Traveling wave solutions in the Burridge-Knopoff model

Muratov, Cyrill B.

doi:10.1103/physreve.59.3847

Cited by 19 publications

(34 citation statements)

References 27 publications

(119 reference statements)

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“…Let us now compare our results to those of previous studies of steady state rupture velocities. In the Amontons-Coulomb case, our solution v c (τ ) takes the exact same form as the one found in [32] for the Burridge-Knopoff model with Amontons-Coulomb friction (compare Eq. (B8) and Eq.…”

Section: Discussionmentioning

confidence: 84%

“…In particular, we also find that the rupture velocity increases with increas-ing shear prestress of the interface and with increasing values of the viscous coefficient. Note that [30][31][32] did not discuss the effect of an interfacial stiffness on rupture speed.…”

Section: Discussionmentioning

confidence: 99%

“…This resolution problem was solved in [31] by introducing a short-wavelength cutoff, obtained by adding Kelvin viscosity to the model. Rupture velocities in the BK model with Amontons-Coulomb friction were studied in Muratov [32], and found to have a unique solution for any given value of the initial shear stress at the interface, with a well-defined continuum limit. We compare our results to those of [30][31][32] in Section IV.…”

Section: Introductionmentioning

confidence: 99%

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Steady-state propagation speed of rupture fronts along one-dimensional frictional interfaces

et al. 2015

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The rupture of dry frictional interfaces occurs through the propagation of fronts breaking the contacts at the interface. Recent experiments have shown that the velocities of these rupture fronts range from quasi-static velocities proportional to the external loading rate to velocities larger than the shear wave speed. The way system parameters influence front speed is still poorly understood. Here we study steady-state rupture propagation in a one-dimensional (1D) spring-block model of an extended frictional interface, for various friction laws. With the classical Amontons-Coulomb friction law, we derive a closed-form expression for the steady-state rupture velocity as a function of the interfacial shear stress just prior to rupture. We then consider an additional shear stiffness of the interface and show that the softer the interface, the slower the rupture fronts. We provide an approximate closed form expression for this effect. We finally show that adding a bulk viscosity on the relative motion of blocks accelerates steady-state rupture fronts and we give an approximate expression for this effect. We demonstrate that the 1D results are qualitatively valid in 2D. Our results provide insights into the qualitative role of various key parameters of a frictional interface on its rupture dynamics. They will be useful to better understand the many systems in which springblock models have proved adequate, from friction to granular matter and earthquake dynamics.

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Section: Discussionmentioning

confidence: 84%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Steady-state propagation speed of rupture fronts along one-dimensional frictional interfaces

et al. 2015

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“…It is expected that some of our experimental results can also be understood in terms of a spring-block model with a Coulomb friction law and periodic boundary conditions. In such a model, Muratov has recently shown [18] that the whole dynamics and, therefore, x m is ruled by the amount of stress accumulated in front of the solitary relaxation. In a mainly elastic medium, such as a gel, with a Coulomb-like friction law, this accumulated stress should depend only on the friction force jump between static and dynamic conditions, and the elastic constant of the problem.…”

mentioning

confidence: 99%

Experimental and theoretical results of stress relaxations in a model of earthquake dynamics

Galeano¹,

Español²,

Rubio³

2000

Europhys. Lett.

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We report on experimental and theoretical results on the dynamical behavior of propagating solitary relaxations in a model system for earthquake-like dynamics. In the experiments, a continuous elastic medium is sheared at very slow rate in between two coaxial cylinders. Solitary propagating relaxations travel at speeds that are proportional to the driving speed, though ∼ 10 3 times faster, and inversely proportional to the number of solitary relaxations in the system. We show that this behavior appears due to the existence of a characteristic length scale which is the typical displacement of a surface element in a relaxation.

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“…With regard to spatially localized travelling waves, the existence of fronts has been established in a continuum limit of the BK model [47] (see also [48][49][50] for numerical studies of rupture fronts in other continuum models based on rate-and-state laws). However, the existence of localized waves was not established so far for the spatially discrete BK model with spinodal friction laws.…”

Section: Introductionmentioning

confidence: 99%

Solitary waves in the excitable Burridge–Knopoff model

2018

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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.

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Traveling wave solutions in the Burridge-Knopoff model

Cited by 19 publications

References 27 publications

Steady-state propagation speed of rupture fronts along one-dimensional frictional interfaces

Steady-state propagation speed of rupture fronts along one-dimensional frictional interfaces

Experimental and theoretical results of stress relaxations in a model of earthquake dynamics

Solitary waves in the excitable Burridge–Knopoff model

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