Rate-and State-dependent Friction Research Articles

Estimation of time-variable friction parameters using machine learning

SUMMARY The laboratory-derived rate- and state-dependent friction (RSF) law governs rock friction. Although a number of studies have investigated the RSF friction parameters, they are not fully understood yet within a physical framework. In this study, we estimated the variation of RSF parameters during stick-slip cycles, in order to have insights into the temporal variation of fault conditions during slipping, which may help understand the relation between the change in friction parameters and the generation of gouge particles. To get a more refined understanding of the evolution of RSF parameters, we estimated these parameters for each of the hundreds of stick-slip events that occurred on laboratory faults during an experiment. We used experiment data for which the gouge particles were removed from the laboratory faults at the beginning of each experiment; this procedure made possible to evaluate the influence of the gouge layer evolution on the variation of the RSF parameters. Since the amount of data was very large, we adopted a random forest (RF) machine learning approach for data analysis. The RF model was trained on simulated friction data and then applied to the experiment stick-slip event data to estimate the RSF parameters. To generate simulated friction data of stick-slip events, a one-degree-of-freedom spring-slider model governed by the RSF law was assumed. From plots of friction change as a function of displacement, some representative features were extracted to account for the RSF parameters and were used as input to the RF algorithm. Using the RF approach, we captured the variation of the RSF parameters a, $b - a$ and ${D}_{\mathrm{c}}\ $defined in the RSF law. The results show that during a first transient phase, the parameter a becomes smaller, while parameters $b - a$ and ${D}_{\mathrm{c}}$ become larger, as the gouge layer becomes thicker. The variation of the RSF parameters becomes less pronounced during the following steady-state phase. These results suggest that the variation of RSF friction parameters may be related to the evolution of the gouge layer.

Tertiary creep behavior for various rate- and state-dependent friction laws

Forecasting the acceleration of slow landslides to the point of catastrophic failure is crucial. It follows the Voight power-law model with the power exponent α, which is typically close to 2 but can be significantly smaller. Understanding the underlying mechanisms may improve landslide warnings. A previous study applied a rate- and state-dependent friction (RSF) law in the form of the aging law to the creep behavior of an underlying shear zone of landslides, and showed an α value of 2. The aging law is one of the conventional forms of RSF law, and we extended the analysis to other representative laws: the slip law, Perrin-Rice-Zheng (PRZ) law, composite law, and Nagata law. We showed that the acceleration is expressed in terms of the slip rate using the state-evolution equation. As the slip rate increases, α decays to 2, regardless of the frictional parameters, following a power law for the aging and Nagata laws and logarithmically for the slip and the composite laws. For the PRZ law, the asymptotic value of α is between 2 and 3 and depends on frictional parameters. In typical RSF laws, the logarithm of the slip rate is proportional to the friction coefficient or normalized shear stress f minus frictional strength Θ. The logarithmic direct effect and a linear increase with slip in f−Θ independent of the slip rate, which is a characteristic of aging and Nagata laws under constant stress conditions, leads to α=2. By contrast, a purely time-dependent increase in f−Θ would lead to α=1. If these two effects coexist, α increases from 1 to 2 with acceleration. In laboratory experiments of investigation of RSF law, constant-load creep tests in the tertiary-creep stage have been rarely conducted, but they could provide new insights on the form of the state-evolution equation and deserve future study.

Slide‐Hold‐Slide Protocols and Frictional Healing in Discrete Element Method (DEM) Simulations of Granular Fault Gouge

AbstractThe empirical constitutive modeling framework of rate‐ and state‐dependent friction (RSF) is commonly used to describe the time‐dependent frictional response of fault gouge to perturbations from steady sliding. In a previous study (Ferdowsi & Rubin, 2020), we found that a granular‐physics‐based model of a fault shear zone, with time‐independent properties at the contact scale, reproduces the phenomenology of laboratory rock and gouge friction experiments in velocity‐step and slide‐hold (SH) protocols. A few slide‐hold‐slide (SHS) simulations further suggested that the granular model might outperform current empirical RSF laws in describing laboratory data. Here, we explore the behavior of the same Discrete Element Method (DEM) model in SH and SHS protocols over a wide range of sliding velocities, hold durations, and system stiffnesses, and provide additional support for this view. We find that, similar to laboratory data, the rate of stress decay during SH simulations is in general agreement with the “Slip law” version of the RSF equations, using parameter values determined independently from velocity step tests. During reslides following long hold times, the model, similar to lab data, produces a nearly constant rate of frictional healing with log hold time, with that rate being in the range of ∼0.5 to 1 times the RSF “state evolution” parameterb. We also find that, as in laboratory experiments, the granular layer undergoes log‐time compaction during holds. This is consistent with the traditional understanding of state evolution under the Aging law, even though the associated stress decay is similar to that predicted by the Slip and not the Aging law.

Transient Slow Slip Characteristics of Frictional‐Viscous Subduction Megathrust Shear Zones

AbstractThe deep roots of subduction megathrusts exhibit aseismic slow slip events, commonly accompanied by tectonic tremor. Observations from exhumed rocks suggest this region of the subduction interface is a shear zone with frictional lenses embedded in a viscous matrix. Here, we use numerical models to explore the transient slip characteristics of finite‐width frictional‐viscous megathrust shear zones. Our model utilizes an invariant, continuum‐based, regularized form of rate‐ and state‐dependent friction (RSF) and simulates earthquakes along spontaneously evolving faults embedded in a 2D heterogeneous continuum. The setup includes two elastic plates bounding a viscoelastoplastic shear zone (subduction interface melange) with inclusions (clasts) of varying distributions and viscosity contrasts with respect to the surrounding weaker matrix. The entire shear zone exhibits the same velocity‐weakening RSF parameters, but the lower viscosity matrix has the capacity to switch between RSF and viscous creep as a function of local stress state. Results demonstrate a mechanism in which stress heterogeneity in these shear zones both (a) sets the “speed limit” for earthquake ruptures that nucleate in clasts such that they propagate at slow velocities; and (b) permits the transmission of slow slip from clast to clast, allowing slow ruptures to propagate substantial distances over the model domain. For reasonable input parameters, modeled events have moment‐duration statistics, stress drops, and rupture propagation rates that overlap with some natural slow slip events. These results provide new insights into how geologic observations from ancient analogs of the slow slip source may scale up to match geophysical constraints on modern slow slip phenomena.

Time to seismic failure induced by repeating SSEs in a single-degree-of-freedom spring-slider model

SUMMARYSome large earthquakes may have been triggered by large slow slip events (SSEs). This paper studies the time it takes from the most recent SSE to seismic failure in a situation where earthquakes are influenced by periodic SSEs, unlike in earlier studies that investigated the impact of a single SSE imposed at an arbitrary timing during a seismic cycle. A single-degree-of-freedom (SDOF) spring-slider model obeying a rate- and state-dependent friction (RSF) law is used to study the time to failure. The results are compared with those in Ohtani et al., which simulates megathrust earthquakes triggered by large, spontaneous SSEs along the deeper extension of a seismogenic zone in a 2-D elastic medium. Suppose a model fault under steady-rate tectonic loading is also impacted by stress steps induced by periodic SSEs. In the absence of SSEs, there is only a unique value of the characteristic slip distance L of RSF for a given seismic return period T. In the presence of SSEs, by contrast, synchronization occurs, and there exists a finite range of L values that corresponds to the same T. The timing (phase difference) of earthquakes relative to the SSEs varies continuously with L within that range. This study focuses on the case where T is triple the SSE return period, TSSE, (T = 3TSSE) to allow comparison with Ohtani et al. For each value of L in that range, disturbance tests that assign random values to T0, the timing of the third SSE within the seismic cycle, are conducted to obtain a probability distribution for tf, the time from that SSE to seismic failure. The distribution is converted into P(t; L), the cumulative probability that seismic failure occurs within time t of the SSE. The P(t; L) is averaged over the different L values to produce $\bar{P}$(t), which takes account of variability in the strength excess on the seismic fault at the time of the SSE, a parameter that is generally unknown. These numerical disturbance tests are conducted for three different values of an SSE size parameter. It is found that the larger the SSEs, the more intensely seismic failure is concentrated within short time intervals following an SSE. For the largest SSE, whereby a single SSE accounts for 3/10 of the total stress accumulated during a seismic cycle, there is a 50 per cent probability that seismic failure occurs within 132 d. These results contrast with those of Ohtani et al., in which tf was found to be concentrated more intensely within even shorter time intervals following an SSE (80 per cent probability that seismic failure occurs within 2.78 d of the SSE). It is suggested the timing of SSE-triggered seismic failure is concentrated more strongly on a fault embedded in a continuum because of a factor that cannot be taken into account by an SDOF model—a spatial structure of stress concentration that is already there on the fault even before the triggering event, the SSE.

EQsimu: a 3-D finite element dynamic earthquake simulator for multicycle dynamics of geometrically complex faults governed by rate- and state-dependent friction

SUMMARY We develop a finite element dynamic earthquake simulator, EQsimu, to model multicycle dynamics of 3-D geometrically complex faults. The fault is governed by rate- and state-dependent friction (RSF). EQsimu integrates an existing finite element code EQdyna for the coseismic dynamic rupture phase and a newly developed finite element code EQquasi for the quasi-static phases of an earthquake cycle, including nucleation, post-seismic and interseismic processes. Both finite element codes are parallelized through Message Passing Interface to improve computational efficiency and capability. EQdyna and EQquasi are coupled through on-fault physical quantities of shear and normal stresses, slip-rates and state variables in RSF. The two-code scheme shows advantages in reconciling the computational challenges from different phases of an earthquake cycle, which include (1) handling time-steps ranging from hundredths of a second to a fraction of a year based on a variable time-stepping scheme, (2) using element size small enough to resolve the cohesive zone at rupture fronts of dynamic ruptures and (3) solving the system of equations built up by millions of hexahedral elements. EQsimu is used to model multicycle dynamics of a 3-D strike-slip fault with a bend. Complex earthquake event patterns spontaneously emerge in the simulation, and the fault demonstrates two phases in its evolution. In the first phase, there are three types of dynamic ruptures: ruptures breaking the whole fault from left to right, ruptures being halted by the bend, and ruptures breaking the whole fault from right to left. As the fault bend experiences more ruptures, the zone of stress heterogeneity near the bend widens and the earthquake sequence enters the second phase showing only repeated ruptures that break the whole fault from left to right. The two-phase behaviours of this bent fault system suggest that a 10° bend may conditionally stop dynamic ruptures at the early stage of a fault system evolution and will eventually not be able to stop ruptures as the fault system evolves. The nucleation patches are close to the velocity strengthening region. Their sizes on the two fault segments are different due to different levels of the normal stress.

Sensitivity of Induced Seismic Sequences to Rate‐and‐State Frictional Processes

AbstractIt is well established that subsurface injection of fluids increases pore fluid pressures that may lead to shear failure along a preexisting fault surface. Concern among oil and gas, geothermal, and carbon storage operators has risen dramatically over the past decade due to the increase in the number and magnitude of induced earthquakes. Efforts to mitigate the risk associated with injection‐induced earthquakes include modeling of the interaction between fluids and earthquake faults. Here we investigate this relationship with simulations that couple a geomechanical reservoir model and RSQSim, a physics‐based earthquake simulator. RSQSim employs rate‐ and state‐dependent friction (RSF) that enables the investigation of the time‐dependent nature of earthquake sequences. We explore the effect of two RSF parameters and normal stress on the spatiotemporal characteristics of injection‐induced seismicity. We perform >200 simulations to systematically investigate the effect of these model components on the evolution of induced seismicity sequences and compare the spatiotemporal characteristics of our synthetic catalogs to observations of induced earthquakes. We find that the RSF parameters control the ability of seismicity to migrate away from the injection well, the total number and maximum magnitude of induced events. Additionally, the RSF parameters control the occurrence/absence of premonitory events. Lastly, we find that earthquake stress drops can be modulated by the normal stress and/or the RSF parameters. Insight gained from this study can aid in further development of models that address best practice protocols for injection operations, site‐specific models of injection‐induced earthquakes, and probabilistic hazard and risk assessments.

Apparent Dependence of Rate- and State-Dependent Friction Parameters on Loading Velocity and Cumulative Displacement Inferred from Large-Scale Biaxial Friction Experiments

We investigated the constitutive parameters in the rate- and state-dependent friction (RSF) law by conducting numerical simulations, using the friction data from large-scale biaxial rock friction experiments for Indian metagabbro. The sliding surface area was 1.5 m long and 0.5 m wide, slid for 400 s under a normal stress of 1.33 MPa at a loading velocity of either 0.1 or 1.0 mm/s. During the experiments, many stick–slips were observed and those features were as follows. (1) The friction drop and recurrence time of the stick–slip events increased with cumulative slip displacement in an experiment before which the gouges on the surface were removed, but they became almost constant throughout an experiment conducted after several experiments without gouge removal. (2) The friction drop was larger and the recurrence time was shorter in the experiments with faster loading velocity. We applied a one-degree-of-freedom spring-slider model with mass to estimate the RSF parameters by fitting the stick–slip intervals and slip-weakening curves measured based on spring force and acceleration of the specimens. We developed an efficient algorithm for the numerical time integration, and we conducted forward modeling for evolution parameters (b) and the state-evolution distances (L_{text{c}}), keeping the direct effect parameter (a) constant. We then identified the confident range of b and L_{text{c}} values. Comparison between the results of the experiments and our simulations suggests that both b and L_{text{c}} increase as the cumulative slip displacement increases, and b increases and L_{text{c}} decreases as the loading velocity increases. Conventional RSF laws could not explain the large-scale friction data, and more complex state evolution laws are needed.

Single slip dynamics

In the present paper we consider a 1-D, single spring-slider analog model of fault and we solve the equation of motion within the coseismic time window. We incorporate in the dynamic problem different rheologic behavior, starting from the Coulomb friction (which postulates a constant value of the dynamic resistance), then the viscous rheology (where the friction resistance linearly depends on the sliding speed), and finally a version of the more refined rate-and state-dependent friction law. We present analytical solutions of the equation of motion for the different cases and we are able to find the common features of the solutions, in terms of the most important physical observables characterizing the solutions of a 1-D dynamic fault problem; the peak slip velocity, the time at which it is attained (or, in other words, the so-called rise time), the total cumulative slip developed at the end of the process (assumed to occur when the sliding speed vanishes or become comparable to its initial value). We also extract some useful dependences of these quantities on the parameters of the models. Finally, we compare the spectral behavior of the resulting sliding velocity and its fall-off at high frequencies.

A microphysical interpretation of rate‐ and state‐dependent friction for fault gouge

The evolution of fault strength during the seismic cycle plays a key role in the mode of fault slip, nature of earthquake stress drop, and earthquake nucleation. Laboratory-based rate- and state-dependent friction (RSF) laws can describe changes in fault strength during slip, but the connections between fault strength and the mechanisms that dictate the mode of failure, from aseismic creep to earthquake rupture, remain poorly understood. The empirical nature of RSF laws remains a drawback to their application in nature. Here we analyze an extensive data set of friction constitutive parameters with the goal of illuminating the microphysical processes controlling RSF. We document robust relationships between: (1) the initial value of sliding (or kinetic) friction, (2) RSF parameters, and (3) the time rates of frictional strengthening (aging). We derive a microphysical model based on asperity contact mechanics and show that these relationships are dictated by: (1) an activation energy that controls the rate of asperity growth by plastic creep, and (2) an inverse relationship between material hardness and the activation volume of plastic deformation. Collectively, our results illuminate the physics expressed by the RSF parameters, and which describe the absolute value of frictional strength and its dependence on time and slip rate. Moreover, we demonstrate that seismogenic fault behavior may be dictated by the interplay between grain properties and ambient conditions controlling the local shear strength of grain-scale asperity contacts.

Dynamics of earthquake nucleation process represented by the Burridge-Knopoff model

Dynamics of earthquake nucleation process is studied on the basis of the one-dimensional Burridge-Knopoff (BK) model obeying the rate- and state-dependent friction (RSF) law. We investigate the properties of the model at each stage of the nucleation process, including the quasi-static initial phase, the unstable acceleration phase and the high-speed rupture phase or a mainshock. Two kinds of nucleation lengths L_sc and L_c are identified and investigated. The nucleation length L_sc and the initial phase exist only for a weak frictional instability regime, while the nucleation length L_c and the acceleration phase exist for both weak and strong instability regimes. Both L_sc and L_c are found to be determined by the model parameters, the frictional weakening parameter and the elastic stiffness parameter, hardly dependent on the size of an ensuing mainshock. The sliding velocity is extremely slow in the initial phase up to L_sc, of order the pulling speed of the plate, while it reaches a detectable level at a certain stage of the acceleration phase. The continuum limits of the results are discussed. The continuum limit of the BK model lies in the weak frictional instability regime so that a mature homogeneous fault under the RSF law always accompanies the quasi-static nucleation process. Duration times of each stage of the nucleation process are examined. The relation to the elastic continuum model and implications to real seismicity are discussed.

Recovery of plate coupling at a ruptured asperity

AbstractEarlier earthquake cycle simulations using rate‐ and state‐dependent friction (RSF) law have revealed that frictional coupling at a ruptured asperity starts immediately after dynamic slip because of logarithmically time‐dependent healing. However, some Global Positioning System (GPS) inversion studies of the 2011 great Tohoku‐Oki earthquake suggest that afterslip continued for 7–8 months or longer in the deep seismogenic interface including asperities where M7 earthquakes repeatedly occurred. Nakatani and Scholz [] introduced the intrinsic cutoff time tcx in the logarithmically time dependent healing and pointed out that a long tcx could lead to delayed start of frictional coupling. Assuming RSF laws into which tcx was incorporated, we conducted a numerical simulation using a block spring model. We defined the time required to restore frictional coupling as Tcpl measured from the instant of dynamic slip. The value of Tcpl for tcx = 10−3 s was almost the same as Tcpl in the conventional law, which was on the order of 10 min in the case of the aging law. It increased approximately in proportion to tcx for short tcx (≤1 s), whereas Tcpl was not so sensitive to tcx for a long tcx (>1 s). Moreover, a long tcx cannot be reasonably assumed because the maximum slip velocity during dynamic slip decreased with increasing tcx, limiting the largest possible Tcpl to be months at most. Friction laws with two state variables, each having a different cutoff time, were also investigated. The maximum slip velocity was determined by the smaller cutoff time, and Tcpl was affected by the larger cutoff time. As a result, longer Tcpl became possible, although the values are still on time scale of months.

RSQSim Earthquake Simulator

As discussed in Tullis et al. (2012b), a promising avenue for improving seismic hazard estimation and reducing the large uncertainties in current assessments is to incorporate more accurate and region‐specific characterizations of the interactions and physical processes that control earthquake occurrence in fault systems. Earthquake simulators are computer models that can contribute to this by carrying out large‐scale simulations of earthquake occurrence to characterize system‐level response of fault systems including processes that control time, place, and extent of earthquake slip. Such simulators were pioneered by work such as Rundle (1988), Robinson and Benites (1995), and Ward (1996). Four such simulators are discussed in the November/December 2012 Seismological Research Letters issue, and their general features are described in Tullis et al. (2012a). This paper describes features specific to one of those four, RSQSim. It also presents results that are relevant to particularly unique features of RSQSim: comparisons with fully dynamic single‐event simulations and the spatial‐temporal clustering of seismicity due to RSQSim’s use of rate‐ and state‐dependent friction (RSF). RSQSim is based on a scheme initially developed by Dieterich (1995) with several enhancements to extend its use in a number of directions including allowing fully general fault system geometries and the attendant normal stress variations, better validation of the quasi‐dynamic part of the seismic cycle against fully dynamic models, and the inclusion of creeping parts of the fault system. Ziv (2003) and Ziv and Rubin (2003) used some elements of the original method to study clustering processes and frequency statistics. RSQSim itself has been used by Dieterich and Richards‐Dinger (2010) to study the effects of both small‐ and large‐scale fault system geometry on recurrence statistics in a idealized but complex fault system. We do not discuss it further here, but RSQSim has also been used successfully to model slow …

Prediction of factor of safety of a slope with an advanced friction model

Landslides and other mass movements are quite prevalent phenomena in hilly areas, and sometimes cause huge losses of both life and property [1]. To mitigate these natural calamities, a precise scientific study that can predict the slope failures is required. Prediction of the mechanical stability of slopes both man made as well as natural is a tedious task after considering the complexity involved in the physical behaviour of the slope forming material as well as their intricate failure mechanisms [1–4]. Traditional approaches to predict stability of a slope are generally based on evaluation of a factor of safety through limit equilibrium analysis (LEA) [1–4]. The factor of safety (FOS) of a slope is defined as the ratio of shear strength to driving stress generally due to gravitational force along the failure plane [1–3]. Physically, FOSo1 signifies the slope to be unstable; however it is stable if FOS41 [1]. The limitation of the LEA is that the analysis is carried out with a single fixed value of shear strength of solids such as rocks and soils, while ignoring the sliding rate dependent failure strength. Experiments on rocks have shown that failure strength (also known as frictional strength) does depend on sliding velocity and time of stationary contact (also known as waiting time) of the sliding interface [5–7]. Moreover, based on these experimental observations, an advanced friction model has been proposed which is known as the rate and state dependent friction (RSF) model. The RSF model is basically a modified form of the Mohr–Coulomb failure criterion [13]. It is to be noted that the RSF model has been extremely successful in recent times for predicting the frictional behaviour of hard and rough solids such as rocks, soils, hard polymers etc. [8–11]. As a result, the RSF model has found widespread applications in the study of mechanics of earthquakes and faulting [10].

A revised rate‐ and state‐dependent friction law obtained by constraining constitutive and evolution laws separately with laboratory data

We propose two major revisions on the rate‐ and state‐dependent friction (RSF) law on the basis of rigorous analysis of friction experiments. First, we find that the direct effect coefficient a, a parameter playing a central role in the RSF constitutive law, is much larger than the traditional, consensual estimate of less than about 0.01. We derive a lower bound of 0.035 for a directly from stress‐velocity relations measured during carefully designed step tests, without relying on any evolution laws as traditional methods do. After correcting for state changes during the steps, inferred indirectly from observed changes in acoustic transmissivities across the interface, we obtain an estimate of a as large as 0.05. Second, we calculate values of the RSF state variable Φ by feeding the measured shear stress and slip velocity values into the constitutive law. The results showed systematic deviations from predictions of the RSF evolution law of the aging type. This leads us to propose a revised evolution law, which incorporates a previously unknown weakening effect related to the shear stress. We also present additional experiment results to corroborate the presence of this new effect. Forward simulations based on our revised evolution law, combined with the larger, revised value of a, very well explain observed variations in both the shear stress and Φ throughout different phases of experiments, including quasi‐static hold, reloading after a hold, and steady state sliding at different velocities, as well as their mutual transitions, all with an identical set of parameter values.

余効すべり人工データを用いたアジョイント法による摩擦パラメータ・初期値の推定

Since variations of slip on plate boundaries depend on frictional properties, it is essential to know frictional parameters on the fault, as well as initial values of simulation variables for earthquake generation prediction. In this study, an adjoint data assimilation method is introduced to a simpli.ed fault model with a rate-and state-dependent friction law as a .rst step toward the goal of estimating the frictional parameters and initial values of simulation variables in a realistic situation. The method is applied to the simpli.ed model which mimics the 2003 Tokachi-oki afterslip. We make synthetic data set, slip velocities on the fault surface by assigning the “true” frictional parameters and initial values artificially. We investigate the feasibility of estimating frictional parameters and initial values through the adjoint data assimilation method on the assumption of knowing “background” values and observations. It is confirmed that the adjoint data assimilation method is computationally efficient to estimate control variables such as frictional parameters and initial values by time-trajectory fitting of observation data, compared to the grid search method. Also, we examine the sensitivity of each frictional parameter and initial value to the observed afterslip velocity data. In the range of our search using afterslip data, we .nd that 1) the initial value of velocity can be constrained, 2) the initial value of state variable cannot be constrained, 3) the frictional parameter value of a-b is well retrieved in the region where aftreslip velocity is observed, and 4) the value of characteristic length L can be retrieved only from the early portion of afterslip velocity data, where the velocity is rapidly changing.

Phase synchronization of slips by periodical (tangential and normal) mechanical forcing in the spring-slider model

In the present study, the character of slip regimes under weak external periodical (tangential and normal) mechanical forcing has been investigated in a laboratory spring-slider system. We report the experimental evidence of phase synchronization in a slip dynamics, induced by the external periodic mechanical impact. At certain conditions, we have a stick-slip effect in the spring-slider system. To describe this effect, we can use rate-and state-dependent friction law.

Frictional response of a thick gouge sample: 2. Friction law and implications for faults

On the basis of experimental results, we propose a new friction law aiming at describing the mechanical behavior of thick gouge layers. As shown in the companion paper, the dominant effect to take into account is a significant slip‐weakening process active over decimetric slip distances. This slip weakening is strongly nonlinear and, formerly, does not involve any characteristic length scale. The decrease of the gouge friction coefficient μ with imposed slip δ is well modeled by a power law: μ = μ0 + αδ−β, with β = 0.4. On this major trend are superimposed second‐order velocity‐weakening and time‐strengthening effects. These effects can be described using classical rate‐ and state‐dependent friction (RSF) laws and are associated with a small length scale dc ≈ 100 μm. Consistent with the general RSF framework, we combine slip‐weakening and second‐order effects in a slip, rate, and state (SRS) friction law with two state variables. We then compute the fracture (or breakdown) energy Gc and the apparent weakening distance Dcapp associated with the slip‐weakening process. Once extrapolated to realistic “geophysical” confining pressures, the obtained values are in excellent agreement with those inferred from real earthquakes: Gc ≈ 5 × 106 J m−2 and Dcapp ≈ 20 cm. We also find that fracture energy scales with imposed slip in our experiments: Gc ∼ δ0.6.

An approach to evaluate ground surface rupture caused by reverse fault movement

An approach for estimating ground surface rupture caused by strong earthquakes is presented in this paper, where the finite element (FE) method of continuous and discontinuous coalescent displacement fields is adopted. The onset condition of strain localization is introduced to detect the formation of the slippage line. In the analysis, the Drucker-Prager constitutive model is used for soils and the rate-and state-dependent friction law is used on the slippage line to simulate the evolution of the sliding. A simple application to evaluate the ground surface rupture induced by a reverse fault movement is provided, and the numerical simulation shows good agreement with failure characteristics observed in the field after strong earthquakes.

ダイラタンシーによる間隙水圧の変動を考慮した地震発生過程のモデル化

Dilatancy and pore fluid pressure changes in a fault zone significantly affect the mode of the nucleation process, and are key factors for understanding the physical mechanisms of some precursory phenomena. We investigate slip processes with a model of a vertical fluid-infiltrated strike-slip fault in a 3-D elastic half space using a Dietrich/Ruina rate-and state-dependent friction law and an evolution equation for plastic porosity. Due to dilatancy, porosity is thought to increase with slip velocity. In this model plastic porosity is assumed to depend logarithmically on the state variable in the friction law. The results of numerical simulations show that porosity in the fault zone increases with increasing slip velocity in the nucleation zone. Then, pore fluid pressure in the fault zone drops significantly, resulting in dilatant hardening. With an increasing dilatancy coefficient, a larger pore fluid pressure drop is observed during the nucleation process, which in turn increases the size of the nucleation zone. In the case of considerably larger dilatancy coefficients, the fault system becomes stable. The various modes of slip processes are reproduced by the spatial variation of dilatancy coefficients. We examine changes in pore-fluid pressure on the fault near the surface. When a slow event occurs in a deeper region of a seismogenic zone in the later stage of an earthquake cycle, it is observed that slip velocity on the fault near the surface increases significantly. Slip velocity also increases slightly around one year before the main rupture begins. These changes in slip velocity lead to a significant drop in the pore-fluid pressure on the fault near the surface.