Appropriate boundary conditions for modeling very long earthquakes and physical consequences

Abstract

We have studied two sets of models to find appropriate boundary conditions for the theoretical modeling of very long earthquakes (i.e., those for which the length L is much larger than the width W of the rectangular aftershock area) and the resulting physical consequences. In the first set, we study the conventional rectangular crack model (for L/W = 2 and 4), i.e., the stress drop is confined to a rectangular region and there is no rupture outside this region ( W model). In the second set of models—which is suggested by geologic data—we constrain the stress drop to a rectangular zone (“seismogenic layer”) but we do not have any constraint on the rupture above and below this layer. The applied stress in this region above and below the seismogenic layer is taken to be exactly equal to the yield stress of the material so that no stress drop occurs when it yields. This implies that the material on either side of the seismogenic layer is shearing aseismically just before the earthquake occurs and also implies that the material in these regions is purely plastic. The major conclusion of our work is that the second set of models, the rupture propagates dynamically for some distance into these nonbrittle zones above and below the seismogenic layer. This result is supported by geological evidence and also by recent seismic evidence that moments of some large earthquakes increase with increasing wavelength at very long periods. The slip in large parts of these zones is found to be nonnegligible. The slip is still confined to a finite width which may be considered as an “effective width” but this width is larger than the width of the seismogenic (brittle) layer. This effective width increases as the length of the region of stress drop increases and may be larger than this length for long faults. Thus, for the second set of models, the aftershock area would underestimate the actual rupture area. The moment and the slip at the center for the second set of models are larger than for the first set for the same L/W ratio of the stress-drop region even if we assume that there is no radiated energy from the regions above and below the seismogenic zone. For L/W = 2, the duration of slip at interior points on the fault for the two models are the same but for L/W = 4, the duration of slip at a given interior point is much larger for the second set of models. However, the particle velocity during this added time of slip for the second class of models is small and almost constant, so that (at least for L/W up to 4) this difference may not be significant for strong ground motion near such a fault. The rise times and slips at any interior point of the fault is mainly width controlled for the first set of models and mainly controlled by the length of the elastic region or the “effective width” for the latter set. Since longer faults are observed to have larger slips than shorter ones of the same width, the first set of models would predict that stress drops increase with rupture length (for the same fault width). However, this observation can be explained by the second set of models even for constant stress drops. For the second set of models, we find that a fault twice the length of another has much less than double the slip for the same average stress drop. We suggest that both classes of models studied are realistic models of earthquake sources but for different cases.