GEOPHYSICAL RESEARCH LETTERS, VOL.19, NO.11, PAGES 1145-1148, JUNE 2, 1992 DISTRIBUTION OF DEFORMATION AROUND A FAULT IN A NON-LINEAR DUCTILE MEDIUM Terence D. Barr and Gregory A. Houseman VIEPS Department of Earth Sciences, Monash University, Melbourne, Australia Abstract.The deformation field surrounding a fault in a non-linear ductile medium is characterized by two com- ponents: a singularity in the stress field at the faulttip which is required by the zero shear stress and discontinuous velocity boundary conditions on the fault, and a non- singular component which is required to satisfy arbitrary stress or velocity boundary conditions on the external boun- dary. For regions closer to the fault tip than to theexternal boundary, the singular component dominates. Near the fault tip thestrain rate components decay as a power-law func- tion of radius with an exponent of -n /(n + 1 ) (where strain rate variesas deviatoric stress to the n th power), and the stress components decay as a power-law function of radius with an exponentof-1/(n + 1 ). The behaviorof the mechanical energy dissipation, however, is independent of therheology; it is inversely proportional to the radialdis- rance from the fault tip for all values of n. These results areconsistent with the prediction of Rice and Rosengren [1968] who obtained the corresponding power-lawrelations for the analogous case of a traction free crack in a non- linear elasticmedium. From both the radial and angular dependences of the components of shear strainrate, a local- ized shear zone will develop alongthe immediate extension of the fault, and the degree of localizationincreases with increasing n. Introduction Deformation in the lower crust is assumed to be dom- inated by a ductile (probably non-linear) rheology as opposed to brittle deformation mechanisms, and faults that cut through the brittleupper crust must either penetrate this ductile medium or terminate at a shear zone or detachment fault within the mid-crust. The transition from localized shearing or faulting to distributed ductile behavior is depen- dent on strain rateamong other things, soit is important to understand the interaction of a fault with the surrounding ductile material to determine how a fault will evolve in the lower crust.For example, it is necessary to determine how strain is partitioned between slipalong thefaultand ductile strain in the mediumaround the fault, and whether the fault can actually propagate into the ductile medium. The problem of how a crustal-scale fault deforms over time provides another impetus for studying theeffects of a fault in a ductile medium. Although locally thecrust may demonstrate brittle behavior, on very long length and time scales, thecrust asa whole canbe approximated asa duc- tile medium. As a large scale fault moves, it deforms over Copyright 1992 by the American Geophysical Union. Paper number 92GL00863 0094-8534/92/92GL-00863503.00 time, rotating and bending. Large transform fault systems such as the SanAndreas Fault system showmarked curva- ture of both the main fault and many of its associated splays. Model Problem We use the finite element method to solve for the distri- bution of stress and strain rate for a simple example in whicha planar fault is cut into a block of non-linear ductile material experiencing simple shear under incompressible, plane-strain. We assume a constitutiverelation [Brace and Kohlstedt, 1980] that relatesthe components of in-plane deviatoric stress and strain rate as follows: zij = BE n •ij (1) where zij is. the deviatoric component of stress, i•ij is the strain rate, E is the second invariant of the strain rate ten- sor,n is the exponent which determines the non-linearity of the rheology, and B is an empiricalconstant. With the con- stitutive relation above, we then solve the force balance equations: + p= 0 (2) wheresummation is over the j index andp is the pressure. We here ignoregravitational body forces. We also assume thatthe material is incompressible: =0 (3) With appropriate boundary conditions, we solve for the velocity and pressure at every point within the medium using the finite element method. Boundary Conditions A fault is introducedinto the homogeneous viscous medium as an internal boundaryacrosswhich slip is allowed. The fault is defined by the boundary conditions: 1) the normal stress acrossthe fault is continuous; 2) the velocity normal to the fault is continuous but otherwise unconstrained; and 3) the shear stress on the fault is equal to zero. A fault with zero shear stressacrossit (i.e. no strength) is not necessarily realistic, but the purpose of this paper is to examine how a very weak slip surface affects the stress and strain rate fields;the effect of increasing shear stress on the fault is briefly discussed below. The initial geometry of the fault is not stablewith time, in the sense thatcontinuing displacement on the fault gen- 1145