Burridge-Knopoff Model Research Articles

Response of random dielectric composites and earthquake models to pulses: prediction possibilities

Following the success of the study of response to local short-duration pulses (of magnetic field, additional ‘sands’, etc.) on magnetic systems and the BTW (sand-pile) model, to locate accurately the respective critical points, the responses to similar short duration pulses (of electric field, of ‘mechanical pushes’ on tectonic plates, etc.) have been studied here numerically for metal-insulator composites before dielectric breakdown and the Burridge-Knopoff (earthquake) model before the critical avalanches. The breakdown susceptibility (defined in the text), obtained from such response behaviour, indicates universal behavior near the catastrophic breakdown or the self-organised critical points. We show that the breakdown (electric) feld for random metal-dielectric composites can be located accurately much before the breakdown, by extrapolating the inverse breakdown susceptibility to its vanishing point. Similarly, the growth of the susceptibility, coming from the stress correlations, in the Burridge-Knopoff model of earthquakes is shown to be exponential in time. Prediction of the earthquake point (in time) is also possible in the model from the study of its inverse logarithm with straight-line extrapolation to its vanishing point.

Growth of breakdown susceptibility in random composites and the stick-slip model of earthquakes: Prediction of dielectric breakdown and other catastrophes.

The responses to short duration pulses (of electric field, of additional ``particles,'' of a mechanical ``push due to blasting'' on any ``tectonic block,'' etc.) have been studied numerically for metal-insulator composites before dielectric breakdown, the Bak-Tang-Weisenfeld (BTW) (sandpile) model before the critical avalanches, and the Burridge-Knopoff stick-slip model of earthquakes. We show that, from the response to weak pulses of appropriate external field, one can estimate the growth of local failure correlations in such systems, giving the breakdown susceptibility. The study of this breakdown susceptibility, contributed to by the correlations of microscopic local failures, indicates universal behavior near the catastrophic (global) breakdown or the self-organized critical points. Its study can thus help in accurately locating the global breakdown or disaster point (much before its occurrence) by extrapolating the inverse breakdown susceptibility to its vanishing point. We have performed numerical studies of Laplace's equation of a dielectric with random bond conductors below its percolation threshold, of the dynamics of the BTW model, and of the dynamical equations of the array of blocks in the stick-slip model of earthquakes. The breakdown susceptibility has a power law growth (with the critical interval from the global breakdown threshold) in both the electric breakdown and in the BTW model. Accurate exponent values for the growth have been obtained in the BTW case. The growth of the susceptibility, coming from stress correlations, in the Burridge-Knopoff model of earthquakes is observed to be exponential in time, as it is for pulse susceptibility in this model. \textcopyright{} 1996 The American Physical Society.

Quasidynamic model for earthquake simulations.

A cellular automaton is constructed that simulates the response of a one-dimensional dynamical Burridge-Knopoff model for earthquake occurrence. The central element of the model involves an approximation to the momentum overshoot of the static equilibrium state in the dynamic case. The model gives satisfactory comparison with, and a reduction in computation by 10N relative to, histories of synthetic seismicity using the dynamic model, when a reasonable choice of overshoot is selected. Solutions that use the ``standard'' quasistatic automaton, in which the final state of stress is set equal to the dynamic friction, give less satisfactory agreement. Thus momentum overshoot is an important factor in regulating earthquake self-organization and cannot be ignored in any accurate simulation of earthquake histories.

Correlation functions and power spectra in a model of an earthquake fault

We study by numerical methods spatial and temporal correlations in a simple version of the Burridge-Knopoff model. We find, for a range of physical parameters, that spatial correlations fall off rapidly at small distances but approximately linearly at large distances. Temporal correlation functions oscillate periodically in sign with an envelope which falls off rapidly for small times and then more slowly. In both cases the decay depends on the stiffness of the system. We argue that the small distance (time) behaviour of the correlation functions is governed by localized earthquakes obeying a Gutenberg-Richter scaling law whereas the long distance (time) behaviour is governed by large earthquakes which may in extreme cases extend over the entire system and do not obey a scaling law. The power spectra of the dynamical variables exhibit a fundamental frequency which can be interpreted as the average time between large earthquakes.

Dynamics of the Burridge-Knopoff Model of Earthquakes

Dynamics of the Burridge-Knopoff Model of Earthquakes

Dynamical origin of spatial order.

We present a numerical study of a one-dimensional version of the Burridge-Knopoff model [Bull. Seismol. Soc. Am. 57, 341 (1967)] with stick-slip dynamics. The solutions of the model in the low velocity regime represent earthquakes in a simple transform fault and have chaotic behavior [J. M. Carlson and J. S. Langer, Phys. Rev. Lett. 62, 2632 (1989); Phys. Rev. A 40, 6470 (1989)]. It has been shown recently that in a higher velocity regime there are solutions of the model with periodic boundary conditions that are solitonlike and not necessarily chaotic [J. Schmittbuhl, J. P. Vilotte, and S. Roux, Europhys. Lett. 21, 374 (1993)]. We show here that stable, nearly periodic solutions also exist in a certain window of parameter space when the model has free boundary conditions. These solutions are periodic in both time and space and display striation effects that are strikingly similar to those seen experimentally by Gollub and co-workers [Phys. Rev. A 43, 811 (1991); Phys. Rev. E 47, 820 (1993)]. For an arbitrary disordered set of initial conditions, the short-time behavior is noisy, but the stable nearly periodic solutions emerge in the long-time limit. We discuss the origin of the window and show that the nature of the solution found depends strongly on the boundary condition. We also discuss the effects of symmetry breaking and disorder and show that even in a highly disordered regime the system can spontaneously organize itself so that very nearly stable noise-free solutions emerge.

Study of the Response to Pulses and Possible Prediction of Catastrophes

We show in a numerical study of the Burridge-Knopoff spring-block model of earthquake, that if one applies very weak puises (fixed amount of momentum) to any particular block at regular intervals and studies the response on the same block (the stress developed on it), then accurate predictions of the timings of the imminent earthquakes are possible. This is shown by studying the (local) puise susceptibility xp, which grows exponentially with the inverse of the time interval from the next (global) earthquake and saturates to a (nonuniversal) value dependent on the model parameters (fixed for any particular serres of earthquakes). Considerable progress bas recently been made in the statistical study of trie (fracture, break- down or earthquake) strength or magnitude distribution of disordered solids (porous media, random composites etc.), of granular packings, or of trie dynamical mortels of earthquake, etc. iii. Several simple mortels (at a semi-microscopic level) bave been introduced for mechan- ical (2) and electrical (3) failure of such randomly disordered media, modelled by randomly disordered lattices with individual bonds breaking irreversibly. These theoretical results about the fracture or breakdown strength distribution etc. have also been checked in several exper- iments (4). Failure processes playmg intrinsic roles in many systems of industrial importance and in many natural catastrophic (disaster) phenomena, the above mentioned statistical stud- ies establishing the nature of (non-self averaging) fracture-strength distribution, its (critical) fluctuations, consequent power laws for trie average strength of such solids, etc. iii, are of extreme importance. Although a significant amount of literature has been developed for these studies of failure strength distribution, not much studies have been made regarding the dynam- ics of microscopic failures. Since the microscopic failures are irreversible and therefore require intermediate redistributions of the forces (potentials), the equations for the dynamics of failure are intrinsically nonlinear and dissipative. The formulation and studies of such equations are necessary for the search of any precursor effect of trie macroscopic failure (if trie macroscopic iailure is at most a critical and not a fully chactic phenomenon). In some recent experiments (5)

Stick-slip dynamics in the relaxation of stresses in a continuous elastic medium.

We report on experimental results on the dynamical behavior of the stress relaxation in a model system for earthquakelike dynamics. In the experiments, a continuous elastic medium, namely, a transparent gel, is sheared at a very slow rate in between two coaxial circular cylinders. The stress relaxations are investigated locally by means of photoelastic techniques. We find four different stick-slip dynamical regimes. Three of them are reminiscent of the unstable solutions of the Burridge-Knopoff model: the sliding, the global relaxation, and the solitonlike propagating solutions. The fourth regime shows complex spatiotemporal behavior involving events of all sizes. For this last type of behavior the statistics of event duration, event separation, and amplitude show exponential behavior.

Propagative slipping modes in a spring-block model

Two transitions are observed in a version of the Burridge-Knopoff model of a tectonic fault. The first transition has already been reported and occurs when the velocity scale of the friction force is varied. We trace the origin of this transition back to what is happening for a single free block. The second transition is observed when varying the speed of sound of the system and concerns the possibility that the system exhibits solitary wave solutions. We provide a necessary condition for the system having a stable solitary wave solution. This condition, which involves a single parameter formed with three of the four parameters of the system, allows one to interpret some of the numerical results.

Dynamics of earthquake faults

The authors present an overview of ongoing studies of the rich dynamical behavior of the uniform, deterministic Burridge-Knopoff model of an earthquake fault, discussing the model's behavior in the context of current seismology. The topics considered include: (1) basic properties of the model, such as the distinction between small and large events and the magnitude vs frequency distribution; (2) dynamics of individual events, including dynamical selection of rupture propagation speeds; (3) generalizations of the model to more realistic, higher-dimensional models; and (4) studies of predictability, in which artificial catalogs generated by the model are used to test and determine the limitations of pattern recognition algorithms used in seismology.

Efficient large-scale simulations of a uniformly driven system.

We present results from an efficient algorithm for simulating systems of locally connected elements that are subject to uniformly increasing stresses and that discharge when these stresses reach some threshold. Previously, large-scale simulations of such systems have been hindered by the very-time-consuming search for those elements that are going to discharge next. We avoid this by using a suitable data structure, reducing computer CPU times by several orders of magnitude in typical cases. In particular, we present simulations for a simple version of the Burridge-Knopoff model introduced by Olami, Feder, and Christensen [Phys. Rev. Lett. 68, 1244 (1992)]. Due to the substantially larger lattices and longer simulation times presently used, we find that the conclusions of these authors have to be modified considerably.

Sweeping of an instability : an alternative to self-organized criticality to get powerlaws without parameter tuning

We show that a notable fraction of numerical and experimental works claiming the observation of self-organized criticality (SOC) rely in fact on a different physical mechanism, which involves the slow sweeping of a control parameter towards a global instability. This slow sweeping (which does not apparently involve a parameter tuning) has been the cause for the confusion with the characteristic SOC situation presenting truly no parameter tuning and functioning persistently in a marginal stability condition due to the operation of a feedback mechanism that ensures a steady state in which the system is marginally stable against a disturbance. The observation of power law distributions of events, often believed to be the hallmark of SOC, can be traced back to the cumulative measurements of fluctuations diverging on the approach of the critical instability. For non-critical instabilities such as first-order transitions, the power law distribution exists on a limiting size range up to a maximum value which is an increasing function of the range of interaction. We discuss the relevance of these ideas on the onset of spinodal decomposition, off-threshold multifractality, an exactly soluble model of rupture, the Burridge-Knopoff model of earthquakes, foreshocks and acoustic emissions, impact ionization breakdown in semiconductors, the Barkhausen effect, charge density waves, pinned flux lattices, elastic string in random potentials and real sandpiles.

Experiments on Earthquakes in a Continuous Elastic Medium

We report on experiments on the dynamical behaviour in a model system for earthquake-like dynamics. Shear is imposed on a transparent gel in between two coaxial circular cylinders. The stress relaxations are investigated locally by means of photoelastic techniques. We found several dynamical regimes; three of them are reminiscent of the unstable solutions of the Burridge-Knopoff model. The fourth regime shows complex spatiotemporal behaviour involving events of many different sizes. For this last type of behaviour the statistics of event duration, event separation, and amplitude show exponential behaviour.

Dynamics of spring-block models: Tuning to criticality.

We have studied the homogeneous Burridge-Knopoff spring-block model with nonlinear friction introduced by Carlson and Langer [Phys. Rev. Lett. 62, 2632 (1989); Phys. Rev. A 40, 6470 (1989)]. There are several different velocity scales that define the model and divide the behavior of the system into distinct regimes. Over much of the parameter space defined by these velocities the system appears to be self-organized to what is reminiscent of a first-order transition. As the friction nonlinearity is varied, there appears to be a continuous (critical) transition to a regime where a global event is continually occurring.

Strongly intermittent chaos and scaling in an earthquake model.

We discuss the relation between scaling laws and dynamical behavior for earthquakes in the framework of a Burridge-Knopoff model. Due to the nontrivial interaction among many degrees of freedom, a new type of strongly intermittent chaos is found. The dynamics is dominated by wild fluctuations, implying exponential tails in the probability distributions. This is caused by very slow relaxation of time correlations, which gives rise to an anomalous behavior for the effective Lyapunov exponent and for the time signal of the earthquake magnitude.

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

We present a detailed numerical study of certain fundamental aspects of a one-dimensional homogeneous, deterministic Burridge-Knopoff model. The model is described by a massive wave equation, in which the key nonlinearity is associated with the stick-slip velocity-weakening friction force at the interface between tectonic plates. In this paper, we present results for the statistical distribution of slipping events in the limit of a very long fault and infinitesimally slow driving rates. Typically, we find that the magnitude distribution of smaller events is consistent with the Gutenberg-Richter law, while the larger events occur in excess of this distribution. The crossover from smaller to larger events is identified with a correlation length describing the transition from localized to delocalized events. We also find that there is a sharp upper cutoff describing the maximum large event. We identify how the correlation length and this upper cutoff scale with the parameters in the model. We find that both are independent of system size, while both do depend on the spatial discretization. In addition to the magnitude distribution, we present a series of measurements of other seismologically relevant quantities, including the event duration, the size of the rupture zone, and the energy release, and discuss the relationship between our measurements and the corresponding empirical laws in seismology.

The mechanism of vertical displacements in a dynamical model of earthquakes

We propose a mechanical model for earthquakes which, unlike the well-known Burridge-Knopoff model, takes into account the transversal vibrations against the background of a longitudinal moving fault. It is shown that the energy of the moving earthquake fault may generate not only longitudinal but also vertical displacements of the ground. The mechanism of parametric instability resulting in stochastic transversal vibrations is responsible for this energy transfer. Numerical experiments show that the amplitudes of transversal and longitudinal vibrations have the same order of magnitude.