9,262 research outputs found

    Avalanches in Breakdown and Fracture Processes

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    We investigate the breakdown of disordered networks under the action of an increasing external---mechanical or electrical---force. We perform a mean-field analysis and estimate scaling exponents for the approach to the instability. By simulating two-dimensional models of electric breakdown and fracture we observe that the breakdown is preceded by avalanche events. The avalanches can be described by scaling laws, and the estimated values of the exponents are consistent with those found in mean-field theory. The breakdown point is characterized by a discontinuity in the macroscopic properties of the material, such as conductivity or elasticity, indicative of a first order transition. The scaling laws suggest an analogy with the behavior expected in spinodal nucleation.Comment: 15 pages, 12 figures, submitted to Phys. Rev. E, corrected typo in authors name, no changes to the pape

    25 Years of Self-Organized Criticality: Numerical Detection Methods

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    The detection and characterization of self-organized criticality (SOC), in both real and simulated data, has undergone many significant revisions over the past 25 years. The explosive advances in the many numerical methods available for detecting, discriminating, and ultimately testing, SOC have played a critical role in developing our understanding of how systems experience and exhibit SOC. In this article, methods of detecting SOC are reviewed; from correlations to complexity to critical quantities. A description of the basic autocorrelation method leads into a detailed analysis of application-oriented methods developed in the last 25 years. In the second half of this manuscript space-based, time-based and spatial-temporal methods are reviewed and the prevalence of power laws in nature is described, with an emphasis on event detection and characterization. The search for numerical methods to clearly and unambiguously detect SOC in data often leads us outside the comfort zone of our own disciplines - the answers to these questions are often obtained by studying the advances made in other fields of study. In addition, numerical detection methods often provide the optimum link between simulations and experiments in scientific research. We seek to explore this boundary where the rubber meets the road, to review this expanding field of research of numerical detection of SOC systems over the past 25 years, and to iterate forwards so as to provide some foresight and guidance into developing breakthroughs in this subject over the next quarter of a century.Comment: Space Science Review series on SO

    25 Years of Self-Organized Criticality: Solar and Astrophysics

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    Shortly after the seminal paper {\sl "Self-Organized Criticality: An explanation of 1/f noise"} by Bak, Tang, and Wiesenfeld (1987), the idea has been applied to solar physics, in {\sl "Avalanches and the Distribution of Solar Flares"} by Lu and Hamilton (1991). In the following years, an inspiring cross-fertilization from complexity theory to solar and astrophysics took place, where the SOC concept was initially applied to solar flares, stellar flares, and magnetospheric substorms, and later extended to the radiation belt, the heliosphere, lunar craters, the asteroid belt, the Saturn ring, pulsar glitches, soft X-ray repeaters, blazars, black-hole objects, cosmic rays, and boson clouds. The application of SOC concepts has been performed by numerical cellular automaton simulations, by analytical calculations of statistical (powerlaw-like) distributions based on physical scaling laws, and by observational tests of theoretically predicted size distributions and waiting time distributions. Attempts have been undertaken to import physical models into the numerical SOC toy models, such as the discretization of magneto-hydrodynamics (MHD) processes. The novel applications stimulated also vigorous debates about the discrimination between SOC models, SOC-like, and non-SOC processes, such as phase transitions, turbulence, random-walk diffusion, percolation, branching processes, network theory, chaos theory, fractality, multi-scale, and other complexity phenomena. We review SOC studies from the last 25 years and highlight new trends, open questions, and future challenges, as discussed during two recent ISSI workshops on this theme.Comment: 139 pages, 28 figures, Review based on ISSI workshops "Self-Organized Criticality and Turbulence" (2012, 2013, Bern, Switzerland

    Avalanches, thresholds, and diffusion in meso-scale amorphous plasticity

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    We present results on a meso-scale model for amorphous matter in athermal, quasi-static (a-AQS), steady state shear flow. In particular, we perform a careful analysis of the scaling with the lateral system size, LL, of: i) statistics of individual relaxation events in terms of stress relaxation, SS, and individual event mean-squared displacement, MM, and the subsequent load increments, Δγ\Delta \gamma, required to initiate the next event; ii) static properties of the system encoded by x=σy−σx=\sigma_y-\sigma, the distance of local stress values from threshold; and iii) long-time correlations and the emergence of diffusive behavior. For the event statistics, we find that the distribution of SS is similar to, but distinct from, the distribution of MM. We find a strong correlation between SS and MM for any particular event, with S∼MqS\sim M^{q} with q≈0.65q\approx 0.65. qq completely determines the scaling exponents for P(M)P(M) given those for P(S)P(S). For the distribution of local thresholds, we find P(x)P(x) is analytic at x=0x=0, and has a value P(x)∣x=0=p0\left. P(x)\right|_{x=0}=p_0 which scales with lateral system length as p0∼L−0.6p_0\sim L^{-0.6}. Extreme value statistics arguments lead to a scaling relation between the exponents governing P(x)P(x) and those governing P(S)P(S). Finally, we study the long-time correlations via single-particle tracer statistics. The value of the diffusion coefficient is completely determined by ⟨Δγ⟩\langle \Delta \gamma \rangle and the scaling properties of P(M)P(M) (in particular from ⟨M⟩\langle M \rangle) rather than directly from P(S)P(S) as one might have naively guessed. Our results: i) further define the a-AQS universality class, ii) clarify the relation between avalanches of stress relaxation and diffusive behavior, iii) clarify the relation between local threshold distributions and event statistics

    A damage model based on failure threshold weakening

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    A variety of studies have modeled the physics of material deformation and damage as examples of generalized phase transitions, involving either critical phenomena or spinodal nucleation. Here we study a model for frictional sliding with long range interactions and recurrent damage that is parameterized by a process of damage and partial healing during sliding. We introduce a failure threshold weakening parameter into the cellular-automaton slider-block model which allows blocks to fail at a reduced failure threshold for all subsequent failures during an event. We show that a critical point is reached beyond which the probability of a system-wide event scales with this weakening parameter. We provide a mapping to the percolation transition, and show that the values of the scaling exponents approach the values for mean-field percolation (spinodal nucleation) as lattice size LL is increased for fixed RR. We also examine the effect of the weakening parameter on the frequency-magnitude scaling relationship and the ergodic behavior of the model

    Self-organized criticality as an absorbing-state phase transition

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    We explore the connection between self-organized criticality and phase transitions in models with absorbing states. Sandpile models are found to exhibit criticality only when a pair of relevant parameters - dissipation epsilon and driving field h - are set to their critical values. The critical values of epsilon and h are both equal to zero. The first is due to the absence of saturation (no bound on energy) in the sandpile model, while the second result is common to other absorbing-state transitions. The original definition of the sandpile model places it at the point (epsilon=0, h=0+): it is critical by definition. We argue power-law avalanche distributions are a general feature of models with infinitely many absorbing configurations, when they are subject to slow driving at the critical point. Our assertions are supported by simulations of the sandpile at epsilon=h=0 and fixed energy density (no drive, periodic boundaries), and of the slowly-driven pair contact process. We formulate a field theory for the sandpile model, in which the order parameter is coupled to a conserved energy density, which plays the role of an effective creation rate.Comment: 19 pages, 9 figure

    Discrete Fracture Model with Anisotropic Load Sharing

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    A two-dimensional fracture model where the interaction among elements is modeled by an anisotropic stress-transfer function is presented. The influence of anisotropy on the macroscopic properties of the samples is clarified, by interpolating between several limiting cases of load sharing. Furthermore, the critical stress and the distribution of failure avalanches are obtained numerically for different values of the anisotropy parameter α\alpha and as a function of the interaction exponent γ\gamma. From numerical results, one can certainly conclude that the anisotropy does not change the crossover point γc=2\gamma_c=2 in 2D. Hence, in the limit of infinite system size, the crossover value γc=2\gamma_c=2 between local and global load sharing is the same as the one obtained in the isotropic case. In the case of finite systems, however, for γ≤2\gamma\le2, the global load sharing behavior is approached very slowly

    Driving rate dependence of avalanche statistics and shapes at the yielding transition

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    We study stress time series caused by plastic avalanches in athermally sheared disordered materials. Using particle-based simulations and a mesoscopic elasto-plastic model, we analyze size and shear-rate dependence of the stress-drop durations and size distributions together with their average temporal shape. We find critical exponents different from mean-field predictions, and a clear asymmetry for individual avalanches. We probe scaling relations for the rate dependency of the dynamics and we report a crossover towards mean-field results for strong driving.Comment: 5 pages, 3 figures, 1 table, supplementary material to be found at http://www-liphy.ujf-grenoble.fr/pagesperso/martens/documents/liu2015-sm.pd
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