We derive the exact equations by which the continuum approximation to the extensional and shear strains can be determined from measurements of fault-lengths or fault-displacement in a faulted domain. ...We develop the theory by which we can infer the extensional and shear strain in a volume of brittlely deformed crust from an incomplete inventory of the faults. To that end, we use empirical power-law relationships between fault-length and fault-displacement, and the power-law cumulative frequency distribution for each of these variables, for sampling domains of one, two, and three dimensions. The theory 1) defines the relationships among the parameters in these power-laws, which allows the self-consistency of results from fault-length and fault-displacement studies in domains of one, two, and three dimensions to be evaluated; 2) defines constraints on the relative sizes of the sampling domain and the largest fault that can be included in an analysis using fault systematics; 3) shows that extensional and shear strains in faulted crust can be inferred knowing only an independent set of the parameters defining the population systematics plus the magnitude of either the displacement or the length for the largest fault in the domain; and 4) defines the constraints on the three-dimensional strain imposed by sampling in one- and two-dimensional domains.
We use the theoretical relations developed in Part I of this work to evaluate the self-consistency of fault-length and fault-displacement data gathered in domains of one and two dimensions from the ...Yucca Mountain area and from the coalfields in south Yorkshire, U.K. These data sets are not all self-consistent. For the Yucca Mt. area, the theory shows that, the volume over which the sampling of the faults must occur should have a horizontal width no smaller than 2.4 times the horizontal length of the largest fault, and a depth no smaller than 1.6 times the vertical extent of the largest vertical-equivalent-fault. It also shows that the volumetric extension must be ≥95% of the extension of a two-dimensional domain and ≥80% of the extension of a one-dimensional domain. The theory successfully accounts for the observed cumulative extensional strain derived from fault-displacement data from a one-dimensional sampling domain at Yucca Mt., Nevada, U.S.A. Faults up to about four orders of magnitude smaller than the largest fault make a significant contribution to the strain. The most robust calculation of cumulative fractional strain requires the parameters inferred from sampling displacement in a one-dimensional domain. This sampling procedure therefore provides the most reliable results.
Fault slip data commonly are used to infer the orientations and relative magnitudes of either the principal stresses or the principal strain rates, which are not necessarily parallel or equal. At the ...local scale of an individual fault, the shear plane and slip direction define the orientations of the local principal strain rate axes but not, in general, the local principal stress axes. At a large scale, the orientations of P and T axes maxima for sets of fault slip data do not provide accurate inversion solutions for either strain rate or stress. The quantitative inversion of such fault slip data, however, provides direct constraints on the orientations and relative magnitudes of the global principal strain rates. To interpret the inversion solution as constraining the global principal stresses requires that (1) the fault slip pattern must have a characteristic symmetry no lower than orthorhombic; (2) the material must be mechanically isotropic; and (3) there must be a linear constitutive relationship between the global stress and the global strain rate. Isotropic linear elastic constitutive equations are appropriate to describe the local deformation surrounding an individual slip discontinuity. Fault slip inversions, however, constrain the characteristics of a large‐scale cataclastic flow, which is described by constitutive equations that are probably, but to an unknown degree, anisotropic and nonlinear. Such material behavior would not strictly satisfy the requirements for the stress interpretation. Thus, at the present state of knowledge, fault slip inversion solutions are most reliably interpreted as constraining the principal strain rates.
Geodetic observations of surface deformation associated with the 1994 Northridge, southern California, earthquake generally are reproduced by simple models of a large‐scale elastic dislocation on a ...blind or buried thrust fault. The smaller‐scale aftershocks of the Northridge earthquake are distributed throughout much of the volume of crust that appears to have deformed elastically during the mainshock. These aftershocks, averaged over volumes that are large relative to their rupture radii, reflect a distributed, permanent deformation that is accommodated by local brittle fracture. We use a micropolar continuum model to invert the aftershocks in such volumes for the average incremental strain, and we compare that deformation both with the elastic strain from the dislocation model of the mainshock and with geodetically measured strain. Aftershock deformation that occurred at depths below about 6 km, and which is associated with the primary rupture zone, is consistent with slow continuation of the southwest‐side‐up reverse slip on the blind Northridge thrust fault. In contrast, aftershock deformation from the upper 5–7 km of the hanging wall block directly above the thrust fault can be characterized by horizontal NE‐SW shortening and horizontal NW‐SE (i.e., fault‐parallel) extension. This pattern of deformation is similar to that associated with the mainshock, as observed geodetically and as calculated from the elastic dislocation model. We interpret that the aftershock activity in the hanging wall represents the quasi‐ductile accommodation by brittle deformation mechanisms of a permanent strain distributed through the hanging wall block. The aftershocks along the mainshock rupture zone are interpreted as resulting from either (1) the time‐dependent release along a weakened fault zone of part of the remaining accumulated elastic strain in the upper crust or (2) the continued slip in the weakened fault zone driven by the deformation of a ductile‐elastic lower crustal layer that relaxes under the stress transferred by the coseismic loss of cohesion in the upper crust. In either case, the aftershock activity suggests that the crust undergoes quasi‐ductile flow as a brittle‐elastic material, and is not a strictly elastic material.
Three single crystal cores of synthetic quartz were experimentally loaded normal to r₁(101̅1) under creep conditions at temperatures of 600°C and 800°C, differential stresses between 0.8 kb and 5.1 ...kb, and at a confining pressure of 3 kb. Study of surface slip bands by refection optical microscopy suggests slip on the following systems, where the braces indicate the slip plane, and the associated square or angular brackets indicate the several slip directions in each slip plane: {m}c, 〈c ± a〉; {z}〈a〉, 〈c ± a〉; {π}〈a〉, {ξ}〈 c + a 〉 {ξʹ}〈c - a〉. No basal slip occurs. Universal stage transmission optical microscopy and transmission electron microscopy reveal two planar deformation features. Subbasal laminas comprise tangles and pile-ups of dislocations on active slip planes but do not indicate major basal slip. Lamellas form subparallel to m (101̅0), z₂ (011̅1) and z₃ (11̅01). They comprise groups of subparallel dislocations having a large edge component to their Burgers vectors and they indicate active slip. This is consistent with the Christie, Griggs, and Carter (1964) model. Optical properties of laminas and lamellas are consistent with the plane phase object theory. The theory of strain induced changes in birefringence could not be tested.
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