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doi: 10.1007/BF00876337pmid: N/A
Recent advances in the theory of fracture and fragmentation are reviewed. Empirical laws in seismology are interpreted from a fractal perspective, and earthquakes are viewed as a self-organized critical phenomenon (SOC). Earthquakes occur as an energy dissipation process in the earth's crust to which the tectonic energy is continuously input. The crust self-organizes into the critical state and the temporal and spatial fractal structure emerges naturally. Power-law relations known in seismology are the expression of the critical state of the crust. An SOC model for earthquakes, which explains the Gutenberg-Richter relation, the Omori's formula of aftershocks and the fractal distribution of hypocenters, is presented. A new view of earthquake phenomena shares a common standpoint with other disciplines to study natural complex phenomena with a unified theory.
Kemeny, John; Hagaman, Roberta
doi: 10.1007/BF00876338pmid: N/A
A model has been developed to simulate the statistical and mechanical nature of rupture on a heterogeneous strike-slip fault. The model is based on the progressive failure of circular asperities of varying sizes and strengths along a fault plane subjected to a constant far-field shear displacement rate. The basis of the model is a deformation and stress intensity factory solution for a single circular asperity under a unidirectional shear stress. The individual asperities are unified through the fault stiffness and the far-field stress and displacement. During fault deformation asperities can fail and reheal, resulting in changes in the local stresses in the asperities, stress drops, and changes in the stiffness of the fault. Depending on how the stress is redistributed following asperity failure and on the strenghts of the neighboring asperities an earthquake event can be the failure of one or more asperities. Following an earthquake event seismic source parameters such as the stress drop, energy change, and moment magnitude are calculated. Results from the model show a very realistic pattern of earthquake rupture, with reasonable source parameters, the proper magnitude-frequency behavior, and the development of characteristic earthquakes. Also the progression ofb-values in the model gives some insight into the phenomenon of ‘self-organized criticality.’
doi: 10.1007/BF00876339pmid: N/A
We present a systematic analysis of the dynamical behavior introduced by fault zone heterogeneities, using a simple mass-spring model with velocity-weakening friction. The model consists of two sliding blocks coupled to each other and to a constant velocity driver by clastic springs. The state of this system can be characterized by the positions of the two blocks relative to the driver. Symmetry stabilizes the system and generates only cyclic behavior. For an asymmetric system where the frictional forces for the two blocks are not equal, the solutions exhibit chaotic behavior. The transition from stable cyclic behavior to chaos is characterized by the period-doubling route to chaos. Lyapunov exponents are computed to quantify the deterministic chaos and to locate the onset of the chaotic evolution in parameter space. In many examples of deterministic chaos, chaotic behavior of a low-order system implies chaos in similar higher order systems. Thus, our results provide substantial evidence that crustal deformation is an example of deterministic chaos.
Hirabayashi, Tadashi; Ito, Keisuke; Yoshii, Toshikatsu
doi: 10.1007/BF00876340pmid: N/A
Multifractal properties of the epicenter and hypocenter distribution and also of the energy distribution of earthquakes are studied for California, Japan, and Greece. The calculatedD q-q curves (the generalized dimension) indicate that the earthquake process is multifractal or heterogeneous in the fractal dimension. Japanese earthquakes are the most heterogeneous and Californian earthquakes are the least. Since the earthquake process is multifractal, a single value of the so-called fractal dimension is not sufficient to characterize the earthquake process. Studies of multifractal models of earthquakes are recommended. Temporal changes of theD q-q curve are also obtained for Californian and Japanese earthquakes. TheD q-q curve shows two distinctly different types in each region; the gentle type and the steep type. The steeptype corresponds to a strongly heterogeneous multifractal, which appears during seismically active periods when large earthquakes occur.D q for smallq or negativeq is considerably more sensitive to the change in fractal structure of earthquakes thanD q forq≥2. We recommend use ofD q at smallq to detect the seismicity change in a local area.
King, Geoffrey; Sammis, Charles
doi: 10.1007/BF00876341pmid: N/A
The mechanical processes that lead from first fracture in an undeformed rock mass to the fault gouge observed in a highly sheared fault zone are outlined. Tensile fracture, dilation, rotation, the collapse of beams and filling of voids are the basic mechanical elements. Repeated many times, over a wide range of scales, they accommodate finite strain and create the complex fabrics observed in highly deformed rocks. Defects that nucleate tensile cracks in the earth are both spatially clustered and occur on a wide range of scales. This inhomogeneity is responsible for features that distinguish deformation of rocks from deformation of laboratory samples. As deformation proceeds, failure at one scale leads to failure at another scale in a process of evolving damage. Abrupt catastrophic failure never extends indefinitely throughout the earth as it does in rock samples. The mechanics of the interactions between scales are investigated. Approximate expressions are modified from engineering damage mechanics for this purpose and their validity is demonstrated by detailed numerical modeling of critical examples.
Kudo, Yozo; Sano, Osam; Murashige, Naokuni; Mizuta, Yoshiaki; Nakagawa, Koji
doi: 10.1007/BF00876342pmid: N/A
The double-torsion test using Aji granite was carried out to investigate the interaction between stress-induced crack path and mineral grains. Crack velocities were controlled at range 10−7 m/s to 10−1 m/s. After the stressed specimens were dyed, we checked the crack path by thin section analysis, using an optical microscope. The stress-induced crack path was divided into two types, transgranular and intergranular cracks, and each path was subdivided with respect to mineral grains. In spite of the extensive range of crack velocities, the ratios between the transgranular and intergranular crack lengths did not change. The crack paths were all jagged, and often showed detour around the grain boundary when faced with obstacles like hard grains or preexisting cracks. That is to say, quartz grain played an important role as an obstacle. Feldspar grain could change the crack path because of its cleavage plane. Biolite grain had a serious effect on the path even if its constitution ratio is very small. Fractal dimensions of the crack paths were calculated by three methods, as indicators of surface roughness. The fractal dimensions were shown in a slight trend with the change of crack velocity. This trend can be explained from the point of limited cracking rate in stress corrosion.
doi: 10.1007/BF00876343pmid: N/A
Double-torsion specimens of two granitic rocks were prepared in several directions with reference to microcracks fabric. Even for the same rock and at the same stress levels, the observed crack velocities in two granitic rocks were dependent on both the propagation direction and the opening direction. The maximum difference by several orders of magnitude was found for both rocks. The highest crack velocity was observed when the subcritical crack was parallel to most of the preexisting cracks. The maximum critical stress intensity factor was about twice as high as the minimum one in different directions. An analysis for a thin plate having anisotropic elasticity under torsional load showed that the observed difference in the crack velocity and the critical stress intensity factor was not an error due to conventional equations derived on the assumption of isotropic elasticity but the true material's property. As the preferred orientation of microcracks has been pointed out for many granitic rocks, we can conclude that the anisotropic nature of the fracture resistance of the two granitic rocks used in this study was not exceptional. A region of a transport-limited velocity was not found for rocks, even at the velocity of 10−2 m/s, that is almost equal to the theoretical limit of the stress corrosion cracking.
Pyrak-Nolte, L.; Myer, L.; Nolte, D.
doi: 10.1007/BF00876344pmid: N/A
The distributions of contact area and void space in single fractures in granite rock have been determined experimentally by making metal casts of the void spaces between the fracture surfaces under normal loads. The resulting metal casts on 52 cm diameter core samples show a complex geometry for the flow paths through the fracture. This geometry is analyzed using finite-size scaling. The spanning probabilities and percolation probabilities of the metal casts are calculted as functions of observation scale. Under the highest stresses of 33 MPa and 85 MPa there is a significant size-dependence of the geometric flow properties for observation scales smaller than 2 mm. Based on this data, the macroscopic percolation properties of the extended fracture can be well represented by relatively small core samples, even under normal stresses larger than 33 MPa. The metal casts also have rich multifractal structure that changes with changing stress.
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