Rock Mass as a Nonlinear Dynamic System. Mathematical Modeling of Stress-Strain State Evolution in the Rock Mass around a Mine Opening

The paper briefly reviews the fundamental (general) evolution properties of nonlinear dynamic systems. The stress-strain state evolution in a rock mass with mine openings has been numerically modeled, including the catastrophic stage of roof failure. The results of modeling the catastrophic failure of rock mass elements are analyzed in the framework of the theory of nonlinear dynamic systems. Solutions of solid mechanics equations are shown to exhibit all characteristic features of nonlinear dynamic system evolution, such as dynamic chaos, self-organized criticality, and catastrophic superfast stress-strain state evolution at the final stage of failure. The calculated seismic events comply with the Gutenberg-Richter law. The cut-off effect has been obtained in numerical computation (downward bending of the recurrence curve in the region of large-scale failure events). Prior to catastrophic failure, change of the probability density functions of stress fluctuations, related to the average trend, occurs, the slope of the recurrence curve of calculated seismic events becomes more gentle, seismic quiescence regions form in the central zones of the roof, and more active deformation begins at the periphery of the opening. These factors point to the increasing probability of a catastrophic event and can be considered as catastrophic failure precursors.     

Blow-up Modes in Fracture of Rock Samples and Earth’s Crust Elements

It is well known that the final stage of macroscopic fracture develops as a catastrophe in a superfast blow-up mode. However, the specific features of this stage are well studied only on large scales of earthquakes. Of particular interest for fracture prediction are both the stage of superfast catastrophic fracture and the mechanical behavior of the medium in the state of self-organized criticality prior to transition of fracture to the blow-up mode in order to reveal precursors of fracture transition to the catastrophic stage. This paper studies experimentally and theoretically the mechanical behavior of the medium prior to the catastrophic stage and transition to the blow-up mode. Rock samples (marble and artificial marble) were tested in three-point bending and uniaxial compression tests. The lateral surface velocities of loaded samples were recorded using a laser Doppler vibrometer. The recording frequency in measurements was 48 kHz, and the determination accuracy of the velocity amplitude was 0.1 μm/s. The estimated duration of the blow-up fracture stage is 10–20 ms. The mechanical behavior of samples in the experimental conditions, including the catastrophic fracture stage, is simulated numerically. The damage accumulation model parameters are determined from a comparison with the experimental data. Certain features of the mechanical response prior to catastrophic fracture are revealed which can be interpreted as fracture precursors.     

Stochastic Porous Media Equations and Self-Organized Criticality

The existence and uniqueness of nonnegative strong solutions for stochastic porous media equations with noncoercive monotone diffusivity function and Wiener forcing term is proven. The finite time extinction of solutions with high probability is also proven in 1-D. The results are relevant for self-organized criticality behavior of stochastic nonlinear diffusion equations with critical states.     

Critical behavior and 1/f noise in lumped systems undergoing two phase transitions

It is shown that the interaction of order parameters when subcritical and supercritical phase transitions take place simultaneously may result in a self-organized critical state and cause a 1/f ^α fluctuation spectrum, where 1≤α≤2. Such behavior is inherent in potential and nonpotential systems of nonlinear Langevin equations. A numerical analysis of the solutions to the proposed systems of stochastic differential equations showed that the solutions correlate with fractional integration and differentiation of white noise. The general behavior of such a system has features in common with self-organized criticality.     

Renormalization group and instantons in stochastic nonlinear dynamics

Stochastic counterparts of nonlinear dynamics are studied by means of nonperturbative functional methods developed in the framework of quantum field theory (QFT). In particular, we discuss fully developed turbulence, including leading corrections on possible compressibility of fluids, transport through porous media, theory of waterspouts and tsunami waves, stochastic magneto-hydrodynamics, turbulent transport in crossed fields, self-organized criticality, and dynamics of accelerated wrinkled flame fronts advancing in a wide canal. This report would be of interest to the broad auditorium of physicists and applied mathematicians, with a background in nonperturbative QFT methods or nonlinear dynamical systems, having an interest in both methodological developments and interdisciplinary applications.     

Low-dimensional behavior and symmetry breaking of stochasticsystems near criticality-can these effects be observed in space and inthe laboratory?

It is demonstrated that nonlinear stochastic systems near criticality (including self-organized criticality) will generally exhibit low-dimensional behavior. A connection is given between the fractal dimensions of finite-dimensional chaotic systems and the anomalous dimensions in stochastic systems near criticality. The effect of additional random noise on stochastic systems will be delineated in terms of the crossover phenomenon between competing criticalities. The possibility of observing such effects in space (such as the onset of substorms) and in the laboratory (such as stochastic particle heating in `noisy' magnetic fields) is discussed     

Spectral and dynamic analysis of plastic instabilities during serrated creep of the aluminum-magnesium alloy

The force response to the development of a macroscopic plastic deformation jump under the conditions of serrated creep of the aluminummagnesium alloy 5456 has been studied using spectral and dynamic analysis methods. The flicker-noise structure of the force response indicating the self-organized criticality state has been revealed. It has been found that a short-term state of plastic instability flatter spontaneously appears during the development of the macroscopic deformation step.     

Granular matter: A wonderful world of clusters in far-from-equilibrium systems

We review several theoretical and experimental methods of modeling and investigating granular matter far from equilibrium. The theoretical methods include an extension of the classical Boltzmann equation to inelastic gases, scalar internal degrees of freedom, and Hamiltonian-like grain–grain interactions; the experimental technique is concerned with thermal properties of electrically conducting clusters. We discuss the results, focusing on phenomena nonexistent in physics of gases, fluids or solids, e.g. anomalous temperature gradients or electric resistance. One of the models is used to study the interplay between classical and self-organized criticality.     

Jerky flow model as a basis for research in deformation instabilities

A phenomenological jerky flow model was developed in which macroscale plastic strain rates are defined by dislocation kinetics. The model takes into account destructive processes governed by shear and bulk defect accumulation. At the heart of the model lie equations of solid mechanics and relaxation-type constitutive equations. A loaded elastoplastic solid is treated as a nonlinear dynamic system whose evolution, according to synergetic laws, is much contributed by negative and positive feedbacks expressed, respectively, through constitutive equations of the first group (relaxation equations) and constitutive equations of the second group (kinetic equations for deformation defect and damage accumulation rates). The negative feedback stabilizes deformation by relaxation, bringing the process to some local dynamic equilibrium. The positive feedback destabilizes deformation, driving the system to a critical state. Numerical experiment was performed in 2D and 3D statements. Statistical analysis of stress fluctuations about the average trend shows that the jerky flow model of an elastoplastic medium demonstrates evolution characteristic of nonlinear dynamic systems: through states of dynamic chaos and self-organized criticality to a global catastrophe.     

Effects of Interactive Function Forms and Refractoryperiod in a Self-Organized Critical Model Based on Neural Networks

Based on the standard self-organizing map neural network model and an integrate-and-tire mechanism, we investigate the effect of the nonlinear interactive function on the self-organized criticality in our model. Based on these we also investigate the effect of the refractoryperiod on the self-organized criticality of the system.     

Effects of Interactive Function Forms and Refractoryperiod in a Self-Organized Critical Model Based on Neural Networks

Based on the standard self-organizing map neural network model and an integrate-and-fire mechanism, we investigate the effect of the nonlinear interactive function on the self-organized criticality in our model. Based on the sewe also investigate the effect of the refractoryperiod on the self-organized criticality of the system.     

Scaling laws in evolution of large computer programs

The approach is based on a paradigm of self-organized criticality proposed for experimental investigation and theoretical modeling of software evolution. The dynamics of modifications is studied for three free, open source programs Mozilla, Free-BSD, and Emacs using the data from version control systems. Scaling laws typical for the self-organization criticality are found. The model of software evolution presenting the natural selection principle is proposed. The results of numerical and analytical investigation of the model are presented. They are in good agreement with the data collected for the real-world software. The text was submitted by the authors in English.     

Self-organization of the critical state in granular superconductors

The critical state in granular superconductors is studied using two mathematical models: systems of differential equations for the gauge-invariant phase difference and a simplified model that is described by a system of coupled mappings and in many cases is equivalent to the standard models used for studying self-organized criticality. It is shown that the critical state of granular superconductors is self-organized in all cases studied. In addition, it is shown that the models employed are essentially equivalent, i.e., they demonstrate not only the same critical behavior, but they also lead to the same noncritical phenomena. The first demonstration of the existence of self-organized criticality in a system of nonlinear differential equations and its equivalence to self-organized criticality in standard models is given in this paper.     

The modify unstable nonlinear Schrödinger dynamical equation and its optical soliton solutions

In this research, we work on a specific class of nonlinear evolution equation which is the modify unstable nonlinear Schrödinger equation. This equation is used to describe a time evolution of disturbances in unstable media. Various solutions have been obtained. The results deduced are of varied types and include bright solution, dark solution, rational dark-bright solution, as well as cnoidal solutions. These solutions might be useful in engineering fields. Some conditions for the stability of these solutions are presented. The method used here is understandable and very powerful for solving the nonlinear problems.     

Acoustic emission from periodically perturbed systems: SOC and predictability

The statistics of acoustic emission (AE) from as diverse processes as volcanic activity (Stromboli, Italy) and martensitic transformations driven by thermal cycles, is shown to verify the paradigm of self-organized criticality. However, catastrophic event predictability both in laboratory (the onset of martensitic transformations) and in on-site applications (volcanic seisms and explosions) through the analysis of historical AE series, is not ruled out as long as the emitting samples are subjected to (quasi)periodic low-frequency/large-scale dynamics.     

Lifetime of bubble rafts: cooperativity and avalanches

We have studied the collapse of pseudo-bi-dimensional foams. These foams are made of uniformly sized soap bubbles packed in an hexagonal lattice sitting at the top of a liquid surface. The collapse process follows the sequence: (1) rupture of a first bubble, driven by thermal fluctuations and (2) a cascade of bursting bubbles. We present a simple numerical model which captures the main characteristics of the dynamics of foam collapse. We show that in a certain range of viscosities of the foaming solutions, the size distribution of the avalanches follows power laws as in self-organized criticality processes.     

Absorbing states and elastic interfaces in random media: two equivalent descriptions of self-organized criticality

We elucidate a long-standing puzzle about the nonequilibrium universality classes describing self-organized criticality in sandpile models. We show that depinning transitions of linear interfaces in random media and absorbing phase transitions (with a conserved nondiffusive field) are two equivalent languages to describe sandpile criticality. This is so despite the fact that local roughening properties can be radically different in the two pictures, as explained here. Experimental implications of our work as well as promising paths for future theoretical investigations are also discussed.     

Spatial evolution character of multi-objective evolutionary algorithm based on self-organized criticality theory

This paper analyzes the spatial evolution character of multi-objective evolutionary algorithms using self-organized criticality theory. The spatial evolution character is modeled by the statistical property of crowding distance, which displays a scale-free feature and a power-law distribution. We propose that the evolutional rule of multi-objective optimization algorithms is a self-organized state transition from an initial scale-free state to a final scale-free state. The target is to get close to a critical state representing the true Pareto-optimal front. Besides, the anti-Matthew effect is the internal incentive factor of most strategies. The final scale-free state reflects the quality of the final Pareto-optimal front. The speed of the state transition reflects the efficiency of the algorithm. We simulate the spatial evolution characters of three typical multi-objective evolutionary algorithms representing three fields, i.e., Genetic Algorithm, Differential Evolution and the Artificial Immune System algorithm. The results prove that the model and the explanation are effective for analyzing the evolutional rule of multi-objective evolutionary algorithms.     

固体破坏的损伤演化诱致突变现象

固体破坏问题在理论上及实际上均极为重要，是涉及力学，物理学及非线性科学等学科的一个十分复杂基本问题，文章介绍了基于细观非线性动力学模型的研究所取得的进展，发现系统显示一种共性特征，称为演化诱致突变，即演化模式从整体稳定向灾变性模式转变，宏观破坏的样本个性行为，以及宏观性质对细观无序性的敏感性。     

Serrated creep and spatio-temporal structures of macrolocalized plastic deformation

The dynamics and morphology of macrolocalized deformation bands have been investigated using a complex of high-speed in situ methods under the conditions of serrated creep of flat samples of the aluminum-magnesium alloy 5456 with different aspect ratios. It has been found that, at the front of a macroscopic plastic deformation jump, a complex structure of propagating deformation bands, which are considered as macrolocalized deformation “quanta,” is spontaneously formed in the material. It has been shown that, with an increase in the sample length, the deformation behavior of the alloy tends to the state of self-organized criticality.