The effect of block rotations on the global seismic moment tensor and the patterns of seismic P and T axes

Abstract

Distributed brittle deformation of the Earth's crust involving block rotations is comparable to the deformation of a granular material, with fault blocks acting like the grains. The deformation of a granular material is not adequately described using classical continuum mechanics because the individual grains within the material rotate in a manner that is not uniquely determined by the large‐scale average deformation. Thus a theoretical link has not existed between the kinematics of deformation involving block rotation and the associated effects on the seismic moment tensor and focal mechanism solutions. We establish this link using micropolar continuum theory (Eringen, 1964, 1966 a, b; Eringen and Suhubi, 1964) and the analysis of the effect of block rotations on fault slickenline patterns by Twiss et al. (1991). This theory takes into account two separate scales of motion: a large‐scale average motion of the material, the macromotion, composed of a macrodeformation rate (i.e., a macrostrain rate) and a macrospin, and a local motion, the microspin, that describes the average rotation rate of grains in the material. The micropolar kinematic theory allows us to predict the orientations of coseismic slip directions V on local shear planes of any orientation in a large‐scale shear zone. We define a local and a global asymmetric micropolar seismic moment tensor in terms of these slip directions. For a restricted kinematic model, the theory shows that two scalar parameters, D and W, determine the symmetry of the global micropolar seismic moment tensor and the pattern of seismic P (shortening) and T (lengthening) axes. The deformation rate parameter D is defined in terms of the principal values of the deformation rate tensor . The deformation is transtensional (constrictional) if 0 ≤ D ≤ 0.5, plane strain if D = 0.5, and transpressional (flattening) if 0.5 < D ≤ 1. The net vorticity parameter W is a normalized value of the difference between microspin and macrospin . It is an objective variable. W = 0 implies that the global micropolar seismic moment tensor is symmetric and that P and T axis patterns have orthorhombic or higher symmetry. W ≠ implies that the global micropolar seismic moment tensor is asymmetric and that P and T axis patterns have monoclinic symmetry. The antisymmetric part of the global micropolar seismic moment tensor is associated with the net vorticity that characterizes the deformation. W has different values for different models of rigid block rotation and thus could serve to identify the rotation mechanism.

Analysis of fault slip inversions: Do they constrain stress or strain rate?

“…Thus, in general, the predicted slip directions for any given plane would be different for the two hypotheses, and both interpretations of the inversion cannot be correct except under very specific conditions. Therefore we ask the following questions: The causes of a mechanical process are determined by the material properties and by the boundary conditions that govern the process [Twiss et al, 1991[Twiss et al, , 1993. If stress boundary conditions are appropriate, then stress is a cause of the process and the deformation rate is an effect; if velocity boundary conditions are appropriate, then the deformation rate is a cause of the process and the stress is an effect.…”

“…We include in the term "deformation rate" both the strain rate tensor and the relative vorticity W that arises in micropolar theory from an independent component of rigid fault-block rotation [Twiss et al, 1991[Twiss et al, , 1993], which we define in more detail in section 3.5.…”

“…In the flow of a continuum, for example, the stress can be considered a cause, and the deformation rate can be considered an effect; conversely, the deformation rate can be considered a cause, and the stress can be considered an effect. The causes of a physical process must be identified by examining the boundary conditions on the physical system, because it is the boundary conditions that represent the externally imposed constraints on the system [Twiss et al , 1991[Twiss et al , , 1993.…”

Kinematics of postseismic relaxation from aftershock focal mechanisms of the 1994 Northridge, California, earthquake

“…According to the micropolar theory, the slip direction on any given surface is determined by two kinematic components: (1) the average large-scale deformation rate represented by the relative motion of the centroids of the rigid blocks, and (2) a local independent rotation rate of the individual blocks about their centroids [Twiss et al, 1993]. In technical terms, the large scale deformation rate (i.e., the rate of change in shape of the crustal volume) is the symmetric part of the velocity gradient tensor (the strain rate) for a continuum defined by the centroids of the rigid blocks that constitute the material.…”

“…Thus use of the micropolar continuum model to interpret fault-slip data provides a better constraint on the characteristics of the strain rate, and also permits the extraction of additional kinematic information about the contributions of block rotations to patterns of slip on fault surfaces (see discussion and examples of Unruh et al [1996]). The micropolar theory is formulated in terms of rates because this is the appropriate form for kinematic variables in the constitutive equations describing ductile deformation [Twiss et al, 1991[Twiss et al, , 1993. The focal mechanisms provide constraints only on the directions of incremental slip on faults but not the actual slip magnitudes.…”