Hiroyuki Nagahama

Rates of aftershock decay and the fractal structure of active fault systems

Abstract Aftershock activity on frequency decay against time is characterized by a power law (the modified Omori formula) of an exponent p, which differs with each aftershock sequence. A theoretical study suggested that p, which is a rate constant of aftershock decay, is related to the fractal dimension of a pre-existing fault system. This has however never been checked. Aftershock activity on size distribution is also characterized by an exponential distribution against magnitude (the Gutenberg—Richter relation) with a slope b. Although p is expected to be related to b, which is related to the partitioning rate of earthquake energy, the relationship has never been established. Here the relation between the p-values and the fractal dimensions of the pre-existing fault systems, and that between the p-values and the b-values are explored, using natural seismicity data and data of the observable fault systems. The p- and b-values were estimated for fifteen aftershock sequences which occurred in Japan. In this paper aftershocks were identified on the basis of a phenomenological definition in the seismicity data. The fractal capacity dimensions D0 are estimated for the pre-existing active fault systems observed on the surface in the aftershock regions. In the present paper the standard box-counting method was adopted to get the D0. Negative correlations between (1) p and D0, and (2) p and b were observed with some scattering. Observation (1) shows that the rate of aftershock decay p decreases systematically with increasing occupancy rate of the pre-existing active fault system D0 and suggests that aftershock decay dynamics is constrained by the pre-existing fracture field. Observation (2) shows that p certainly has a relation with b. Moreover, we offer possible interpretation on these negative correlations and some scatters in both observations: the scatters are interpreted as the scatter of the difference of two fractal dimensions between 3-D fracture construction in the crust and 2-D cross-sectional surface (observed active fault system). Supported by further tests, this paper strongly suggests that the scaling for a natural fracture system is self-affine (with different fractal scalings in different directions) rather than self-similar, which would be a manifestation of regional anisotropy of the fracture system, and that the seismic parameters p and b depend on the 3-D construction of the fracture system in the crust.

Voltage changes induced by stick‐slip of granites

Tri-axial compression tests of dry granites with a pre-cut shear surface were conducted with an electrode to detect electrical potential. The samples were shaped 19.5 mm in diameter and 60 mm in length, and deformed at a strain rate of 10 -3 s -1 and confining pressure of 78.5 MPa. Pulse-like electrode voltage changes were detected several times just before (0.5-5 ms) stick-slip motion during compression. Magnitudes of stick-slips (e.g. normal stress drop of 5-160 MPa) and intensities of electrode voltage changes (0.1-10 V) increased whenever stick-slip occurred. Here we propose some scaling laws on the number of stick-slips and relate those statistics to the intensity of the electrical phenomena. Maximum electrode voltage change is proportional to normal stress drop that depends on real contact area of the slip plane. Local charge distributions, fractoemission or both may due to failure of asperities just before stick-slip.

Spatial distribution of aftershocks and the fractal structure of active fault systems

—The relationship between the fractal dimensions of aftershock spatial distribution and of pre-existing fracture systems is examined. Fourteen main shocks occurring in Japan were followed by aftershocks, and the aftershocks occurred in swarms around the main shock. Epicentral distributions of the aftershocks exhibit fractal properties, and the fractal dimensions are estimated by using correlation integral. Observable pre-existing active fault systems in the fourteen aftershock regions have fractal structures, and the fractal dimensions are estimated by using the box-counting method. The estimated fractal dimensions derive positive correlation, showing independence from the main-shock magnitude. The correlation shows that aftershock distributions become less clustered with increasing fractal dimensions of the active fault system. That is, the clusters of the aftershocks are constrained under the fractal properties of the pre-existing active fault systems. If the fractal dimension of the active fault system is the upper limit value of the fractal dimension of the actual fracture geometries of rocks, then the clustering aftershocks manifest completely random and unpredictable distribution.

Micromorphic Continuum and Fractal Fracturing in the Lithosphere

—It seems that internal structures and discontinuities in the lithosphere essentially influence the lithospheric deformation such as faulting or earthquakes. The micromorphic continuum provides a good framework to study the continuum with microstructure, such as earthquake structures. Here we briefly introduce the relation between the theory of micromorphic continuum and the rotational effects related to the internal microstructure in epicenter zones. Thereafter the equilibrium equation, in terms of the displacements (the Navier equation) in the medium with microstructure, is derived from the theory of the micromorphic continuum. This equation is the generalization of the Laplace equation in terms of displacements and can lead to Laplace equations such as the local diffusion-like conservation equations for strains. These local balance/stationary state of strains under the steady non-equilibrium strain flux through the plate boundaries bear the scale-invariant properties of fracturing in the lithospheric plate with microstructure.

HODGE DUALITY AND CONTINUUM THEORY OF DEFECTS

Dual material space-time with defect field is presented in the language of differential forms: one is the strain space-time whose basic equation is the continuity equation for the dislocation 2-form; the other is the stress space-time whose basic equation is the continuity equation for the couple-stress and angular momentum 2-form. Continuity and kinematic equations in each space can be derived by the transformation from p-form to (p + 1)-form. Moreover, several constitutive equations can be recognized as the transformation between the p-form of the strain space-time and the (4-p)-form of the stress space-time. These kinematic, continuity and constitutive equations can be interpreted geometrically as Cartan structure equations, Bianchi identities and Hodge duality transformations, respectively.

The sections’ fractal dimension of grain boundary

Abstract The fractal dimensional increment of the experimentally dynamic recrystallized grain boundary is proportional to logarithm of Zener–Hollomon parameter. The fractal dimensional increment is defined as the fractal dimension of the grain shape minus the Euclidean dimension of certain transection. To draw the geometrical image of the fractal dimensional increment, the basic rule of the sections’ fractal dimension is introduced. The geometrical implication of the fractal dimensional increment is concluded as the fractal dimension of the crossing point distribution on the grain boundary transected by the circumscribing circle or ellipse with the equivalent-area of the grain, and a power law relationship between the Zener–Hollomon parameter and the number of crossing points is found. Therefore, summarizing power laws among the Zener–Hollomon parameter, the differential stress and the number of the crossing points on the grain boundary, the number of crossing points could respond to the differential stress.

FRACTAL GRAIN BOUNDARY MIGRATION

Fractal analysis on experimentally recrystallized quartz grain boundaries has been employed to relate the grain boundary complexities with deformation conditions, such as strain rate and temperature. The fractal dimensional increment of the grain boundaries, defined as (D-1), and the degree of irregularity in grain boundaries, is proportional to the logarithmic of the Zener–Hollomon parameter that is defined by strain rate and temperature (the Arrhenius term). The physical mean of the empirical relationship can be explained theoretically by a new grain boundary migration model (GBM or cell dynamics model) extended by the fractal concepts and the dimension analysis. This is a more general model than the migration growth model for the fractal grain boundaries.