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A dissipation-based analysis of an earthquake fault model
Abstract
We analyze the dynamics of a discrete dynamical model of earthquake faulting, derived from the Burridge-Knopoff model. The system is shown to exhibit a characteristic event size (L*=2nu2fl2) that separates two distinct regimes in the statistical distribution of event sizes and magnitudes. The dynamics of the system exhibits scaling laws that are in agreement with observed seismic laws. Influence of the frictional rate of dissipation, of the elastic stiffness coupling, and of the system size is investigated. The dynamics of the system is rather insensitive to the numerical treatment of the nonsmooth friction. In contrast, the exact form of the velocity weakening friction law is shown to have a major effect on the dynamics. For friction laws allowing local reversal backslipping, the distribution of the magnitudes does not exhibit a Gutenberg and Richter (GR) distribution, while by precluding backslipping a GR distribution is observed. Two populations of events can be characterized based on dissipation: weakly dissipative events that allow the mechanical energy of the system to increase; and strongly dissipative events that release a large fraction of the elastic potential energy of the system and introduce large stress heterogeneities. A coarse grain analysis in terms of the stored elastic energy and the magnitude of the disorder provides new interesting insights on the dynamics of the model. Weakly dissipative events, which reproduce the seismic laws, are shown to follow a deterministic evolution. A statistical criterion for the initiation of big dissipative events is proposed.
... We quantitatively illustrate such synchronization effect by computing the evolution of the interfacial elastic energy, E h , and of the bulk elastic energy, E t , following the definition of Schmittbuhl et al. (1996). The interfacial elastic energy, E h , is quantified through the sum of the relative distance between two asperities over all the spatial links defined by the two-dimensional Delaunay triangulation: ...
Plain Language Summary
Earthquakes are the results of a slip along a rough fault on which a complex and discrete set of asperities establish the interfacial contacts and control the frictional stability of the fault. We propose a novel experimental setup capable of measuring directly the subtle motion of individual asperities on an analog faulting interface. By capturing the temporal evolution of the slip of each asperity, we link the mechanical behavior of the global fault with the collective behavior of local asperities. Many destabilizing events at the local asperity scale are found during the globally stable stage of the fault. We prove that the interseismic coupling of asperities is affected by the normal load, the peak height of asperities, and the interactions between asperities. The spatiotemporal interactions of asperities are quantified as slip episodes to mimic the ruptures including both stable and unstable slips. With the catalog of slip episodes, we reproduce the significant characteristics and scaling laws observed in natural faults, such as the magnitude‐frequency distribution and the moment‐duration scaling. Such upscaling suggests that our results can be extrapolated to natural faults and provide insights into fault physics and mechanics.
... In these simulations, a simplified friction law, a velocity-weakening law, was employed as a friction law, where the friction force is assumed to be a single-valued decreasing function of the sliding velocity. These earlier simulations were further extended in several ways, computing various observables [13][14][15], extending the model to two dimensions [11,[16][17][18][19][20], taking account of the effect of the viscosity [21][22][23][24][25], modifying the form of the friction force [21,23,[26][27][28][29][30], considering the long-range interactions between blocks [18,31], driving the system only at one end of the system (the train model) [32][33][34] and examining the continuum limit [24], etc. ...
The nature of the high-speed rupture or the main shock of the Burridge–Knopoff spring-block model in two dimensions obeying the rate- and state-dependent friction law is studied by means of extensive computer simulations. It is found that the rupture propagation in larger events is highly anisotropic and irregular in shape on longer length scales, although the model is completely uniform and the emergent rupture-propagation velocity is nearly constant everywhere at the rupture front. The manner of the rupture propagation sometimes mimics the successive ruptures of neighbouring ‘asperities’ observed in real, large earthquakes. Large events tend to be unilateral, with its epicentre lying at the rim of its rupture zone. The epicentre site of a large event is also located next to the rim of the rupture zone of some past event. Event-size distributions are computed and discussed in comparison with those of the corresponding one-dimensional model. The magnitude distribution exhibits a power-law behaviour resembling the Gutenberg–Richter law for smaller magnitudes, which changes over to a more characteristic behaviour for larger magnitudes. For very large events, the rupture-length distribution exhibits mutually different behaviours in one dimension and in two dimensions, reflecting the difference in the underlying geometry.
This article is part of the theme issue ‘Statistical physics of fracture and earthquakes’.
... In these simulations, as a friction law, a simplified friction law, a velocity-weakening law, was employed where the friction force is assumed to be a singlevalued decreasing function of the sliding velocity. These earlier simulations were further extended in several ways, computing various observables [13][14][15] , extending the model to two dimensions 11, [16][17][18] , taking account of the effect of the viscosity [19][20][21][22] , modifying the form of the friction force 19,21,[23][24][25] , considering the long-range interactions between blocks 18,26 , driving the system only at one end of the system (the train model) 27 , and examining the continuum limit 22 , etc. ...
The nature of the high-speed rupture or the main shock of the Burridge-Knopoff spring-block model in two dimensions obeying the rate-and-state dependent friction law is studied by means of extensive computer simulations. It is found that the rupture propagation in larger events is highly anisotropic and irregular in shape on longer length scales, although the model is completely uniform and the rupture-propagation velocity is kept constant everywhere at the rupture front. The manner of the rupture propagation sometimes mimics the successive ruptures of neighboring "asperities" observed in real large earthquakes. Large events tend to be unilateral, with its epicenter lying at the rim of its rupture zone. The epicenter site is also located next to the rim of the rupture zone of some past event. Event-size distributions are computed and discussed in comparison with those of the corresponding one-dimensional model. The magnitude distribution exhibits a power-law behavior resembling the Gutenberg-Richter law for smaller magnitudes, which changes over to a more characteristic behavior for larger magnitudes. The behavior of the rupture length for larger events is discussed in terms of the strongly anisotropic rupture propagation of large events reflecting the underlying geometry.
... Depending on the choice of parameters and system size, this regime can lead to chaotic dynamics or to the propagation of nonlinear wavetrains [57,63,64]. 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 [57,62,[64][65][66]. ...
This thesis analyses localized travelling waves for some classes of nonlinearlattice differential equations describing excitable mechanical systems. Thesesystems correspond to an infinite chain of blocks connected by springs and sliding on a surface in the presence of a nonlinear velocity-dependent friction force. We investigate both the Burridge-Knopoff model (with blocks attached to springs pulled at constant velocity) and a chain of free blocks sliding on an inclined plane under the effect of gravity. For a class of non-monotonic friction functions, both systems display a large response to perturbations above a threshold, one of the main properties of excitable systems. This response induces the propagation of either solitary waves orfronts, depending on the model and parameter regime. We study these localized waves numerically and theoretically for a broad range of friction laws and parameter regimes, which leads to the analysis of nonlinear advance-delay differential equations. Phenomena of propagation failure and oscillations of the travelling wave profile are also investigated. The introduction of a piecewise linear friction function allows one to construct localized waves explicitly in the form of oscillatory integrals and to analyse some of their properties such as shape and wave speed. An existence proof for solitary waves is obtained for the excitable Burridge-Knopoff model in the weak coupling regime.
... Depending on the choice of parameters and system size, this regime can lead to chaotic dynamics or to the propagation of nonlinear wavetrains [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]. ...
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.
... 1. models of tectonic stress generation, localization and transfer in a fault zone or its surroundings (e.g., King et al., 1994;Ismail-Zadeh et al., 2005Aoudia et al., 2007); 2. dynamic system models reproducing "universal" features of seismicity (including large events, main shocks, and aftershocks) common to a wide class of non-linear systems (e.g., Burridge and Knopoff, 1967;Ogata, 1988;Bak and Tang, 1989;Rundle and Klein, 1993;Hainzl et al., 1999;Shnirman and Blanter, 2003;Zaliapin et al., 2003;Turcotte et al., 2007;Vere-Jones and Zhuang, 2008;Lennartz et al., 2011), and 3. Earth-specific models reproducing the features of seismicity at -a single fault (e.g., Dieterich, 1994;Schmittbuhl et al., 1996;Lyakhovsky et al., 2001;Zöller et al., 2004Zöller et al., , 2006Ben-Zion and Lyakhovsky, 2006;Lapusta and Liu, 2009;Nodal and Lapusta, 2010) or a fault system (e.g., Wang et al., 1983;Gabrielov et al., 1990Gabrielov et al., , 2007Ward, 1992Ward, , 1996Ward, , 2000Panza et al., 1997;Soloviev andIsmail-Zadeh, 2003, Rundle et al., 2006;, 2012aPeresan et al., 2007;Zöller and Hainzl, 2007;Pollitz, 2009;Bielak et al., 2010;Vorobieva et al., 2014). ...
Understanding of lithosphere dynamics, tectonic stress localization, earthquake occurrences, and seismic hazards has significantly advanced during the last decades. Meanwhile, despite the major advancements in geophysical sciences, yet we do not see a decline in earthquake disaster impacts and losses. Although earthquake disasters are mainly associated with significant vulnerability of society, comprehensive seismic hazards assessments and earthquake forecasting could contribute to preventive measures aimed to reduce impacts of earthquakes. Modeling of lithosphere dynamics and earthquake simulations coupled with a seismic hazard analysis can provide a better assessment of potential ground shaking due to earthquakes. Here we present a block-and-fault dynamics (BAFD) model, which simulates earthquakes due to lithosphere dynamics and allows for studying the influence of fault network properties and regional movements on seismic patterns. The model's performance is analyzed in terms of reproduction of basic features of the observed seismicity such as the frequency-magnitude relationship, clustering of earthquakes, occurrences of large events, fault slip rates, and earthquake mechanisms. Several studies related to the application of the BAFD model to the following earthquake-prone regions are reviewed: the southeastern Carpathians, Caucasus, the western India, Tibet-Himalaya, and the Sunda Arc. We examine then a new approach to seismic hazard analysis, which is based on instrumentally recorded, historical and BAFD-simulated earthquakes, and analyze how earthquake modeling can assist in hazard assessment. Finally, we discuss perspectives in modeling of earthquake occurrences and improvements in hazard assessment.
The block-and-fault dynamics model was developed using the following assumptions: a region is a structure of perfectly rigid blocks separated by infinitely thin flat faults; all deformations occur in fault planes and on the block bottoms. Blocks move as a consequence of a prescribed movement of the structure boundaries and the underlying medium. The displacements of the blocks are determined so that the structure is in a quasi-static equilibrium state. The interaction of blocks along the fault planes is viscous-elastic, as long as the ratio of stress to pressure remains below a certain strength level. When this critical level is exceeded, a stress-drop (“failure”) occurs. These failures are considered as earthquakes. A review of the results obtained by numerically simulating for various block structures is given, including the results recently obtained for the Altai-Sayany-Baikal region. The use of the model for seismic hazard assessment is discussed.
Earth deformation is a multi-scale process ranging from seconds (seismic deformation) to millions of years (tectonic deformation). Bridging short- and long-term deformation and developing seismotectonic models has been a challenge in experimental tectonics for more than a century. Since the formulation of Reid's elastic rebound theory 100 years ago, laboratory mechanical models combining frictional and elastic elements have been used to study the dynamics of earthquakes. In the last decade, with the advent of high-resolution monitoring techniques and new rock analogue materials, laboratory earthquake experiments have evolved from simple spring-slider models to scaled analogue models. This evolution was accomplished by advances in seismology and geodesy along with relatively frequent occurrences of large earthquakes in the past decade. This coincidence has significantly increased the quality and quantity of relevant observations in nature and triggered a new understanding of earthquake dynamics. We review here the developments in analogue earthquake modelling with a focus on those seismotectonic scale models that are directly comparable to observational data on short to long timescales. We lay out the basics of analogue modelling, namely scaling, materials and monitoring, as applied in seismotectonic modelling. An overview of applications highlights the contributions of analogue earthquake models in bridging timescales of observations including earthquake statistics, rupture dynamics, ground motion, and seismic-cycle deformation up to seismotectonic evolution.
We analyze the dynamics of two interactive parallel faults on the basis of a discrete dynamical model. This fault model can be regarded as an extension of the Burrige-Knopoff model. Our main concert is about the temporal variation of characteristic events on the two interactive faults. The tendency is observed that the characteristic events continue to occur only on one of the faults for a period of time; the occurence is quasiperiodic in this period as generally found for the Burridge-Knopoff model. However, the activity is suddenly transferred to the other fault at some time. The period of characteristic even occurrnece tends to alternate on the two faults. This length of this period is generally lager when the interaction are weak between the two faults. This suggests that the recurrence of large events on neighboring faults is highly complex. It is also indicate from our analysis that long apparently quasiperiodic recurrence of large events on a faults may cease at some time if weak fault interaction exist.
Starting with a Green's function representation of the solution of the elastic field equations for the case of a prescribed displacement discontinuity on a fault surface, it is shown that a shear fault (relative displacement parallel to the fault plane) is rigorously equivalent to a distribution of double-couple point sources over the fault plane. In the case of a tensile fault (relative displacement normal to the fault plane) the equivalent point source distribution is composed of force dipoles normal to the fault plane with a superimposed purely compressional component. Assuming that the fault break propagates in one direction along the long axis of the fault plane and that the relative displacement at a given point has the form of a ramp time function of finite duration, T, the total radiated P and S wave energies and the total energy spectral densities are evaluated in closed form in terms of the fault plane dimensions, final fault displacement, the time constant T, and the fault propagation velocity. Using fault parameters derived principally from the work of Ben-Menahem and Toksöz on the Kamchatka earthquake of November 4, 1952, the calculated total energy appears to be somewhat low and the calculated energy spectrum appears to be deficient at short periods. It is suggested that these discrepancies are due to over-simplification of the assumed model, and that they may be corrected by (1) assuming a somewhat roughened ramp for the fault displacement time function to correspond to a stick-slip type of motion, and (2) assuming that the short period components of the fault displacement wave are coherent only over distances considerably smaller than the total fault length.
A laboratory and a numerical model have been constructed to explore the role of friction along a fault as a factor in the earthquake mechanism. The laboratory model demonstrates that small shocks are necessary to the loading of potential energy into the focal structure; a large part, but not all, of the stored potential energy is later released in a major shock, at the end of a period of loading energy into the system. By the introduction of viscosity into the numerical model, aftershocks take place following a major shock. Both models have features which describe the statistics of shocks in the main sequence, the statistics of aftershocks and the energy-magnitude scale, among others.
An explicit expression is derived for the body force to be applied in the absence of a dislocation, which produces radiation identical to that of the dislocation. This equivalent force depends only upon the source and the elastic properties of the medium in the immediate vicinity of the source and not upon the proximity of any reflecting surfaces. The theory is developed for dislocations in an anisotropic inhomogeneous medium; in the examples isotropy is assumed. For displacement dislocation faults, the double couple is an exact equivalent body force.
This chapter presents general numerical methods for treating dynamical problems involving unilateral contact and dry friction. Some examples of applications related to the structural response of rigid or deformable geomaterials, such as rocks, soils, collections of blocks, granular materials, are illustrated. The chapter explains the Coulomb's dry friction law. This law is relevant for a large class of applications to geomaterials. It accounts for the main features of dry friction. It may be easily improved without drastic changes in the proposed methods. The frictional problems appear to be strongly non linear, and call for the techniques of nonsmooth mechanics. Convex analysis is widely used to formulate friction equations and numerical algorithms.
Attempt is made at a classification of models of brittle fracture mechanism and at tracing their interdependence with rheological properties of medium. Definitions of brittle body and ideally brittle body are given. Based on the above, some models of fracture are classified, and the corresponding formal criteria of fracture are discussed. Detailed examination of energy criterion by Griffith is made, and relation is derived which expresses this criterion for an arbitrary rheological model of medium. This expression is simplified for the case of ideally brittle material. General considerations are illustrated by an examination of cracks in elastic and linearly visco-elastic media.
The failure of an asperity, ie the dynamic rupture of a small fault area with finite stress drop surrounded by a broken or weak fault area which has no stress drop but which slips after the asperity fails, is proposed as a model for the rupture process of a subevent in a composite earthquake. The rupture area of the composite earthquake surrounding the subevent is modeled by the weak fault area surrounding the asperity in the subevent model. The resulting seismic moment of the subevent is proportional to the stress drop and the rupture area of the subevent, as well as the radius of the composite earthquake. By setting the stress drops of the asperity models equal to the dynamic stress drops of the subevents, the composite earthquake can be modeled as the sum of a set of subevents which cover the rupture area of the composite earthquake.-from Author
A numerical boundary integral method, relating slip and traction on a plane in an elastic medium by convolution with a discretized Green function, can be linked to a slip-dependent friction law on the fault plane. Such a method is developed here in two-dimensional plane-strain geometry. Spontaneous plane-strain shear ruptures can make a transition from sub-Rayleigh to near-P propagation velocity. Results from the boundary integral method agree with earlier results from a finite difference method on the location of this transition in parameter space. The methods differ in their prediction of rupture velocity following the transition. The trailing edge of the cohesive zone propagates at the P-wave velocity after the transition in the boundary integral calculations. Refs.