Statistical physics of fault patterns self-organized by repeated earthquakes

This work presents at attempt to model brittle ruptures and slips in a continental plate and its spontaneous organization by repeated earthquakes in terms of coarse-grained properties of the mechanical plate. A statistical physics model, which simulates anti-plane shear deformation of a thin plate with inhomogeneous elastic properties, is thus analyzed theoretically and numerically in order to study the spatio-temporal evolution of rupture patterns in response to a constant applied strain rate at its borders, mimicking the effect of neighboring plates. Rupture occurs when the local stress reaches a threshold value. Broken elements are instantaneously healed and retain the original material properties, enabling the occurrence of recurrent earthquakes. Extending previous works (Cowieet al., 1993;Miltenbergeret al., 1993), we present a study of the most startling feature of this model which is that ruptures become strongly correlated in space and time leading to the spontaneous development of multifractal structures and gradually accumulate large displacements. The formation of the structures and the temporal variation of rupture activity is due to a complex interplay between the random structure, long-range elastic interactions and the threshold nature of rupture physics. The spontaneous formation of fractal fault structures by repeated earthquakes is mirrored at short times by the spatio-temporal chaotic dynamics of earthquakes, well-described by a Gutenberg-Richter power law. We also show that the fault structures can be understood as pure geometrical objects, namely minimal manifolds, which in two dimensions correspond to the random directed polymer (RDP) problem. This mapping allows us to use the results of many studies on the RDP in the field of statistical physics, where it is an exact result that the minimal random manifolds in 2D systems are self-affine with a roughness exponent 2/3. We also present results pertaining to the influence of the degree β of stress release per earthquake on the competition between faults. Our results provide a rigorous framework from which to initiate rationalization of many, reported fractal fault studies.

[1]  Moore,et al.  Chaotic nature of the spin-glass phase. , 1987, Physical review letters.

[2]  Didier Sornette,et al.  The dynamical thermal fuse model , 1992 .

[3]  Obukhov,et al.  Self-organized criticality in a crack-propagation model of earthquakes. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[4]  M. Zoback,et al.  In situ stress measurements to 3.5 km depth in the Cajon Pass Scientific Research Borehole: Implications for the mechanics of crustal faulting , 1992 .

[5]  Sornette,et al.  Fault self-organization as optimal random paths selected by critical spatiotemporal dynamics of earthquakes. , 1993, Physical review letters.

[6]  Grinstein,et al.  Conservation laws, anisotropy, and "self-organized criticality" in noisy nonequilibrium systems. , 1990, Physical review letters.

[7]  Didier Sornette,et al.  Critical phase transitions made self-organized : a dynamical system feedback mechanism for self-organized criticality , 1992 .

[8]  Sornette,et al.  Growth of fractal fault patterns. , 1990, Physical review letters.

[9]  Self-organized ruptures in an elastic medium: a possible model for earthquakes , 1992 .

[10]  Didier Sornette,et al.  Statistical physics model for the spatiotemporal evolution of faults , 1993 .

[11]  Didier Sornette,et al.  Self-Organized Criticality and Earthquakes , 1989 .

[12]  Keisuke Ito,et al.  Earthquakes as self-organized critical phenomena , 1990 .

[13]  D. Sornette,et al.  Some consequences of a proposed fractal nature of continental faulting , 1990, Nature.

[14]  Critical phase transitions made self-organized: proposed experiments , 1993 .

[15]  Sornette,et al.  Dynamics and memory effects in rupture of thermal fuse networks. , 1992, Physical review letters.

[16]  D. Turcotte.  Fractals in geology and geophysics , 2009, Encyclopedia of Complexity and Systems Science.

[17]  Hwa,et al.  Dissipative transport in open systems: An investigation of self-organized criticality. , 1989, Physical review letters.

[18]  L. Knopoff,et al.  Model and theoretical seismicity , 1967 .

[19]  A. Crisanti,et al.  Strongly intermittent chaos and scaling in an earthquake model. , 1992, Physical Review A. Atomic, Molecular, and Optical Physics.

[20]  Grinstein,et al.  Generic scale invariance and roughening in noisy model sandpiles and other driven interfaces. , 1991, Physical review letters.

[21]  D. Huse,et al.  Huse, Henley, and Fisher respond. , 1985, Physical review letters.

[22]  D. Sornette,et al.  Organization of Rupture , 1993 .

[23]  P. Bak,et al.  Earthquakes as a self‐organized critical phenomenon , 1989 .

[24]  D. Sornette.  Self-Organized Criticality in Plate Tectonics , 1991 .

[25]  B. Derrida,et al.  Interface energy in random systems , 1983 .

[26]  M. Mézard,et al.  Spin Glass Theory and Beyond , 1987 .

[27]  D. Sornette,et al.  Structuration of the lithosphere in plate tectonics as a self‐organized critical phenomenon , 1990 .

[28]  A. Hansen,et al.  Perfect plasticity in a random medium , 1992 .

[29]  Didier Sornette,et al.  Linking short-timescale deformation to long-timescale tectonics , 1992, Nature.

[30]  P. Bak,et al.  Self-organized criticality. , 1988, Physical review. A, General physics.

[31]  Benoit B. Mandelbrot,et al.  Fractals in Geophysics , 1989 .

[32]  H. Stanley,et al.  Introduction to Phase Transitions and Critical Phenomena , 1972 .

[33]  Zhang.  Ground state instability of a random system. , 1987, Physical review letters.

[34]  Sornette,et al.  Statistical model of earthquake foreshocks. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[35]  Carlson,et al.  Mechanical model of an earthquake fault. , 1989, Physical review. A, General physics.

[36]  Zhang,et al.  Scaling of directed polymers in random media. , 1987, Physical review letters.