The following HTML text is provided to enhance online
readability. Many aspects of typography translate only awkwardly to HTML.
Please use the page image
as the authoritative form to ensure accuracy.
Proceedings of the National Academy of Sciences of the United States of America
inappropriate modeling of the transfer of stress from the fault regions outside it into the lattice segment. A throughgoing or lattice-wide fracture is not stopped by the same mechanism as that which stops the smaller fractures whose spans lie wholly within the lattice. There are thus two arguments against the applicability of homogeneous lattice models of self-organization: first, it is not possible to prevent the “runaway” or lattice-wide event, which is not stopped by the same mechanism as the smaller earthquakes, and second, there is no mechanism for discriminating between small and large earthquakes, as demanded by the finiteness of the energy budget for earthquakes (3). The same arguments apply against the development of scaling on an otherwise homogeneous system in the tensor (in-plane) case as well.
The dynamics of the fracture process represents an escape from the tyranny of the resolute increase of stress. A redistribution argument based on conservation laws no longer applies in this case, since the stresses are no longer solutions to Laplace’s equation but are solutions to the elastic wave equation. The loss of energy in elastic wave radiation reduces the amount of energy available to promote further slip. However, dynamics does not provide an escape from the failure of a homogeneous system to develop an internally derived scale size that separates large from small fractures; absence of scaling is a powerful argument that self-organization of dynamic fractures on a single homogeneous fault must also lead to the abyss of the lattice-wide event.
We illustrate these remarks by a consideration of the self-organization of dynamic fractures on a single homogeneous fault model with periodic edge conditions through the vehicle of the Burridge-Knopoff (B-K) (12) spring-block model in one dimension. We do not elaborate on the dynamic B-K model, which has been discussed in detail elsewhere, except to remark that, in order that changes in the stress due to fracture be redistributed via elastic wave transmission, we guarantee that the (supersonic) dispersion, due to the local influence of the transverse springs in some dynamic versions of this model (13, 14), is moderated by fine-tuning a radiation damping term (15) that is equivalent to introducing a local viscosity as a frictional damping of the slip (12). It can be shown that the tunable radiation term merely represents a scaling of the size of the fracture with respect to the lattice spacing. Laboratory measurements of the nature of sliding friction in the range of slip velocities that occur in earthquakes have not been made: in our model, we do not invoke a velocity-dependent sliding friction; instead, the strength of the bonds at the crack edges drops instantly to the dynamic friction at the instant that the critical threshold stress is reached (15). Although the B-K models do not generate redistributed stresses that are scaled by the crack length, nevertheless these B-K models with critical radiation damping generate slip pulses whose ranges are of the order of a(l/k)1/2 where a is the lattice spacing and l and k are the transverse and longitudinal spring constants and hence appropriate to simulations of self-healing pulses (16) in large earthquakes wherein slip is concentrated mainly near the growing edge of a crack. This model is more appropriate to the modeling of large earthquakes than small ones.
For arbitrary initial conditions on the homogeneous one-dimensional (1D) dynamic B-K fault model, the system quickly organizes itself into fractures with a power-law distribution of sizes (17). The events with power-law distribution of fracture sizes is only a transient state. Power-law transiency in the self-organization of other nonlinear systems has also been identified recently (18). Ultimately, a runaway event takes place that spans the entire lattice and hence has an infinite length on the periodic lattice. After the first runaway, subsequent seismicity displays only periodic runaways to the exclusion of smaller events, but this is a consequence of the smoothness of the stress after each runaway; other scenarios after the first runaway are possible, but they too lead to further runaways, and so on indefinitely.
In the transient state, the power-law distribution for the sequence of seismicity is not Poissonian but is rather a distribution that is dominated by (almost) periodic localized clusters. There is an intense clustering of repetitive events of almost the same size at the same locations, at almost equal time intervals (Fig. 1). These persistent clusters disappear after some time and others appear elsewhere. Such persistence is due to the smoothness of the stress across the extent of a fracture in this model (19). If the stress is smooth after fracture, the restoration of stress by the external loading mechanism brings the extent of a previous fracture to the uniform critical threshold at the same time, and hence local recurrence dominates this phase. Because of the nearest-neighbor property of the 1D B-K model, any changes in the stress can only take place at the edges of the fractures. Thus, the length of a fracture differs from its immediate predecessor only at its edges, and hence the persistence of a cluster is approximately scaled by the length of the fractures. The power-law distribution that results from this model describes the number of clusters of a given length weighted by the number of repetitions within the cluster. Thus, the spatial localization is evanescent, and the pattern has an overriding imprint of a periodicity imposed by the coupling of the smoothness of the postfracture stress and the homogeneity of the threshold fracture strength. The probability that any point along this fault will experience a large earthquake is the same for all points along the fault over the long term; over the long term, there can be no spatial localization.
A rolloff that is observed in the distribution (17) does not define a characteristic length; the scale size that is implied is an artifact of the fact that the count is terminated at the time of the first lattice-wide event; events that are slightly less than lattice-wide in size are undercounted compared with expectations for a larger lattice. The magnitude that corresponds to the rolloff corresponds to the parameter that mimics the seismic radiation; the larger the energy loss in the parameterized radiation, the longer the time to inevitable runaway.
Even if long-range forces are taken into account, the self-organization of fractures on a homogeneous landscape must ultimately develop a dynamic crack that is larger than any given size. A similar conclusion is reached if one introduces a weakening of strength into continuum quasistatic models with long-range redistribution of stress (20).
The Influence of Geometry
We turn to inhomogeneous faults or fault zones as systems with intrinsic scale sizes to explore the differences between large
FIG. 1. Portion of the slip history of the transient phase on a homogeneous 1D B-K model with periodic end conditions. Vertical strokes define the linear extent of a fracture; fractures with greater length release more energy. Evanescent persistent clusters can be identified. After a long time, this pattern is replaced by lattice-wide periodic fractures. [Reproduced with permission from ref. 17 (copyright 1994, American Institute of Physics).]