Properties of earthquakes generated by fault dynamics

Dynamics of a brake system governed by modified Burridge-Knopoff-Pad model

Despite widespread investigations to improve on the performance of automobile brake systems, the undesirable phenomenon of disc brake squeal noise remains a very challenging problem to solve. Consequently, we thus propose a basic theoretical brake system based on the modified Burridge-Knopoff-Pad model in this study; in order to elucidate on brake squeal noise and other nonlinear vibrations. Linear stability analysis clearly depicts the stability/instability of some spatial-homogeneous steady states, when some parameters of the system are varied. Our investigation brings into sharp focus the role of nonlinearity, and strongly suggests that the instability of perturbed continuous wave is physically manifested as brake squeal noise. Result of numerical simulations clearly underscores the displacement and velocity profiles of the block-pad brake dynamical system.

Nonequilibrium dynamics of coupled oscillators under the shear-velocity boundary condition

Deterministic and stochastic coupled oscillators with inertia are studied on the rectangular lattice under the shear-velocity boundary condition. Our coupled oscillator model exhibits various nontrivial phenomena and there are various relationships with wide research areas such as the coupled limit-cycle oscillators, the dislocation theory, a block-spring model of earthquakes, and the nonequilibrium molecular dynamics. We show numerically several unique nonequilibrium properties of the coupled oscillators. We find that the spatial profiles of the average value and variance of the velocity become nonuniform when the dissipation rate is large. The probability distribution of the velocity sometimes deviates from the Gaussian distribution. The time evolution of kinetic energy becomes intermittent when the shear rate is small and the temperature is small but not zero. The intermittent jumps of the kinetic energy cause a long tail in the velocity distribution.

Inhomogeneity effects on earthquake fault events

We present a detailed analysis of the dynamical behavior of an inhomogeneous Burridge-Knopoff model, a simplified mechanical model of an earthquake. Regardless of the size of seismic faults, a soil element rarely has a continuous appearance. Instead, their surfaces have complex structures. Thus, the model we suggest keeps the full Newtonian dynamics with inertial effects of the original model, while incorporating the inhomogeneities of seismic fault surfaces in stick-slip friction force that depends on the local structure of the contact surfaces as shown in recent experiments. The numerical results of the proposed model show that the cluster size and the moment distributions of earthquake events are in agreement with the Gutenberg-Richter law without introducing any relaxation mechanism. The exponent of the power-law size distribution we obtain falls within a realistic range of value without fine tuning any parameter. On the other hand, we show that the size distribution of both localized and delocalized events obeys a power law in contrast to the homogeneous case. Thus, no crossover behavior between small and large events occurs.

A scalable electronic analog of the Burridge-Knopoff model of earthquake faults

The Burridge-Knopoff model implements an earthquake fault as a mechanical block-spring chain. While numerical studies of the model are abundant, experimental investigations are limited to a two-blocks, analog electronic implementation that was proposed by drawing an analogy between mechanical and electrical quantities. Although elegant, this approach is not versatile, mostly because of its heavy reliance on inductors. Here, we propose an alternative, inductorless implementation of the same system. The experimental characterization of the proposed circuit shows very good agreement with theoretical predictions. Besides periodic oscillations, the circuit exhibits a chaotic regime: the corresponding markers of chaoticity, namely, the correlation dimension and the maximum Lyapunov exponent, were experimentally assessed to be consistent with those provided by numerical simulations. The improved versatility and scalability of the circuit is expected to allow for experimental implementations of the Burridge-Knopoff model with a large number of blocks. In addition, the circuit can be used as the basic element of scalable platforms to investigate the dynamics of networks of oscillators and related phenomena.

Shear Strain-Induced Two-Dimensional Slip Avalanches in Rhombohedral MoS2

Slip avalanches are ubiquitous phenomena occurring in three-dimensional materials under shear strain, and their study contributes immensely to our understanding of plastic deformation, fragmentation, and earthquakes. So far, little is known about the role of shear strain in two-dimensional (2D) materials. Here we show some evidence of 2D slip avalanches in exfoliated rhombohedral MoS2, triggered by shear strain near the threshold level. Utilizing interfacial polarization in 3R-MoS2, we directly probe the stacking order in multilayer flakes and discover a wide variety of polarization domains with sizes following a power-law distribution. These findings suggest that slip avalanches can occur during the exfoliation of 2D materials, and the stacking orders can be changed via shear strain. Our observation has far-reaching implications for the development of new materials and technologies, where precise control over the atomic structure of these materials is essential for optimizing their properties as well as for our understanding of fundamental physical phenomena.

Hidden jerk in universal creep and aftershocks

Most materials exhibit creep memory under the action of a constant load. The memory behavior is governed by Andrade's creep law, which also has an inherent connection with the Omori-Utsu law of earthquake aftershocks. Both empirical laws lack a deterministic interpretation. Coincidentally, the Andrade law is similar to the time-varying part of the creep compliance of the fractional dashpot in anomalous viscoelastic modeling. Consequently, fractional derivatives are invoked, but since they lack a physical interpretation, the physical parameters of the two laws extracted from curve fit lack confidence. In this Letter, we establish an analogous linear physical mechanism that underlies both laws and relates its parameters with the material's macroscopic properties. Surprisingly, the explanation does not require the property of viscosity. Instead, it necessitates the existence of a rheological property that relates strain with the first order time derivative of stress, which involves jerk. Further, we justify the constant quality factor model of acoustic attenuation in complex media. The obtained results are validated in light of the established observations.

State-dependent driving: a route to non-equilibrium stationary states

We study three different experiments that involve dry friction and periodic driving, and which employ both single- and many-particle systems. These experimental set-ups, besides providing a playground for investigation of frictional effects, are relevant in broad areas of science and engineering. Across all these experiments, we monitor the dynamics of objects placed on a substrate that is being moved in a horizontal manner. The driving couples to the degrees of freedom of the substrate and this coupling in turn influences the motion of the objects. Our experimental findings suggest emergence of stationary-states with non-trivial features. We invoke a minimalistic phenomenological model to explain our experimental findings. Within our model, we treat the injection of energy into the system to be dependent on its dynamical state, whereby energy injection is allowed only when the system is in its suitable-friction state. Our phenomenological model is built on the fact that such a state-dependent driving results in a force that repeatedly toggles the frictional states in time and serves to explain our experimental findings.

Universal dynamo paradigm for solar activity, Higgs fields and disasters

There is still a problem of a correct and accurate description of the dynamo and its uses in various fields of physics. To solve this problem, a special and universal representation of dynamo is proposed. The magnetic induction equation of dynamo is presented in the form of a Lienard relaxation oscillator with cubic nonlinear restoring force corresponding to the Mexican hat or champagne bottle potential which is used to determine the Higgs fields which are considered here in its general sense. Universal dynamo paradigm in field theory which can be used to describe disasters is proposed. Using solar activity as an example, it is shown how a dynamo induces a magnetic analogue of the Higgs fields with a broken symmetry of the magnetic field. Various dynamo modes are considered and different dynamo numbers are estimated. The dynamo effect can be used in field theory as an alternative to spontaneous symmetry breaking. Opportunities for the promotion of the new dynamo paradigm are discussed.

Numerical evidences of almost convergence of wave speeds for the Burridge–Knopoff model

This paper deals with the numerical approximation of a stick–slip system, known in the literature as Burridge–Knopoff model, proposed as a simplified description of the mechanisms generating earthquakes. Modelling of friction is crucial and we consider here the so-called velocity-weakening form. The aim of the article is twofold. Firstly, we establish the effectiveness of the classical Predictor–Corrector strategy. To our knowledge, such approach has never been applied to the model under investigation. In the first part, we determine the reliability of the proposed strategy by comparing the results with a collection of significant computational tests, starting from the simplest configuration to the more complicated (and more realistic) ones, with the numerical outputs obtained by different algorithms. Particular emphasis is laid on the Gutenberg–Richter statistical law, a classical empirical benchmark for seismic events. The second part is inspired by the result by Muratov (Phys Rev 59:3847–3857, 1999) providing evidence for the existence of traveling solutions for a corresponding continuum version of the Burridge–Knopoff model. In this direction, we aim to find some appropriate estimate for the crucial object describing the wave, namely its propagation speed. To this aim, motivated by LeVeque and Yee (J Comput Phys 86:187–210, 1990) (a paper dealing with the different topic of conservation laws), we apply a space-averaged quantity (which depends on time) for determining asymptotically an explicit numerical estimate for the velocity, which we decide to name LeVeque–Yee formula after the authors’ name of the original paper. As expected, for the Burridge–Knopoff, due to its inherent discontinuity of the process, it is not possible to attach to a single seismic event any specific propagation speed. More regularity is expected by performing some temporal averaging in the spirit of the Cesàro mean. In this direction, we observe the numerical evidence of the almost convergence of the wave speeds for the Burridge–Knopoff model of earthquakes.

Complexity and self-organized criticality in liquid foams. A short review

This short review deals with the work done on liquid foams within the framework of the physics of complexity. It aims to stimulate new theoretical and experimental work on foam dynamics as complex dynamical systems. In particular, it examines these systems in relation to Self-Organized Criticality (SOC), for which foams could be used as an accessible experimental model system.

Debonding waves in gel thin films

We develop a mathematical model for the sliding of a gel sheet adhered to a moving substrate. The sliding takes place by the motion of detached region between the gel sheet and the substrates, i.e. the propagation of a Schallamach wave. Efficient numerical methods are developed to solve the problem. Numerical examples illustrate that the model can describe the Schallamach wave and are consistent with the existing experiments qualitatively.

Detecting depinning and nonequilibrium transitions with unsupervised machine learning

Using numerical simulations of a model disk system, we demonstrate that a machine learning generated order-parameter-like measure can detect depinning transitions and different dynamic flow phases in systems driven far from equilibrium. We specifically consider monodisperse passive disks with short range interactions undergoing a depinning phase transition when driven over quenched disorder. The machine learning derived order-parameter-like measure identifies the depinning transition as well as different dynamical regimes, such as the transition from a flowing liquid to a phase separated liquid-solid state that is not readily distinguished with traditional measures such as velocity-force curves or Voronoi tessellation. The order-parameter-like measure also shows markedly distinct behavior in the limit of high density where jamming effects occur. Our results should be general to the broad class of particle-based systems that exhibit depinning transitions and nonequilibrium phase transitions.

On the Statistical Significance of the Variability Minima of the Order Parameter of Seismicity by Means of Event Coincidence Analysis

Natural time analysis has led to the introduction of an order parameter for seismicity when considering earthquakes as critical phenomena. The study of the fluctuations of this order parameter has shown that its variability exhibits minima before strong earthquakes. In this paper, we evaluate the statistical significance of such minima by using the recent method of event coincidence analysis. Our study includes the variability minima identified before major earthquakes in Japan and Eastern Mediterranean as well as in global seismicity.

Nature of the high-speed rupture of the two-dimensional Burridge-Knopoff model of earthquakes

The nature of the high-speed rupture or the main shock of the Burridge-Knopoff spring-block model in two dimensions obeying the rate- and state-dependent friction law is studied by means of extensive computer simulations. It is found that the rupture propagation in larger events is highly anisotropic and irregular in shape on longer length scales, although the model is completely uniform and the emergent rupture-propagation velocity is nearly constant everywhere at the rupture front. The manner of the rupture propagation sometimes mimics the successive ruptures of neighbouring 'asperities' observed in real, large earthquakes. Large events tend to be unilateral, with its epicentre lying at the rim of its rupture zone. The epicentre site of a large event is also located next to the rim of the rupture zone of some past event. Event-size distributions are computed and discussed in comparison with those of the corresponding one-dimensional model. The magnitude distribution exhibits a power-law behaviour resembling the Gutenberg-Richter law for smaller magnitudes, which changes over to a more characteristic behaviour for larger magnitudes. For very large events, the rupture-length distribution exhibits mutually different behaviours in one dimension and in two dimensions, reflecting the difference in the underlying geometry.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

Clogging and depinning of ballistic active matter systems in disordered media

We numerically examine ballistic active disks driven through a random obstacle array. Formation of a pinned or clogged state occurs at much lower obstacle densities for the active disks than for passive disks. As a function of obstacle density, we identify several distinct phases including a depinned fluctuating cluster state, a pinned single-cluster or jammed state, a pinned multicluster state, a pinned gel state, and a pinned disordered state. At lower active disk densities, a drifting uniform liquid forms in the absence of obstacles, but when even a small number of obstacles are introduced, the disks organize into a pinned phase-separated cluster state in which clusters nucleate around the obstacles, similar to a wetting phenomenon. We examine how the depinning threshold changes as a function of disk or obstacle density and find a crossover from a collectively pinned cluster state to a disordered plastic depinning transition as a function of increasing obstacle density. We compare this to the behavior of nonballistic active particles and show that as we vary the activity from completely passive to completely ballistic, a clogged phase-separated state appears in both the active and passive limits, while for intermediate activity, a readily flowing liquid state appears and there is an optimal activity level that maximizes the flux through the sample.

Avalanche precursors in a frictional model

We present a one-dimensional numerical model based on elastically coupled sliders on a frictional incline of variable tilt. This very simple approach makes it possible to study the precursors to the avalanche and to provide a rationalization of different features that have been observed in experiments. We provide a statistical description of the model leading to master equations describing the state of the system as a function of the angle of inclination. Our central results are the reproduction of large-scale regular events preceding the avalanche, on the one hand, and an analytical approach providing an internal threshold for the outbreak of rearrangements before the avalanche in the system, on the other hand.

Statistical properties of the one-dimensional Burridge-Knopoff model of earthquakes obeying the rate- and state-dependent friction law

Statistical properties of the one-dimensional spring-block (Burridge-Knopoff) model of earthquakes obeying the rate- and state-dependent friction law are studied by extensive computer simulations. The quantities computed include the magnitude distribution, the rupture-length distribution, the main shock recurrence-time distribution, the seismic-time correlations before and after the main shock, the mean slip amount, and the mean stress drop at the main shock, etc. Events of the model can be classified into two distinct categories. One tends to be unilateral with its epicenter located at the rim of the rupture zone of the preceding event, while the other tends to be bilateral with enhanced "characteristic" features resembling the so-called "asperity." For both types of events, the distribution of the rupture length L_{r} exhibits an exponential behavior at larger sizes, ≈exp[-L_{r}/L_{0}] with a characteristic "seismic correlation length" L_{0}. The mean slip as well as the mean stress drop tends to be rupture-length independent for larger events. The continuum limit of the model is examined, where the model is found to exhibit pronounced characteristic features. In the continuum limit, the characteristic rupture length L_{0} is estimated to be ∼100 [km]. This means that, even in a hypothetical homogenous infinite fault, events cannot be indefinitely large in the exponential sense, the upper limit being of order ∼10^{3} kilometers. Implications to real seismicity are discussed.

Inertia and universality of avalanche statistics: The case of slowly deformed amorphous solids

By means of a finite elements technique we solve numerically the dynamics of an amorphous solid under deformation in the quasistatic driving limit. We study the noise statistics of the stress-strain signal in the steady-state plastic flow, focusing on systems with low internal dissipation. We analyze the distributions of avalanche sizes and durations and the density of shear transformations when varying the damping strength. In contrast to avalanches in the overdamped case, dominated by the yielding point universal exponents, inertial avalanches are controlled by a nonuniversal damping-dependent feedback mechanism, eventually turning negligible the role of correlations. Still, some general properties of avalanches persist and new scaling relations can be proposed.

Depinning and nonequilibrium dynamic phases of particle assemblies driven over random and ordered substrates: a review

We review the depinning and nonequilibrium phases of collectively interacting particle systems driven over random or periodic substrates. This type of system is relevant to vortices in type-II superconductors, sliding charge density waves, electron crystals, colloids, stripe and pattern forming systems, and skyrmions, and could also have connections to jamming, glassy behaviors, and active matter. These systems are also ideal for exploring the broader issues of characterizing transient and steady state nonequilibrium flow phases as well as nonequilibrium phase transitions between distinct dynamical phases, analogous to phase transitions between different equilibrium states. We discuss the differences between elastic and plastic depinning on random substrates and the different types of nonequilibrium phases which are associated with specific features in the velocity-force curves, fluctuation spectra, scaling relations, and local or global particle ordering. We describe how these quantities can change depending on the dimension, anisotropy, disorder strength, and the presence of hysteresis. Within the moving phase we discuss how there can be a transition from a liquid-like state to dynamically ordered moving crystal, smectic, or nematic states. Systems with periodic or quasiperiodic substrates can have multiple nonequilibrium second or first order transitions in the moving state between chaotic and coherent phases, and can exhibit hysteresis. We also discuss systems with competing repulsive and attractive interactions, which undergo dynamical transitions into stripes and other complex morphologies when driven over random substrates. Throughout this work we highlight open issues and future directions such as absorbing phase transitions, nonequilibrium work relations, inertia, the role of non-dissipative dynamics such as Magnus effects, and how these results could be extended to the broader issues of plasticity in crystals, amorphous solids, and jamming phenomena.

Power-law rheology controls aftershock triggering and decay

The occurrence of aftershocks is a signature of physical systems exhibiting relaxation phenomena. They are observed in various natural or experimental systems and usually obey several non-trivial empirical laws. Here we consider a cellular automaton realization of a nonlinear viscoelastic slider-block model in order to infer the physical mechanisms of triggering responsible for the occurrence of aftershocks. We show that nonlinear viscoelasticity plays a critical role in the occurrence of aftershocks. The model reproduces several empirical laws describing the statistics of aftershocks. In case of earthquakes, the proposed model suggests that the power-law rheology of the fault gauge, underlying lower crust, and upper mantle controls the decay rate of aftershocks. This is verified by analysing several prominent aftershock sequences for which the rheological properties of the underlying crust and upper mantle were established.

Stable and unstable development of an interfacial sliding instability

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

Phase response curves for models of earthquake fault dynamics

We systematically study effects of external perturbations on models describing earthquake fault dynamics. The latter are based on the framework of the Burridge-Knopoff spring-block system, including the cases of a simple mono-block fault, as well as the paradigmatic complex faults made up of two identical or distinct blocks. The blocks exhibit relaxation oscillations, which are representative for the stick-slip behavior typical for earthquake dynamics. Our analysis is carried out by determining the phase response curves of first and second order. For a mono-block fault, we consider the impact of a single and two successive pulse perturbations, further demonstrating how the profile of phase response curves depends on the fault parameters. For a homogeneous two-block fault, our focus is on the scenario where each of the blocks is influenced by a single pulse, whereas for heterogeneous faults, we analyze how the response of the system depends on whether the stimulus is applied to the block having a shorter or a longer oscillation period.

Rupture dynamics in model polymer systems

In this paper we explore the rupture dynamics of a model polymer system to capture the microscopic mechanism during relative motion of surfaces at the single polymer level. Our model is similar to the model for friction introduced by Filippov, Klafter, and Urbakh [Filippov et al., Phys. Rev. Lett., 2004, 92, 135503]; but with an important generalization to a flexible transducer (modelled as a bead spring polymer) which is attached to a fixed rigid planar substrate by interconnecting bonds (modelled as harmonic springs), and pulled by a constant force FT. Bonds are allowed to rupture stochastically. The model is simulated, and the results for a certain set of parameters exhibit a sequential rupture mechanism resulting in rupture fronts. A mean field formalism is developed to study these rupture fronts and the possible propagating solutions for the coupled bead and bond dynamics, where the coupling excludes an exact analytical treatment. Numerical solutions to mean field equations are obtained by standard numerical techniques, and they agree well with the simulation results which show sequential rupture. Within a travelling wave formalism based on the Tanh method, we show that the velocity of the rupture front can be obtained in closed form. The derived expression for the rupture front velocity gives good agreement with the stochastic and mean field results, when the rupture is sequential, while propagating solutions for bead and bond dynamics are shown to agree under certain conditions.

An empirically based steady state friction law and implications for fault stability

Empirically based rate-and-state friction laws (RSFLs) have been proposed to model the dependence of friction forces with slip and time. The relevance of the RSFL for earthquake mechanics is that few constitutive parameters define critical conditions for fault stability (i.e., critical stiffness and frictional fault behavior). However, the RSFLs were determined from experiments conducted at subseismic slip rates (V < 1 cm/s), and their extrapolation to earthquake deformation conditions (V > 0.1 m/s) remains questionable on the basis of the experimental evidence of (1) large dynamic weakening and (2) activation of particular fault lubrication processes at seismic slip rates. Here we propose a modified RSFL (MFL) based on the review of a large published and unpublished data set of rock friction experiments performed with different testing machines. The MFL, valid at steady state conditions from subseismic to seismic slip rates (0.1 µm/s < V < 3 m/s), describes the initiation of a substantial velocity weakening in the 1-20 cm/s range resulting in a critical stiffness increase that creates a peak of potential instability in that velocity regime. The MFL leads to a new definition of fault frictional stability with implications for slip event styles and relevance for models of seismic rupture nucleation, propagation, and arrest.

Aftershocks and Omori's law in a modified Carlson-Langer model with nonlinear viscoelasticity

A modified Carlson-Langer model for earthquakes is proposed, which includes nonlinear viscoelasticity. Several aftershocks are generated after the main shock owing to the damping of the additional viscoelastic force. Both the Gutenberg-Richter law and Omori's law are reproduced in a numerical simulation of the modified Carlson-Langer model on a critical percolation cluster of a square lattice.

Foreshock and aftershocks in simple earthquake models

Many models of earthquake faults have been introduced that connect Gutenberg-Richter (GR) scaling to triggering processes. However, natural earthquake fault systems are composed of a variety of different geometries and materials and the associated heterogeneity in physical properties can cause a variety of spatial and temporal behaviors. This raises the question of how the triggering process and the structure interact to produce the observed phenomena. Here we present a simple earthquake fault model based on the Olami-Feder-Christensen and Rundle-Jackson-Brown cellular automata models with long-range interactions that incorporates a fixed percentage of stronger sites, or asperity cells, into the lattice. These asperity cells are significantly stronger than the surrounding lattice sites but eventually rupture when the applied stress reaches their higher threshold stress. The introduction of these spatial heterogeneities results in temporal clustering in the model that mimics that seen in natural fault systems along with GR scaling. In addition, we observe sequences of activity that start with a gradually accelerating number of larger events (foreshocks) prior to a main shock that is followed by a tail of decreasing activity (aftershocks). This work provides further evidence that the spatial and temporal patterns observed in natural seismicity are strongly influenced by the underlying physical properties and are not solely the result of a simple cascade mechanism.

Dynamics of intermittent force fluctuations in vesicular nanotubulation

Irregular force fluctuations are seen in most nanotubulation experiments. The dynamics behind their presence has, however, been neither commented upon nor modeled. A simple estimate of the mean energy dissipated in force drops turns out to be several times the thermal energy. This coupled with the rate dependent nature of the deformation reported in several experiments point to a dynamical origin of the serrations. We simplify the whole process of tether formation through a three-stage model of successive deformations of sphere to ellipsoid, neck-formation, and tubule birth and extension. Based on this, we envisage a rate-softening frictional force at the neck that must be overcome before a nanotube can be pulled out. Our minimal model includes elastic and visco-elastic deformation of the vesicle, and has built-in dependence on pull velocity, vesicle radius, and other material parameters, enabling us to capture various kinds of serrated force-extension curves for different parameter choices. Serrations are predicted in the nanotubulation region. Other features of force-extension plots reported in the literature such as a plateauing serrated region beyond a force drop, serrated flow region with a small positive slope, an increase in the elastic threshold with pull velocity, force-extension curves for vesicles with larger radius lying lower than those for smaller radius, are all also predicted by the model. A toy model is introduced to demonstrate that the role of the friction law is limited to inducing stick-slip oscillations in the force, and all other qualitative and quantitative features emerging from the model can only be attributed to other physical mechanisms included in the deformation dynamics of the vesicle.

Non-monotonic dependence of the friction coefficient on heterogeneous stiffness

The complexity of the frictional dynamics at the microscopic scale makes difficult to identify all of its controlling parameters. Indeed, experiments on sheared elastic bodies have shown that the static friction coefficient depends on loading conditions, the real area of contact along the interfaces and the confining pressure. Here we show, by means of numerical simulations of a 2D Burridge-Knopoff model with a simple local friction law, that the macroscopic friction coefficient depends non-monotonically on the bulk elasticity of the system. This occurs because elastic constants control the geometrical features of the rupture fronts during the stick-slip dynamics, leading to four different ordering regimes characterized by different orientations of the rupture fronts with respect to the external shear direction. We rationalize these results by means of an energetic balance argument.

Aftershocks in a frictional earthquake model

Inspired by spring-block models, we elaborate a "minimal" physical model of earthquakes which reproduces two main empirical seismological laws, the Gutenberg-Richter law and the Omori aftershock law. Our point is to demonstrate that the simultaneous incorporation of aging of contacts in the sliding interface and of elasticity of the sliding plates constitutes the minimal ingredients to account for both laws within the same frictional model.

Effect of inertia on sheared disordered solids: critical scaling of avalanches in two and three dimensions

Molecular dynamics simulations with varying damping are used to examine the effects of inertia and spatial dimension on sheared disordered solids in the athermal quasistatic limit. In all cases the distribution of avalanche sizes follows a power law over at least three orders of magnitude in dissipated energy or stress drop. Scaling exponents are determined using finite-size scaling for systems with 10(3)-10(6) particles. Three distinct universality classes are identified corresponding to overdamped and underdamped limits, as well as a crossover damping that separates the two regimes. For each universality class, the exponent describing the avalanche distributions is the same in two and three dimensions. The spatial extent of plastic deformation is proportional to the energy dissipated in an avalanche. Both rise much more rapidly with system size in the underdamped limit where inertia is important. Inertia also lowers the mean energy of configurations sampled by the system and leads to an excess of large events like that seen in earthquake distributions for individual faults. The distribution of stress values during shear narrows to zero with increasing system size and may provide useful information about the size of elemental events in experimental systems. For overdamped and crossover systems the stress variation scales inversely with the square root of the system size. For underdamped systems the variation is determined by the size of the largest events.

Dynamics of polymer molecules with sacrificial bond and hidden length systems: towards a physically-based mesoscopic constitutive law

We investigate the entropic force-elongation behavior of a polymer chain in the presence of the sacrificial bond and hidden length (SBHL) system observed experimentally in many biomaterials. We show that in most cases the SBHL system leads to a significant increase in toughness. However, the presence of a large number of bonds or relatively strong bonds in the SBHL system can reduce the net gain in toughness. We also incorporate the polymer model into a network of polymers with random properties (e.g., contour length, number and strength of sacrificial bonds, length of hidden loops). This allows us to derive a physically-based mesoscopic force-displacement law that governs the collective behavior.

Chaos on the conveyor belt

The dynamics of a spring-block train placed on a moving conveyor belt is investigated both by simple experiments and computer simulations. The first block is connected by a spring to an external static point and, due to the dragging effect of the belt, the blocks undergo complex stick-slip dynamics. A qualitative agreement with the experimental results can be achieved only by taking into account the spatial inhomogeneity of the friction force on the belt's surface, modeled as noise. As a function of the velocity of the conveyor belt and the noise strength, the system exhibits complex, self-organized critical, sometimes chaotic, dynamics and phase transition-like behavior. Noise-induced chaos and intermittency is also observed. Simulations suggest that the maximum complexity of the dynamical states is achieved for a relatively small number of blocks (around five).

Dynamical model of faulting in two dimensions and self-healing of large fractures

We describe a model for the simulation of extended two-dimensional in-plane dynamical ruptures and for the rapid calculation of statistical properties of repeated model-seismicity events. The discretization involves first- and second-nearest neighbors and is isotropic in both compression and shear properties. All rupture events obey a fracture criterion in the appropriate coordinate frame and numerical oscillations in slip velocity at crack tips due to discretization are minimized. The rupture velocities of fractures, in cases of homogeneous stress drop equal to the strength, are the supershear P-wave velocity in the direction of the prestress and the S-wave velocity in the perpendicular direction. We use the model to study the growth and healing of individual faults to understand the formation of propagating slip pulses. We confirm two mechanisms for the generation of isolated rupture pulses that have been proposed, namely, (1) a decrease in the dynamical friction with accelerating slip and (2) the encounter of the growing crack with extended regions of large difference between the threshold fracture stress and the prestress. We describe a third mechanism which is that of a velocity-dependent friction that operates equally on both the phases of increasing and decreasing slip velocities and has a characteristic length scale. It is a proxy for energy loss by radiation in a three-dimensional medium. In the case of an elongated rectangular model fault with an upper free surface and lower rigid boundary, pulses develop due to the influence of stress waves reflected from the rigid bottom boundary. In general, the excess of strength over stress drop controls crack fracture speeds; if it is too large, the crack stops. Under homogeneous stress conditions, isolated slip pulses are controlled by the spatial distribution of heterogeneities and by the velocity-dependent friction parametrization.

Avalanches in strained amorphous solids: does inertia destroy critical behavior?

Simulations are used to determine the effect of inertia on athermal shear of amorphous two-dimensional solids. In the quasistatic limit, shear occurs through a series of rapid avalanches. The distribution of avalanches is analyzed using finite-size scaling with thousands to millions of disks. Inertia takes the system to a new underdamped universality class rather than driving the system away from criticality as previously thought. Scaling exponents are determined for the underdamped and overdamped limits and a critical damping that separates the two regimes. Systems are in the overdamped universality class even when most vibrational modes are underdamped.

Observation of spatio-temporal structure in stick-slip motion of an adhesive gel sheet

We studied the sliding friction between an adhesive gel sheet and a glass substrate. In this system, the probability distribution of the force drop obeys a power law similar to that found in earthquakes and granular systems. We observed the motion of the slip regions at the frictional interfaces and obtained the spatial distributions of shear strain by image analysis. The frictional force evaluated by the image analysis is in good agreement with the actual force measured by a load cell. This indicates that the present method provides a powerful tool to study the spatio-temporal structure in the heterogeneous stick-slip motions in sliding friction.

Master equation approach to friction at the mesoscale

At the mesoscale friction occurs through the breaking and formation of local contacts. This is often described by the earthquakelike model which requires numerical studies. We show that this phenomenon can also be described by a master equation, which can be solved analytically in some cases and provides an efficient numerical solution for more general cases. We examine the effect of temperature and aging of the contacts and discuss the statistical properties of the contacts for different situations of friction and their implications, particularly regarding the existence of stick-slip.

Plasticity of ductile metallic glasses: a self-organized critical state

We report a close correlation between the dynamic behavior of serrated flow and the plasticity in metallic glasses (MGs) and show that the plastic deformation of ductile MGs can evolve into a self-organized critical state characterized by the power-law distribution of shear avalanches. A stick-slip model considering the interaction of multiple shear bands is presented to reveal complex scale-free intermittent shear-band motions in ductile MGs and quantitatively reproduce the experimental observations. Our studies have implications for understanding the precise plastic deformation mechanism of MGs.

Asperity characteristics of the Olami-Feder-Christensen model of earthquakes

Properties of the Olami-Feder-Christensen (OFC) model of earthquakes are studied by numerical simulations. The previous study indicated that the model exhibited "asperity"-like phenomena, i.e., the same region ruptures many times near periodically [T. Kotani, Phys. Rev. E 77, 010102(R) (2008)]. Such periodic or characteristic features apparently coexist with power-law-like critical features, e.g., the Gutenberg-Richter law observed in the size distribution. In order to clarify the origin and the nature of the asperity-like phenomena, we investigate here the properties of the OFC model with emphasis on its stress distribution. It is found that the asperity formation is accompanied by self-organization of the highly concentrated stress state. Such stress organization naturally provides the mechanism underlying our observation that a series of asperity events repeat with a common epicenter site and with a common period solely determined by the transmission parameter of the model. Asperity events tend to cluster both in time and in space.

Dynamics of transition from static to kinetic friction

We propose a model for a description of dynamics of cracklike processes that occur at the interface between two blocks prior to the onset of frictional motion. We find that the onset of sliding is preceded by well-defined detachment fronts initiated at the slider trailing edge and extended across the slider over limited lengths smaller than the overall length of the slider. Three different types of detachment fronts may play a role in the onset of sliding: (i) Rayleigh (surface sound) fronts, (ii) slow detachment fronts, and (iii) fast fronts. The important consequence of the precursor dynamics is that before the transition to overall sliding occurs, the initially uniform, unstressed slider is already transformed into a highly nonuniform, stressed state. Our model allows us to explain experimental observations and predicts the effect of material properties on the dynamics of the transition to sliding.

Binding site models of friction due to the formation and rupture of bonds: state-function formalism, force-velocity relations, response to slip velocity transients, and slip stability

We present a model describing friction due to the thermally activated formation and rupture of molecular bonds between two surfaces, with long molecules on one surface attaching to discrete or continuous binding sites on the other. The physical assumptions underlying this model are formalized using a continuum approximation resulting in a class of master-equation-like partial differential equations that is a generalization of a friction model due to Persson [Phys. Rev. B 51, 13568 (1995)] and is identical to the equations used to describe muscle contraction, first proposed by A. F. Huxley. We examine the properties of this friction model in the continuous binding site limit noting that this model is capable of producing both monotonically increasing and an increasing-decreasing force dependence on slip velocity. When monotonically increasing, the force dependence on velocity is (asymptotically) logarithmic. The model produces a transient increase in friction in response to a sudden velocity increase, whether or not the steady-state friction force is a decreasing or increasing function of steady slip velocity. The model also exhibits both stable steady slip and stick-slip-like oscillatory behavior, in the presence of compliance in the loading machine, even when the steady-state friction force is a decreasing function of steady-state slip velocity.