Intrinsic properties of a Burridge-Knopoff model of an earthquake fault

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.

Viscoelastic Slider Blocks as a Model for a Seismogenic Fault

In this work, a model is proposed to examine the role of viscoelasticity in the generation of simulated earthquake-like events. This model serves to investigate how nonlinear processes in the Earth's crust affect the triggering and decay patterns of earthquake sequences. These synthetic earthquake events are numerically simulated using a slider-block model containing viscoelastic standard linear solid (SLS) elements to reproduce the dynamics of an earthquake fault. The simulated system exhibits elements of self-organized criticality, and results in the generation of avalanches that behave similarly to naturally occurring seismic events. The model behavior is analyzed using the Epidemic-Type Aftershock Sequence (ETAS) model, which suitably represents the observed triggering and decay patterns; however, parameter estimates deviate from those resulting from natural aftershock sequences. Simulated aftershock sequences from this model are characterized by slightly larger p-values, indicating a faster-than-normal decay of aftershock rates within the system. The ETAS fit, along with realistic simulated frequency-size distributions, supports the inclusion of viscoelastic rheology to model the seismogenic fault dynamics.

Generic mechanism for remote triggering of earthquakes

"Remote triggering" refers to the inducement of earthquakes by weak perturbations that emanate from faraway sources, typically intense earthquakes that happen at much larger distances than their nearby aftershocks, sometimes even around the globe. Here, we propose a mechanism for this phenomenon; the proposed mechanism is generic, resulting from the breaking of Hamiltonian symmetry due to the existence of friction. We allow a transition from static to dynamic friction. Linearly stable stressed systems display giant sensitivity to small perturbations of arbitrary frequency (without a need for resonance), which trigger an instability with exponential oscillatory growth. Once nonlinear effects kick in, the blow up in mean-square displacements can reach 15-20 orders of magnitude. Analytic and numerical results of the proposed model are presented and discussed.

Unified view of avalanche criticality in sheared glasses

Plastic events in sheared glasses are considered an example of so-called avalanches, whose sizes obey a power-law probability distribution with the avalanche critical exponent τ. Although the so-called mean-field depinning (MFD) theory predicts a universal value of this exponent, τ_{MFD}=1.5, such a simplification is now known to connote qualitative disagreement with realistic systems. Numerically and experimentally, different values of τ have been reported depending on the literature. Moreover, in the elastic regime, it has been noted that the critical exponent can be different from that in the steady state, and even criticality itself is a matter of debate. Because these confusingly varying results have been reported under different setups, our knowledge of avalanche criticality in sheared glasses is greatly limited. To gain a unified understanding, in this work, we conduct a comprehensive numerical investigation of avalanches in Lennard-Jones glasses under athermal quasistatic shear. In particular, by excluding the ambiguity and arbitrariness that has crept into the conventional measurement schemes, we achieve high-precision measurement and demonstrate that the exponent τ in the steady state follows the prediction of MFD theory, τ_{MFD}=1.5. Our results also suggest that there are two qualitatively different avalanche events. This binariness leads to the nonuniversal behavior of the avalanche size distribution and is likely to be the cause of the varying values of τ reported thus far. To investigate the dependence of criticality and universality on applied shear, we further study the statistics of avalanches in the elastic regime and the ensemble of the first avalanche event in different samples, which provide information about the unperturbed system. We show that while the unperturbed system is indeed off-critical, criticality gradually develops as shear is applied. The degree of criticality is encoded in the fractal dimension of the avalanches, which starts from zero in the off-critical unperturbed state and saturates in the steady state. Moreover, the critical exponent τ is consistent with the prediction of the MFD τ_{MFD} universally, regardless of the amount of applied shear, once the system becomes critical.

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.

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'.

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.

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.

Speed of fast and slow rupture fronts along frictional interfaces

The transition from stick to slip at a dry frictional interface occurs through the breaking of microjunctions between the two contacting surfaces. Typically, interactions between junctions through the bulk lead to rupture fronts propagating from weak and/or highly stressed regions, whose junctions break first. Experiments find rupture fronts ranging from quasistatic fronts, via fronts much slower than elastic wave speeds, to fronts faster than the shear wave speed. The mechanisms behind and selection between these fronts are still imperfectly understood. Here we perform simulations in an elastic two-dimensional spring-block model where the frictional interaction between each interfacial block and the substrate arises from a set of junctions modeled explicitly. We find that material slip speed and rupture front speed are proportional across the full range of front speeds we observe. We revisit a mechanism for slow slip in the model and demonstrate that fast slip and fast fronts have a different, inertial origin. We highlight the long transients in front speed even along homogeneous interfaces, and we study how both the local shear to normal stress ratio and the local strength are involved in the selection of front type and front speed. Last, we introduce an experimentally accessible integrated measure of block slip history, the Gini coefficient, and demonstrate that in the model it is a good predictor of the history-dependent local static friction coefficient of the interface. These results will contribute both to building a physically based classification of the various types of fronts and to identifying the important mechanisms involved in the selection of their propagation speed.

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.

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.

State-variable friction for the Burridge-Knopoff model

This work shows the relationship of the state variable rock-friction law proposed by Dieterich to the Carlson and Langer friction law commonly used in the Burridge-Knopoff (BK) model of earthquakes. Further to this, the Dieterich law is modified to allow slip rates of zero magnitude yielding a three parameter friction law that is included in the BK system. Dynamic phases of small scale and large scale events are found with a transition surface in the parameter space. Near this transition surface the event size distribution follows a power law with an exponent that varies as the transition is approached contrasting with the invariant exponent observed using the Carlson and Langer friction. This variability of the power-law exponent is consistent with the range of exponents measured in real earthquake systems and is more selective than the range observed in the Olami-Feder-Christensen model.

Strain waves, earthquakes, slow earthquakes, and afterslip in the framework of the Frenkel-Kontorova model

The one-dimensional Frenkel-Kontorova (FK) model, well known from the theory of dislocations in crystal materials, is applied to the simulation of the process of nonelastic stress propagation along transform faults. Dynamic parameters of plate boundary earthquakes as well as slow earthquakes and afterslip are quantitatively described, including propagation velocity along the strike, plate boundary velocity during and after the strike, stress drop, displacement, extent of the rupture zone, and spatiotemporal distribution of stress and strain. The three fundamental speeds of plate movement, earthquake migration, and seismic waves are shown to be connected in framework of the continuum FK model. The magnitude of the strain wave velocity is a strong (almost exponential) function of accumulated stress or strain. It changes from a few km/s during earthquakes to a few dozen km per day, month, or year during afterslip and interearthquake periods. Results of the earthquake parameter calculation based on real data are in reasonable agreement with measured values. The distributions of aftershocks in this model are consistent with the Omori law for temporal distribution and a 1/r for the spatial distributions.

Simulation study of earthquakes based on the two-dimensional Burridge-Knopoff model with long-range interactions

Spatiotemporal correlations of the two-dimensional (2D) spring-block (Burridge-Knopoff) models of earthquakes with the long-range interblock interactions are extensively studied by means of numerical computer simulations. The long-range interaction derived from an elastic theory, which takes account of the effect of the elastic body adjacent to the fault plane, falls off with distance r as 1r;{3} . Comparison is made with the properties of the corresponding short-range models studied earlier. Seismic spatiotemporal correlations of the long-range models generally tend to be weaker than those of the short-range models. The magnitude distribution exhibits a "near-critical" behavior, i.e., a power-law-like behavior close to the Gutenberg-Richter law, for a wide parameter range with its B -value, B approximately 0.55 , insensitive to the model parameters, in sharp contrast to that of the 2D short-range model and those of the 1D short-range and long-range models where such a near-critical behavior is realized only by fine tuning the model parameters. In contrast to the short-range case, the mean stress drop at a seismic event of the long-range model is nearly independent of its magnitude, consistent with the observation. Large events often accompany foreshocks together with a doughnutlike quiescence as their precursors, while they hardly accompany aftershocks with almost negligible seismic correlations observed after the main shock.

Near-mean-field behavior in the generalized Burridge-Knopoff earthquake model with variable-range stress transfer

Simple models of earthquake faults are important for understanding the mechanisms for their observed behavior in nature, such as Gutenberg-Richter scaling. Because of the importance of long-range interactions in an elastic medium, we generalize the Burridge-Knopoff slider-block model to include variable range stress transfer. We find that the Burridge-Knopoff model with long-range stress transfer exhibits qualitatively different behavior than the corresponding long-range cellular automata models and the usual Burridge-Knopoff model with nearest-neighbor stress transfer, depending on how quickly the friction force weakens with increasing velocity. Extensive simulations of quasiperiodic characteristic events, mode-switching phenomena, ergodicity, and waiting-time distributions are also discussed. Our results are consistent with the existence of a mean-field critical point and have important implications for our understanding of earthquakes and other driven dissipative systems.

Interoccurrence time statistics in the two-dimensional Burridge-Knopoff earthquake model

We have numerically investigated statistical properties of the so-called interoccurrence time or the waiting time, i.e., the time interval between successive earthquakes, based on the two-dimensional (2D) spring-block (Burridge-Knopoff) model, selecting the velocity-weakening property as the constitutive friction law. The statistical properties of frequency distribution and the cumulative distribution of the interoccurrence time are discussed by tuning the dynamical parameters, namely, a stiffness and frictional property of a fault. We optimize these model parameters to reproduce the interoccurrence time statistics in nature; the frequency and cumulative distribution can be described by the power law and Zipf-Mandelbrot type power law, respectively. In an optimal case, the b value of the Gutenberg-Richter law and the ratio of wave propagation velocity are in agreement with those derived from real earthquakes. As the threshold of magnitude is increased, the interoccurrence time distribution tends to follow an exponential distribution. Hence it is suggested that a temporal sequence of earthquakes, aside from small-magnitude events, is a Poisson process, which is observed in nature. We found that the interoccurrence time statistics derived from the 2D BK (original) model can efficiently reproduce that of real earthquakes, so that the model can be recognized as a realistic one in view of interoccurrence time statistics.

Structure of fluctuations near mean-field critical points and spinodals and its implication for physical processes

We analyze the structure of fluctuations near critical points and spinodals in mean-field and near-mean-field systems. Unlike systems that are non-mean-field, for which a fluctuation can be represented by a single cluster in a properly chosen percolation model, a fluctuation in mean-field and near-mean-field systems consists of a large number of clusters, which we term fundamental clusters. The structure of the latter and the way that they form fluctuations has important physical consequences for phenomena as diverse as nucleation in supercooled liquids, spinodal decomposition and continuous ordering, and the statistical distribution of earthquakes. The effects due to the fundamental clusters implies that they are physical objects and not only mathematical constructs.

Burridge-Knopoff model: Exploration of dynamic phases

Slider-block models are often used to simulate earthquake dynamics. However, the models' origins are more conceptual than analytical. This study uses Navier's equations of an elastic bulk to derive a one-dimensional slider-block model, the Burridge-Knopoff model. This model exhibits a critical phase transition by varying the friction parameter. Accurate analytical estimates are made of event size limits for the small scale, large scale, and intermediate dynamic phases. The absence of large scale quasiperiodic delocalized events is noted for the parameter set investigated here. The time intervals between large scale events are approximately exponentially distributed for the system in its critical state, in agreement with the theory of nonequilibrium critical systems and earthquake dynamics.

Simulation of the Burridge-Knopoff model of earthquakes with variable range stress transfer

Simple models of earthquake faults are important for understanding the mechanisms for their observed behavior, such as Gutenberg-Richter scaling and the relation between large and small events, which is the basis for various forecasting methods. Although cellular automaton models have been studied extensively in the long-range stress transfer limit, this limit has not been studied for the Burridge-Knopoff model, which includes more realistic friction forces and inertia. We find that the latter model with long-range stress transfer exhibits qualitatively different behavior than both the long-range cellular automaton models and the usual Burridge-Knopoff model with nearest-neighbor springs, depending on the nature of the velocity-weakening friction force. These results have important implications for our understanding of earthquakes and other driven dissipative systems.

Criticality in the Burridge-Knopoff model

Criticality is a potential origin of the scale invariance observed in the Gutenberg-Richter law for earthquakes. In support of this hypothesis, the Burridge-Knopoff (BK) model of an earthquake fault system is known to exhibit a dynamic phase transition, but the critical nature of the transition is uncertain. Here it is shown that the BK model exhibits a dynamic transition from large-scale stick-slip to small-scale creep motion and through a finite size scaling analysis the critical nature of this transition is established. The order parameter describing the critical transition suggests that the Olami-Feder-Christensen model may be tuned to criticality through its assumptions describing the relaxation of the system.

Simulation study of spatiotemporal correlations of earthquakes as a stick-slip frictional instability

Spatiotemporal correlations of earthquakes are studied numerically on the basis of the one-dimensional spring-block (Burridge-Knopoff) model. As large events approach, the frequency of smaller events gradually increases, while, just before the mainshock, it is dramatically suppressed in a close vicinity of the epicenter of the upcoming mainshock, a phenomenon closely resembling the "Mogi doughnut."

Fluid limit of nonintegrable continuous-time random walks in terms of fractional differential equations

The fluid limit of a recently introduced family of nonintegrable (nonlinear) continuous-time random walks is derived in terms of fractional differential equations. In this limit, it is shown that the formalism allows for the modeling of the interaction between multiple transport mechanisms with not only disparate spatial scales but also different temporal scales. For this reason, the resulting fluid equations may find application in the study of a large number of nonlinear multiscale transport problems, ranging from the study of self-organized criticality to the modeling of turbulent transport in fluids and plasmas.

Giant stress fluctuations at the jamming transition

We study the stress response to a steady imposed shear rate in a concentrated suspension of colloidal particles. We show that, in a small range of concentrations and shear rates, stress exhibits giant fluctuations. The amplitude of these fluctuations obeys a power-law behavior, up to the apparition of a new branch of flow, leading to an excess of high amplitude fluctuations which exhibit a well-defined periodicity.

Scale invariance and invariant scaling in a mixed hierarchical system

We consider a mixed hierarchical model with heterogeneous and monotone conditions of destruction. We investigate how scaling properties of defects in the model are related with heterogeneity of rules of destruction, determined by concentration of the mixture. The system demonstrates different kinds of criticality as a general form of system behavior. The following forms of critical behavior are obtained: stability, catastrophe, scale invariance, and invariant scaling. Different slopes of the magnitude-frequency relation are realized in areas of critical stability and catastrophe. A simple relation between the slope of magnitude-frequency relation and parameters of the mixture is established.

Spinodals, scaling, and ergodicity in a threshold model with long-range stress transfer

We present both theoretical and numerical analyses of a cellular automaton version of a slider-block model or threshold model that includes long-range interactions. Theoretically we develop a coarse-grained description in the mean-field (infinite range) limit and discuss the relevance of the metastable state, limit of stability (spinodal), and nucleation to the phenomenology of the model. We also simulate the model and confirm the relevance of the theory for systems with long- but finite-range interactions. Results of particular interest include the existence of Gutenberg-Richter-like scaling consistent with that found on real earthquake fault systems, the association of large events with nucleation near the spinodal, and the result that such systems can be described, in the mean-field limit, with techniques appropriate to systems in equilibrium.

Slip complexity in dynamic models of earthquake faults

We summarize recent evidence that models of earthquake faults with dynamically unstable friction laws but no externally imposed heterogeneities can exhibit slip complexity. Two models are described here. The first is a one-dimensional model with velocity-weakening stick-slip friction; the second is a two-dimensional elastodynamic model with slip-weakening friction. Both exhibit small-event complexity and chaotic sequences of large characteristic events. The large events in both models are composed of Heaton pulses. We argue that the key ingredients of these models are reasonably accurate representations of the properties of real faults.

Dynamic friction and the origin of the complexity of earthquake sources

We study a simple antiplane fault of finite length embedded in a homogeneous isotropic elastic solid to understand the origin of seismic source heterogeneity in the presence of nonlinear rate- and state-dependent friction. All the mechanical properties of the medium and friction are assumed homogeneous. Friction includes a characteristic length that is longer than the grid size so that our models have a well-defined continuum limit. Starting from a heterogeneous initial stress distribution, we apply a slowly increasing uniform stress load far from the fault and we simulate the seismicity for a few 1000 events. The style of seismicity produced by this model is determined by a control parameter associated with the degree of rate dependence of friction. For classical friction models with rate-independent friction, no complexity appears and seismicity is perfectly periodic. For weakly rate-dependent friction, large ruptures are still periodic, but small seismicity becomes increasingly nonstationary. When friction is highly rate-dependent, seismicity becomes nonperiodic and ruptures of all sizes occur inside the fault. Highly rate-dependent friction destabilizes the healing process producing premature healing of slip and partial stress drop. Partial stress drop produces large variations in the state of stress that in turn produce earthquakes of different sizes. Similar results have been found by other authors using the Burridge and Knopoff model. We conjecture that all models in which static stress drop is only a fraction of the dynamic stress drop produce stress heterogeneity.