Spectral-element modeling of spontaneous earthquake rupture on rate and state faults: Effect of velocity-strengthening friction at shallow depths

We develop a spectral-element methodology (SEM) for simulating dynamic rupture on rate and state faults and use it to study how the rupture is affected by a shallow fault region of steady-state velocity-strengthening friction. Our comparison of the developed SEM and a spectral boundary-integral method (BIM) for an anti-plane (two-dimensional) test problem shows that the two methods produce virtually identical solutions for the finest resolution we use and that the convergence with grid reduction of the developed SEM methodology is comparable to that of BIM. We also use the test problem to compare numerical resolution required for different state evolution laws and for linear slip-weakening friction. Using our three-dimensional implementation of the methodology, we find that a shallow velocity-strengthening fault region can significantly alter dynamic rupture and ground motion. The velocity-strengthening region suppresses supershear propagation at the free surface occurring in the absence of such region, which could explain the lack of universally observed supershear rupture near the free surface. In addition, the velocity-strengthening region promotes faster fall-off of slip velocity behind the rupture front and decreases final slip throughout the entire fault, causing a smaller average stress drop. The slip decrease is largest in the shallow parts of the fault, resulting in the depth profile of slip qualitatively consistent with observations of shallow co-seismic slip deficit. The shallow velocity-strengthening region also reduces the amplification of strong ground motion due to a low-velocity bulk structure

Recent Milestones in Unraveling the Full-Field Structure of Dynamic Shear Cracks and Fault Ruptures in Real-Time: From Photoelasticity to Ultrahigh-Speed Digital Image Correlation

The last few decades have seen great achievements in dynamic fracture mechanics. Yet, it was not possible to experimentally quantify the full-field behavior of dynamic fractures, until very recently. Here, we review our recent work on the full-field quantification of the temporal evolution of dynamic shear ruptures. Our newly developed approach based on digital image correlation combined with ultrahigh-speed photography has revolutionized the capabilities of measuring highly transient phenomena and enabled addressing key ques- tions of rupture dynamics. Recent milestones include the visualization of the complete displacement, particle velocity, strain, stress and strain rate fields near growing ruptures, capturing the evolution of dynamic friction during individual rupture growth, and the detailed study of rupture speed limits. For example, dynamic friction has been the big- gest unknown controlling how frictional ruptures develop but it has been impossible, until now, to measure dynamic friction during spontaneous rupture propagation and to understand its dependence on other quantities. Our recent measurements allow, by simul- taneously tracking tractions and sliding speeds on the rupturing interface, to disentangle its complex dependence on the slip, slip velocity, and on their history. In another application, we have uncovered new phenomena that could not be detected with previous methods, such as the formation of pressure shock fronts associated with “supersonic” propagation of shear ruptures in viscoelastic materials where the wave speeds are shown to depend strongly on the strain rate

Spectral-element simulations of long-term fault slip: Effect of low-rigidity layers on earthquake-cycle dynamics

We develop a spectral element method for the simulation of long-term histories of spontaneous seismic and aseismic slip on faults subjected to tectonic loading. Our approach reproduces all stages of earthquake cycles: nucleation and propagation of earthquake rupture, postseismic slip and interseismic creep. We apply the developed methodology to study the effects of low-rigidity layers on the dynamics of the earthquake cycle in 2-D. We consider two cases: small (M ~ 1) earthquakes on a fault surrounded by a damaged fault zone and large (M ~ 7) earthquakes on a vertical strike-slip fault that cuts through shallow low-rigidity layers. Our results indicate how the source properties of repeating earthquakes are affected by the presence of a damaged fault zone with low rigidity. Compared to faults in homogeneous media, we find (1) reduction in the earthquake nucleation size, (2) amplification of slip rates during dynamic rupture propagation, (3) larger recurrence interval, and (4) smaller amount of aseismic slip. Based on linear stability analysis, we derive a theoretical estimate of the nucleation size as a function of the width and rigidity reduction of the fault zone layer, which is in good agreement with simulated nucleation sizes. We further examine the effects of vertically-stratified layers (e.g., sedimentary basins) on the nature of shallow coseismic slip deficit. Our results suggest that low-rigidity shallow layers alone do not lead to coseismic slip deficit. While the low-rigidity layers result in lower interseismic stress accumulation, they also cause dynamic amplification of slip rates, with the net effect on slip being nearly zero

Variability of earthquake nucleation in continuum models of rate-and-state faults and implications for aftershock rates

Using two continuum models of rate-and-state faults, one with a weaker patch and the other with rheological transition from steady state velocity-weakening to velocity-strengthening friction, we simulate several scenarios of spontaneous earthquake nucleation plausible for natural faults, investigate their response to static shear stress steps, and infer the corresponding aftershock rates. Overall, nucleation processes at weaker patches behave similarly to theories based on spring-slider models, with some notable deviations. In particular, nucleation and aftershock rates are affected by normal stress heterogeneity in the nucleation zone. Nucleation processes at rheological transitions behave differently, producing complex slip velocity histories, nonmonotonic responses to static stress changes, and aftershock rates with pronounced peaks and seismic quiescence. For such processes, positive stress steps sometimes delay nucleation of seismic events by inducing aseismic transients that relieve stress and postpone seismic slip. Superposition of the complex aftershock response for spatially variable stress changes results in Omori's law for a period of time followed by seismic quiescence. Such behavior was observed at the base of the seismogenic zone near the 1984 Morgan Hill earthquake. We show that the computed aftershock rates are linked to unperturbed slip velocity evolution in the nucleation zone and construct simplified analytical scenarios that explain some features of the response. The qualitative differences that we find between the two nucleation models indicate that aftershock response of rate-and-state faults to static stress changes would depend on the conditions under which nucleation occurs on natural faults and may be different from predictions based on spring-slider models

Stability of pulse-like earthquake ruptures

Pulse-like ruptures arise spontaneously in many elastodynamic rupture simulations and seem to be the dominant rupture mode along crustal faults. Pulse-like ruptures propagating under steady-state conditions can be efficiently analysed theoretically, but it remains unclear how they can arise and how they evolve if perturbed. Using thermal pressurisation as a representative constitutive law, we conduct elastodynamic simulations of pulse-like ruptures and determine the spatio-temporal evolution of slip, slip rate and pulse width perturbations induced by infinitesimal perturbations in background stress. These simulations indicate that steady-state pulses driven by thermal pressurisation are unstable. If the initial stress perturbation is negative, ruptures stop; conversely, if the perturbation is positive, ruptures grow and transition to either self-similar pulses (at low background stress) or expanding cracks (at elevated background stress). Based on a dynamic dislocation model, we develop an elastodynamic equation of motion for slip pulses, and demonstrate that steady-state slip pulses are unstable if their accrued slip $b$ is a decreasing function of the uniform background stress $\tau_\mathrm{b}$. This condition is satisfied by slip pulses driven by thermal pressurisation. The equation of motion also predicts quantitatively the growth rate of perturbations, and provides a generic tool to analyse the propagation of slip pulses. The unstable character of steady-state slip pulses implies that this rupture mode is a key one determining the minimum stress conditions for sustainable ruptures along faults, i.e., their ``strength''. Furthermore, slip pulse instabilities can produce a remarkable complexity of rupture dynamics, even under uniform background stress conditions and material properties

Understanding dynamic friction through spontaneously evolving laboratory earthquakes

Friction plays a key role in how ruptures unzip faults in the Earth’s crust and release waves that cause destructive shaking. Yet dynamic friction evolution is one of the biggest uncertainties in earthquake science. Here we report on novel measurements of evolving local friction during spontaneously developing mini-earthquakes in the laboratory, enabled by our ultrahigh speed full-field imaging technique. The technique captures the evolution of displacements, velocities and stresses of dynamic ruptures, whose rupture speed range from sub-Rayleigh to supershear. The observed friction has complex evolution, featuring initial velocity strengthening followed by substantial velocity weakening. Our measurements are consistent with rate-and-state friction formulations supplemented with flash heating but not with widely used slip-weakening friction laws. This study develops a new approach for measuring local evolution of dynamic friction and has important implications for understanding earthquake hazard since laws governing frictional resistance of faults are vital ingredients in physically-based predictive models of the earthquake source

Pulse-like and crack-like ruptures in experiments mimicking crustal earthquakes

Theoretical studies have shown that the issue of rupture modes has important implications for fault constitutive laws, stress conditions on faults, energy partition and heat generation during earthquakes, scaling laws, and spatiotemporal complexity of fault slip. Early theoretical models treated earthquakes as crack-like ruptures, but seismic inversions indicate that earthquake ruptures may propagate in a self-healing pulse-like mode. A number of explanations for the existence of slip pulses have been proposed and continue to be vigorously debated. This study presents experimental observations of spontaneous pulse-like ruptures in a homogeneous linear-elastic setting that mimics crustal earthquakes; reveals how different rupture modes are selected based on the level of fault prestress; demonstrates that both rupture modes can transition to supershear speeds; and advocates, based on comparison with theoretical studies, the importance of velocity-weakening friction for earthquake dynamics

Pressure shock fronts formed by ultra-fast shear cracks in viscoelastic materials

Spontaneously propagating cracks in solids emit both pressure and shear waves. When a shear crack propagates faster than the shear wave speed of the material, the coalescence of the shear wavelets emitted by the near-crack-tip region forms a shock front that significantly concentrates particle motion. Such a shock front should not be possible for pressure waves, because cracks should not be able to exceed the pressure wave speed in isotropic linear-elastic solids. In this study, we present full-field experimental measurements of dynamic shear cracks in viscoelastic polymers that result in the formation of a pressure shock front, in addition to the shear one. The apparent violation of classic theories is explained by the strain-rate-dependent material behavior of polymers, where the crack speed remains below the highest pressure wave speed prevailing locally around the crack tip. These findings have important implications for the physics and dynamics of shear cracks such as earthquakes