parameter space of fracture energy, upper yield stress, and crack length. A solution for this model can be represented as a convolution of a singular solution having abrupt stress drop with a 'rupture distribution function.' The convolution eliminates the singularity and spreads out the rupture front in space-time. If the solution for abrupt stress drop has an inverse square root singularity at the crack tip, as it does for sub-Rayleigh rupture velocity, then the rupture velocity of the convolved solution is independent of the rupture distribution function and depends only on the fracture energy and crack length. On the other hand, a crack with abrupt stress drop propagating faster than the shear wave velocity has a lower-order singularity. A supershear rupture front must necessarily be spread out in space-time if a finite fracture energy is absorbed as stress drops.

[1] Energy loss in a fault damage zone, outside the slip zone, contributes to the fracture energy that determines rupture velocity of an earthquake. A nonelastic two-dimensional dynamic calculation is done in which the slip zone is modeled as a fault plane and material off the fault is subject to a Coulomb yield condition. In a mode 2 crack-like solution in which an abrupt uniform drop of shear traction on the fault spreads from a point, Coulomb yielding occurs on the extensional side of the fault. Plastic strain is distributed with uniform magnitude along the fault, and it has a thickness normal to the fault proportional to propagation distance. Energy loss off the fault is also proportional to propagation distance, and it can become much larger than energy loss on the fault specified by the fault constitutive relation. The slip velocity function could be produced in an equivalent elastic problem by a slip-weakening friction law with breakdown slip D c increasing with distance. Fracture energy G and equivalent D c will be different in ruptures with different initiation points and stress drops, so they are not constitutive properties; they are determined by the dynamic solution that arrives at a particular point. Peak slip velocity is, however, a property of a fault location. Nonelastic response can be mimicked by imposing a limit on slip velocity on a fault in an elastic medium.

can be generated as a secondary effect of a droi• of shear stress in an asperity. The pulse shape is unstable, and the initial slip pulse can change during propagation into a collection of sharper pulses. Such a pulse enables slip to occur with little loss of energy to friction, while at the same time increasing irregularity of stress and slip at the source.

Rupture propagation in antiplane strain is investigated by using both analytic and numerical methods. Under the assumption that a solid will absorb energy irreversibly when it is strained at a sufficiently large shear stress it is found that energy must be absorbed at the rupture front in addition to the work done against the sliding friction stress. The energy absorbed increases with propagation distance, so it is not negligible at any length scale and is much larger than the ideal surface energy of molecular cohesion. The concept of a critical crack length carries over to the case of a finite stress-slip law on a fault plane but does not carry over to a homogeneous inelastic medium. In a dynamic slip event, while a typical value of particle velocity is proportional to stress drop, the peak value near the fault is proportional to material strength. \ \

[1] The heat generated in a slip zone during an earthquake can raise fluid pressure and thereby reduce frictional resistance to slip. The amount of fluid pressure rise depends on the associated fluid flow. The heat generated at a given time produces fluid pressure that decreases inversely with the square root of hydraulic diffusivity times the elapsed time. If the slip velocity function is crack-like, there is a prompt fluid pressure rise at the onset of slip, followed by a slower increase. The stress drop associated with the prompt fluid pressure rise increases with rupture propagation distance. The threshold propagation distance at which thermally induced stress drop starts to dominate over frictionally induced stress drop is proportional to hydraulic diffusivity. If hydraulic diffusivity is 0.02 m 2 /s, estimated from borehole samples of fault zone material, the threshold propagation distance is 300 m. The stress wave in an earthquake will induce an unknown amount of dilatancy and will increase hydraulic diffusivity, both of which will lessen the fluid pressure effect. Nevertheless, if hydraulic diffusivity is no more than two orders of magnitude larger than the laboratory value, then stress drop is complete in large earthquakes.