The statistical properties of avalanches in a dissipative particulate system under slow shear are investigated using molecular dynamics simulations. It is found that the magnitude-frequency distribution obeys the Gutenberg-Richter law only in the proximity of a critical density and that the exponent is sensitive to the minute changes in density. It is also found that aftershocks occur in this system with a decay rate that follows the Modified Omori law. We show that the exponent of the magnitude-frequency distribution and the time constant of the Modified Omori law are decreasing functions of the shear stress. The dependences of these two parameters on shear stress coincide with recent seismological observations [D. Schorlemmer et al. Nature 437, 539 (2005); C. Narteau et al. Nature 462, 642 (2009)].
Individual earthquakes may be regarded as large scale ruptures involving a wide range of structural and compositional heterogeneities in the crust. However, the statistical properties of a population of earthquakes are often described by simple power-laws. Among them, two laws are ubiquitous and occupy a central position in statistical seismology: the Gutenberg-Richter (GR) law [1] and the Modified Omori law (MOL) [2,3]. The GR law describes the earthquake magnitude-frequency distribution
where M w is the moment magnitude and b a constant with a value around 1 along active fault zones. The MOL describes the aftershock occurrence rate
where t is the elapsed time from the triggering event (the so-called mainshock), p a positive non-dimensional constant with a typical value of 1, and c is a time constant. The parameters in these two laws are believed to bear some information on the physical state of the crust. Indeed, Schorlemmer et al. find that the b-value of the GR law is decreasing going from normal (extension) over strike-slip (shear) to thrust (compression) earthquakes [4]. Narteau et al. also find that the time constant c in the MOL has the same dependence on the faulting mechanism [5]. These two observations indicate that, under a simple assumption, b and c are decreasing functions of shear stress. Although the underlying mechanism needs further investigation, this may reflect a common time-dependent behavior of fracturing in rocks during the propagation of earthquake ruptures and the nucleation of aftershocks.
Because the shear stress along an active fault is not directly measurable, a solution to address stress dependences in earthquake statistics is to analyze models that implement a restricted set of physical processes. There are indeed many models that resemble seismicity, ranging from rock fracture experiments [6][7][8] to computer simulations on cellular automata [9]. Among them, sheared granular media [10][11][12][13][14] fit our purpose perfectly because it is a simple representation of a granular fault gouge for which both the energy and the stress can be easily defined. Here, we perform numerical simulations showing that, as for real seismicity, avalanches in sheared granular matter obeys the GR law and the MOL with b and cvalues which are decreasing functions of the shear stress.
Our granular system is made of frictionless spheres with diameters of d and 0.7d (the ratio of populations is 1 : 1). For the sake of simplicity, we assume that the mass M of these particles are the same. We limit ourselves to a rather small-size system (N = 1500) for computational efficiency. Using the radius and the position of particle i, which are denoted by R i and r i , respectively, the force between particles i and j is written as
, particles i and j are not in contact so that the force vanishes. Throughout this study, we adopt the units in which d = 1, M = 1, and k = 1. We choose ζ = 2.0, which corresponds to the vanishing coefficient of restitution. A constant shear rate γ is applied to the system through the Lees-Edwards boundary conditions [15]. Note that under these boundary conditions the system volume is constant. Thus, the important parameters are the shear rate γ and the packing fraction φ. A steady state of uniform shear rate γ can be realized starting from a class of special initial conditions. Here, we investigate such uniform steady states.
As observed in many previous works on amorphous systems [12][13][14][16][17][18][19][20][21], the temporal fluctuations of the energy becomes volatile if the shear rate is sufficiently low and the density is sufficiently high. From the temporal fluctuation of the energy E(t), we define an energy drop event as follows. The beginning of an event is defined as the time t = t 1 at which the energy E(t) starts decreasing; i. e. , Ė(t 1 ) = 0 and Ë(t 1 ) < 0. The definition of the end of an event is the opposite; t = t 2 at which Ė(t 2 ) = 0 and Ë(t 2 ) > 0. Such an energy drop event is referred to as an avalanche throughout this paper. We also calculate the (global) shear stress σ(t) using the virial [22]. Important quantities that characterize an avalanche are the magnitude M ≡ log 10 [E(t 1 ) -E(t 2 )] + 11, the initial stress σ ≡ σ(t 1 ), the stress drop ∆σ ≡ σ(t 1 ) -σ(t 2 ), and the duration T ≡ t 2 -t 1 . Note that, with this definition of magnitude, the GR law reads P (M ) ∝ 10 -2/3bM , as M w ≃ 2/3 log 10 [E(t 1 ) -E(t 2 )] + const. Hereafter, we use β ≡ 2/3b instead of b. Note also that here the spatial information of avalanches is discarded. In this case, one might overlook simultaneous avalanches occurring at different places. However, this is unlikely because the present system is small (i. e. the characteristic length is approximately of 9d). First, we discuss the nature of the avalanche magnitude-frequency distribution. Figures 1 (a) and (b) show these distributions at several shear rates for φ = 0.644 and φ = 0.650, respectively. The distribu
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