PUBLICATIONS. Journal of Geophysical Research: Solid Earth. A friction to flow constitutive law and its application to a 2-D modeling of earthquakes

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1 PUBLICATIONS Journal of Geophysical Research: Solid Earth RESEARCH ARTICLE Key Points: We propose a constitutive law describing transition from friction to flow The law merges a strength profile and a velocity-dependent fault model The law was applied to a 2-D earthquake sequence modeling across a lithosphere Correspondence to: T. Shimamoto, shima_kyoto@yahoo.co.jp Citation: Shimamoto, T., and H. Noda (2014), A friction to flow constitutive law and its application to a 2-D modeling of earthquakes, J. Geophys. Res. Solid Earth, 119, , doi: / 2014JB0111. Received 4 APR 2014 Accepted 7 OCT 2014 Accepted article online 12 OCT 2014 Published online 15 NOV 2014 Corrected 19 DEC 2014 This article was corrected on 19 DEC See the end of the full text for details. A friction to flow constitutive law and its application to a 2-D modeling of earthquakes Toshihiko Shimamoto 1 and Hiroyuki Noda 2 1 State Key Laboratory of Earthquake Dynamics, Institute of Geology, China Earthquake Administration, Beijing, China, 2 Department of Mathematical Science and Advanced Technology, Japan Agency for Marine-Earth Science and Technology, Yokohama, Japan Abstract Establishment of a constitutive law from friction to high-temperature plastic flow has long been a challenging task for solving problems such as modeling earthquakes and plate interactions. Here we propose an empirical constitutive law that describes this transitional behavior using only friction and flow parameters, with good agreements with experimental data on halite shear zones. The law predicts steady state and transient behaviors, including the dependence of the shear resistance of fault on slip rate, effective normal stress, and temperature. It also predicts a change in velocity weakening to velocity strengthening with increasing temperature, similar to the changes recognized for quartz and granite gouge under hydrothermal conditions. A slight deviation from the steady state friction law due to the involvement of plastic deformation can cause a large change in the velocity dependence. We solved seismic cycles of a fault across the lithosphere with the law using a 2-D spectral boundary integral equation method, revealing dynamic rupture extending into the aseismic zone and rich evolution of interseismic creep including slow slip prior to earthquakes. Seismic slip followed by creep is consistent with natural pseudotachylytes overprinted with mylonitic deformation. Overall fault behaviors during earthquake cycles are insensitive to transient flow parameters. The friction-to-flow law merges Christmas tree strength profiles of the lithosphere and rate dependency fault models used for earthquake modeling on a unified basis. Strength profiles were drawn assuming a strain rate for the flow regime, but we emphasize that stress distribution evolves reflecting the fault behavior. A fault zone model was updated based on the earthquake modeling. 1. Introduction There are two common ways of characterizing the lithosphere or fault rheology across it (Figure 1). One is a strength profile consisting of the frictional and flow strengths (dashed line in Figure 1a), sometimes referred to as Christmas tree, which has been used widely to discuss thickness and internal structures of the lithosphere [e.g., Sibson, 1977, 1982; Bird, 1978; Goetze and Evans, 1979]. Conceptual fault models have been proposed in association with similar revised strength profiles [Scholz, 1988; Shimamoto, 1989; Kawamoto and Shimamoto, 1998]. However, such strength profiles cannot be directly used in modeling large earthquakes since they miss transient properties. After rate-and-state friction laws [Dieterich, 1979; Ruina, 1983] were proposed, a rate dependency fault model [Tse and Rice, 1986; Scholz, 1998] was proposed for seismogenicaseismic transition (e.g., Figure 1b), and it has been highly successful in modeling the entire seismic cycles and their sequences [e.g., Lapusta and Rice, 2003; Shibazaki and Shimamoto, 2007]. However, such friction models did not include the flow property of the lower part of the lithosphere. The two approaches are attempting to describe the same lithosphere properties and should be described by a unified law. Then how can the two models be reconciled? A linear combination of friction and flow laws was proposed as an empirical law [Reinen et al., 1992], but it cannot describe the transitional behavior of a halite shear zone from friction to plastic flow [Reinen et al., 1992, Figure 3; Noda and Shimamoto, 2010, Figure 4]. On the other hand, Shimamoto [2004] and Shimamoto and Noda [2010] proposed an empirical friction to flow law that smoothly connects those behaviors. A fault stability analysis with the law was reported previously by Noda and Shimamoto [2012]. This paper demonstrates that the law describes the experimental data for the brittle-plastic transition of halite shear zones very well, that the law can describe the strength profile and the rate dependency of a fault on a unified basis, and that the law can be used in modeling earthquake sequences on a fault across the lithosphere. SHIMAMOTO AND NODA American Geophysical Union. All Rights Reserved. 8089

2 Figure 1. Two widely used lithosphere models. (a) The strength profile and (b) the rate dependency model of frictional properties. Frictional and flow strengths in Figure 1a were drawn using a friction coefficient f of 0.6 and quartzite flow law [Hirth et al., 2001] with an assumed stain rate of (1/3) s 1 and a geothermal gradient of 25 C/km. The friction to flow law gives strength profiles consisting of blue and red curves for frictional and plastic regimes, respectively, and black curves for the transitional regime. Thick and thin curves correspond to hydrostatic and twice hydrostatic pore pressures, respectively. The stippled region in Figure 1b with negative (a b) is a potentially unstable, velocity-weakening regime. 2. A Friction to Flow Constitutive Law Our friction to flow law is given by τ=τ flow ¼ tanhðτ friction =τ flow Þ ¼ tanhðfσ=τ flow Þ (1) where τ is the shear stress or shear resistance on a fault, τ friction is friction as determined by a friction law, f is a friction coefficient, σ is an effective normal stress (normal stress minus pore fluid pressure), and τ flow is a flow stress given by a flow law. At shallow depths σ is small and τ flow is very large since temperature is low, then fσ/τ flow << 1andtanh(fσ/τ flow ) fσ/τ flow,andafrictionlaw(τ τ friction = fσ ) is recovered. Whereas σ is very large and τ flow is small at greater depths (i.e., fσ/τ flow >> 1 and tanh(μσ/τ flow ) 1), and equation (1) gives a flow law (τ τ flow ). Halite (NaCl) is still the only material for which a complete transition from friction to fully plastic flow was produced by shearing experiments, including the effect of normal stress, slip rate (or shear strain rate in the flow regime), and temperature though restricted to steady state [Shimamoto, 1986; Kawamoto and Shimamoto, 1997, 1998]. A complete transition from frictional behavior to fully plastic shearing deformation was produced by shearing experiments on halite using a triaxial apparatus, at confining pressures to 2 MPa and slip rates ranging from to 300 μm/s [Shimamoto, 1986]. The work was extended to include the effect of temperature using a high-temperature biaxial friction apparatus [Kawamoto and Shimamoto, 1997], and the results from this work were reproduced in Figure 2 for comparisons with the friction-to-flow law. A strength profile is obtained when T is increased in proportion to σ like a geotherm (Figure 2a). At constant temperatures, steady state shear resistance τ ss is almost proportional to σ with f ~ 0.6, it deviates from the linear dependency with increasing σ, and pressure-insensitive fully plastic flow is attained at temperatures of 3 C and 0 C (Figure 2b). The curves across the friction to flow transition are nearly identical to hyperbolic tangent curves, and this motivated us to propose equation (1). The rate dependency of τ ss was studied at several (T, σ) conditions along the horizontal axes of Figure 2a, and results are shown in Figures 2c and 2d with only the temperature values labeled. A power law flow law holds at pressure-insensitive fully plastic flow regime [Kawamoto and Shimamoto, 1997] (Figure 2c), whereas stick slip occurred in association with velocity-weakening behaviors at temperatures below 200 C (Figure 2d). Velocity strengthening at slow slip rates changes to velocity weakening with increasing V at T = 200 C near the peak friction. SHIMAMOTO AND NODA American Geophysical Union. All Rights Reserved. 8090

3 Journal of Geophysical Research: Solid Earth Figure 2. Friction-to-flow law compared with the steady state shear resistance τ ss of halite [Kawamoto and Shimamoto, 1997]. (a) Shear stress plotted against the normal stress and temperature when temperature was increased linearly with increasing normal stress, (b) shear stress versus normal stress at four temperatures, (c) shear stress versus shear strain rate in the pressureinsensitive fully plastic regime, and (d) rate dependency of τ ss at several temperatures. Solid curves exhibit the best fit curves to the whole data, using the friction to flow law with a least squares method. Slip rate for Figures 2a and 2b was 3 μm/s. Cross symbols connected with vertical bars in Figure 2d indicate ranges of shear stress during stick slip. The results in Figure 2 illuminate a full spectrum of mechanical properties across an analogue lithosphere, not only in terms of strength profile but also in terms of the temperature and rate dependencies. To fit the friction to flow law to the data, we used a logarithmic steady state friction law of a form [Dieterich, 1979; Ruina, 1983] f ¼ f ss ðv Þ ¼ f 0 þ ða bþlnðv=v 0 Þ; (2) where a b represents the rate dependency of friction and f0 is the steady state friction coefficient at a reference slip rate V0 (selected here as 1 μm/s). We also used a power law for the plastic flow τ flow ¼ τ flow ss ¼ ½ð γ =AÞexpðQ=RT Þ 1=n ¼ τ r ½ðV=wC ÞexpðQ=RT Þ 1=n (3) 1 n with a steady state flow stress τ flow ss, a preexponential factor A having a dimension of [time stress ], an activation energy Q, the gas constant R, an absolute temperature T, and a stress exponent n. For the simplicity in the dimensional consistency, we used an equation on the right side with an arbitrary reference shear stress τ r, a constant C, a slip rate V and shear zone width w (C = Aτ rn depends on the selection of τ r). Then the friction-to-flow law for the steady state is given by τ ss ¼ τ flow ss tanhðf ss σ=τ flow ss Þ; (4) which is a function of T, V, effective normal stress σ, and the above mentioned parameters if the shear zone thickness w is known; w was 0.7 mm in the experiments used here [Kawamoto and Shimamoto, 1997]. All data in Figure 2 were fit with equations (2) (4) by solving a nonlinear least squares problem by LevenbergMarquardt method [e.g., Levenberg, 1944; Marquardt, 1963], and the best fit curves are shown as solid lines in SHIMAMOTO AND NODA American Geophysical Union. All Rights Reserved. 8091

4 Table 1. Parameters and Boundary Conditions Used in Our Lithosphere Model and 2-D Numerical Simulations of Earthquake Sequences on a Strike-Slip Fault (z Is a Depth in km) Quantities [References] Symbols Values Frictional Properties Reference friction f Direct effect [Rice et al., 2001] a T Rate dependency [Blanpied et al., 1998] a b (1 z/3 km) at z < 6km at z > 6km State-evolution distance L 5mm Flow Parameters Reference stress τ r 1MPa Steady state power [Hirth et al., 2001] n 4 Instantaneous power [Noda and Shimamoto, 2010; Hirth et al., 2001] m 4/0.6 Activation energy [Hirth et al., 2001] Q 135 kj mol 1 Preexponential factor [Hirth et al., 2001] C s 1 MPa 1 State-evolution strain [Noda and Shimamoto, 2010] γ c 0.3 Shear zone thickness w 300 m Elastodynamic Parameters S wave speed [Harris et al., 2009] c s 3464 m s 1 Rock density [Harris et al., 2009] ρ 26 kg m 3 Boundary Condition Acceleration of gravity g 9.8 m s 2 Effective normal stress σ 5MPa+ρgz P p Pore pressure P p 1000 kg/m 3 gz (Hydrostatic) 2000 kg/m 3 gz (Twice hydrostatic) Absolute temperature T K + z 25Kkm 1 Plate or fault velocity V pl 10 9 ms 1 Depth at which V = V pl km Interval of mirrors.96 km Figure 2. The best fit curves slightly deviate from some experimental data (e.g., shear stress versus strain rate curve at 0 C in Figure 2c, and shear stress versus slip rate curves at C and 100 C in Figure 2d). However, the overall trends are captured surprisingly well with a simple constitutive law with constants: f 0 = 0.56 ± 0.016, a b = ± , log 10 (A/1 (MPa) n s 1 ) = 2.14 ± 0.73, n =7.±0.75,Q =138±23(kJ)(mol) 1 (the errors are standard errors). Rate dependency of friction a b determined from data at C and 100 C in Figure 2d gives a b of ± Kawamoto and Shimamoto [1997] report f 0 =0.6,A = (MPa) n s 1, n =7.6,andQ = 179 (kj) (mol) 1 that were determined separately for the frictional and pressure-insensitive fully plastic regimes. This n value is high for power law creep [e.g., Nicolas and Poirier, 1976] probably because the plastic deformation is in the low-temperature end of the power law creep or between the exponential and power law regimes. Note that plastic flow data at room temperature in Shimamoto [1986] were fit by an exponential law or by power law with n ~17. Compared with this shearing deformation at room temperature, the deformation at elevated temperature in Figure 2 is close to the deformation in the power law regime. Our best fit valuesforf 0, a b, n agree well with the reported values, our Q value is somewhat smaller than the originally reported value, and our A value has a large error and is considerably different from the original values. The shearing experiments focused on the transitional behavior from friction to plastic flow, and the reported data are not exhaustive in the frictional and fully plastic regimes. This is probably a reason that some parameters such as A were not recovered well from the fitting with the friction-to-flow law. But we consider that the overall agreement is reasonable and that equation (4) describes the transitional behavior from friction to plastic flow remarkably well. 3. A Lithosphere Model With a Friction to Flow Law We now apply our friction to flow law to a simple lithosphere with f 0 = 0.6 with V 0 =10 9 m/s and quartzite flow law [Hirth et al., 2001] to represent a continental crust (Table 1 gives a list of all parameters used in our lithosphere model, including references). Values of the frictional constitutive parameters are those typically used in earthquake modeling [Blanpied et al., 1998; Lapusta et al., 2000; Rice et al., 2001; Harris et al., 2009]. SHIMAMOTO AND NODA American Geophysical Union. All Rights Reserved. 8092

5 Figure 3. (a and b) The rate dependency of the steady state shear resistance τ ss with respect to the natural logarithm of slip rate V, as expected from the friction to flow law. The τ ss is normalized with an effective normal stress σ in Figure 3a. State-evolution distance L was selected as 5 mm to make the earthquake simulation numerically tractable (see section 5 below). The preexponential factor A in equation (3) for quartzite is given by A ¼ Cf H2Oτ r n (5) where f H2O is the water fugacity [Hirth et al., 2001]. We calculated f H2O from pore pressure using the equation of state of water [Pitzner and Sterner, 1994]. The width of shear zone w in equation (3) is poorly constrained in connecting frictional and flow properties. But we assumed that w = 300 m because major ductile shear zones (mylonite zones) are typically several hundreds of meters wide and because a very small w causes a too wide seismogenic zone for crustal earthquakes [Sibson, 1980]. Selection of transient flow parameters is commented in the next section. Figures 1a gives strength profiles for hydrostatic pore pressure P p (thick solid curve) and twice hydrostatic P p (thin solid curve), as calculated using equation (4) with parameters in Table 1. The curves clearly reveal transitional regimes shown in black curves with a peak strength at depths of about 17 km and 20 km, between blue and red curves for the friction and flow regimes, respectively (Figure 1a). Throughout this paper we define the transitional regime such that the shear resistance τ is smaller than shear stress τ friction predicted from a friction law and the shear stress τ flow determined by a flow law by more than 1%; that is, τ is smaller than (0.99 τ friction ) and (0.99 τ flow ) in the transitional regime. The strength profile is similar to those in fault models [Scholz, 1988; Shimamoto, 1989; Kawamoto and Shimamoto, 1998]. A high P p reduces the amplitude of the peak strength and increases its depth. A small difference in the flow stress between the hydrostatic and twice hydrostatic P p in Figure 1a is due to the effect of pore pressure on A (we assumed that water pressure is equal to P p in calculating the water fugacity in equation (5)). We now illustrate how the friction to flow law predicts the rate dependency of τ ss, using an example in Figure 3 where derivatives of τ ss /σ and τ ss with respect to ln(v) are shown in Figures 3a and 3b, respectively. The vertical axis of Figure 3a gives a b value in the rate-and-state friction law, equation (2). The rate dependency of friction is assumed to change along the blue dashed line (velocity weakening below 3 km to a depth of km). The rate dependency for the fully plastic flow with a power law is derived from equation (3) as follows: dτ flow ss =dlnðvþ ¼ τ flow ss =n (6) where τ flow ss and n denote steady state flow stress and stress exponent in the flow law, respectively. Equation (6) is shown by a series of red dashed curves, and the rate dependency in the transitional regime is shown by black lines connecting the blue and red curves. The frictional regime, shown by blue solid lines, markedly expands with increasing V. Below the frictional regime, the rate dependency sharply increases toward peaks in the transitional regime. Such a change in the rate dependency from the friction to transitional SHIMAMOTO AND NODA American Geophysical Union. All Rights Reserved. 8093

6 Figure 4. Transient response (a) in friction f, and (b and c) in flow stress τ flow of faults with rate-and-state friction and flow properties, respectively, upon an e-fold step increase in the slip rate. Changes in natural logarithm of τ flow are plotted against shear strain γ and slip in Figures 4b and 4c, respectively. regimes is very similar to experimental results for granite under hydrothermal conditions [Blanpied et al., 1995], which have given a basis for the rate dependency model for modeling earthquakes (e.g., Figure 1b). But in such models the increase in the rate dependency was regarded as a change in the frictional parameters, whereas our friction to flow law indicates that the sharp rise in the rate dependency is due to the involvement of plastic flow. Fitting the experimental data with a friction law with multiple state variables would be possible [e.g., Blanpied et al., 1998], but the ultimate transition to flow at great depths is not elucidated with such approaches. Our friction to flow law, equation (1), merges to a friction law on the low-temperature/low-pressure side and to a flow law on the high-temperature/ high-pressure side, with a smooth connection between the two. Thus, the law is bounded by the friction and flow laws, and the prediction from our law cannot be totally wrong unless the transitional behavior deviates from a smooth connection. Our calculated result on the rate dependency of the steady state shear resistance in Figure 3a agrees with the data on granite and quartz gouge by Blanpied et al. [1995] and with the data on gabbro gouge by He et al. [2007], although a complete fitting of data with the friction to flow cannot be done because flow parameters of their samples are not determined. However, a comparison of Figure 10 in Blanpied et al. [1995] with our Figure 3a suggests that theratedependencyofgraniteandquartz gouge in their experiments changes from the rate dependency in the frictional regime toward the large rate dependency (or a high (a b) value) near the peak strength. Those experimental results on granite, quartz, and gabbro gouge suggest that nothing unusual happens in the transitional regime that deviates completely from a smooth transition from friction to flow. 4. Full Formulation of the Friction to Flow Constitutive Law With Transient Properties We now expand the friction to flow law to include transient friction and flow behaviors. For the friction law, we use a rate-and-state friction law with the aging law [Dieterich, 1979; Ruina, 1983], but other evolution laws can be used as well. The friction coefficient f is given by the following where θ is the state variable which evolves as f ¼ f 0 þ a lnðv=v 0 Þþb lnðv 0 θ=lþ (7) dθ Vθ ¼ 1 dt L (8) Equation (2) is the steady state solution to those equations. Upon an e-fold step increase in V applied to a fault in a steady state sliding, f increases by a, and decreases asymptotically by b, approaching a new steady state with a characteristic state-evolution slip L (Figure 4a). SHIMAMOTO AND NODA American Geophysical Union. All Rights Reserved. 8094

7 For the flow law, we use a rate-and-state flow law with a transient behavior [Noda and Shimamoto, 2010]. τ flow ðv; ΨÞ ¼ τ r ½ðV=wCf H2OÞexpðQ=RTÞŠ 1=m expðψþ ¼ τ r ½ð γ=cf H2OÞexpðQ=RTÞŠ 1=m expðψþ; (9) where Ψ is the state variable for flow, τ r is a reference shear stress (selected here as 1 MPa) introduced for the dimensional consistency such that exp( Ψ) is nondimensional, and m is the power exponent describing the instantaneous effect as discussed later. Note that V/w is an average engineering shear-strain rate γ (called simply strain rate hereafter). As in the case of friction law, the state variable for flow is considered to be a quantity that cannot change instantly with time but can evolve with shear strain or with time. One can imagine that the dislocation density and other microstructural properties, for example, can be quantities affecting Ψ because they cannot change instantly and some inelastic deformation is needed to create new dislocations and to change microstructures. Experimentally determined characteristic shear strain of the state evolution γ c for halite shear zone is 0.3 without significant dependency on the strain rate [Noda and Shimamoto, 2010]andsucha large γ c suggests that fabric changes (e.g., changes in grain shape and crystallographic preferred orientation) also affect the transient behavior. On the other hand, dislocation density cannot increase instantly upon a step increase in the strain rate γ, so that stress τ flow also increasesabruptlytoaccommodatetheincreasein γ by increasingthevelocityofdislocationmotion(see Orowan s equation[e.g., Nicolas and Poirier, 1976]). This can be a possible process for the instantaneous response in τ flow upon a step increase in γ. Noda and Shimamoto [2010] proposed that the state Ψ decays to its steady state value Ψ ss with a characteristic shear strain γ c for halite shear zones: dψ=dt ¼ ð γ=γ c ÞðΨ ss Ψ Þ ¼ ðv=l flow where L flow = wγ c is the state-evolution distance for flow and Ψ ss is given by ÞðΨ ss ΨÞ; (10) Ψ ss ðvþ ¼ ð1=n 1=mÞ½Q=RT þ lnðv=wcf H2OÞŠ: (11) Equation (9) becomes a familiar power law, equation (3), at a steady state as confirmed by τ flow ðv; Ψ ss Þ ¼ τ r ½ðV=wCf H2OÞexpðQ=RTÞŠ 1=m expðψ ss Þ ¼ τ r ½ðV=wCf H2OÞexpðQ=RTÞŠ 1=m ½ðV=wCf H2OÞexpðQ=RTÞ ¼ τ r ½ðV=wCf H2OÞexpðQ=RTÞŠ 1=n ¼ τ flow ss ðvþ Š 1=n 1=m (12) Consider an e-fold step increase in V in equation (9) from V 1 to ev 1 where the state variable is Ψ 1.Thenfrom equation (9) the instantaneous change in the flow stress on the logarithmic scale, Δ(ln(τ flow )), is simply given by Δðlnðτ flow Þ ¼ lnðτ flow ðv 1 ; Ψ 1 ÞÞ lnðτ flow ðev 1 ; Ψ 1 Þ ¼ 1=m (13) Thus, m specifies the instantaneous response upon a step change in slip rate or strain rate. Moreover, equations (9) (11) for V = ev 1 (constant) yields Ψ ¼ Ψ ss þ ðψ 1 Ψ ss Þexpð Vt=wγ c Þ ¼ Ψ ss þ ðψ 1 Ψ ss Þexpð γ=γ c Þ ¼ Ψ ss þ ðψ 1 Ψ ss Þexpð δ=l flow Þ (14) where δ is displacement across the shear zone from the velocity step. Thus, upon an abrupt e-fold increase in V, ln(τ flow )increasesby1/m and then asymptotically approaches a new steady state over a characteristic shear strain γ c (Figure 4b) or over a state-evolution distance for flow L flow (Figure 4c). We had to use the transient flow parameters (m and characteristic shear strain γ c ) for halite [Noda and Shimamoto, 2010] (γ c = 0.3 and n/m = 0.6) in our modeling because those have not been determined for other rocks and minerals (Table 1). Those parameters and a shear zone width w of 300 m yields state-evolution distance, L flow = wγ c = 90 m, which is much longer than the state-evolution distance for friction L = 5 mm we assumed. The friction to flow law, equation (1), indicates that if either of fσ or τ flow is much smaller than the other then τ is given approximately by the smaller one not only at the steady state but also during the transient flow. Figure 5 exhibits fault behaviors upon velocity steps from 0.3 to 3 nm/s at various depths. Figure 1a gives a strength profile corresponding to a fixed slip rate of 1 nm/s, so that the velocity step is across the slip rate for SHIMAMOTO AND NODA American Geophysical Union. All Rights Reserved. 8095

8 Figure 5. (a and b) Fault responses at different depths upon a velocity step from 0.3 to 3 nm/s (see Table 1 for the constitutive parameters of the fault). The fault is assumed to be at the steady state before applying the velocity step. Friction-dominated behaviors can be seen better in Figure 5a with a short displacement scale, whereas transient behaviors caused by plastic flow are revealed better in Figure 5b with a long displacement scale. this strength profile. The state-evolution distances, L and L flow, are so different that the transient behaviors in the friction- and flow-dominated regimes cannot be displayed clearly with the same scale of displacement. Thus, we used short and long displacement scales in Figures 5a and 5b, respectively, to exhibit the friction- and flow-dominated transient behaviors. Note that the boundaries between frictional, transitional, and fully plastic regimes for the hydrostatic P p are at 14 and 23 km in depths for the steady stateat1nm/s(figure1a).atadepth of 7 km in the frictional regime, a stepwise increase in τ is followed by an exponential decay toward a new steady state (the upper curve in Figure 5a). At depths of 16 and 21 km in the transitional regime, abrupt increases in τ are followed by nearly exponential decays with a short state-evolution distance for friction L (the second and third curves from the top in Figure 5a) and then τ increases toward a new steady state over a long state-evolution distance for flow L flow (the second and third curves from the top in Figure 5b). Such a peak-decay followed by an opposite decay behavior was observed for granite gouge under hydrothermal conditions [Blanpied et al., 1998] and for halite shear zone [Noda and Shimamoto, 2010]. In the plastic regimes (25 and 30 km in depths), step increases in τ are followed by gradual increases toward the steady states over L flow (the bottom two curves in Figure 5b). Earthquake modeling below was done including those transient behaviors from friction to fully plastic regimes. 5. A Two-Dimensional Earthquake Modeling Simulations were conducted using a spectral boundary integral equation method (BIEM) for elastodynamics [Lapusta et al., 2000] which allows efficient computations of slow tectonic loading, nucleation of earthquakes, and dynamic rupture propagation with inertial effects fully accounted for. This method implements an efficient calculation of the convolution of Green s function with current and previous slip history, using a fast Fourier transform technique. In order to model a sequence of earthquakes, the ability to take logarithmically wide range of time steps is critically important since we need to resolve dynamic earthquake rupture propagation as well as slow tectonic loading processes. The adaptive stepper [Lapusta et al., 2000] together with the state-evolution scheme [Noda and Lapusta, 2010] controls the time steps so that the stability condition is always satisfied for the rate-and-state friction laws with only one state variable. We neglect the evolution of Ψ in the stability criterion and set a safety factor of 2 in determining the length of time step because the state-evolution distance for flow is much longer than that for friction. A strike-slip fault was modeled as a planer fault in an antiplane two-dimensional problem with a mirror at the ground surface. Another mirror is set at a depth of.96 km to produce spatial periodicity which is needed in the BIEM used here. The fault is assumed to be sliding steadily by a plate velocity below km in depth. A time window within which the inertial effects are accounted for is set as the time for the S wave to travel km; a wave radiated from the bottom of the rupture at about 20 km can reflect at the ground surface and return to its origin safely. The earthquake modeling was done with the parameters in Table 1. Note that 5 MPa is added to the effective normal stress σ to avoid the divergence of solution on the surface (z = 0). SHIMAMOTO AND NODA American Geophysical Union. All Rights Reserved. 8096

9 Journal of Geophysical Research: Solid Earth Figure 6. Two-dimensional modeling of earthquake sequences on a strike-slip fault across the lithosphere using the friction to flow law. (a and b) Cumulative slip distribution for hydrostatic and twice hydrostatic pore pressure distributions, respectively. Red and blue curves indicate dynamic rupture propagation at 1 s intervals and interseismic creep at year intervals, respectively. (c) Blue and red curves exhibit stress profiles just before and just after the fifth earthquake in Figure 6a, 9 respectively. The black curve in Figure 6c shows the steady state strength profile for V = 10 m/s, the long-term plate rate in the model. (d) This strength profile is compared with fault properties. For the hydrostatic Pp case, the recurrence interval of earthquakes is about 376 years on average, the seismic displacement is at maximum (~12 m) at a depth of km where no creep occurs during interseismic periods, and the depth limit of the rupture is at 19 km (Figure 6a). With twice hydrostatic Pp, the recurrence interval is ~183 years, the maximum coseismic displacement is ~6 m at a depth of 7 15 km, and the rupture reaches to ~22 km (Figure 6b). Higher Pp causes more frequent earthquakes with deeper downward propagation. We now look at fault motion closely for the case of hydrostatic Pp. The strength profile for V = 1 nm/s = 10 9 m/s roughly falls in between the stress profiles just before and after an earthquake (cf. black, blue, and red curves in Figure 6c) partly because we did not include dynamic weakening such as thermal pressurization during seismic fault motion [Noda and Lapusta, 2013, and references therein]. Depths of 3 14 km correspond to the rate-weakening regime at a steady state at a plate rate (V = 1 nm/s), where microseismicity could occur in conventional earthquake models (Figure 3a). Earthquake ruptures span from the surface to the depth of 19 km well within the transitional regime at the steady state and below the peak strength by about 2 km (Figure 6d). Note that the stress builds up below about 18 km during seismic fault motion with a sharp maximum at 19 km where the rupture was arrested (peak in the red curve in Figure 6c). The depth of the maximum velocity strengthening is 21 km at the steady state (Figure 3a), and this barrier along with the negative dynamic stress drop probably stopped the rupture effectively. The transitional regime was defined as a regime with τ smaller than (0.99 τ friction) and (0.99 τ flow) as in Figures 1a, 3a, and 3b. Note that the shear resistance τ was referred to as the shear traction on the fault in the elastodynamics, whereas τ friction and τ flow SHIMAMOTO AND NODA American Geophysical Union. All Rights Reserved. 8097

10 Journal of Geophysical Research: Solid Earth Figure 7. Simulated fault behavior during the interseismic period following the fifth event. (a and b) Evolutions of the spatiotemporal distribution of the slip rate and of the deviation of shear stress τ from its steady state value τ ss, in color scales shown at the top. (c) An enlarged view of the first 20 years of the slip distribution with the same color scale, highlighting the afterslip distribution, and (d) fault slip distributions at the depths of 15 km for the same period. Vertical lines at a depth of 19 km in Figures 7a and 7c show the depth of rupture arrest, and the white-black dashed lines and the black dashed lines in the same figures indicate the boundaries between the frictional and transitional regimes and between the transitional and fully plastic regimes, respectively. SSEs denote slow slip events prior to the next event. The slip rate was 9 fixed at 10 m/s = 1 nm/s at a depth of km as a boundary condition. at each point at a given time, used to define the deformation regimes, were calculated from the friction and flow laws (equations (7) and (9)) and the same σ, V, θ, and Ψ that evolve in the simulation. Figures 7a and 7b show how the slip rate and the deviation of shear stress from its steady state value (τ τ ss) at each depth evolve with time during the interseismic period. Figures 7c and 7d exhibit a close-up view of the slip rate distribution with the same color scale and postseismic slip curves, respectively, in a period of 20 years after the event. The changes in the boundaries between deformation regimes with time are shown by two types of dashed lines in Figures 7a and 7c. Inside ongoing ruptures, τ is eventually given by τ friction since τ flow at such a high strain rate is tremendously large; note a nearly linear increase in τ with increasing depth down to a depth of 19 km, reflecting the frictional behavior (red curve in Figure 6c). But the lower part of the seismic slip zone changes within a year to the transitional regime due to low slip rates after the earthquake (see the distribution of the transitional regime between the frictional and plastic regimes in Figures 7a and 7c). This is because the power law dependency of the plastic flow law yields much faster decrease in the strength than the logarithmic dependency of the friction law. The fault is fully coupled at depths of km and occasionally coupled at depths of km and km where total slip is partially accommodated by interseismic creep. The creep initiated at very shallow depths and propagated to 8 km prior to the nucleation of the next earthquake there (Figure 7a); note that the fifth event nucleated at about 13 km in depth (Figure 6a). Creep also initiated at a depth near the SHIMAMOTO AND NODA American Geophysical Union. All Rights Reserved. 8098

11 rupture arrest and reached to a depth of 15 km after about 1 years, and an increased slip rate caused an emergence of the frictional regime within a wedge-shaped area in Figure 7a. But the bottom margin of the locked zone remained transitional due to the low slip rate in the locked interface. Comparison of stress profiles before and after the fifth event reveals that the shear stress drops by the earthquake at depths shallower than 18 km, but it builds up below this depth with a sharp maximum at a depth of 19 km where the earthquake rupture was arrested (a zone marked with Negative stress drop in Figure 6c). The rupture can travel in a region of negative stress drop as long as the energy release rate is higher than the fracture energy. The high-stress zone at depths around 20 km persists throughout the interseismic period although the depths of high stress become gradually shallow to 15 ~ 19 km (yellowish orange zone in Figure 7b). The shift of the high-stress zone can be recognized in the difference in the strength profiles just before and after the fifth event (blue and red curves in Figure 6c), as compared with the strength profile showing τ ss. The blue curve gives the stress profile just before the fifth event and does not correspond to the distribution of (τ τ ss ) following the fifth event in Figure 7b. But the stress given by the blue curve is greater than τ ss shown by the black curve in the depth range of km. The high stress at depths of km during the interseismic period must have driven the interseismic creep penetrating into the seismogenic zone, eventually leading to the slow slip events prior to the next earthquake (SSEs in Figure 7a; sharp peaks in blue curve in Figure 6c). SSEs in this case are slow slip events with slip rates much slower than seismic slip rates but faster than the creep rates. But the sixth event was triggered by creep in shallow depth, and a SSE acted as the nucleation of the event. The earthquake modeling revealed the most prominent afterslip with a decelerating rate at around a depth of 19 km, where the rupture was arrested, due to a sharp buildup of stress there (yellowish orange zone in Figure 7c; see also slip distribution in Figure 7d). An average slip rate at a depth of 19 km in the 8 months after the event, corresponding to the two contours in Figure 7d, was 3 mm/yr and was larger than the plate velocity we assigned (10 9 m/s ~31.5 mm/yr) by more than 1 order of magnitude. With increasing depth from the arrested rupture front, the slip rate just after the event first decreased to about 24 km in depth and then gradually increased to around 39 km (see color changes in Figure 7c and changes in the distances between the contour lines in Figure 7d). An average slip rate at 24 km depth in the 5.4 years after the event was 46 mm/yr, 46% greater than the plate velocity. An average slip rate at 39 km depth in the 3 years was 84 mm/yr, 2.7 times as large as the plate velocity. This fairly rapid afterslip at depths is probably due to decreasing rate dependency with increasing T in the plastic regime (equation (6) and Figures 3a and 3b). The slip rate decreases to the plate rate toward the bottom ( km) where we constrained V to that rate. 6. Discussion We proposed an empirical friction to flow law that agrees with the results from shearing experiments on halite (Figures 2, 4, and 5), demonstrated that the law can construct a strength profile across the lithosphere and velocity-dependent fault model on a unified basis (Figures 1 and 3), and solved earthquake cycles along a fault across the lithosphere (Figures 6 and 7). Our results warrant further studies at least in the following four aspects Shearing Experiments on Friction to Flow Our friction to flow law is confirmed so far only for halite data for a full transition from friction to plastic flow. It has been known for a long time that exactly the same laws hold for frictional and flow properties of halite and other rocks and minerals [e.g., Nicolas and Poirier, 1976]. Thus, a good agreement between the law and halite data probably implies general applicability of the law to other rocks and minerals, although this has to beconfirmed in the future. An advantage of the law is that full spectrum of properties in the transitional regime from the friction to flow can be predicted using only friction and flow parameters. However, almost no experimental data are reported on the transient flow properties, and we had to use flow parameters m and γ c, specifying the instantaneous and transient flow behaviors (equations (9) and (10); Figure 4), in our earthquake modeling in Figures 6 and 7. In particular, the average shear strain during interseismic periods in our earthquake modeling was about 0.04 (slip of about 12 m during the interseismic periods for a 300 m wide shear zone; Figure 6a), and this is much smaller than the characteristic shear strain γ c = 0.3 for halite [Noda and Shimamoto, 2010]. Thus, the steady state flow is not achieved along deep fault or plate interface, unless γ c and/or w are much smaller than those we used. Transient flow properties for important rocks are definitely needed. SHIMAMOTO AND NODA American Geophysical Union. All Rights Reserved. 8099

12 Figure 8. Friction to fully plastic transition at several temperatures shown on each curve, as calculated with the frictionto-flow law. (a) Effect of temperature at a strain rate of 10 6 /s and (b) the effect of strain rate at a temperature of 1000 C on the friction to plastic flow transition for diabase. A friction coefficient of 0.7 was assumed, and we used diabase flow parameters (n = 3.05, Q = 276 kj/mol, A = (MPa) n /s) after Caristan [1982]. The dashed vertical line indicates roughly an upper limit of normal stress in shearing experiments along a precut specimen with triaxial gas apparatuses at confining pressures less than about 300 MPa. Higher normal stresses can be achieved with solid pressure-medium apparatuses and anvil-type rotary-shear apparatuses. However, the friction to flow law can be a useful guide for planning future experiments because friction to plastic transition in the steady state, similar to Figure 2, can be constructed easily if the friction coefficient and steady state flow parameters in equation (3) are known. Figure 8 exhibits an example of such calculation with a friction coefficient of 0.7 and flow law parameters for diabase [Caristan, 1982]. If an apparatus is capable of producing a plate velocity, say 1 nm/s, an assumed strain rate of 10 6 /s for the figure is achieved for a 1 mm thick shear zone. Then a high-pressure gas apparatus can probably produce friction to plastic flow transition at temperatures of C, without melting a diabase specimen (Figure 8a). But when a slip rate is on the order of 1 μm/s at a temperature of 1000 C, normal stresses over several GPa would be needed to achieve the transition (Figure 8b). This will not be an easy experiment even with a solid pressure-medium apparatus or a rotary-shear anvil-type apparatus. Moreover, apparatuses to be used should have a capability for measuring transient behaviors. The friction to flow law has to be evaluated by such future experiments. It is encouraging that the experimental results on the friction of granite under hydrothermal conditions by Blanpied et al. [1995] are very similar to a change in the rate dependency from friction to transitional regimes, expected from the friction to flow law (Figure 3a). But reproducing the whole brittle to fully plastic transition for important rocks is still a long way to go Strength Profiles and Rate Dependency Fault Models Simple brittle-plastic strength profiles have been used in huge number of literatures to characterize the thickness and internal structures of the lithosphere [e.g., Bird, 1978; Goetze and Evans, 1979; Kohlstedt et al., 1995]. Another highly successful model is the rate dependency fault model such as Figure 1b, used in the modeling of earthquake cycles [e.g., Tse and Rice, 1986; Lapusta and Rice, 2003; Shibazaki and Shimamoto, 2007]. Those are different characterization of the lithosphere, complementing each other. Our friction to flow law merged those two approaches on a unified basis, and the strength profiles and fault models would be two important areas where our law is applicable. However, it should be kept in mind that a strength profile gives a profile for the steady state shear resistance τ ss for assumed slip and strain rates, whereas in reality the distribution of the shear resistance τ and the deformation regimes evolve with time and those can be constructed only by solving the fault or plate boundary motion. In the present example, the steady state strength profile works as a reasonable approximation to the overall distribution of τ (cf. red, blue, and black curves in Figure 6c). However, if one considers dramatic weakening of the fault during coseismic slip [e.g., Tsutsumi and Shimamoto, 1997; Noda and Shimamoto, 2005; Wibberley and Shimamoto, 2005; Rice, 2006; Brantutetal., 2008; Noda et al., 2009; Di Toro et al., 2011], the stress distribution and its evolution will be totally different from simple strength profiles. Whether Christmas tree-type strength profiles exist or not in such cases is still an open question. An advantage of using the friction to flow law over the previous rate dependency models is that the former includes flow properties in the middle to lower parts of the lithosphere. SHIMAMOTO AND NODA American Geophysical Union. All Rights Reserved. 8100

13 6.3. Earthquake Modeling The overall fault behaviors during seismic cycles with the friction to flow law in Figure 6 are quite similar to those with the rate-and-state friction law assuming rate dependency as a function of depth [Tse and Rice, 1986; Lapusta and Rice, 2003]. This is not surprising because the friction to flow law predicts a similar rate dependency in the seismogenic zone as in the friction model (cf. Figures 1b and 3a). However, notable differences exist in the functional form of the constitutive law for the middle to lower parts of the lithosphere. The friction to flow law is realistic because it includes high-temperature flow properties at depths, and the interseismic behaviors in Figure 7 should be different from that with the rate dependency models. However, behaviors cannot be compared since the details of the interseismic behaviors have not been reported in previous studies. Whether an enhanced afterslip similar to that recognized at a depth of 39 km (Figure 7d) occurs or not in natural earthquakes is an interest point of investigation. An interplay between coseismic slip and interseismic creep is of interest. Fault creep begins to occur at depths shallower than a few kilometers and near the bottom of the coseismic zone, and the interseismic creep invades into the locked zone, eventually leading to the onset of the next event either at the shallow or deep end of the locked zone (Figures 6a and 7a). The creep zone below the locked zone was in the transitional regime for about 1 years after the event, after which a frictional regime emerged in a wedge-like zone at the shallow part of the deep creep zone (Figure 7a). The episodic slow slip events occurred 5 times in our modeling only in the frictional regime (encircled areas with SSEs in Figure 7a), and two of them occurred just before an earthquake. The SSE in the shallow part acted as the nucleation of the sixth event. The SSEs may be relevant to the slow slip inferred from tremors about 3 months prior to the 2004 Parkfield earthquake [Shelly, 2009]. Since the fault below about 15 km was in the transitional regime during the interseismic period, fault rocks at km depths experienced a seismic event overprinted by deformation in the transitional regime. In particular, the interseismic creep is even greater than the coseismic slip at depths of km (Figure 6a). Shearing experiments on halite revealed that mylonitic textures formed not only in the fully plastic regime but also in the transitional regime [Hiraga and Shimamoto, 1987;Shimamoto, 1989;Kawamoto and Shimamoto, 1997, 1998]. Thus, the repeated seismic slip followed by the interseismic creep in the transitional regime in about lower one fifth of the seismogenic zone is consistent with the occurrence of natural pseudotachylytes overprinted by mylonitic deformation, reported at various places [e.g., Sibson, 1980; Hobbs et al., 1986; Lin et al., 2005, and references quoted therein] Earthquake Modeling With Different Transient Flow Parameters We used the steady state flow parameters for quartzite after Hirth et al. [2001] in the 2-D earthquake modeling in section 5, but for transient flow parameters we had to use the values determined for halite [Noda and Shimamoto, 2010] because no such data are available for important rocks constituting the crust and upper mantle (Table 1). Another uncertainty in our modeling is the width of shear zone w which we assumed as 300 m. One may argue that the results from the earthquake modeling may dramatically change if different values are used in the modeling. Thus, we discuss here possible or likely ranges of the transient flow parameters and report results from additional modeling of earthquakes using the same fault model and the same values for other parameters (Figures 9 and 10). However, this paper is focused on friction to flow law and its application to a 2-D earthquake modeling, and full analyses of transient behaviors of various rocks under various tectonic setting are beyond the scope of this paper. Moreover, we tuned the shear zone width w to have a typical seismogenic width in a continental crust (Figures 1a and 3a and section 3) following an approach of Sibson [1980], and we did not conduct earthquake modeling with different w values here because changing the width of the seismogenic zone will not change the essence of fault behaviors. We first examine the characteristic strain for the state evolution of flow γ c using recent papers reporting stress-strain curves upon step change(s) in strain rates. A recent paper by Hansen et al. [2012] reports results from ring shear experiments on olivine in a grain boundary sliding regime, using a Paterson gas apparatus. Figure 7 of their paper reports stress-strain curves from five runs with shear strains up to almost 15, but the curves for the transient behaviors upon step changes in strain rate are so small that detailed analysis of the transient behaviors could not be done. However, the duration of step change experiments was 0.2 to 0.4 in terms of shear strain, and hence, γ c would be on the order of 0.1 (this is about one third of the value for halite). Rutter and Brodie [2004] report detailed triaxial compression experiments on synthetic quartzite at high SHIMAMOTO AND NODA American Geophysical Union. All Rights Reserved. 8101