Introduction
One-dimensional dynamical models of the Burridge-Knopoff (BK) type [1] have been used extensively in recent years to investigate the self-organization of earthquakes [2,3]. Much has been learned through the use of BK models, such as the importance of geometrically induced heterogeneity, the destabilization of seismicity due to fault interaction, the role of dynamics, healing,and the nature of friction as influences on the seismic development of a region.
However BK models suffer from two flaws that limit their usefulness as a model of real-earth phenomena. First, stress redistribution after fracture is not scaled by the size of the crack or by some other dimension intrinsic to the nature of the fracture or the geometry; rather, stresses are redistributed by nearest-neighbor coupling in the region beyond the edge of fracture in its own plane, and over a constant distance in the direction transverse to the fracture. Both these features introduce arbitrary and unrealistic scale sizes into the problem.The second problem is equally serious: BK models do not account properly for the loss of fracture energy due to seismic wave radiation away from the fault.The loss of energy by radiation is the principal dissipative process that determines the final state of stress on the fault and hence determines the nature of evolutionary seismicity on a network of faults.
The modeling of dynamic fracturing of cracks embedded in an infinite elastic continuum provides a solution to both problems: it displays both appropriate scaling and takes energy loss by seismic radiation properly into account.Therefore, we have constructed a dynamic, anti plane,continuum crack model to explore the problems of the arrest of motion and the final state of stress on a fault and to investigate the organization of earthquakes both on a single fault and on a network of faults.Chatterjee and Knopoff [4] have presented a modification to Kostrov's [5] solution to the problem of the dynamic growth of a 2-D homogeneous, anti plane crack, to take into account crack growth in the presence of arbitrary fracture threshold and pre-stress. Because the method is a boundary integral procedure, none of the difficulties associated with dispersion due to numerical integration of the wave equation arise.We have written an iterative computational code to solve for the seismic history of a fault system.In these models, the stresses are redistributed beyond the edge of fracture with a fall-off rate that is scaled roughly by the length of the fracture in the case of small fractures, and by a scaling that depends on some characteristic distance for larger events, with details of the scaling that depend on the actual distribution of stress drops and displacements. We emphasize that this is a fully dynamical model of repetitive fractures in the presence of arbitrary pre-stress and arbitrary fracture threshold, using a two-dimensional model of ruptures in a three-dimensional elastic medium.
Healing
As one application of this model, we have investigated the problem of arrest of motion and of the final state of stress on a homogeneous fault segment bounded by unbreakable barriers at the ends. If we adopt the usual stopping criterion that particle motion ceases when the velocity reaches zero, we find that motion at a particular location on the fault is arrested only after reflected stress waves from two barriers converge at the point; the stress waves emanating from one barrier are not strong enough by themselves to arrest slip motions. As a consequence, healing does not begin at the instant of encounter of a growing crack with a single unbreakable barrier.If a homogeneous earthquake fracture is localized between two unbreakable barriers,healing begins in the interior of the crack and spreads bilaterally at the shear wave speed. This pattern of healing is exactly what is needed to ensure that the final slip on a crack of length 2L is proportional to the usual elliptical distribution (L^2-x^2)^1/2 and that the final stress is homogeneous on the ruptured portion of the crack; the overshoot stress drop is 27%greater than the dynamical stress drop.These results are changed if we assume that there is a large enough positive characteristic velocity below which motion is arrested; in the latter case, we can indeed develop healing extending into the crack from the barriers, and may even develop slip pulses where healing takes place in the absence of any encounter with a barrier.
Slip at Landers
Suppose that a barrier of finite width is not unbreakable but only slows down the rate of growth of the crack.Once the fracture erodes this barrier, breakthrough into the segment beyond will severely impact the final state of stress frozen into both the old and the new segments; large kinks can be frozen into the final slip when stress waves from the barrier interact with sites which are already in the process of healing. Whether the healing process (i.e. the restoration of bond strength) takes place quickly or slowly on the time scale of earthquakes will greatly influence the final slip and stress expected for a given earthquake. Until now, little information has been available on the time scale of the restoration of bond strength.
We have compared the theoretical results of two slip models with the slip measured after the Landers earthquake which, according to inversion$^6$, ruptured in two distinct segments with a notable time delay between the portions.In Fig. 1a we show the fracture history of a two-segment fault whose parameters are chosen to fit the Landers earthquake. Healing initiates in the southern (left) portion of the Landers fault; the stress waves from the emergence of the crack through the Landers/Homestead Valley junction may or may not reactivate the healed portion of the fault, depending on the strength of the newly healed bonds.
Figure 1b shows the final slip both for the case in which rupture is reactivated (+)and for the case in which the frozen sites remain frozen (diamond).The comparison with slip observed after the Landers earthquake indicates that bond strength is re-established extremely rapidly, a rather surprising but nevertheless important result for future modeling of stress histories.
The final stress caused by the large kink in slip is shown in Fig. 1c (Not Available). Beside the obvious stress differential which has developed between the southern and northern segments of the fracture, a large kink in stress has also developed at the northernmost tip of the fracture that will also act as a barrier to the growth of future ruptures. We predict that proximate activity on this fault will be on the southern segment and that the Landers/HV junction will not be breached in this subsequent earthquake.
Heaton Pulses, Geometrical In homogeneity
We have shown above that fluctuations in strength or pre-stress can generate stress kinks that strongly influence segmentation. This is true even if the actual fracture strength is homogeneous.As an example, we consider the seismic history of a fault with homogeneous fracture strength but highly inhomogeneous initial stress(Fig. 2a). Large pockets of space and time develop in which earthquakes do not take place; these are the consequence of large negative stress kinks that were frozen into these regions by previous earthquakes, as in the specific case of Fig. 1. Eventually these barriers are overcome by later fractures, and other barriers appear elsewhere. Even for these complex histories on homogeneous systems, we find that the earthquake history quickly organizes itself into a runaway event; we have remarked elsewhere that runaway events are non-physical [7].
Since the long range interactions of a 2-D model accentuate the possibility of developing runaways on homogeneous systems,we have inquired whether a velocity dependent strength model such as those introduced by Beeler and Tullis [8], and Cochard and Madariaga [9], which can develop self-healing pulses [10], will limit runaways on homogeneous systems. We find that self-healing pulses do not solve the problem of runaways.
As a consequence of this last result, we turn to spatial in homogeneity of fracture strength as mechanism to limit the undesirable feature of runaways.We have earlier introduced the proposal that in homogeneity is necessary to prevent runaways in cases with nearest neighbor interactions [2]. In one example of many experiments, we show the evolutionary history for the heterogeneous fracture threshold distribution shown at the right of Fig. 2b. The evolutionary seismicity with these long range stress redistributions for this case of spatially varying fracture thresholds, now displays what appears to be a stable pattern with properties much as we have observed in the nearest-neighbor cases, as can be seen in Fig 2c.
Fault Interactions
We have some preliminary results on the evolution of seismicity on a fault network through our studies of the interactive effects of long-range stress redistributions on two fault segments on a single fault plane that are separated by a creeping zone. If the size of the creeping zone is sufficiently large compared to the size of the fault segments, then each segment breaks as an isolated system and both segments quickly organize into a history of characteristic earthquakes which invariably rupture the entire length of the segment. As the size of the creeping zone shrinks, the interaction between theta segments due to the long-range forces breaks up the earthquake sequences into smaller earthquakes, thereby interrupting,delaying, or even inhibiting altogether the onset of large characteristic earthquakes. Thus the development of characteristic earthquakes seems to depend on the existence of long reaches of relatively smooth fault structures without significant interaction with activity on neighboring faults. This issue and others demand further exploration.
The asperity model
There are numerous other problems, having to do both with the problem of rupture and healing of single earthquake events, as well as with the organization of earthquake histories, which we have and can explore using the dynamic continuum crack model. We have used our model to investigate how much of the great range of stress drops of earthquakes in Southern California could be explained by the geometry of the fault. We have taken as extremes an asperity model of an earthquake fault and a homogeneous fault bounded by unbreakable barriers. We have found that the energy release to stress drop ratio could be up to three times greater in the asperity model as compared to the barrier model. While significant, this falls far short of the 1.5 to 2 orders of magnitude difference in ratios that are observed in nature.Thus we have used our model to elucidate the fact that we must look beyond fault geometry alone to solve the stress drop riddle.
Proposed Research, 1998
We propose to attack the following issues with the long-range stress redistribution model:
1. The slip in the 1992 Landers earthquake is the only clear-cut example thus far of a multi-segment earthquake that we have tried to fit using our dynamic continuum crack model. Our conclusion is that bond strength between the walls of an earthquake fault, i.e. healing, may at times be restored very rapidly. We propose to study other multi segment earthquakes for which we have sufficient data, such as the 1984 Morgan Hill earthquake and possibly the Superstition Hills earthquakes, to see if this conclusion holds.
2. We have begun work on understanding seismic history on a network of faults by investigating the interactive influence on seismic histories of two fault segments on a single plane separated by a creeping zone. We propose to expand this effort to investigating the seismic histories of multiple faults distributed in a 2-D space.
3. We propose to use these models to develop more realistic``seismograms" than are presently available from BK models, for application to strong motion studies.
4. We have already begun to explore the development of seismicity on the fault system of Southern California using these two-dimensional antiplane, dynamical models. We are using as test structures those given by Ward [11] to provide a priori assumptions about the lengths, fracture strengths and loading rates of fault segments. We will develop an iterative seismicity history for comparison with the quasistatic results given by Ward.
5. Our model is currently limited to conditions of anti-plane slip. We propose to initiate a long-term proposal to converting the present program to calculate rupture dynamics under conditions of in-plane slip.
6. One of the defects of the 2-D models is that their Green's functions have infinitely long tails. Cochard and Madariaga [9] attack this problem by allowing the dynamic friction to increase with decreasing velocity, imposing a low, finite velocity at which locking takes place; a Heaton pulse is then generated. We consider this an it ad hoc although computationally attractive solution to the nonphysical problem of the infinite tail in the 2-D case. We propose instead to start development of a fully 3-D dynamic model,using the 3-D Green's function for a double-couple point source in the fracture plane embedded in the elastic medium; again this program will take much time to develop.
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