Spring-block System Research Articles

Numerical Simulation of Seismic Cycles with a Composite Rate- and State-Dependent Friction Law

Properties of rate- and state-dependent friction laws are examined through numerical simulations. Among several versions of the rate- and state-dependent friction laws, we use the slowness law, the slip law, and the composite law. The last was recently proposed by Kato and Tullis (2001) for better explaining experimental observations of rock friction in the laboratory. Two kinds of simulations are done. One is for cyclic stick-slip motion of a single degree of freedom spring-block system. The other is for seismic cycles at a subduction zone, where a 2D model of uniform elastic half-space is used. Assuming the same values of friction parameters, we compare the simulation results for the three friction laws. From the two kinds of simulations, we find that the recurrence interval of simulated earthquakes is the longest, and the magnitude of preseismic sliding is the smallest, for the composite law.

CHAOS IN PIECEWISELY INTEGRABLE TRAIN MODEL FOR TWO BLOCKS

The train model of two blocks with stick-slip dynamics (M. de Sousa Vieira, 1995) is believed to be the simplest spring-block system, which displays chaos. Here we simplify it even more by linearizing the velocity dependence of the friction force. In this way, the nonlinearity of the equations of motion is reduced to the time moments, when a block starts to move or stops, and when the analytical solutions are to be sewn together. We demonstrate, that for small values of the velocity of blocks, the character of motion is not changed. This is observed on the bifurcation diagrams, the Lyapunov exponents, the phase portraits and the power spectra.

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.

差分を基礎としたバネーブロック系を用いた弾性体変形の計算手法の開発

差分を基礎としたバネーブロック系を用いた弾性体変形の計算手法の開発

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.

Complex Behavior in Earthquake Dynamics

Recent progresses in understanding earthquake dynamics with the aid of a simple spring-block system is reviewed from a physicists' point of view. Dynamical instability due to negative dynamical friction amplifies any perturbation and leads to a chaotic behavior for almost any initial configurations. It is also pointed out that static friction gives another source of the complex behavior which is characteristic to a threshold element system.

Nonlinear dynamics of dry friction

The dynamical behaviour caused by dry friction is studied for a spring-block system pulled with constant velocity over a surface. The dynamical consequences of a general type of phenomenological friction law (stick-time-dependent static friction, velocity-dependent kinetic friction) are investigated. Three types of motion are possible: stick - slip motion, continuous sliding, and oscillations without sticking events. A rather complete discussion of local and global bifurcation scenarios of these attractors and their unstable counterparts is present.

Increase of creep interevent intervals: a conceptual model

Creep event sequences on shallow faults are studied here by means of a one-dimensional fault model assuming that the aseismic section is extended from the surface down to few hundreds metres depth. The sequences of creep events on a fault are qualitatively described in terms of sequences of stick-slip movements (relaxation oscillations) of a spring-block system, with very low velocity peaks. The model is based on the nonlinear rate- and state-dependent frictional properties of faults, where time-dependent coefficients can be considered. During this study it was necessary to define the properties of friction at low velocity, thereby avoiding the singularity implied by previous formulations. In order to study the deformations near the fault, the effect of rock elastic nonlinearity is modeled by ascribing a nonlinear elastic characteristic to the ‘spring’ of the model. If the system parameters do not depend explicitly on time, the system undergoes relaxation oscillations with a constant period. A temporal variation of the fault parameters can be successfully used in order to explain increasing intervals of time between velocity peaks in creep event sequences. In particular, allowing the sliding section to extend with time provides an increase in the intervals between creep events.

Tricritical Behavior in Rupture Induced by Disorder

We discover a qualitatively new behavior for systems where the load transfer has limiting stress amplification as in real fiber composites. We find that the disorder is a relevant field leading to tricriticality, separating a first-order regime where rupture occurs without significant precursors from a second-order regime where the macroscopic elastic coefficient exhibits power law behavior. Our results are based on analytical analysis of fiber bundle models and numerical simulations of a two-dimensional tensorial spring-block system in which stick-slip motion and fracture compete.

Dynamic phases in a spring-block system

When a block is pulled via a spring across a surface, there appear to be several different dynamic ``phases.'' These can be characterized by the pulling velocity ${\mathit{v}}_{0}$ and a dynamic velocity ${\mathit{v}}_{\mathit{D}}$=g \ensuremath{\surd}m/k , where g is the acceleration of gravity, m is the mass of the block, and k is the spring constant. For sufficiently small ${\mathit{v}}_{0}$, the block displays stick-slip (relaxation) motion. Then, as a function of decreasing ${\mathit{v}}_{\mathit{D}}$, this stick-slip motion is first nearly periodic, then aperiodic with approximately an exponential slip size distribution, and then aperiodic with possibly a power-law slip size distribution. When ${\mathit{v}}_{0}$ is increased, the block eventually ceases to stick and just slides across the surface. The motion can then be adequately described by a Langevin model.