Spring-block Model Research Articles

Nontrivial temporal scaling in a Galilean stick-slip dynamics

We examine the stick-slip fluctuating response of a rough massive nonrotating cylinder moving on a rough inclined groove which is submitted to weak external perturbations and which is maintained well below the angle of repose. The experiments presented here, which are reminiscent of Galileo's works with rolling objects on inclines, have brought in the last years important insights into the friction between surfaces in relative motion and are of relevance for earthquakes, differing from classical block-spring models by the mechanism of energy input in the system. Robust nontrivial temporal scaling laws appearing in the dynamics of this system are reported, and it is shown that the time-support where dissipation occurs approaches a statistical fractal set with a fixed value of dimension. The distribution of periods of inactivity in the intermittent motion of the cylinder is also studied and found to be closely related to the lacunarity of a random version of the classic triadic Cantor set on the line.

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."

On the separation of timescales in spring-block earthquake models

Abstract. One of the most widespread spring-block earthquake models, the Olami-Feder-Christensen model, is investigated without making the assumption that the duration of individual earthquakes is negligible. While the Gutenberg-Richter law for the size distribution of earthquakes is preserved qualitatively for earthquakes of finite duration, the b-value decreases with increasing earthquake duration. The effect decreases with increasing lattice size, although it is not clear whether it completely vanishes in the limit of infinite grid size. For realistic values of earthquake duration, the effect is negligible, so that the original model with zero earthquake duration is appropriate for most applications.

Stick‐slip motion in simulated granular layers

Two‐dimensional numerical simulations of shear in a gravity‐free layer of circular grains were conducted to illuminate the basic mechanics of shear of granular layers (such as layers of fault gouge). Our simulated granular layers exhibit either stable (steady state) or unstable (stick slip) motion. The transition from steady to stick‐slip sliding depends on loading velocity and applied confining stress in a way similar to a simple model of a block on a frictional surface. We investigate the conditions which lead to naturally occurring stick‐slip behavior and study in detail the systems behavior prior to and during slip events. Matching our numerical results to a spring block model, the system of grains was found to have bulk static and dynamic coefficients of friction that differ by about 0.1. This differing static and dynamic friction emerged spontaneously, from the collective behavior of grains, and was not prescribed a priori via a frictional rule between grain contacts. Results show that the micromechanics of contact forces is responsible for stick‐slip behavior: During the “stuck” phase, and in preparation for slip, more and more grain contacts which carry low forces slide, resulting in accelerating internal stress release. When enough of the low‐force contacts frictionally slide, the granular layer weakens and losses rigidity, leading to motion of contacts that carry larger forces and large‐scale slip. Our results may have implications to the understanding of the stability of gouge layers and are thus related to the underlying physics of earthquakes.

Interpretation of various slip modes on a plate boundary based on laboratory and numerical experiments

This paper discusses various slip modes on a plate boundary on the basis of a two-degree-of-freedom block-spring model and large-scale biaxial experiments, including a new experimental result on afterslip. We conducted slip experiments using large granite blocks with a pre-existing fault surface of 100 cm in length. Velocity-strengthening friction was given over a half of the fault length by inserting a thin Teflon sheet, while the other half retained velocity-weakening friction of the bare rock surface. Under a loading at a constant velocity, dynamic stick-slip repeated on the velocity-weakening region, causing afterslip on the velocity-strengthening region. The velocity-strengthening region experienced small coseismic slip as well, with the magnitude decreasing with the distance from the velocity-weakening region. The behaviors observed in the laboratory experiments were quantitatively simulated by a two-degree-of-freedom block-spring model, in which two blocks (Block 1 and Block 2) are connected by a liner spring and driven by a slowly moving driver. The friction on each block was assumed to obey rate and state dependent friction law. When a - b was assumed to be negative for Block 1, and positive for Block 2, afterslip occurred at Block 2. This model can also reproduce wide spectrum of slip modes by adjusting frictional parameters.

Comparison between thermal–viscous coupling and frictional sliding

We investigated the similarity between thermal–viscous coupling (TVC) and frictional sliding, proposed by Kameyama and Kaneda [Pure Appl. Geophys. 159 (2002) 2011]. We consider a one-dimensional layer composed of viscous material, which is sandwiched and sheared by two thick elastic layers. The rate of viscous deformation depends on the temperature T c in the viscous layer as well as shear stress τ. The temperature T c changes owing to heating by viscous dissipation and conductive cooling. We carried out velocity-stepping tests for the steady-state deformation both numerically and analytically, and compared the temporal evolution of small perturbations with that of the spring-block model with rate- and state-dependent friction (RSF). We found that, as is the case of frictional slip stability, the manner of temporal evolution is classified into four regimes depending on whether it is stable or not and whether it is monotonous or oscillatory with time. By further interpreting TVC in terms of general RSF theory by Ruina [J. Geophys. Res. 88 (1983) 10359], we obtained the relations between the parameters appearing in the phenomenological RSF law and the nondimensional parameters which characterize the nature of TVC. A further improvement of this approach might be important for estimating the actual values of frictional constitutive parameters at the deeper portion of seismogenic faults of interplate or inland earthquakes where a ductile deformation is expected to be significant.

A numerical method of viscoelastic finite element coupled with spring-block model to simulate the interaction of faults and crustal blocks

A numerical method of viscoelastic finite element coupled with spring-block model is developed to study temporal processes from the slow tectonic motion of large-scale crust to the rapid failure of small-scale faults. Our modeling demonstrates that the motion of crustal blocks is driven by forces from tectonic plate boundaries, and the deformation is distributed on faults for the stress accumulating. The coupling model generates earthquake sequences that display a magnitude-frequency scaling consistent with Gutenberg-Richter law. The frictional heterogeneities affeci earthquakes occurrence and stresses distribution of crustal blocks. Rupture of earthquakes starts at the nucleation node, and propagates bilaterally along faults with the stress triggering, release and redistribution. The failure of faults releases part of crustal stresses, the stress state of crustal blocks near fault is affected by the rupture of local segments on the fault, and the stress state of crustal blocks far away from the fault is controlled by the seismic activity of the whole fault.

Some further analogies between the Bak-Sneppen model for biological evolution and the spring-block earthquake model

In recent years, a great deal of attention has been devoted to so-called self-organized critical (SOC) systems, which are open, extended systems that organize themselves into steady metastable states without any temporal- or spatial-predominant scale (except those imposed by the finite size of the system). The SOC concept has been used to describe the statistical properties of several physical systems through numerical models based on cellular automaton. In particular, Bak and Sneppen proposed a SOC model for biological evolution at the level of entire species or faunas that exhibit punctuated equilibrium behavior. On the other hand, Olami, Feder and Christensen suggested that a two-dimensional spring-block earthquake model can explain some properties of real seismicity. In this paper, we show that there exist several further interesting analogies between these SOC models, besides those remarked on by other authors. Both of them exhibit punctuated equilibrium in the long term, which leads us to suggest an equivalent characterization of seismic and "evolutionary" provinces through the long-term slopes of the stair-shaped graphs of cumulative activity in the course of time.PACS Nos.: 91.30P, 05.65, 87.23

Parallel and distributed simulations and visualizations of the Olami–Feder–Christiensen earthquake model

In this work we present some numerical results and visualizations obtained from parallel and distributed simulations of the Olami–Feder–Christiensen earthquake model. This model is a two-dimensional cellular automata, continuous in space and discrete in time, constructed in order to simulate earthquake activity and it was introduced as a generalization of the one-dimensional spring block model defined by Burridge–Knopoff. For this model we determine with high accuracy its exponents of the power-law behavior of the histogram of the size of earthquakes; we study the final state for several values of the parameters of the model; and we determine the path to criticality by visualizations of the site frequency histogram

The dynamic friction term in the spring-block models for earthquakes: a constraint from seismic moment and energy catalogue

In the spring-block models of earthquakes, one of the key factors is the dynamic friction term which determines the complexity of the faulting process. Generally, two kinds of friction, namely velocity-dependent friction and slip-dependent friction, are used in the modelling. But until now there has still been a lack of information on which kind of friction term is more suitable for modelling the phenomenology of earthquakes. Based on the numerical studies of Shaw (1998 Bull. Seismol. Soc. Am. 88 1457), we have examined the ratio of the broadband radiated energy and the scalar seismic moment of shallow earthquakes worldwide from 1987 to 1998. The result shows that for earthquakes with strike-slip mechanisms, velocity-dependent friction seems to be predominant, while for thrust and normal events, slip-dependent friction seems predominant. This suggests that in the spring-block models for earthquakes, the type of focal mechanism has to be accounted for and different types of earthquakes require different dynamic friction terms in the corresponding spring-block model.

Effect of frictional heating on pre-seismic sliding: a numerical simulation using a rate-, state- and temperature-dependent friction law

Summary Using a simple spring-block model and a rate- and state-dependent friction law incorporating the effect of temperature variation, I perform a numerical simulation of aseismic sliding prior to unstable (seismic) faulting. I compare the simulation result of pre-seismic sliding in the case where the temperature effect due to frictional heating on frictional strength is taken into consideration (heated fault) with the result in the case where the temperature effect is neglected (unheated fault), assuming model parameters appropriate for the nucleation site of a large interplate earthquake. It is found that an earthquake occurrence time is a little earlier on the heated fault than on the unheated fault. This is because the frictional strength is weakened by temperature rise due to frictional heating during pre-seismic sliding. The amplitude of precursory crustal deformation caused by pre-seismic sliding on the heated fault is estimated to be about 20 per cent smaller than that on the unheated fault.

Power injected in dissipative systems and the fluctuation theorem

We consider three examples of dissipative dynamical systems involving many degrees of freedom, driven far from equilibrium by a constant or time dependent forcing. We study the statistical properties of the injected and dissipated power as well as the fluctuations of the total energy of these systems. The three systems under consideration are: a shell model of turbulence, a gas of hard spheres colliding inelastically and excited by a vibrating piston, and a Burridge-Knopoff spring-block model. Although they involve different types of forcing and dissipation, we show that the statistics of the injected power obey the “fluctuation theorem" demonstrated in the case of time reversible dissipative systems maintained at constant total energy, or in the case of some stochastic processes. Although this may be only a consequence of the theory of large deviations, this allows a possible definition of “temperature" for a dissipative system out of equilibrium. We consider how this “temperature" scales with the energy and the number of degrees of freedom in the different systems under consideration.

Pattern formation and selection in quasistatic fracture.

Fracture in quasistatically driven systems is studied by means of a discrete spring-block model. Developed from close comparison with desiccation experiments, it describes crack formation induced by friction on a substrate. The model produces cellular, hierarchical patterns of cracks, characterized by a mean fragment size linear in the layer thickness, in agreement with experiments. The selection of a stationary fragment size is explained by exploiting the correlations prior to cracking. A scaling behavior associated with the thickness and substrate coupling, derived and confirmed by simulations, suggests why patterns have similar morphology despite their disparity in scales.

Further seismic properties of a spring-block earthquake model

Within the context of self-organized critical systems, Olami (OFC) (1992) proposed a spring-block earthquake model. This model is non-conservative and reproduces some seismic properties such as the Gutenberg-Richter law for the size distribution of earthquakes. In this paper we study further seismic properties of the OFC model and we find the stair-shaped curves of the cumulative seismicity. We also find that in the long term these curves have a characteristic straight-line envelope of constant slope that works as an attractor of the cumulative seismicity, and that these slopes depend on the system size and cannot be arbitrarily large. Finally, we report that in the OFC model the recurrence time distribution for large events follows a log-normal behaviour for some non-conservation levels.

Frequency-Size Distribution of Model Seismicity in a Statistical Mechanical Model of Earthquakes

The frequency-size distribution of model earthquakes in a one-dimensional Burridge-Knopoff spring-block model [Phys. Rev. Lett. 22 (1989) 2632] is studied. Analytical approach is taken to avoid the limitations on the size of models and computation time in numerical calculations. By using the Hamiltonian of the spring-block system a concept of `seismic phonon' is proposed which is similar to the concept of phonon in solid state physics. The result of the occurrence frequency of model earthquakes shows a distribution similar to the modified gamma distribution proposed by Main and Burton using information theory.

Tilted sandpiles, interface depinning, and earthquake models

A model of a slowly tilted one-dimensional sandpile with dynamically varying critical slopes is studied. The energy dissipation events are found to be power-law distributed with exponent $\ensuremath{\tau}=1.11\ifmmode\pm\else\textpm\fi{}0.02$ and obey finite size scaling. The model represents a different universality class than the corresponding point driven models. However, the finite size scaling exponent D and the surface roughness exponent are the same for both driving conditions. The pile height exhibits periodic oscillations, but the oscillation period measured in tilt events diverges as system size increases because the average tilt increment between avalanches is inversely proportional to system size. We demonstrate that the model can be mapped onto a slowly driven interface depinning model and a one-dimensional spring-block model with dynamically varying friction thresholds, supporting a broad universality relating different models of self-organized critical phenomena.

Influence of crustal viscosity on earthquake energy distribution in a viscoelastic Spring‐Block System

We show that viscosity of crust‐forming rocks makes the frequency curve of earthquake energy deviate from an empirical power law relation. To estimate the effect of viscosity, we consider a modified Burridge‐Knopoff type spring‐block model. The model consists of the blocks connected by Maxwell springs which represent the material property of crustal rocks. We applied a method of cellular automaton model to the system. The resulting earthquake sequences depend on the relaxation time τ of the Maxwell spring. For relatively small 1/τ, where elasticity dominates, the earthquake energy frequencies show the power law relation and deviate from the relation at large earthquake energies. The separation shifts to smaller energy with increase of 1/τ. For large 1/τ, the distribution does not show the power law relation. The decay of earthquake frequencies after major mainshocks is not observed.

Solitary wave in a Burridge-Knopoff model with slip-dependent friction as a clue to understanding the mechanism of the self-healing slip pulse in an earthquake rupture process

Abstract. A one-dimensional Burridge-Knopoff spring-block model with slip-dependent friction was studied to explore the possibility of a solitary wave solution existing for the problem of earthquake faulting. The result may be used as an alternative case of the crack model (e.g., Madariaga and Cochard, 1996) and the spring-block model with velocity-dependent friction (e.g., Carlson and Langer, 1989) in the understanding of the mechanism of the self-healing slip pulse proposed by Heaton (1990). In general, the conditions for a solitary wave solution to exist are discussed by the trajectory in the phase space. By taking the first order approximation, it is demonstrated that a solitary wave solution exists in which the slip behaves as a propagating solitary wave with the propagation velocity less than that of an acoustic (seismic) wave, and the source time function at each position remains the same. As an alternative approach other than numerical calculations, the analytical solution, although simple, sheds light on some of the properties of the self-healing slip pulse. From the solution it is seen that the width of the pulse depends on its propagation velocity and the friction, consistent with experience in physics. It is pointed out that the self-healing slip pulse may exist for a broad class of frictional constitutive laws which, to some extent, explains the fact that the self-healing slip pulse may be observed for a variety of earthquakes occurring within different seismogenic environments.

Self-organized criticality in an isotropically driven model approaching equilibrium

Self-organized criticality (SOC) in systems governed by stick-slip dynamics is studied by means of a spring-block model. Contrary to conventional models which are loaded additively in their stationary states, our model is driven towards equilibrium in a multiplicative and isotropic manner via a temporally increasing coupling strength. A coarsening process builds up correlations in the bulk and establishes power-law avalanche size distributions, independent of boundary conditions and system parameters. The observed behavior is robust and suggests that some experimental systems approaching equilibrium via a punctuated threshold dynamics may well show SOC.

Self-Organized Criticality in Stick-Slip Models with Periodic Boundaries

A spring-block model governed by threshold dynamics and driven by temporally increasing spring constants is investigated. Due to its novel multiplicative driving, criticality occurs even with periodic boundary conditions via a mechanism distinct from that of previous models. This mechanism is dictated by a coarsening process. The results show a high degree of universality. The observed behavior should be relevant to a class of systems approaching equilibrium via a punctuated threshold dynamics.