Solitary waves in the excitable Burridge–Knopoff model

Highlights

• We show the existence of solitary waves in the Burridge–Knopoff model for spinodal friction laws.

• Shock-like profiles are obtained near a continuum limit and propagation failure occurs at weak coupling.

• A closed-form expression of travelling waves is obtained for a class of piecewise-linear friction laws.

Abstract

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.

Introduction

A significant body of work has been devoted to elucidating nonlinear mechanisms of earthquakes [[1], [2], [3]]. Almost fifty years ago, Burridge and Knopoff [4] introduced a nonlinear lattice model to investigate the generation of earthquakes along faults, or more generally the occurrence of dynamical instabilities at frictional interfaces. The Burridge–Knopoff (BK) model formally describes a chain of blocks connected by springs and pulled over a surface, each block being attached to a spring pulled at constant velocity and subject to a friction force with the surface. When considering two plates under friction, the blocks can either correspond to a discretization of a plate or account for an existing microstructure [5]. It has been shown that this simple slider-block model is able to reproduce some statistical features of earthquakes generated by fault dynamics, see [[6], [7], [8]] and references therein.

A key feature of the BK model lies in the friction force between the blocks and the fixed surface, which depends nonlinearly, often non-monotonically, on sliding velocity. Non-monotone behaviours of the average kinetic friction can be inferred from microscopic models of sliding surfaces, in which interactions between asperities or atoms are incorporated through a local periodic potential [[9], [10]] (see also [11] for more references on the modelling of friction at small scales). Along the same line, experiments performed at the macroscale with a broad range of materials have revealed that the steady-state kinetic friction coefficient is non-monotone versus sliding velocity (see [12] for a review). Friction is velocity-strengthening (i.e. frictional resistance increases with sliding velocity) for high enough velocities; a phenomenon which can originate from radiation of waves within the bulk [[4], [10]], frictional heating [13], or hydrodynamic viscous friction for interfaces containing fluids [14] (see [12] for additional mechanisms). Conversely, friction becomes velocity-weakening in a regime of lower velocities [[12], [13], [15]]; a behaviour intimately linked with the occurrence of stick–slip instabilities and earthquake phenomena [[2], [14]] (additional fundamental mechanisms involved in such instabilities are reviewed in [16]).

Different types of friction laws have been introduced in order to model such velocity-strengthening or weakening regimes at the macroscale and describe stick–slip phenomena. Rate-and-state laws incorporating state variables for the frictional interface are frequently used, see [[2], [13], [17], [18]] for reviews. Detailed bifurcation studies have been performed for the BK model with a single block and several rate-and-state laws (see e.g. [[17], [19]] and references therein), and quantitative agreement with experimental data on Hopf bifurcations have been reported [20]. For chains of blocks, BK models with rate-and-state friction laws have been investigated in [[18], [21], [22]] through numerical simulations, but their mathematical analysis remains a delicate task. Indeed, the study of large amplitude excitations in such models is made more difficult by the additional degrees of freedom introduced to describe the state of the interface (but small amplitude waves are simpler to address, see section 2 of [23]). Another type of friction law has been developed to account for the weakening of fault gouges by acoustic fluidization [24], a phenomenon which can be at the origin of slip instabilities [25]. This higher-dimensional model involves a nonlinear diffusion equation for the elastic energy density in the fault core. In particular, analysing the different regimes of steady-state friction requires to solve a one-dimensional nonlinear boundary value problem and study the associated stationary bifurcations [24].

A simpler class of widely used friction laws is given by generalized Coulomb laws with velocity-dependent kinetic friction coefficient [26]. In that case, the set-valued character of the friction laws requires an adapted numerical treatment [[27], [28], [29]] and can lead to analytical complications in dynamical studies (see [23], section 1.4). Alternatively, single-valued laws can be used to approximate set-valued friction laws [6] or account for additional physical effects leading to velocity-strengthening at small enough velocities, such as internal dissipation induced by shear deformations [4] or adhesive friction [[30], [31]].

The dynamics of the BK model has been extensively studied in the case of steady-state velocity-weakening friction. Depending on the choice of parameters and system size, this regime can lead to chaotic dynamics or to the propagation of nonlinear wave trains [[18], [32], [33], [34]]. In particular, periodic travelling waves close to solitary waves (with highly localized slipping events propagating at constant velocity) have been reported in numerical and analytical studies [[18], [23], [33], [35], [36]].

In this paper, we consider a different situation corresponding to ’spinodal’ friction laws [[4], [17], [31], [37]], where steady-state friction is velocity-strengthening both for small and large enough velocities and an intermediate velocity-weakening region exists. This situation has been reported in a number of friction experiments performed with rocks, rubber and hydrogels [[31], [38], [39]]. In that case, the dynamics of a single block can be described by a Van der Pol type equation, a situation frequently encountered in the modelling of excitable media [[40], [41], [42], [43]]. Numerical studies of the BK model with spinodal friction laws (either generalized Coulomb laws or regularizations thereof) have revealed different types of wave patterns, ranging from synchronous oscillations, periodic travelling waves and phase fronts [[44], [45]] to chimera-like states (see Fig. 8 of reference [46] for an early observation of this phenomenon). These different studies were focused on the oscillatory regime where the pulling velocity lies within the velocity-weakening region.

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. The analysis of finite amplitude travelling waves in the discrete system is more delicate because it leads to a nonlinear advance-delay differential equation for the wave profiles. In order to tackle this problem, an interesting inverse approach was described in reference [51], where particular friction laws were computed in order to fit prescribed explicit wave profiles. An analytical moving kink solution (with block displacements given by an odd function of the moving frame coordinate $z$) was proposed but turns out to be erroneous (the velocity dependent friction force induced by the kink is even in $z$, and cannot be balanced by the odd inertial and stress terms in the dynamical equation (10) of [51]).

In the present paper, we analyse in detail the existence of localized waves in the discrete BK model with spinodal friction laws, using both extensive numerical simulations and analytical techniques. We restrict our attention to solitary waves and treat the case of fronts in a companion paper [52]. We consider the excitable regime where the pulling velocity lies within a velocity-strengthening domain of the spinodal friction laws (above velocity-weakening). In that case, each single block admits a stable state of continuous slip but displays a large response to perturbations above some threshold. When blocks become coupled, our numerical simulations reveal that this response generates a solitary wave for a broad range of friction laws (smooth or nonsmooth) and parameter values. In the case of regularized Coulomb laws, the solitary wave consists of a moving localized region where sticking occurs, in contrast to the propagation of localized slipping events previously reported for velocity-weakening friction [33].

We analyse in detail the influence of parameters on the existence and qualitative properties of solitary waves. For low enough coupling between blocks, we observe a phenomenon of propagation failure corresponding to the rapid extinction of initially propagating pulses. We find that solitary waves develop shock-like profiles in the opposite limit of large coupling, a phenomenon connected with the existence of weak solutions in a formal continuum limit. When the pulling velocity is decreased towards the boundary of the velocity-strengthening domain, the dynamics of the blocks becomes underdamped and we observe solitary waves with oscillatory tails, while propagation failure takes place above some critical pulling velocity. For certain friction laws (near the transition to a velocity-weakening law), we also observe bistability between continuous slip and limit-cycle oscillations and the existence of propagating fronts connecting these two stable states.

In order to obtain analytical expressions for solitary waves and explain some of their qualitative properties, we introduce a simplified piecewise linear friction law with two velocity-strengthening regions separated by a (negative) jump discontinuity. The discontinuity in the friction force can be interpreted as a rough approximation for the existence of a small intermediate velocity-weakening region. Alternatively, we propose a possible physical realization of this piecewise linear model as a chain of impulsively forced mechanical oscillators (stiffness and damping are assumed linear). In this system, excitability arises from an impulsive force applied in the direction of motion above some critical deflection. Such types of piecewise linear nonlinearities have been extensively used for the mathematical study of travelling waves in different types of PDEs [[53], [54], [55], [56]] and spatially discrete systems [[57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68]]. Following this approach allows us to obtain explicit solitary waves in the form of oscillatory integrals and numerically compute their profiles and wave speed. We explain the occurrence of propagation failure below some critical coupling by the existence of a saddle–node bifurcation of solitary waves.

It is worthwhile to stress that we obtain fully localized solitary waves, i.e. blocks lie in the state of stable slip at infinity on both sides of the chain (this constitutes another important difference with the previous works [[23], [33]]). Similar solutions have been previously obtained for other types of excitable lattice dynamical systems where the coupling is diffusive rather than elastic. These systems correspond to spatially discrete FitzHugh–Nagumo equations with either smooth [69] or piecewise linear [61] bistable nonlinearities. In our case, the properties of solitary waves are quite different at large coupling with the occurrence of shocks in the BK model.

The outline of this paper is as follows. In Section 2, we introduce the excitable BK model and different types of spinodal friction laws. In Section 3, we use numerical simulations to study the existence and shape of solitary waves depending on parameters. We also consider a continuum limit of the model and illustrate the bistable dynamics near the transition to a velocity-weakening law. Section 4 provides analytical results for the idealized piecewise linear friction law. Section 5 summarizes the main findings and points out interesting open problems. The chain of impulsively forced mechanical oscillators leading to the piecewise linear excitable BK model is described in Appendix.