ON ANTIPLANE DISLOCATION MODELS OF STRIKE-SLIP FAULTS IN LAYERED MEDIA
Maurizio Bonafede and Eleonora Rivalta University of Bologna, Italy

Faults generally cut across elastic (e.g. basement-sediment) and rheological (e.g. brittle-ductile) boundaries. A layered medium is described mathematically as a welded sequence of homogeneous layers separated by interfaces on which continuous displacements and tractions are imposed. Results within the theory of generalized Cauchy equations are employed to obtain uniformly convergent solutions for the stress field close to the fault plane of a strike-slip fault. An important result is that the dislocation density is bounded but discontinuous at any layer boundary, the jump depending only on the rigidity ratio, not on the crack lengths and, apparently, not on the stress drop (but see later). From the knowledge of the dislocation density and its singular properties, the stress and displacement fields can be evaluated by straightforward integration of elementary dislocation solutions. From these results it can be shown that, on a vertical planar strike-slip fault, the stress-drop must be discontinuous at any welded interface between layers endowed with different rigidities. The jump in stress drop values across an interface is proportional to the jump in the rigidity values. The stress-drop discontinuity is strictly related to the boundary conditions imposed on layer discontinuities and on the fault plane. However, the stress on a fault plane is determined by tectonic processes and fault rheology. Is the stress-drop discontinuity a necessary corollary of tectonic processes ? The answer is YES in several cases, such as:
(Y1) elastic layers strained uniformly from a common stress-free configuration, which undergo faulting with complete release of stress, (leading to zero residual stress);
(Y2) elastic layers strained uniformly from the configuration left after the previous earthquake in which the residual stress was present on the fault plane (repetitive earthquake cycle). But the answer is NO in a few important cases, e.g.:
(N1) fresh fractures developing in presence of friction: the initial stress is proportional to the local value of rigidity but the residual stress is not; the stress drop may even be negative in the softer layers !
(N2) Recent sediments on top of pre-stressed basement rock: in presence of uniform tectonic strain rate, the initial stress before failure in the sedimentary layer can be much less than required from the discontinuity condition.
(N3) Rheological discontinuity: the initial stress in a ductile (e.g. viscoelastic) layer is not proportional to strain. In presence of uniform tectonic strain rate, the initial stress in the brittle layer can be typically much higher than in the ductile layer; much higher than in the ductile layer;
(N4) Previous earthquakes with slip terminating at the interface: these would leave the residual stress in the faulted layer, but a very high (theoretically infinite) stress in the layer beyond the crack tip. In cases (N1-N4), the actual mechanisms of rupture cannot obey the stress-drop discontinuity condition. If fractures take place across layer boundaries, one or more of the model assumptions must be violated:
(a) the interface cannot remain welded after crack slip; a horizontal shear crack may develop along the boundary between layers; this is certainly a possibility, but it does not seem to be feasible at large depth (where friction is exceedingly high).
(b) The displacement cannot be antiplane, i.e. the displacement cannot be in one and the same direction in both media; This second possibility conflicts with the welded bondary condition,
(c) the fault cannot be planar}, i.e. it cannot lie on the same plane in both media; It seems that this is the only generally feasible way to produce strike- slip fractures across a layered medium, with arbitrary stress drop values. As discussed above, this applies particularly to
(c1) fresh fractures in presence of friction (the stress drop discontinuity is greater than prescribed for a planar fault);
(c2) faults extending from the basement rock into a surface layer of recent sediments (stress drop discontinuity is greater than prescribed for a planar fault);
(c3) faults extending into the ductile crust (the stress drop discontinuity is less than prescribed for a planar fault). In such cases the two-dimensional approximation is inappropriate. In cases (c1) and (c2) throughgoing strike slip faults are possible if the fault surface rotates in the direction of the tensile axis, resulting in left-stepping secondary fractures in the surface layers, associated with right-lateral faults. Similar secondary faults are very common in the south Iceland Seismic Zone. These secondary faults then represent the branching and upward continuation of the main fault at depth. Laboratory experiments with clay cakes or sand boxes also suggest the same complex style of fracture. This mechanism of segmentation plays an increasing role in the presence of recent sediments (with less initial stress than assumed above) or in presence of uniform tectonic tensional strain (decreasing the normal stress more in the hard layer than in the softer one). A tectonic compressional strain plays an opposite role. In case (c3) the stress drop is typically higher in the softer layer. A further possibility for having stress drop values larger in the softer medium may be one in which slip has taken place in the past over the deep fault section, below the interface. Even in this case, a planar fault across the interface cannot develop without violating the welded boundary condition. Both mechanisms operate if a stable- sliding fault section is present below the interface (as inferred from geodetic data for the San Andreas fault). A similar situation may be present below the South Iceland Seismic Zone. In similar cases, a troughgoing non-planar fault surface is theoretically possible, with its normal rotated toward the compressive axis. But this is energetically disfavoured with respect to antithetic faulting taking place in the softer brittle layer, on conjugate planes (with respect to the deep shear zone).

Laboratory experiments and in-field data actually show that a nascent transform boundary typically develops as an elongated domain of parallel conjugate faults, before a mature wrench fault develops in the direction of prevailing motion. Large strike-slip earthquakes have occurred in the SISZ in the recent past, with magnitudes above 7 and inter-event time of approximately 80 years. GPS measurements show that the SISZ is undergoing left lateral shear in the ENE- WSW direction at a rate of 2 cm/y. In apparent contradiction with the kinematic framework, major earthquakes in the SISZ take place on N-S trending, dextral faults. The SISZ appears to be a transform zone under formation (Bjarnason et al. 1993), whose origin can be ascribed to the southward migration of the Eastern rift zone. The present results offer a key to understand the apparent kinematic paradox mentioned above: steady shear motion under the seismogenic crust may actually take place in the ENE-WSW direction under the SISZ, but displacement continuity across major structural interfaces denies the possibility of fulfilling the stress-drop discontinuity condition for planar fresh fractures taking place in the shallow crust. A similar zone of steady slip at depth is well documented below the San Andreas fault in California. Dextral faulting in the conjugate direction is then a feasible alternative.