Fig. 7 by the broken lines. Let us add to this distribution some heterogeneity and look at the slip and stress distribution, as shown in Fig. 7 with the solid lines. We observe that a modest change of the slip distribution produces large variations in the residual stress pattern. Since stress can only change inside a limited range (0 < T < T_{u}), these lateral variations put some fault elements closer to rupture than others. As a consequence, the future seismicity of the fault is completely determined by these heterogeneities in the residual stress field. Thus the mechanism that generates complexity in our model is clearly identified, although we do not know what is the bifurcation value for the control parameter T_{sp}. We propose that the key to the creation of heterogeneity in our models is partial stress drop. This is not at all a new concept in seismology; Brune (5) proposed from observational arguments that most earthquakes presented partial stress drop and suggested that the dynamic stress drop (T_{u} in our case) was much larger than the static stress drop. The main difference between our results and the suggestion by Brune (5) is that he considered a model where partial stress drop was uniform along the fault, while in our models a partial stress drop triggers strong lateral variations of the static stress drop.
We have demonstrated that, for certain highly rate-dependent friction laws, a simple antiplane fault embedded in a homogeneous medium can spontaneously become complex. This complexity has several interesting features:
Premature locking of the fault, so that slip duration at any point of the fault is independent of the total size of the fault. Premature healing produces partial stress drop, so that stress heterogeneity may be simply due to the extreme sensitivity of fault stress to very small changes in the slip distribution.
Self-healing slip pulses are spontaneously generated for large values of T_{sp}. These pulses were proposed by Heaton (35) and they were explained in our previous work (10).
Stress heterogeneity and partial stress drop are manifestations of the same underlying instability. Partial stress drop occurs for all friction models that have a strong rate-dependent friction. Partial stress drop disorganizes the fault for the simple reason that stress drop of neighboring points will be highly variable.
Slip gradient (dislocation density) and stress heterogeneity appear when small scale modes of slip on the fault can express themselves. For full stress drop models like rate-independent friction laws, these small scale modes are suppressed by the requirement that the stress drop be fixed, uniform, and determined only by constitutive parameters. In that case, only material heterogeneity can produce complexity.
Seismic events (i.e., events whose length is greater than the length of the slip-weakening zone) follow an ℓ^{2} scaling law in which seismic moment scales like the product of partial stress drop and the square of the length of the zone that actually slipped during the event. Thus, the regularization length that is included in our slip-weakening model has no influence on the properties of large seismic events. We find that the size of the slip pulse that traverses the fault determines the actual final size of the rupture. If the slip pulse is large, it simply reaches further along the fault.
Seismic-like events in our simulations are both periodic and simple for small values of the control parameter T_{sp}. For values larger than a critical value situated around T_{sp}=0.6, we observe events of all sizes and a much higher rate of seismicity.
In conclusion, we have shown that a rate-dependent friction can spontaneously produce heterogeneity for large values of a control parameter. This limit corresponds to that of the friction law used by Carlson and Langer (14) in their study of the Burridge and Knopoff model. At the other extreme, the rate-independent friction suppresses these instabilities for the very simple reason that partial stress drop is eliminated from the outset. It is very likely that both material heterogeneities and dynamically generated complexity play a role in determining the observed complexity of faulting and seismic events. Given the little knowledge that we currently have about the friction laws at high slip rates, we firmly believe that heterogeneity should be explored without preconceived assumptions about which of material and dynamically generated heterogeneity dominates in the earth.
J.P.Vilotte provided many useful and timely discussions. This work was initiated with support from the European Union under Contract N ERBSC1 from the “Science” Program.
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