Burridge-Knopoff Model Research Articles

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.

Noise-induced chaos in an electronic Burridge-Knopoff model analog

Electronic analog circuits are valuable platforms to experimentally address physical systems whose implementation would otherwise be unwieldy. Besides validating numerical predictions, electronic analogs of nonlinear systems are also ideal tools to study the effect of noise injection in realistic conditions and are often more scalable than numerical approaches. In this work, we address the effect of noise injection in an electronic analog of the Burridge-Knopoff mechanical model of earthquake faults. Noise is injected into the friction term of the mechanical system: interestingly, the presence of a sufficient level of noise can trigger the onset of chaos in otherwise periodic regimes.

NUMERICAL SIMULATION OF DYNAMICS OF BLOCK MEDIA BY MOVABLE LATTICE AND MOVABLE AUTOMATA METHODS

Two versions of modified Burridge-Knopoff model including state dependent friction, elastic force and thermal conductivity are derived. The friction model describes a velocity weakening of friction and elasticity between moving blocks and an increase of both static friction and rigidity during stick periods as well their weakening during motion. It provides a simplified but qualitatively correct behavior including the transition from smooth sliding to stick-slip behavior, which is often observed in various tribological and tectonic systems. Attractor properties of the model dynamics is studied also. The alternative versions of the model are proposed which apply a simulation of the motion of interacting elastically connected mesh elements and motion of relatively large solid blocks, utilizing technique of the movable cellular automata. First version of the model was already basically studied before. Its advanced version here involves all components of the real system: state-depending friction and changeable rigidity, as well as heat production and thermal conductivity. Model based on the movable automata also involves the components included into traditional lattice model. It has its own ad-vantages and disadvantages which are also discussed in the paper.

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.

Statistical analogies between earthquakes, micro-quakes in metals and avalanches in the 1D Burridge-Knopoff model

Universalities and intriguing analogies in the statistics of avalanches are revealed for three physical systems defined on largely different length and energy scales. Earthquakes induced by tectonic scale dynamics, micro-scale level quakes observed from slipping crystallographic planes in metals and a one-dimensional, room-scale spring-block type Burridge-Knopoff model is studied from similar statistical viewpoints. The validity of the Gutenberg-Richter law for the probability density of the energies dissipated in the avalanches is proven for all three systems. By analysing data for three different seismic zones and performing acoustic detection for different Zn samples under deformation, universality for the involved scaling exponent is revealed. With proper parameter choices the 1D Burridge-Knopoff model is able to reproduce the same scaling law. The recurrence times of earthquakes and micro-quakes with magnitudes above a given threshold present again similar distributions and striking quantitative similarities. However, the 1D Burridge-Knopoff model cannot account for the correlations observed in such statistics.

On the possibility of rogue wave generation based on the dynamics of modified Burridge-Knopoff model of earthquake fault

In this study we propose a modified Burridge‐Knopoff model of earthquake fault, in which two tectonic plates are strongly coupled by nonlinear springs. By minimizing the effects of the veloci‐ ty‐weakening stick‐slip friction force between the masses and the moving surface, and in the limit of low amplitude oscillations; the system exhibits both stick‐slip and damped oscillatory motions as the values of some parameters are varied. Such motions usually characterize the dynamics of an earthquake fault, even though it is not always felt because of the low amplitude of vibrations. However when enough stress builds up in the subduction zones to overcome the frictional forces between tectonic plates, the oceanic rocks suddenly slip and there is violent release of energy at the epicentre. This outburst of energy simply signifies the generation of a very large amplitude and localized nonlinear wave. Such wave profile exactly fits the Peregrine solution of the damped/ forced nonlinear Schrodinger amplitude equation, derived from the modified one‐dimensional Burridge‐Knopoff equation of motion. In the regime of minimal or no frictional forces, these mon‐ ster waves suddenly appear and disappear without traces as shown by the numerical investigations. Our results strongly suggest that rogue waves emanates from the dynamics of nonlinearly coupled tectonic plates in subduction zones. This is further complemented by the fact that these giant waves were initially observed in Pacific and Atlantic oceans, which play hosts to the world’s largest oceanic subduction zones.

Sensitivity of a simple earthquake nucleation model to small parameter perturbation: Conditions for the occurrence of deterministic chaos

A new mechanism that may account for the onset of chaotic dynamics of earthquake faults is proposed and analyzed. The concept is to build on the Burridge-Knopoff model, which integrates the spring-block setup with the Dieterich-Ruina's rate- and state-dependent friction law to interpolate for the key aspects of earthquake episodes, including the seismic nucleation, fracture propagation and arrest, as well as the rupture healing. Results obtained indicate that deterministic chaos occur in case frictional parameters exhibit small oscillations about their equilibrium values. Based on the construction of appropriate phase portraits, power spectra and the Lyapunov exponents it could be concluded that a single time-dependent parameter is sufficient for the chaotic behavior to emerge, while the fully developed chaos is found when two perturbed parameters are brought into play.

Coupled mechano-electrokinetic Burridge-Knopoff model of fault sliding events and transient geoelectric signals

We introduce the first fully self-consistent numerical model combining seismic micro-ruptures, occurring within a generalized Burridge-Knopoff spring-block model, with the nucleation and propagation of electric charge pulses within a coupled mechano-electrokinetic system. This model provides a general theoretical framework for modeling and analyzing geoelectric anomalies to earthquakes. In particular, it is able to reproduce the unipolar pulses that have often been reported before large seismic events, as well as various previously reported anomalies in the statistical moments of the ambient electric fields and the shape of their power spectra.

A new type of relaxation oscillation in a model with rate-and-state friction

In this paper, we prove the existence of a new type of relaxation oscillation occurring in a one-block Burridge–Knopoff model with Ruina rate-and-state friction law. In the relevant parameter regime, the system is a slow-fast ordinary differential equation with two slow variables and one fast. The oscillation is special for several reasons: firstly, its singular limit is unbounded, the amplitude of the cycle growing like as . As this estimate reflects, the unboundedness of the cycle—for this non-polynomial system—cannot be captured by a simple -dependent scaling of the variables, in contrast to e.g. Gucwa and Szmolyan (2009 Discrete Continuous Dyn. Syst. S 2 783–806). We therefore obtain its limit on the Poincaré sphere. Here we find that the singular limit consists of a slow part on an attracting critical manifold, and a fast part on the equator (i.e. at ) of the Poincaré sphere, which includes motion along a center manifold. The reduced flow on this center manifold runs out along the manifold’s boundary, in a special way, leading to a complex return to the slow manifold. We prove the existence of the limit cycle by showing that a return map is a contraction. The main technical difficulty lies in the fact that the critical manifold loses hyperbolicity at an exponential rate at infinity. We therefore use the method in Kristiansen (2017 Nonlinearity 30 2138–84), applying the standard blowup technique in an extended phase space. In this way, we identify a singular cycle, consisting of 12 pieces, all with desirable hyperbolicity properties, that enables the perturbation into an actual limit cycle for . The result proves a conjecture in Bossolini et al (2017 Nonlinearity 30 2805–34). The Bossolini et al (2017 Nonlinearity 30 2805–34) also includes a preliminary analysis based on the approach in Kristiansen (2017 Nonlinearity 30 2138–84) but several details were missing. We provide all the details in the present manuscript and lay out the geometry of the problem, detailing all of the many blowup steps.

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'.

Slow-Slip Phenomena Represented by the One-Dimensional Burridge–Knopoff Model of Earthquakes

Slow-slip phenomena, including afterslips and silent earthquakes, are studied using a one-dimensional Burridge--Knopoff model that obeys the rate-and-state dependent friction law. By varying only a few model parameters, this simple model allows reproducing a variety of seismic slips within a single framework, including main shocks, precursory nucleation processes, afterslips, and silent earthquakes.

Sliding contact on the interface of elastic body and rigid surface using a single block Burridge-Knopoff model

Burridge and Knopoff proposed a mass-spring model to explore interface dynamics along a fault during an earthquake. The Burridge and Knopoff (BK) model is composed of a series of blocks of equal mass connected to each other by springs of same stiffness. The blocks also are attached to a rigid driver via another set of springs that pulls them at a constant velocity against a rigid substrate. They studied dynamics of interface for an especial case with ten blocks and a specific set of fault properties. In our study effects of Coulomb and rate-state dependent friction laws on the dynamics of a single block BK model is investigated. The model dynamics is formulated as a system of coupled nonlinear ordinary differential equations in state-space form which lends itself to numerical integration methods, e.g. Runge-Kutta procedure for solution. The results show that the rate and state dependent friction law has the potential of triggering dynamic patterns that are different from those under Coulomb law.

Nonlinear dynamics behind the seismic cycle: One-dimensional phenomenological modeling

In present paper, authors examine the dynamics of a spring-slider model, considered as a phenomenological setup of a geological fault motion. Research is based on an assumption of delayed interaction between the two blocks, which is an idea that dates back to original Burridge–Knopoff model. In contrast to this first model, group of blocks on each side of transmission zone (with delayed interaction) is replaced by a single block. Results obtained indicate predominant impact of the introduced time delay, whose decrease leads to transition from steady state or aseismic creep to seismic regime, where each part of the seismic cycle (co-seismic, post-seismic and inter-seismic) could be recognized. In particular, for coupling strength of order 102 observed system exhibit inverse Andronov–Hopf bifurcation for very small value of time delay, τ≈0.01, when long-period (T = 12) and high-amplitude oscillations occur. Further increase of time delay, of order 10−1, induces an occurrence of a direct Andronov–Hopf bifurcation, with short-period (T = 0.5) oscillations of approximately ten times smaller amplitude. This reduction in time delay could be the consequence of the increase of temperature due to frictional heating, or due to decrease of pressure which follows the sudden movement along the fault. Analysis is conducted for the parameter values consistent with previous laboratory findings and geological observations relevant from the seismological viewpoint.

Solitary waves in the excitable Burridge–Knopoff model

The Burridge–Knopoff model is a lattice differential equation describing a chain of blocks connected by springs and pulled over a surface. This model was originally introduced to investigate nonlinear effects arising in the dynamics of earthquake faults. One of the main ingredients of the model is a nonlinear velocity-dependent friction force between the blocks and the fixed surface. For some classes of non-monotonic friction forces, the system displays a large response to perturbations above a threshold, which is characteristic of excitable dynamics. Using extensive numerical simulations, we show that this response corresponds to the propagation of a solitary wave for a broad range of friction laws (smooth or nonsmooth) and parameter values. These solitary waves develop shock-like profiles at large coupling (a phenomenon connected with the existence of weak solutions in a formal continuum limit) and propagation failure occurs at low coupling. We introduce a simplified piecewise linear friction law (reminiscent of the McKean nonlinearity for excitable cells) which allows us to obtain an analytical expression of solitary waves and study some of their qualitative properties, such as wave speed and propagation failure. We propose a possible physical realization of this system as a chain of impulsively forced mechanical oscillators. In certain parameter regimes, non-monotonic friction forces can also give rise to bistability between the ground state and limit-cycle oscillations and allow for the propagation of fronts connecting these two stable states.

Traveling waves in a spring-block chain sliding down a slope.

Traveling waves are studied in a spring slider-block model. We explicitly construct front waves (kinks) for a piecewise-linear spinodal friction force. Pulse waves are obtained as the matching of two traveling fronts with identical speeds. Explicit formulas are obtained for the wavespeed and the wave form in the anticontinuum limit. The link with localized waves in a Burridge-Knopoff model of an earthquake fault is briefly discussed.

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.

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.

Implicit Euler simulation of one-dimensional Burridge-Knopoff model of earthquakes with set-valued friction laws

In the simulations of Burridge-Knopoff (BK) model of earthquakes, the friction force laws are important to produce earthquake-like stick-slip behaviors. Some friction force laws are set-valued and the BK model with them can produce consistent results with observed data of earthquakes in some aspects. However, it is cumbersome to simulate the BK model with set-valued laws by conventional explicit integration methods. In the presence of set-valued laws, the explicit integration methods can easily lead to the numerical chattering, violated constraints on the velocity of force laws, and the difficulty of identifying the states of blocks of the BK model. This paper employs an implicit Euler integration method to simulate the BK model with symmetric and asymmetric set-valued laws. This method removes the numerical chattering in the BK model, even in the cases of large time step sizes. It can easily detect the stuck or slipping state of a block element. Comparing to previous results integrated by explicit integration methods in the literature, the results integrated by this implicit method show smoother curves and lower irregularities in the magnitude distribution of events.

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.

Nucleation process in the Burridge-Knopoff model of earthquakes

The nucleation process of the one-dimensional Burridge-Knopoff model of earthquakes obeying the rate- and state-dependent friction law is studied both analytically and numerically. The properties of the nucleation dynamics, the nucleation lengths and the duration times are examined together with their continuum limits.