The aim of this project is to advance the physical basis for understanding the earthquake process. We continue fully inertial elastodynamic investigations of fault models to understand what model features control event sequences and population statistics, partly to resolve current debates in earthquake source theory. Also, progress has been made on understanding what controls whether the mode of rupture propagation on a velocity-weakening fault is a self-healing pulse or a crack, in 2D and now in 3D, as well as on understanding mechanisms by which self-healing ruptures can occur between dissimilar materials.
(1) Elastodynamic simulations of earthquake sequences in a 2D model of a faulted plate coupled to a moving substrate: A 2D elastodynamic model is adopted of a faulted crustal plate, coupled to a steadily moving substrate, and we study its predictions of earthquake sequences. The model is based on the Lehner et al. (JGR, 1981) elastic generalization of Elsasser coupling to a substrate, as extended to dynamics by Johnson (GJI, 1992) but simplified in a scalar wave sense like in Langer et al. (Proc. NAS, 1996) and Myers et al. (Phys. Rev. Lettr., 1996). The nondimensional description for along-fault displacement , averaged over seismogenic depth, is, where x and y are in the plane of the plate, the fault is on , and is the substrate loading rate. The depth-averaged stress, proportional to , and slip are related by rate- and state- dependent friction laws. We have developed a spectral numerical procedure for this class of problems, similar to that of Perrin et al. (J. Mech. Phys. Solids, 1995) as used in 2D dynamic simulations of earthquake sequences in a depth-variable fault model by Rice and Ben-Zion (Proc. NAS, 1996). The fault segment studied is replicated periodically along strike.
We put velocity strengthening regions at the ends of the fault segment which serve as barriers to ruptures. This is done to prevent the ruptures that tend to span the whole fault segment from wrapping around and corrupting the stress and slip history. Our goal is to investigate within a rigorous numerical approach whether the part of the fault segment with the uniform properties can produce G-R-like event complexity for physically justifiable values of parameters. Rice and Ben-Zion (1996) and Cochard and Madariaga (JGR, 1996) have found a broad distribution of event sizes in uniform fault models, rigorously analyzed within a continuum limit, only for special choice of constitutive parameters, and have not found a G-R-like small event distribution (unless the uniform fault is replaced by one with strong local heterogeneities; Ben-Zion and Rice, JGR, 1995, 1997). In contrast, Langer et al. (1996) and Myers et al. (1996), in analyses of the crustal plane model studied here, report generically complex response and a G-R type small event distribution, although it has been suggested that the latter may be related to the abrupt strength drop procedure used in their numerical algorithm (Rice and Ben-Zion, 1996).
In the simulations done so far, no desired slip complexity was observed. A typical sequence of instabilities is shown in Figure 1, top (in terms of real, dimensional variables), where full black lines are printed every 5 years, and the dashed lines are printed after equal increments of moment release during a dynamic event. There are few small events located at the ends of the creeping barriers and most likely caused by high stress concentration resulting from the creep. All the other events have lengths of the order of the crustal thickness, which is taken to be 15 km, and most of the events are comparable in size to the whole fault segment studied.
The individual model earthquakes produced by the simulations have many realistic features like rupture velocities of order kilometers per second, slip velocity of order meters per second, and long preceding nucleation phase of quasistatic growth (Figure 1, bottom). This rigorous treatment has become possible only recently, after major improvements in the numerical scheme employed, which are discussed in the progress report on the companion project New methodology in computational seismology for dynamic rupture along complex fault systems. The improvements have made it possible to consider much smaller state-transition slip distances of the constitutive law, which in turn allowed to make the nucleation size of an instability small compared to the crustal thickness. Problems were experienced previously in the simulations, like slow rupture propagation and essentially quasistatic development of instabilities up to sizes comparable to the crustal thickness. These were most likely caused by selection of too large slip-weakening distance of the friction law. We are currently studying whether taking longer fault segments and still smaller nucleation sizes (which has become numerically tractable now) will complexify slip history in this model.
The model earthquakes in the sequences studied exhibit both crack-like and pulse-like modes of propagation. An instability develops in a crack-like manner (that is, duration of slip at each point participating in the event is comparable to the total duration of the event) if it starts far from the velocity-strengthening regions. Propagation of an instability is pulse-like (that is, duration of slip at a point is just a fraction of the total duration of the event) if it starts close to the regions. Thus, mode of rupture propagation in this model is mainly controlled by the geometric heterogeneity introduced. Cochard and Madariaga (JGR, 1996) and Zheng and Rice (AGU, 1996) have shown, however, that pulse-like ruptures can be obtained just from dynamics, without any geometric heterogeneity involved, under appropriate conditions and enhanced velocity weakening at seismic slip rates, and that such ruptures promote complexification of slip history by leaving highly heterogeneous stress distribution. The studies are under way to learn if modifications of the constitutive law in the crustal plane model to allow enhanced velocity weakening at rapid slip rates will produce dynamically generated pulse-like ruptures and/or alter our present conclusions on lack of dynamically generated slip complexity, and especially lack of a G-R type small event distribution.
(2) Mode of rupture, self-healing slip pulse versus enlarging shear crack, for a velocity-weakening fault in a 3D solid: We study the mode of earthquake rupture on a spatially uniform velocity-weakening fault embedded in a 3D elastic solid. The fault is subjected to a uniform applied shear stress to , which we perturb only in a localized region to nucleate failure. Our aim is to establish which conditions of stressing and constitutive response lead to rupture propagation in a self-healing slip pulse mode, and which in an enlarging shear crack mode. Previous investigations of this issue involved 2D elastodynamic modeling (Perrin et al., J. Mech. Phys. Solids, 1995; Beeler and Tullis, BSSA, 1996; Cochard and Madariaga, JGR, 1996; Zheng and Rice, EOS, 1996; Zheng, Ph.D. Thesis, 1997).
Zheng and Rice, however, gave a general theorem, applicable to 3D as well as 2D modeling, identifying a stress level tpulse such that no solution in the form of an indefinitely enlarging shear crack, concentrating stress ahead of itself, could exist when . Let V be slip rate and the steady state strength (of a rate/state friction formulation, in which the slip distance L for state evolution is much smaller than typical seismic slip). Then tpulse is defined as the largest to satisfying for all V > 0. (See Figure 1a of last year's progress report.)
For stresses , Zheng and Rice also identified a parameter , with V being the largest solution of . They found from their 2D simulations that for T near 1 (which means to only slightly above tpulse) the rupture mode remains self-healing, whereas for T near 0 the mode is of enlarging shear crack form, with transition in the vicinity of .
Here we use 3D simulations to illustrate the theorem of no crack-like rupture when , and we explore the usefulness of T in sorting the rupture mode in 3D when .
The elastodynamic model used for the surrounding medium is a 3D scalar model (see Figure 2 of last year's progress report), which is simplified compared to real vectorial elastodynamics in that displacement occurs in only one direction and there is only one wave speed. The computational methodology is based on the spectral formulation of elastodynamics (Geubelle and Rice, J. Mech. Phys. Solids, 1995; Cochard and Rice, ibid, 1997)
In 2D, the initial, overstressed, nucleation patch was a small segment. In 3D, for symmetry reasons, it is chosen to be a circular region.
The constitutive law used agrees with the Dieterich-Ruina form based on lab studies at low slip rate V, i.e., rate/state with logarithmic dependence, but extrapolates such studies with a much stronger steady-state velocity dependence at typical seismic slip rate (as suggested by a limited number of recent lab data), i.e., in the form 1/(1+V/Vweak).
We find that T is still a useful parameter to predict the rupture mode. But we find that the transition between the crack mode and the self-healing mode is around T ~ 0.25, with self-healing occurring for T > 0.25 and crack occurring for T < 0.25. In other words, a self-healing slip pulse is more likely to occur in 3D than in 2D, everything else being the same. Figure 2 shows the slip velocity history for Vweak = 100 and T = 0.30 (top, pulse like) and T = 0.15 (bottom, crack like).
Within this 3D physical model, our long-standing goal is to determine whether or not a homogeneous, tectonically loaded, fault segment will spontaneously self-organize and give rise to a broad range of events. We are interested in addressing this issue for well-established laboratory-derived friction laws as well as for enhanced weakening laws of the type described above, since previous 2D simulations (Cochard and Madariaga, JGR, 1996) suggested that complexity is more likely to occur in this case. With previously used procedures, the computational requirements were too demanding for the study of any realistically large problem. Presumably, however, the recent numerical improvements that we have made (as explained in the companion progress report on new methodologies) will soon allow us to achieve this goal. We plan also to do these calculations in the framework of vectorial elastodynamics, with distinct mode II and III effects, rather than just for the scalar 3D theory.
(3) Self-healing ruptures along dissimilar material interfaces: Dissimilarity of material properties between elastic solids bounding a fault plane allows inhomogeneous slip to induce a change in normal stress. As shown by Adams (J. Appl. Mech., submitted, 1997), such effect provides a mechanism for self-healing slip pulses that propagate along a frictional interface at a generalized Rayleigh speed, under a remote driving shear stress that is arbitrarily less than the nominal shear strength of the interface based on the remote compressive normal stress . Adams gives such solutions for constant or velocity-dependent friction coefficient f. Related recent studies are by Adams (ibid, 1995), Andrews and Ben-Zion (JGR, 1997) and Harris and Day (BSSA, 1997). Here a formulation for steadily traveling dislocation distributions on the interface between elastic half-spaces by Weertman (JGR, 1980) is used to give a simple re-derivation of the Adams solution, and to extend it to a wide class of friction models (rate and state, possible temperature variation from shear heating, friction with cohesion). In these solutions, slip d and the local interface stresses and depend only on , so that (at t = 0)
, ,
where B is dislocation density, and and are functions of disturbance speed c. Because , one has where is the slip velocity. Hence if the disturbance moves at the Rayleigh speed c, for which =0 (such c exists for modest dissimilarity of properties, typically for shear wave speeds different by less than 30%), then and [where , in general] along the interface. Since t is unaltered from its remote value, the mechanism of low-stress sliding in this class of solutions is that s is reduced, however much it must be, by appropriate selection of V, to allow frictional slip at the remotely applied . If this mechanism, with lab-like f values, is the source of the self-healing rupture mode implied by seismic inversion studies (Heaton, PEPI, 1990), then average slip rates in the pulse must be more than an order of magnitude greater than is now generally assumed (above 10 m/s rather than of order 1 m/s), and pulse widths must be much more narrow. Further limitations of the mechanism are that it acts for in-plane but not for anti-plane slip, and requires a unique direction of rupture propagation along the interface.
Publications Fully or Partially Supported by 1997 SCEC Studies
Papers:
Ben-Zion, Y. and J. R. Rice, Dynamic simulations of slip on a smooth fault in an elastic solid, J. Geophys. Res., vol. 102, pp. 17,771-17,784, 1997.
Cochard, A., and J. R. Rice, A spectral method for numerical elastodynamic fracture analysis without spatial replication of the rupture event, J. Mech. Phys. Solids, vol. 45, pp. 1393-1418, 1997.
Cochard, A., A new, faster, numerical method to evaluate the stress contribution of past slip history on a fault plane, involved in earthquake simulations, to be submitted to Bull. Seismol. Soc. Am., 1998.
Lapusta, N., G. Zheng, Y. Ben-Zion, J. Morrissey and J. R. Rice, Elastodynamic analysis for slow loading processes with episodes of rapid rupture on rate and state dependent faults, to be submitted to Bull. Seismol. Soc. Am., 1998.
Zheng, G., Dynamics of the earthquake source: An investigation of conditions under which velocity-weakening friction allows a self-healing versus crack-like mode of rupture, Ph.D. thesis, Harvard University, 1997.
Abstracts:
Cochard, A., and J. R. Rice, Mode of rupture, self-healing slip pulse versus enlarging shear crack, for a velocity-weakening fault in a 3D solid, EOS Trans. Amer. Geophys. Union, vol. 78, Fall Meeting Supplement, in press, 1997.
Lapusta, N., and J. R. Rice, Elastodynamic simulations of earthquake sequences in a 2D model of a faulted plate coupled to a moving substrate, EOS Trans. Amer. Geophys. Union, vol. 78, Fall Meeting Supplement, in press, 1997.
Rice, J. R., Slip pulse at low driving stress along a frictional fault between dissimilar media, EOS Trans. Amer. Geophys. Union, vol. 78, Fall Meeting Supplement, in press, 1997.
Figure 1: Top: Typical sequence of events (see text). Bottom: Development of a crack-like event from the sequence.
Figure 2: Slip-velocity as a function of time along a line going through the center of the initial circular path. Top: self-healing slip-pulse behavior (T = 0.30); Bottom: crack-like behavior (T = 0.15). In both cases, Vweak = 100.
