We use a finite difference method to study crack propagation in a three-dimensional continuum, for conditions of both uniform and nonuniform prestress. The rupture criterion employed satisfies two fundamental physical requirements: it ensures finite stresses in the continuum and finite energy dissipation in crack extension. The finite stress numerical simulations exhibit abrupt jumps in rupture velocity when sharp changes in prestress are encountered on the crack plane, behavior analogous to that predicted theoretically for two-dimensional, singular cracks. For uniform prestress conditions, the slip velocity function is approximately a low-pass filtered version of that of a singular, constant rupture velocity crack. For nonuniform prestress, spatial variations of peak slip velocity are strongly coupled to spatial variations of rupture velocity. For uniform prestress and low cohesion, rupture velocity is predicted to exceed the S -wave velocity in directions for which mode II (inplane) crack motion dominates. A subshear rupture velocity is predicted for directions of predominantly mode III (antiplane) crack motion. Introduction of stress heterogeneities is sufficient, in each of the three cases studied, to reduce average rupture velocity to less than the S velocity, but local supershear rupture velocities can still occur in regions of high prestress. Rupture models with significant segments of supershear propagation velocities may be consistent with seismic data for some large earthquakes, even where average rupture velocity can be reliably determined to be subshear.

Three-dimensional simulation of spontaneous rupture: The effect of nonuniform prestress

Laboratory studies of large-scale simulated faults show that physical properties of the fault, specifically normal stress and fault roughness, significantly influence the unstable shear failure behavior of the fault. In addition, the experiments provide insights into important length-scaling effects that are useful for assessing concepts such as critical crack length or rupture nucleation dimension. Stick-slip shear failures have been generated along a 2-m-long simulated fault in a block of Sierra white granite. Experiments were conducted at normal stresses between 0.6 and 4 MPa with two different prepared roughnesses of the simulated fault: a smooth fault with a profilometer measured roughness of 0.2 μm and a rough fault with a measured roughness of ∼80 μm. High-speed records of shear strains and slip velocities clearly show both the propagation of the stick-slip failures and details of the local deformations and motions from which slip weakening of the fault at the onset of sliding can be resolved. Both the dynamic stress drops of the stick-slip events and the apparent fracture energies of the events determined from the records of the fault slip weakening increase with increasing normal stress. Although critical slip-weakening displacements are insensitive to normal stress, they increase with increased fault roughness such that the apparent fracture energies are also larger on the rough fault than on the smooth fault. Slip velocities in these experiments are influenced by roughness and are proportional to stress drop. Although the onset of slip is sharply defined, the termination of slip is more gradual without a clear stopping phase. Measured rupture velocities on the rough fault are lower than those on the smooth fault. When a length scale based on rupture nucleation dimensions, directly proportional to critical displacements, is introduced, the data suggest that at fault lengths close to the nucleation lengths, rupture velocity and slip velocity increase with increasing fault length.

Effects of physical fault properties on frictional instabilities produced on simulated faults

We analyze three-dimensional finite difference solutions for a simple shear-crack model of faulting to determine the effects of fault length and width on the earthquake slip function. The fault model is dynamic, with only rupture velocity, fault dimensions, and dynamic stress-drop prescribed. The numerical solutions are accurate for frequencies up to 5 Hz, and are combined with asymptotic results for shear cracks in order to characterize the slip function at higher frequencies. 

Near the hypocenter, the slip velocity exhibits a square-root singularity whose intensity increases with hypocentral distance. At distances greater than the fault width, w , growth of the velocity intensity ceases, and the slip function becomes nearly invariant with distance along the fault length. Closed-form expressions are developed for the dependence of static slip ( s ∞), slip rise time ( TR ), and slip velocity intensity ( V ) on fault geometry. Along the center line of a long, narrow fault, at hypocentral distances exceeding w , these expressions reduce to s ∞ ≈ w Δτ/μ, TR ≈ 0.5 w/vR , and V ≈ √ w /2 vR Δτ/μ, where Δτ is the dynamic stress drop, μ the shear modulus, and vR the rupture velocity. 

The numerical results imply that uniform-dislocation kinematic earthquake models in which slip is represented by a ramp time function will underpredict high-frequency ground motion relative to low-frequency ground motion. A further implication of the numerical solutions is that the nature of inelastic processes at the advancing edge of a long fault will depend on fault width, but will be independent of rupture length.

Three-dimensional finite difference simulation of fault dynamics: Rectangular faults with fixed rupture velocity

Dynamic behaviour of mining-induced fault slip

Supershear earthquake ruptures propagate faster than the shear wave velocity. Although there is evidence that this occurs in nature, it has not been experimentally demonstrated with the use of crustal rocks. We performed stick-slip experiments with Westerly granite under controlled upper-crustal stress conditions. Supershear ruptures systematically occur when the normal stress exceeds 43 megapascals (MPa) with resulting stress drops on the order of 3 to 25 MPa, comparable to the stress drops inferred by seismology for crustal earthquakes. In our experiments, the sub-Rayleigh–to–supershear transition length is a few centimeters at most, suggesting that the rupture of asperities along a fault may propagate locally at supershear velocities. In turn, these sudden accelerations and decelerations could play an important role in the generation of high-frequency radiation and the overall rupture-energy budget.

https://science.sciencemag.org/content/sci/340/6137/1208.full.pdf

From sub-Rayleigh to supershear ruptures during stick-slip experiments on crustal rocks.

Earthquakes, at least the shallow ones, take place along pre-existing fault planes. The controlling factor is, therefore, friction, and the fault growth process resembles that of stick-slip propagation. We have simulated this process in a two-dimensional model. It is found that propagation velocity can range from sub-shear to 1.1 V s (the latter is not a limiting value).

K. C. Thomson

Papers

Stick-slip propagation velocity and seismic source mechanism