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Rupture Dynamics on Nonplanar Faults with Strongly Rate-Weakening Friction and Off-Fault Plasticity

AbstractNatural fault surfaces are nonplanar at all scales. Slip on such faults induces local stress perturbations that lead to irregular rupture propagation and potentially activate off-fault inelastic deformation. Using 2D plane-strain finite difference simulations, we study rupture phenomenology on nonplanar faults in elastic-plastic (Drucker-Prager) media. Motived by high-velocity friction experiments, we use a strongly rate-weakening friction law formulated in a rate-and-state framework. Studies of ruptures on planar faults in elastic media show how strongly rate-weaking friction laws can lead to rupture propaga-tion in the form of slip pulses, provided that the background stress level on the fault is below a critical value. At higher stress levels, ruptures generally take the form of cracks, which can produce several times more slip than slip pulses, even though the differences in dynamic stress drop are minor. This phe-nomenology holds also for ruptures in elastic-plastic media, though the minimum background stress re-quired for slip pulses to be self-sustaining is increased relative to that in an elastic medium. Plastic de-formation generally occurs in the extensional quadrants unless the maximum principal compressive stress is oriented at a low angle to the fault. This is consistent with Templeton and Rice's (JGR, 2008) re-sults with slip-weakening friction laws (which produce crack-like ruptures). However, the extent of plastic deformation (and amount of slip) is reduced when rupture occurs in the slip-pulse instead of crack-like mode. The addition of fault roughness, having amplitude-to-wavelength ratios of 10-3 to 10-2, does not greatly change the overall distribution or extent of plastic deformation, but instead modulates its amplitude.This spatial variability of plastic deformation enhances the irregularity of rupture propaga-tion due to fault roughness.

SCEC 2009

Flat Faults: Rupture Mode and Plastic Strain Distribution

Coordinate Mapping

Irregular geometries are typically handled using finite element methods, which may suffer from numerical oscillations and often require special treatment of nonphysical hourglass modes. But finite difference meth-ods can also be used via a global mapping between irregular physical domain and Cartesian (rectangular) computational domain. Governing equations are transformed and solved in computational domain.

At low Ψ, plastic strain in compressional quadrants; otherwise in extensional quadrants. Consistent with Templeton and Rice’s [2008] study with slip-weakening friction.

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Strongly Rate-Weakening Friction and Slip PulsesFor strongly rate-weakening friction laws, background stress levels (τb) comparable to τpulse associated with slip pulses; higher τb causes crack-like ruptures [Zheng and Rice, 1998]. We use rate-and-state slip law formulation:

Model Formulation

Off-Fault Inelastic ResponseNonassociative Drucker–Prager plasticity

Remote Loading and Initial ConditionsUniform initial stress field with Ψ being angle between maximum compressive principal stress and fault:

Rough Faults

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Influence of Orientation of Maximum Principal Compressive Stress, Ψ

Influence of Background Stress, τb

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Increasing τb switches rupture mode from pulse to crack; total slip and extent of plastic strain strain increases.

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γ = 0 (plastic)γ = 10−2.5 (plastic) γ = 10−2 (plastic)

Fluctuations in Rupture Velocity, vr

High Frequency Ground Motion

Rupture speed diminishes when off-fault plasticity is consid-ered. Variability in rupture speed increases as parameters (τb and γ) approach marginal stability boundary at which self-sustaining propagation is just barely possible—conditions at which natural earthquakes are expected to occur. Response becomes highly nonlinear near critical conditions.

Increasing fault roughness decreases both total slip and extent of plastic strain, though general location of plasticity remains same. Roughness modulates amplitude of plastic strain. Ruptures thus experience larger fluctuations in dissipated energy, leading to increased variability in propa-gation speed. Normal stress perturbations as large as initial stress level.

Self-Similar Fractal Fault Surfaces

Eric M. Dunham (edunham@stanford.edu)Department of Geophysics, Stanford University

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1Approximate peak and residual strengths, τp and τr (held constant across all simulations):

Nondimensionalization

plastic dilatancy:

internal friction coefficient chosen to match τp/σ0 on fault:no cohesion: c = 0

Length of state-evolution region at rupture front, approximated using Palmer and Rice [1973] expression assuming exponential weakening with slip:

(good resolution)Relate strength drop, Δτ, to peak slip velocity, Vp, at rupture front by assuming radiation-damping response:

linear taper of fault shear stress to promote unilateral propagation

Nucleation by applying Gaussian-shaped shear stress perturbation on fault, centered at x = 0, at t = 0.

Initial stresses resolved onto rough fault

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Measurements (e.g., LiDAR and profilometer) of fault surfaces [Power and Tullis, 1991; Renard et al., 2006; Sagy et al., 2007] show faults are rough at all scales in a self-similar manner: amplitude-to-wavelength ratio of roughness, γ, is independent of scale of observation. Values of γ in slip direction for natural faults range from γ=10−2 (immature faults) to γ=10−3 (mature faults).

Influence of Amplitude-to-Wavelength Ratio of Roughness, γ [Sagy et al., 2007] [Candela et al., 2008]

fault normal fault parallel

[Shakal et al., 2006]

Synthetic velocity seismograms from rupture closest to stability boundary have comparable levels of high frequency waves to those in actual seismograms (compare with 2004 M 6.0 Parkfield records to left).

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Poor resolution excites waves, eikx+st, having wave-numbers close to Nyquist wavenumber, π/h. Such waves have numerical phase speeds, -Im(s)/k, less than desired cs. Upwind differencing effectively damps these modes at rate -Re(s). Properties of 5th order method used in this work shown below.

dispersion

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