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Mikaberidze G, D'Souza RM. Sandpile cascades on oscillator networks: The BTW model meets Kuramoto. CHAOS (WOODBURY, N.Y.) 2022; 32:053121. [PMID: 35649989 DOI: 10.1063/5.0095094] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/08/2022] [Accepted: 04/11/2022] [Indexed: 06/15/2023]
Abstract
Cascading failures abound in complex systems and the Bak-Tang-Weisenfeld (BTW) sandpile model provides a theoretical underpinning for their analysis. Yet, it does not account for the possibility of nodes having oscillatory dynamics, such as in power grids and brain networks. Here, we consider a network of Kuramoto oscillators upon which the BTW model is unfolding, enabling us to study how the feedback between the oscillatory and cascading dynamics can lead to new emergent behaviors. We assume that the more out-of-sync a node is with its neighbors, the more vulnerable it is and lower its load-carrying capacity accordingly. Also, when a node topples and sheds load, its oscillatory phase is reset at random. This leads to novel cyclic behavior at an emergent, long timescale. The system spends the bulk of its time in a synchronized state where load builds up with minimal cascades. Yet, eventually, the system reaches a tipping point where a large cascade triggers a "cascade of larger cascades," which can be classified as a dragon king event. The system then undergoes a short transient back to the synchronous, buildup phase. The coupling between capacity and synchronization gives rise to endogenous cascade seeds in addition to the standard exogenous ones, and we show their respective roles. We establish the phenomena from numerical studies and develop the accompanying mean-field theory to locate the tipping point, calculate the load in the system, determine the frequency of the long-time oscillations, and find the distribution of cascade sizes during the buildup phase.
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Affiliation(s)
- Guram Mikaberidze
- Department of Mathematics, University of California, Davis, Davis, California 95616, USA
| | - Raissa M D'Souza
- Department of Computer Science and Department of Mechanical and Aerospace Engineering, University of California, Davis, Davis, California 95616, USA
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Vallianatos F, Papadakis G, Michas G. Generalized statistical mechanics approaches to earthquakes and tectonics. Proc Math Phys Eng Sci 2017; 472:20160497. [PMID: 28119548 DOI: 10.1098/rspa.2016.0497] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Despite the extreme complexity that characterizes the mechanism of the earthquake generation process, simple empirical scaling relations apply to the collective properties of earthquakes and faults in a variety of tectonic environments and scales. The physical characterization of those properties and the scaling relations that describe them attract a wide scientific interest and are incorporated in the probabilistic forecasting of seismicity in local, regional and planetary scales. Considerable progress has been made in the analysis of the statistical mechanics of earthquakes, which, based on the principle of entropy, can provide a physical rationale to the macroscopic properties frequently observed. The scale-invariant properties, the (multi) fractal structures and the long-range interactions that have been found to characterize fault and earthquake populations have recently led to the consideration of non-extensive statistical mechanics (NESM) as a consistent statistical mechanics framework for the description of seismicity. The consistency between NESM and observations has been demonstrated in a series of publications on seismicity, faulting, rock physics and other fields of geosciences. The aim of this review is to present in a concise manner the fundamental macroscopic properties of earthquakes and faulting and how these can be derived by using the notions of statistical mechanics and NESM, providing further insights into earthquake physics and fault growth processes.
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Affiliation(s)
- Filippos Vallianatos
- UNESCO Chair on Solid Earth Physics and Geohazards Risk Reduction, Laboratory of Geophysics and Seismology , Technological Educational Institute of Crete , Chania , Greece
| | - Giorgos Papadakis
- UNESCO Chair on Solid Earth Physics and Geohazards Risk Reduction, Laboratory of Geophysics and Seismology , Technological Educational Institute of Crete , Chania , Greece
| | - Georgios Michas
- UNESCO Chair on Solid Earth Physics and Geohazards Risk Reduction, Laboratory of Geophysics and Seismology , Technological Educational Institute of Crete , Chania , Greece
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Kale S, Ostoja-Starzewski M. Elastic-plastic-brittle transitions and avalanches in disordered media. PHYSICAL REVIEW LETTERS 2014; 112:045503. [PMID: 24580467 DOI: 10.1103/physrevlett.112.045503] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2013] [Indexed: 06/03/2023]
Abstract
A spring lattice model with the ability to simulate elastic-plastic-brittle transitions in a disordered medium is presented. The model is based on bilinear constitutive law defined at the spring level and power-law-type disorder introduced in the yield and failure limits of the springs. The key parameters of the proposed model effectively control the disorder distribution, significantly affecting the stress-strain response, the damage accumulation process, and the fracture surfaces. The model demonstrates a plastic strain avalanche behavior for perfectly plastic as well as hardening materials with a power-law distribution, in agreement with the experiments and related models. The strength of the model is in its generality and ability to interpolate between elastic-plastic hardening and elastic-brittle transitions.
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Affiliation(s)
- Sohan Kale
- Department of Mechanical Science and Engineering, Institute for Condensed Matter Theory and Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, Illinois 61820, USA
| | - Martin Ostoja-Starzewski
- Department of Mechanical Science and Engineering, Institute for Condensed Matter Theory and Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, Illinois 61820, USA
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Lennartz-Sassinek S, Zaiser M, Main IG, Manzato C, Zapperi S. Emergent patterns of localized damage as a precursor to catastrophic failure in a random fuse network. Phys Rev E 2013; 87:042811. [PMID: 23679477 DOI: 10.1103/physreve.87.042811] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/16/2013] [Indexed: 11/07/2022]
Abstract
We study the failure of disordered materials by numerical simulations of the random fuse model. We identify emergent patterns of localized damage prior to catastrophic failure by statistically averaging the density of damage around the eventual failure nucleation point. The resulting pattern depends on fracture density and obeys the same scaling relations as would be expected for the stress field generated by a critical crack nucleating in a finite, disorder-free effective medium of varying size. The growth of this critical crack absorbs preexisting clusters according to a well-defined scaling relation. Unfortunately, in single model runs such precursory signals are not obvious. Our results imply that reliable and accurate prediction of failure in time-independent, microscopically brittle random materials in a real case is inherently problematic, and degrades with system size.
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Affiliation(s)
- S Lennartz-Sassinek
- School of Engineering and School of Geosciences, University of Edinburgh, Edinburgh, EH9 3JL, United Kingdom.
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Senatorski P. Dynamics of a zone of four parallel faults: A deterministic model. ACTA ACUST UNITED AC 2012. [DOI: 10.1029/95jb02624] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
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Ouillon G, Sornette D. Segmentation of fault networks determined from spatial clustering of earthquakes. ACTA ACUST UNITED AC 2011. [DOI: 10.1029/2010jb007752] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
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Picallo CB, López JM, Zapperi S, Alava MJ. From brittle to ductile fracture in disordered materials. PHYSICAL REVIEW LETTERS 2010; 105:155502. [PMID: 21230919 DOI: 10.1103/physrevlett.105.155502] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/09/2010] [Revised: 07/01/2010] [Indexed: 05/30/2023]
Abstract
We introduce a lattice model able to describe damage and yielding in heterogeneous materials ranging from brittle to ductile ones. Ductile fracture surfaces, obtained when the system breaks once the strain is completely localized, are shown to correspond to minimum energy surfaces. The similarity of the resulting fracture paths to the limits of brittle fracture or minimum energy surfaces is quantified. The model exhibits a smooth transition from brittleness to ductility. The dynamics of yielding exhibits avalanches with a power-law distribution.
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Affiliation(s)
- Clara B Picallo
- Instituto de Física de Cantabria, CSIC-UC, E-39005 Santander, Spain
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Picallo CB, López JM, Zapperi S, Alava MJ. Optimization and plasticity in disordered media. PHYSICAL REVIEW LETTERS 2009; 103:225502. [PMID: 20366105 DOI: 10.1103/physrevlett.103.225502] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/18/2009] [Indexed: 05/29/2023]
Abstract
We study the plastic yielding of disordered media using the perfectly plastic random fuse model. The yield surfaces are shown to be different from those obtained minimizing the sum of the local yield thresholds, i.e., the so-called minimum "energy" surfaces. As a result, the global yield stress is lower than expected from naive optimization and the difference persists as the sample size increases. At variance with minimum energy surfaces, height-height fluctuations of yield surfaces exhibit multiscaling. We provide a theoretical argument that explains how this behavior arises from the very different nature of the optimization problem in both cases.
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Affiliation(s)
- Clara B Picallo
- Instituto de Física de Cantabria, CSIC-UC, E-39005 Santander, Spain.
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Im MY, Lee SH, Kim DH, Fischer P, Shin SC. Scaling behavior of the first arrival time of a random-walking magnetic domain. PHYSICAL REVIEW LETTERS 2008; 100:167204. [PMID: 18518241 DOI: 10.1103/physrevlett.100.167204] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/03/2008] [Indexed: 05/26/2023]
Abstract
We report a universal scaling behavior of the first arrival time of a traveling magnetic domain wall into a finite space-time observation window of a magneto-optical microscope enabling direct visualization of a Barkhausen avalanche in real time. The first arrival time of the traveling magnetic domain wall exhibits a nontrivial fluctuation and its statistical distribution is described by universal power-law scaling with scaling exponents of 1.34+/-0.07 for CoCr and CoCrPt films, despite their quite different domain evolution patterns. Numerical simulation of the first arrival time with an assumption that the magnetic domain wall traveled as a random walker well matches our experimentally observed scaling behavior, providing an experimental support for the random-walking model of traveling magnetic domain walls.
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Affiliation(s)
- M-Y Im
- Department of Physics and Center for Nanospinics of Spintronic Materials, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea
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Baret JC, Vandembroucq D, Roux S. Extremal model for amorphous media plasticity. PHYSICAL REVIEW LETTERS 2002; 89:195506. [PMID: 12443126 DOI: 10.1103/physrevlett.89.195506] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/26/2002] [Indexed: 05/24/2023]
Abstract
An extremal model for the plasticity of amorphous materials is studied in a simple two-dimensional antiplane geometry. The steady state is analyzed through numerical simulations. Long-range spatial and temporal correlations in local slip events are shown to develop, leading to nontrivial and highly anisotropic scaling laws. In particular, the plastic strain is shown to concentrate statistically over a region which tends to align perpendicular to the displacement gradient. By construction, the model can be seen as giving rise to a depinning transition, the threshold of which (i.e., the macroscopic yield stress) also reveals scaling properties reflecting the localization of the activity.
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Affiliation(s)
- Jean-Christophe Baret
- Unité Mixte CNRS/Saint-Gobain Surface du Verre et Interfaces, 39 Quai Lucien Lefranc, 93303 Aubervilliers cedex, France
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Abstract
Crackling noise arises when a system responds to changing external conditions through discrete, impulsive events spanning a broad range of sizes. A wide variety of physical systems exhibiting crackling noise have been studied, from earthquakes on faults to paper crumpling. Because these systems exhibit regular behaviour over a huge range of sizes, their behaviour is likely to be independent of microscopic and macroscopic details, and progress can be made by the use of simple models. The fact that these models and real systems can share the same behaviour on many scales is called universality. We illustrate these ideas by using results for our model of crackling noise in magnets, explaining the use of the renormalization group and scaling collapses, and we highlight some continuing challenges in this still-evolving field.
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Affiliation(s)
- J P Sethna
- Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853-2501, USA
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Grasso JR, Sornette D. Testing self-organized criticality by induced seismicity. ACTA ACUST UNITED AC 1998. [DOI: 10.1029/97jb01344] [Citation(s) in RCA: 72] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
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Zapperi S, Vespignani A, Stanley HE. Plasticity and avalanche behaviour in microfracturing phenomena. Nature 1997. [DOI: 10.1038/41737] [Citation(s) in RCA: 183] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
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Schmittbuhl J, Vilotte JP, Roux S. A dissipation-based analysis of an earthquake fault model. ACTA ACUST UNITED AC 1996. [DOI: 10.1029/96jb02294] [Citation(s) in RCA: 35] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
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Gil L, Sornette D. Landau-Ginzburg theory of self-organized criticality. PHYSICAL REVIEW LETTERS 1996; 76:3991-3994. [PMID: 10061164 DOI: 10.1103/physrevlett.76.3991] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Sornette D. Power laws without parameter tuning: An alternative to self-organized criticality. PHYSICAL REVIEW LETTERS 1994; 72:2306. [PMID: 10055844 DOI: 10.1103/physrevlett.72.2306] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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