Earthquake Ruptures with Strongly

Rate-Weakening Friction and Off-Fault

Plasticity: 1. Planar Faults

Eric M. Dunham, David Belanger, Lin Cong, and Jeremy E. Kozdon

in preparation for BSSA, draft of March 16, 2010

Abstract

We study dynamic rupture propagation on flat faults using two-dimensionalplane strain models, which feature strongly rate-weakening fault friction (in arate-and-state framework) and off-fault Drucker–Prager plasticity. Account-ing for plasticity bounds stresses near the rupture front and limits slip ve-locities to ∼10 m/s. As originally shown for ruptures in an elastic medium(Zheng and Rice, 1998), a consequence of strongly rate-weakening frictionis the existence of a critical background stress level at which self-sustainingrupture propagation (in the form of self-healing slip pulses) is just barelypossible. At higher background stress levels, ruptures are crack-like. Thisphenomenology remains unchanged when accounting for off-fault plasticity,but the critical stress level is increased. The increase depends on the extentand amount of plastic deformation, which is influenced by the orientation ofthe initial stress field and the proximity of the initial stress state to the yieldsurface.

Introduction

As recent laboratory experiments have shown, the frictional resistance toslip decreases dramatically at coseismic slip rates (Tsutsumi and Shimamoto,1997; Tullis and Goldsby, 2003a,b; Prakash and Yuan, 2004; Hirose and Shi-mamoto, 2005; Beeler et al., 2008). This is referred to as dynamic weakening.

1

One consequence of strongly rate-weakening friction is that for a range of ini-tial stress conditions, ruptures occur as self-healing slip pulses instead of ascracks (Cochard and Madariaga, 1994, 1996; Beeler and Tullis, 1996; Zhengand Rice, 1998; Lapusta and Rice, 2003; Noda et al., 2009).

For uniform prestress conditions and a given friction law, the rupturemode is primarily determined by the background shear stress on the fault,τ b. According to the understressing theory developed by Zheng and Rice(1998) for ruptures in elastic solids, there exists a critical background stresslevel, τpulse, below which ruptures cannot take the form of cracks. No con-clusive statement can be made about rupture mode for τ b > τpulse, but it isseen in numerical experiments that self-sustaining slip pulses occur within anarrow range of τ b around τpulse (Cochard and Madariaga, 1994, 1996; Beelerand Tullis, 1996; Zheng and Rice, 1998; Lapusta and Rice, 2003; Noda et al.,2009). This phenomenology is also observed for shear ruptures in the lab-oratory (Lykotrafitis et al., 2006). It is expected that natural faults aremost likely to host ruptures soon after stresses on the fault first reach theminimum level that permits propagation, provided that nucleation eventsare sufficiently frequent. The occurrence of ruptures at higher stress levels,while certainly possible, becomes increasingly less likely as time progressesand stress levels rise. The probability of rupture as a function of τ b couldpotentially be quantified, given constraints on the rate of nucleation eventsfrom, say, background seismicity, or perhaps from earthquake cycle simula-tions that produce a realistic distribution of event sizes.

Several lines of reasoning suggest that natural earthquakes occur in theself-healing rupture mode. Seismic inversions show that the duration of slipat a particular point on the fault is much shorter than expected from crack-like models (Heaton, 1990). Noda et al. (2009) have also found, in theirsimulations of ruptures with laboratory-based friction law parameters, thatonly pulse-like ruptures exhibit the observed scaling between slip and rupturelength; crack-like ruptures produce about an order of magnitude too muchslip for a given rupture length.

Additional evidence supporting dynamic weakening of natural faults comesfrom measurements of stress levels and heat flow around major faults. Forlaboratory-based parameters, the critical stress level separating pulses andcracks in an elastic medium is ∼0.2–0.3 times the effective normal stress, farless than initial stress levels that are conventionally assumed in rupture mod-eling (e.g., Harris et al., 2009). Furthermore, almost all slip occurs at evenlower stress levels than the already low τ b, thus minimizing the production of

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heat during friction sliding. This offers a possible explanation for the lack ofmeasurable heat flow anomalies around the San Andreas Fault (Brune et al.,1969; Lachenbruch and Sass, 1980; Saffer et al., 2003) and is consistent withdirect measurements and inferences of the stress state surrounding the SanAndreas Fault (Zoback et al., 1987; Hardebeck and Michael, 2004; Hickmanand Zoback, 2004). A more detailed discussion of these issues is given byNoda et al. (2009).

Existing theory and models dealing with strongly rate-weakening frictionlaws have thus far been restricted to ruptures in an elastic medium. But thosemodels predict extreme stress conditions and unreasonably high slip velocitiesnear the rupture front. For example, the simulations by Noda et al. (2009)predict peak slip velocities exceeding 100 m/s and fault-parallel strains oforder 0.1. It is almost certainly the case that prior to these conditions beingreached, the material surrounding the fault will begin deform inelasticallywith associated dissipation of energy.

The purpose of this study is to investigate how rupture dynamics withstrongly rate-weakening friction laws, and in particular the understressingtheory of Zheng and Rice (1998), generalizes when accounting for off-faultinelastic deformation. This is done in the framework of continuum plastic-ity, as has been used recently in a number of earthquake models (Andrews,2005; Duan and Day, 2008; Templeton and Rice, 2008; Viesca et al., 2008).Furthermore, while in this study our focus is on rupture propagation onflat faults, we consider in a companion study (Earthquake Ruptures withStrongly Rate-Weakening Friction and Off-Fault Plasticity: 2. NonplanarFaults; Eric M. Dunham, David Belanger, Lin Cong, and Jeremy E. Koz-don, submitted to BSSA) the influence of fault roughness. When modelingslip on nonplanar surfaces, it is essential to consider plasticity to bound theotherwise unreasonable stress perturbations around bends in the fault sur-face. The present study thus provides the basic framework upon which theinvestigation of rough faults can be built.

Model Description

We study rupture propagation using two-dimensional plane-strain models.The fault surface is the plane y = 0, which obeys a strongly rate-weakeningfriction law encapsulated in a rate-and-state framework. We account forinelastic deformation of the material surrounding the fault using Drucker–

3

Prager viscoplasticity. The medium is assumed to be homogeneous and infi-nite in extent, and is governed by momentum conservation,

ρ∂vi

∂t=

∂σij

∂xj, (1)

for density ρ, particle velocity vi, and stress σij ; and the constitutive response,

∂σij

∂t= Lijkl (ǫkl − ǫp

kl) . (2)

Lijkl is the tensor of elastic moduli, ǫij = (1/2)(∂vi/∂xj + ∂vj/∂xi) is thetotal strain rate, and ǫp

ij is the plastic strain rate (their difference being theelastic strain rate, such that (2) is simply the time derivative of Hooke’s law).Assuming isotropic elastic properties, Lijklǫkl = Kǫkkδij + 2G (ǫij − δij ǫkk/3)

for bulk modulus K and shear modulus G. The S-wave speed is cs =√

G/ρ.

We consider a Poisson solid, for which Poisson’s ratio is ν = 1/4 and theP-wave speed is cp =

√3cs. The equations governing the evolution of plastic

strain will be discussed shortly.

Strongly Velocity-Weakening Fault Friction

The steady state friction coefficient used in this work features the extremevelocity-weakening response seen in recent experiments (Tsutsumi and Shi-mamoto, 1997; Tullis and Goldsby, 2003a,b; Prakash and Yuan, 2004; Hiroseand Shimamoto, 2005; Beeler et al., 2008). The particular form we use is

fss(V ) = fw +fLV (V ) − fw

[1 + (V/Vw)n]1/n, (3)

with slip velocity V , weakening velocity Vw, fully weakened friction coefficientfw, and a conventional low-velocity friction coefficient

fLV (V ) = f0 − (b − a) ln(V/V0). (4)

Here, a is the direct effect parameter, b is the state evolution parameter, andf0 and V0 are the reference friction coefficient and slip velocity, respectively.The form of (3) is such that fss(V ) ≈ fLV (V ) for V ≪ Vw and fss(V ) ≈ fw forV ≫ Vw. Laboratory experiments suggest that Vw ∼ 0.1 m/s and fw ∼ 0.2.The abruptness of the transition between the two limits is controlled by theparameter n; the original flash-heating model (Rice, 1999, 2006; Beeler and

4

Tullis, 2003; Beeler et al., 2008) emerges in the n → ∞ limit. Finite valuesof n smooth the onset of strongly rate-weakening behavior; we prefer, fornumerical accuracy, to have a smooth transition, so we choose n = 8 in thiswork.

Purely rate-weakening friction laws cannot be used since they lead tomathematically ill-posed problems (Rice et al., 2001). Instead, we encapsu-late the steady state response (3) in the experimentally motivated frameworkof rate-and-state friction (in the slip law form (Rice, 1983)):

f(V, Θ) = a ln(V/V0) + Θ (5)

dΘ

dt= −V

L[f(V, Θ) − fss(V )] , (6)

in which Θ is the state variable. The fault strength τ , which is always equalto the shear stress acting on the fault, is

τ = f(V, Θ)σ, (7)

where σ is the normal stress on the fault (taken to be positive in compression).

Off-Fault Inelastic Response

The off-fault material is idealized as a Drucker–Prager elastic-plastic solid,as in recent rupture dynamics studies (Andrews, 2005; Ben-Zion and Shi,2005; Duan and Day, 2008; Templeton and Rice, 2008; Viesca et al., 2008;Ma and Beroza, 2008); we consider both rate-independent plasticity andviscoplasticity. The Drucker–Prager model is closely related to the Mohr–Coulomb model, as discussed by Templeton and Rice (2008). The shearyield stress depends on the mean stress, with sensitivity determined by theinternal friction coefficient of the material. It is also possible to includecohesion, but we neglect it in this study under the assumption that thematerial surrounding the fault has been highly damaged in previous events.The yield function is thus

F (σij) = τ + µσkk/3, (8)

in which τ =√

sijsij/2 is the second invariant of the deviatoric stress tensor

sij = σij−(σkk/3) δij , σkk/3 is the mean stress, and µ is related to the internal

5

friction coefficient. When F (σij) < 0, the material response is elastic. SeeFigure 1a.

A scalar measure of the shear plastic strain rate is the equivalent plastic

strain rate, λ, which is defined as λ =√

2epij e

pij for deviatoric plastic strain

rate epij = ǫp

ij − (ǫpkk/3) δij. The equivalent plastic strain, γp, is thus defined

though λ = dγp/dt. During rate-independent plastic flow, λ is determinedsuch that stresses lie exactly on the yield surface, i.e., F (σij) = 0. Plasticflow is partitioned between the various components of the plastic strain ratetensor by the flow rule

ǫpij = λPij(σij), (9)

with Pij(σij) = sij/(2τ) + (β/3)δij; β determines the degree of plastic dila-tancy, that is, the ratio of volumetric to shear plastic strain.

Under certain stress conditions, a material undergoing plastic deforma-tion may become unstable to shear localization (Rudnicki and Rice, 1975).This is a phenomenon that is commonly observed in nature, and it has beensuggested by Templeton and Rice (2008) as a possible explanation for fre-quently observed auxiliary cracks branching off of the main fault. However,the rate-independent form of Drucker–Prager plasticity is known to be math-ematically ill-posed if conditions permit localization, with the consequencethat numerical solutions of these equations will not converge with mesh re-finement. Andrews (2005) used a numerical viscoplastic-type regularizationalgorithm to obtain solutions free of localization. Others have simply em-ployed the rate-independent form and acknowledged that localization in theirnumerical solutions was limited by the employed grid spacing (Duan and Day,2008; Templeton and Rice, 2008; Viesca et al., 2008).

In our work, we adopt a rigorous formulation of viscoplasticity that isknown to be well-posed mathematically (Perzyna, 1966; Loret and Prevost,1990; Sluys and de Borst, 1992). The formulation permits localization, butlimits the localization process so that numerical solutions converge with meshrefinement. In this model, stresses are permitted to exceed the yield surface,and the additional stress above the yield surface (a scalar measure of which isF (σij) > 0) obeys a viscous rheology. Stresses that exceed the yield conditionrelax back toward the yield surface in the absence of additional loading, andasymptotically flow at a stress level that increases with the applied strainrate. At sufficiently high strain rate, the response is effectively elastic. Moreprecisely, during plastic flow the yield condition F (σij) = 0 is replaced with

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shear strain, γ

time, tη (2G)

(a)

(b)

shea

r st

ress

, sh

ear

stre

ss,

(c)

Figure 1: (a) Drucker–Prager yield condition with no cohesion. The initialstress state is marked with a filled circle. (b) Viscoplastic stress-strain historyat a constant strain rate, γ, for the case of pure shear deformation at constantmean stress. Response is linear elastic until stresses reach the yield stress,−µσ0

kk/3. Stresses then exceed yield; asymptotically, the additional stressabove yield is ηγ for viscosity η. (c) Relaxation of stress to the yield surface,over a time scale ∼ η/G (G is shear modulus), in the absence of additionalstrain.

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ηλ = F (σij) (10)

for viscosity η. Figure 1 illustrates properties of the viscoplastic response.Substituting (9) into (2), and writing the resulting equation along with

(10), gives the complete set of equations that determine the evolution ofstress for a given strain rate history:

∂σij

∂t= Lijkl [ǫkl − 〈λ〉Pkl(σij)] (11)

ηλ = F (σij), (12)

where 〈λ〉 = λ for λ > 0 and zero otherwise (this formulation simply accountsfor the lack of plastic flow when stress are below yield).

Initial Conditions and Model Parameters

We assume a spatially uniform prestress field, σ0ij . We fix σ0

yy = −σ0, thestress component responsible for the normal stress acting on a planar fault,and vary both σ0

xy = τ b and σ0xx. The latter is specified in terms of the angle,

Ψ, between the maximum principal compressive stress and the fault surfacey = 0, defined through the relation

σ0

xx =

[

1 − 2σ0xy

σ0yy tan(2Ψ)

]

σ0

yy. (13)

Following Templeton and Rice (2008), the initial out-of-plane normal stressis taken to be the average of the two in-plane normal stresses: σ0

zz = (σ0xx +

σ0yy)/2. This stress component is only relevant during plastic flow and, for

this particular choice, the Drucker–Prager yield function coincides with theMohr–Coulomb condition (a correspondence that generally ceases once plas-tic flow ensues).

The parameters of the friction law are similar to those used by Nodaet al. (2009), who selected them from a compilation of laboratory data, butwe regard the state-evolution distance L as a free parameter that is selectedto model rupture propagation at a desired scale. These values are given inTable 1. For the steady state friction law (3) with our chosen parameters,the critical background stress τpulse, as defined by Zheng and Rice (1998), isτpulse = 0.2429σ0 (= 30.6059 MPa for σ0 = 126 MPa). The corresponding

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Material Propertiesshear modulus G 32.04 GPa

shear wave speed cs 3.464 km/sPoisson’s ratio ν 0.25

Drucker–Prager internal friction parameter µ sin(arctan(0.7)) = 0.5735Drucker–Prager plastic dilatancy parameter β µ/2 = 0.2867

Drucker–Prager viscosity η (GR0/cs)/10 = 0.2775 GPa sFriction Law Parameters

direct effect parameter a 0.016evolution effect parameter b 0.02

reference slip velocity V0 1 µm/ssteady state friction coefficient at V0 f0 0.7

state evolution distance L 0.2572 mweakening slip velocity Vw 0.17 m/s

fully weakened friction coefficient fw 0.13Initial Conditions

effective normal stress on flat fault σ0 126 MPainitial state variable Θ(t = 0) variable

background shear stress τ b variableOther

characteristic extent of state-evolution region R0 0.3 kmcritical background shear stress τpulse 0.2429σ0 = 30.6059 MPa

Table 1: Physical properties and model parameters.

characteristic slip velocity during steady sliding, V pulse, is that at the inter-section of fss(V )σ0 and the radiation damping line just tangent to this curve:V pulse = 1.5474 m/s.

While the rate-and-state friction law does not have well-defined peak andresidual strengths, τp and τ r, we can estimate them as follows. At the rupturefront, the shear stress is assumed to increase from the background value, τ b,to the peak strength, τp, with negligible state evolution. The direct effectresponse gives

τp ≈ τ b + aσ0 ln (V p/Vini)

= σ0 [a ln (V p/Vini) + Θini] , (14)

9

where Vini and Θini are the initial slip velocity and initial state variable,respectively, and V p is the peak slip velocity at the rupture front. Theresidual strength is approximated as

τ r = fss(Vpulse)σ0, (15)

giving τ r = 0.1864σ0 (= 23.4864 MPa). Next we relate V p to the strengthdrop, τp − τ r, assuming a radiation-damping elastodynamic response:

V p ≈ 2cs(τp − τ r)/G. (16)

As τ b is varied relative to τpulse in the simulations, we keep the static frictioncoefficient, τp/σ0, fixed at 0.7 by varying Vini. The initial velocity field in themedium is piecewise constant (it is discontinuous across y = 0) and solely inthe x-direction; that is, we set vx(t = 0) = ±Vini/2 for the material aboveand below the fault, respectively, and vy(t = 0) = 0. The initial slip velocity,together with the resolved shear and normal stress, is required to satisfy thefriction law (7). That equation is solved for the initial value of the statevariable.

Assuming an exponential slip-weakening process at the rupture front (afairly good approximation for the slip law), we can estimate the fractureenergy as Γ ≈ (τp − τ r)L. By holding τp and τ r fixed as we vary τ b, we alsoroughly preserve the spatial extent of the state-evolution region at the rupturefront in the limit of vanishing rupture velocity, R0, which we approximateusing the formula given by Palmer and Rice (1973):

R0 ≈ 3π

4

GΓ

(τp − τ r)2(17)

≈ 3π

4

GL

τp − τ r; (18)

We nondimensionalize length and time scales using R0 and R0/cs, respec-tively, but we also provide dimensional scales in our plots by choosing L suchthat R0 = 300 m. Particle and slip velocities are normalized by cs∆τ/G and2cs∆τ/G, respectively, where ∆τ = τpulse − τ r.

We select a spatial discretization, quantified by the uniform grid spacing∆x, of ∆x = R0/16 (and in some cases R0/32 or R0/64 as part of refinementtests); this ensures numerical accuracy of about 1% (see insets in Figure 2)and prevents the contamination of our results by spurious oscillations. The

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computational domain, at the lowest resolution, is spanned by 4801 × 2402grid points.

We use a nonzero η in the viscoplastic rheology with the dual objectivesof ensuring well-posed numerical simulations and having a relaxation time,η/G, that is a small fraction of the characteristic wave transit time across thestate-evolution region, R0/cs; specifically, we choose η = (GR0/cs)/10. Thismeans that plasticity is important even during the rapid weakening processat the rupture front.

Ruptures are nucleated by instantaneously applying, at t = 0, a Gaussian-shaped shear stress perturbation on the fault centered at x = 0. The ampli-tude of the nucleating perturbation is 2τ pulse; its width (the standard devia-tion of the Gaussian profile) is R0. Ruptures are made unilateral by makingthe fault velocity-strengthening some distance to the left of the nucleationzone. Specifically, we increase a linearly with distance from 0.016 to 0.024in the region −40R0 < x < −20R0 and set a = 0.024 for x < −40R0; Vw issimilarly increased by 5 m/s over the same region.

Numerical Method

The governing equations, (1), (11), and (12), form a system of first-order par-tial differential equations with an algebraic constraint; for uniform materialproperties they can be written as

∂q

∂t+

∂Gx(q)

∂x+

∂Gy(q)

∂y= 〈λ〉S(q) (19)

ηλ = F (q). (20)

Here, q is a vector containing velocities and stresses. The “fluxes” Gi (socalled because ∂Gx(q)/∂x + ∂Gy(q)/∂y is the divergence of the flux) arefunctions of material properties and q, and 〈λ〉S(q) is a source/sink termthat is nonzero only during plastic flow.

The domain is discretized with a uniform mesh, and all components ofvelocity and stress are defined at each grid point (i.e., we do not use a stag-gered grid). Having all of the stress components at the location greatlysimplifies integration of the plasticity equations. The spatial derivatives areapproximated using a summation-by-parts finite difference method (Kreissand Scherer, 1974, 1977; Strand, 1994) and boundary conditions are enforced

11

weakly using the simultaneous approximation term technique (Carpenteret al., 1994). Artificial dissipation is added in the form of upwind differ-entiation operators (Mattsson et al., 2004) to control spurious oscillationswithout compromising the high-order accuracy of the difference scheme (allplots in this work are generated using unfiltered simulation data). Thismethod is provably stable, even with nonlinear rate-and-state friction laws,and the scheme is high-order accurate (Kozdon et al., 2009). Specifically, themethod is fifth order accurate in the interior and third order accurate near theboundaries; the entire scheme is globally fourth order accurate. A detaileddescription of the method, as well as proofs of stability and accuracy, is givenby J. E. Kozdon, E. M. Dunham, and J. Nordstrom, Higher-order treatmentof nonlinear boundary conditions for elastodynamics using summation-by-parts operators, in preparation.

Time stepping equations (19) and (20) presents a particular challengedue to the algebraic constraint. For simplicity of notation, we write theseequations, after spatial discretization and incorporation of the boundary con-ditions through penalty terms, as

dQ

dt+ E(Q) = 〈Λ〉S(Q) (21)

ηΛ = F (Q). (22)

where Q is now a vector containing the values of velocity and stress atall grid points and similarly for Λ. The equations are identified as a sys-tem of differential-algebraic equations (DAE). The particular form for rate-independent plasticity, which arises when η = 0, is that of an index 2 DAE(Hairer and Wanner, 2004). The choice of η > 0 can be viewed mathe-matically as a regularization procedure (Ascher and Lin, 1997) that reducesthe index of the DAE to 1; this permits (22) to be solved explicitly for thealgebraic variable Λ. Substituting the result in (21) yields the ordinary dif-ferential equation (ODE)

dQ

dt+ E(Q) =

〈F (Q)〉η

S(Q). (23)

The η → 0 limit is a singular one and the ODE becomes increasingly stiffin this limit, necessitating the use of at least partially implicit time-steppingmethods. Such methods are, of course, required for the rate-independentequations; otherwise the constraint equation could not be strictly enforced.

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However, we very much desire to keep the evaluation of the elastic term,E(Q), explicit, as this involves calculating spatial derivatives and would be-come prohibitively costly if done implicitly.

There are a variety of approaches to solve equations (21) and (22). Wefocus entirely on one-step procedures, which update the fields from time tn

to tn+1 = tn +∆t, namely, from Qn to Qn+1. The first, and perhaps simplest,method to do this is based on a first-order operator-splitting procedure. Inthis method, one first solves the elastic problem

dQ

dt+ E(Q) = 0 (24)

over the time interval tn to tn+1 to update Qn to Q∗. This is done mostefficiently using an explicit time-stepping method. We use both third- andfourth-order low-storage Runge–Kutta methods (Williamson, 1980; Kennedyet al., 2000). “Low-storage” refers to the fact that only two storage unitsare required for each field; these are repeatedly overwritten at each inter-nal Runge–Kutta stage. This substantially reduces memory requirements ascompared to traditional Runge–Kutta methods.

We remark that it is necessary to use Runge–Kutta methods with at leastthree internal stages; this is the minimum that is required to have a stabilityregion that contains the imaginary axis (e.g., Kreiss and Scherer, 1992).This is needed since several of the eigenvalues of the Jacobian of the ODE,−∂E/∂Q, are almost entirely imaginary (as is typical for wave-propagationproblems with minimal dissipation).

Next, using Q∗ as the initial condition on Q at time tn, one solves

dQ

dt= 〈Λ〉S(Q) (25)

ηΛ = F (Q) (26)

over the time interval tn to tn+1 to obtain Qn+1. Note that this equationinvolves only field quantities defined at a particular grid point, which greatlysimplifies the solution. The simplest numerical method to solve this problemis the first-order implicit Euler discretization:

Qn − Q∗

∆t= 〈Λn+1〉S

(

Qn+1)

(27)

ηΛn+1 = F(

Qn+1)

, (28)

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which requires the solution of these coupled nonlinear equations for Qn+1

and Λn+1. For the Drucker–Prager model, this is readily done in closed form,permitting the efficient updating of the fields.

An alternative to the operator-splitting procedure that retains high-orderaccuracy during either elastic or plastic response (but has overall first-orderaccuracy due to the abrupt switching between elastic and plastic behavior),is to use coupled implicit-explicit (or additive) Runge–Kutta methods. Forrate-independent plasticity, we have found that a highly efficient choice isthe third-order half-explicit Runge–Kutta method (Brasey and Hairer, 1993),which is designed specifically for index 2 DAEs. We have also successfullyused several of the high-order methods proposed by Kennedy and Carpenter(2003), but they are not written in a low-storage form and hence increasememory requirements.

We have found that for the grid spacings we use in this study, the predomi-nant source of error comes from approximating the spatial derivatives of fieldsin the term E(Q), and that it makes little difference which time-integrationmethod we use. Hence, we default to the simplest operator-splitting methodin this work, but we do note that on certain test problems at high resolution,the coupled implicit-explicit methods give superior performance.

Ruptures on Flat Faults

In simulations of rupture propagation in linear elastic solids with stronglyrate-weakening friction laws (e.g., Noda et al., 2009), slip velocities canbecome extremely large (∼ 300 m/s) at the rupture front; these are associ-ated with fault-parallel strains of ∼ 0.1 and unreasonably large fault-parallelstresses. Accounting for off-fault plasticity limits these stresses, and peak slipvelocities drop to ∼ 10 m/s, consistent with the estimates of Sleep (2010).Figure 2 shows representative time histories of slip velocity and shear stress ata point on the fault for both elastic and elastic-viscoplastic off-fault response.

We next explore the role of the prestress orientation, Ψ, and backgroundstress, τ b, on the distribution of plastic strain and rupture mode. Figure3 confirms the findings of Templeton and Rice (2008), who studied rupturepropagation with slip-weakening friction; namely, that unless Ψ is quite low,plastic strain occurs on the extensional side of the fault (specifically, the sideof the fault for which the direction of fault-parallel displacement is oppositeto the propagation direction). This is well explained in terms of properties of

14

Δx = R0/64

Δx = R0/16

elasticviscoplastic

viscoplastic

Δx = R0/64

Δx = R0/16

viscoplastic

viscoplastic

elastic

t−trup(x) (s) t−t

rup(x) (s)

V (

m/s

)

τ/σ0

τ/σ0

V (

m/s

)

t−trup(x) (s) t−t

rup(x) (s)

(a) (b)

V/(2

c sΔτ/

G)

0

3.24

6.47

9.71

12.95

−0.5 0 0.5 1 1.5 20

5

10

15

20

−0.1 0 0.1 0.20

5

10

−0.5 0 0.5 1 1.5 20.1

0.2

0.3

0.4

0.5

0.6

0.7

−0.1 0 0.1 0.20.1

0.2

0.3

0.4

0.5

0.6

0.7

Figure 2: Time histories of (a) slip velocity and (b) shear stress for self-healing slip pulses in elastic and elastic-viscoplastic solids, at x = 87.4R0

(= 26.25 km), for Ψ = 50◦ and τ b = 0.2857σ0. Time is relative to rupturefront arrival time trup(x) (defined as the time at which slip first exceedsthe state evolution distance). Insets compare elastic-viscoplastic solutions atdifferent levels of numerical resolution; the L2 (root-mean-square) error ofthe lower resolution solution is about 1%.

the stress field surrounding a propagating rupture, determined by both theprestress and the stress perturbations due to slip. Extensive discussions ofthis can be found in Poliakov et al. (2002), Rice et al. (2005), and Templetonand Rice (2008).

As τ b is increased, self-sustaining ruptures are first seen for τ b slightlylarger than τpulse; they take the form of slip pulses. At higher levels of τ b,the rupture mode is expected to switch to crack-like. This transition doesoccur, but is preceded by another form of slip pulse that is generated inresponse to arrest waves triggered by the forced termination of the rupturefront propagating in the −x-direction as it enters the velocity-strengtheningportion of the fault. This mechanism for generating pulse-like ruptures iswell known (Johnson, 1990). These rupture modes are illustrated in Figure4.

Finally, we quantify how the critical stress level required for rupture prop-agation is altered by plasticity. This is done by varying Ψ and determining,for each Ψ, the minimum τ b (to an accuracy of about 0.5 MPa) at whichself-sustaining propagation is first observed. The results, presented in Figure5, illustrate that the critical stress level increases with Ψ, particularly for

15

2.0

1.0

0.0

γp

×10−3

Ψ=10°

τb/σ0=0.25400.00.10.2

−0.1−0.2

0 10 20 30 40 50 60−10x (km)

y (km)

x/R00 100 20050 150

0.5

−0.5

0.0 y/R0

0.00.10.2

−0.1−0.2

0 10 20 30 40 50 60−10x (km)

y (km)

x/R00 100 20050 150

0.5

−0.5

0.0 y/R0

2.0

1.0

0.0

γp

×10−3

(a)

(b)

Ψ=30°

τb/σ0=0.2540

Figure 3: Distribution of plastic strain, γp, for (a) Ψ = 10◦ and (b) Ψ = 30◦,at time t = 216.5R0/cs (= 18.75 s). Except for very small Ψ, plastic strainoccurs exclusively in the extensional quadrants (slip is right-lateral). Theextent of plastic strain increases for Ψ > 30◦ (not shown). Plastic straincan be normalized by τp/2G = 1.38 × 10−3. For both cases, the backgroundstress, τ b, is slightly above that required for self-sustaining slip-pulse rupturepropagation. Note factor of 20 exaggeration in y-direction.

Ψ > 40◦. This can be understood as a combination of two effects.First, varying Ψ alters the proximity of the prestress state to the yield sur-

face. Following Templeton and Rice (2008), we quantify this as the closeness-to-failure ratio τ 0/ (−µσ0

kk), as illustrated graphically in Figure 1a. A plotof this ratio for σ0

xy = τpulse is shown in Figure 6. For values of Ψ below∼10◦ or above ∼70◦, the prestress state violates the yield condition; suchprestress states cannot exist. Numerical experiments indicate that as Ψ ap-proaches ∼70◦, the extent of the region experiencing plastic strain increasesgreatly. This is because only slight stress perturbations (of the proper sign,of course) are required to induce plastic flow. The associated increase ofenergy dissipation during plastic flow explains why τ b must be substantiallyhigher for self-sustaining rupture propagation at high values of Ψ. By con-sidering the prestress state alone, one might expect this effect to also occurfor very low values of Ψ; however, this is not seen. This is because the stressperturbations due to slip must also be taken into account, and for these lowvalues of Ψ, they are far less effective at driving the stress state closer to theyield condition. This is closely related to why the location of plastic strainswitches from the extensional side of the fault to the compressional side at

16

τb/σ0 =

0.4127

0.3492

0.2857

0.2698

x (km)

slip

, δ (

m)

Ψ = 50˚

−30 −15 0 15 30 45 600

10

20

30

0

38.88

77.76

116.6−100 −50 0 50 100 150 200

x/R0

δ/L

(a)

(b)

−9 −6 −3 0 30

1

2

3×10

−3

y (km)

pla

stic

str

ain, γp

y/R0

−10−20 100

γp/(τp/2

G)

0

0.73

1.45

2.18Ψ = 50˚τb/σ0 =

0.4127

0.3492

0.2857

x = 30 km

= 100R0

Figure 4: (a) Profiles of slip with time interval 17.32R0/cs (= 1.5 s) forΨ = 50◦, illustrating the transition in rupture mode as the background stress,τ b, is increased. Self-sustaining propagation is first possible around τ b/σ0 ≈0.28; for values slightly above this, ruptures take the form of self-healingslip pulses (τ b/σ0 = 0.2857 and 0.3492 in figure). At higher τ b/σ0, ruptureswould be crack-like for uniform friction-law parameters, but arrest wavesfrom the cessation of slip in the velocity-strengthening region (x < −9 km)cause pulse-like ruptures (τ b/σ0 = 0.4127 in figure). At even higher τ b/σ0,ruptures become cracks (not shown). (b) Cross-sectional profile of plasticstrain at x = 100R0 (= 30 km); the spatial extent of plastic strain is roughlyproportional to slip.

17

τpulse/σ0

τb /σ0

arresting

slip pulsegrowing

slip pulse

0.24

0.26

0.28

0.3

0.32

0.34

angle of maximum compression, Ψ0° 20° 40° 60° 80°

Figure 5: Minimum background stress, τ b, required for self-sustaining rupturepropagation as a function of prestress orientation, Ψ.

0

0.5

1

angle of maximum compression, Ψ0° 20° 40° 60° 80°

Figure 6: Proximity of prestress state to yield surface, as defined by Tem-pleton and Rice (2008); shown for σ0

xy = τpulse. Prestress states having Ψless than ∼10◦ or greater than ∼70◦ violate the yield condition and are thusprecluded.

low values of Ψ.As found by Templeton and Rice (2008) and Viesca et al. (2008) in their

studies with slip-weakening friction, plastic strain accumulates almost instan-taneously across a discontinuity surface emanating from the rupture front.This is also true for our models with strongly velocity-weakening rate-and-state friction, though the jump is smoothed by the finite viscoplastic relax-ation time.

18

Conclusions

Accounting for off-fault plasticity influences rupture dynamics on stronglyrate-weakening faults in several ways. Stresses and slip velocities are boundedto reasonable values at the rupture front, and there is an associated increasein the amount of energy dissipated during propagation. The location andextent of plastic strain is consistent with that found by Templeton and Rice(2008), in that it occurs almost exclusively on the extensional side of thefault except when the maximum principal compressive stress is inclined atvery shallow angles to the fault surface.

Strongly rate-weakening friction laws give rise to self-healing slip pulsesin elastic solids for background stresses around a critical level (Zheng andRice, 1998). We find that this remains the case when plasticity is takeninto account, but the critical stress levels are increased by plasticity. Themagnitude of the increase depends strongly on the prestress orientation, inparticular when the angle between the maximum principal compressive stressand the fault surface exceeds 50–60◦.

This study is the first of a two-part exploration of rupture dynamicson strongly rate-weakening faults with off-fault plasticity. The second part(Earthquake Ruptures with Strongly Rate-Weakening Friction and Off-FaultPlasticity: 2. Nonplanar Faults; Eric M. Dunham, David Belanger, Lin Cong,and Jeremy E. Kozdon, submitted to BSSA) extends this work to nonplanarfaults, with a particular emphasis on understanding the connection betweenfault roughness and high-frequency ground motion.

Data and Resources

The numerical simulations were conducted at the Stanford Center for Com-putational Earth and Environmental Science with computational support byDennis Michael and Robert Clapp.

Acknowledgments

This work was initiated when EMD, DB, and LC were at Harvard Uni-versity. Work there was supported by the Southern California EarthquakeCenter (SCEC) as funded by Cooperative Agreements NSF EAR-0106924

19

and USGS 02HQAG0008, the Harvard Faculty Aide program for undergrad-uate research (under the supervision of Prof. James R. Rice), matchingfunds from Harvard’s Department of Earth and Planetary Sciences, and anaward from the Highbridge Undergraduate Research fund through Harvard’sDepartment of Mathematics to DB. Later work at Stanford by EMD andJEK was supported by NSF-EAR award 0910574 and by SCEC (SCEC con-tribution number XXXX). Discussions with James R. Rice, Elizabeth L.Templeton, and Robert C. Viesca throughout this project were invaluable.

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Department of GeophysicsStanford University397 Panama MallStanford, CA 94305edunham@stanford.edujkozdon@stanford.edu(E.M.D., J.E.K.)

BBN Technologies10 Moulton StreetCambridge, MA 02138dbelange@bbn.com(D.B.)

Stanford Graduate School of Business518 Memorial Way Stanford, CA 94305wtsung@stanford.edu(L.C.)

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