Robert C. Viesca

Stable and unstable development of an interfacial sliding instability.

Examining a nonlinear instability of sliding rate on a frictional interface of elastic bodies, we investigate whether laboratory-constrained frictional relations suggest universal scaling under even the simplest of configurations. We find blowup solutions by solving an equivalent, classical problem in fracture mechanics. The solutions are fixed points of a dynamical system and we show that their stability is lost by a cascade of Hopf bifurcations as a single problem parameter is increased, leading to chaotic dynamics.

Modeling slope instability as shear rupture propagation in a saturated porous medium

When a region of intense shear in a slope is much thinner than other relevant geometric lengths, this shear failure may be approximated as localized slip like in faulting, with strength determined by frictional properties of the sediment and effective stress normal to the failure surface. Peak and residual frictional strengths of submarine sediments indicate critical slope angles well above those of most submarine slopes—in contradiction to abundant failures. Because deformation of sediments is governed by effective stress, processes affecting pore pressures are a means of strength reduction. However, common methods of examining slope stability neglect dynamically variable pore pressure during failure. We examine elastic-plastic models of the capped Drucker-Prager type and derive approximate equations governing pore pressure about a slip surface when the adjacent material may deform plastically. In the process we identify an elastic-plastic hydraulic diffusivity with an evolving permeability and plastic storage term analogous to the elastic term of traditional poroelasticity. We also examine their application to a dynamically propagating subsurface rupture and find indications of downslope directivity.

Steadily propagating slip pulses driven by thermal decomposition

Geophysical observations suggest that mature faults weaken significantly at seismic slip rates. Thermal pressurization and thermal decomposition are two mechanisms commonly used to explain this dynamic weakening. Both rely on pore fluid pressurization with thermal pressurization achieving this through thermal expansion of native solids and pore fluid and thermal decomposition by releasing additional pore fluid during a reaction. Several recent papers have looked at the role thermal pressurization plays during a dynamically propagating earthquake, but no previous models have studied the role of thermal decomposition. In this paper we present the first solutions accounting for thermal decomposition during dynamic rupture, solving for steady state self-healing slip pulses propagating at a constant rupture velocity. First, we show that thermal decomposition leads to longer slip durations, larger total slips, and a distinctive along–fault slip rate profile. Next, we show that accounting for more than one weakening mechanism allows multiple steady slip pulses to exist at a given background stress, with some solutions corresponding to different balances between thermal pressurization and thermal decomposition, and others corresponding to activating a single reaction multiple times. Finally, we study how the rupture properties depend on the fault properties and show that the impact of thermal decomposition is largely controlled by the ratio of the hydraulic and thermal diffusivities χ = αhy/αth and the ratio of pore pressure generated to temperature rise buffered by the reaction Pr/Er.

Self-similar slip instability on interfaces with rate- and state-dependent friction

We examine the development of a frictional instability, with diverging sliding rate, at the interface of elastic bodies in contact. Evolution of friction is determined by a slip rate and state dependence. Following Viesca (2016 Phys. Rev. E 93, 060202(R). (doi:10.1103/PhysRevE.93.060202)), we show through an appropriate change of variable, the existence of blow-up solutions that are fixed points of a dynamical system. The solutions show self-similarity of the simple variety: separable dependence of time and space. For an interface with uniform frictional properties, there is a single-problem parameter. We examine the linear stability of these fixed points, as this problem parameter is varied. Specifically, we consider two archetypical elastic settings of the slip surface, in which interactions between points on the surface are either local or non-local. We show that, independent of the nature of elastic interactions, the fixed-points lose stability in the same matter as the parameter is increased towards a limit value: an apparently infinite sequence of Hopf bifurcations. However, for any value of the parameter, the nonlinear development of the instability is attraction, if not asymptotic convergence, towards these fixed points, owing to the existence of stable eigenmodes. For comparison, we perform numerical solutions of the original evolution equations and find precise agreement with the results of the analysis.

Earthquake nucleation in intact or healed rocks

Earthquakes are generated because faults lose strength with increasing slip and slip rate. Among the simplest representations of slip-dependent strength is the linear slip-weakening model, characterized by a linear drop to a residual friction. However, healed fault rocks often exhibit some slip strengthening before the onset of weakening. Here we investigate the effect of such a slip-hardening phase on the initial growth of a slip patch and on the nucleation of rupture instabilities. We assume a piecewise linear strength versus slip constitutive relation. We compute stress and slip distributions for in-plane or antiplane rupture configurations in response to an increasing, locally peaked (parabolic with curvature κ) stress profile. In contrast with the strictly linear slip-weakening case, our calculations show that the curvature of the loading profile and the level of background stress strongly influence the nucleation size. Even for small amounts of slip hardening, we find that the critical nucleation size scales with inline image for κ[RIGHTWARDS ARROW]0, i.e., crack growth remains stable up to very large crack sizes for sufficiently smooth loading profiles. Likewise, when the background stress τb is very close to the initial strength τc, the critical crack size scales with inline image. An eigenvalue analysis shows that the nucleation length increases as the proportion of the crack undergoing slip hardening increases, irrespective of the details of the loading profile. Overall, our results indicate that earthquake nucleation sizes can significantly increase due to slip hardening (e.g., in healed fault rocks), especially when the background loading is smooth.

The fracture energy of ruptures driven by flash heating

We present a model for dynamic weakening of faults based on local flash heating at microscopic asperity contacts coupled to bulk heating at macroscopic scale. We estimate the fracture energy G associated with that rheology and find that for constant slip rate histories G scales with slip δ as math formula at small slip, while math formula at large slip. This prediction is quantitatively consistent with data from laboratory experiments conducted on dry rocks at constant slip rate. We also estimate G for crack-like ruptures propagating at constant speed and find that math formula in the large slip limit. Quantitative estimates of G in that regime tend to be several orders of magnitude lower than seismologically inferred values of G. We conclude that while flash heating provides a consistent explanation for the observed dynamic weakening in laboratory experiments with kinematically imposed slip, its contribution to the energy dissipation during earthquakes becomes negligible for large events when considering the elastodynamic coupling between strength and slip evolution.

Earthquake nucleation on faults with heterogeneous frictional properties, normal stress.†

We examine the development of an instability of fault slip rate. We consider a slip rate and state dependence of fault frictional strength, in which frictional properties and normal stress are functions of position. We pose the problem for a slip rate distribution that diverges quasi-statically within finite time in a self-similar fashion. Scenarios of property variations are considered and the corresponding self-similar solutions found. We focus on variations of coefficients, a and b respectively, controlling the magnitude of a direct effect on strength due to instantaneous changes in slip rate and of strength evolution due to changes in a state variable. These results readily extend to variations in fault normal stress, σ, or the characteristic slip distance for state evolution, Dc. We find that heterogeneous properties lead to a finite number of self-similar solutions, located about critical points of the distributions: maxima, minima, and between them. We examine the stability of these solutions and find that only a subset is asymptotically stable, occurring at just one of the critical point types. Such stability implies that, during instability development, slip rate and state evolution can be attracted to develop in the manner of the self-similar solution, which is also confirmed by solutions to initial value problems for slip rate and state. A quasi-static slip-rate divergence is ultimately limited by inertia, leading to the nucleation of an outward expanding dynamic rupture: asymptotic stability of self-similar solutions then implies preferential sites for earthquake nucleation, which are determined by distribution of frictional properties.