Avalanches in breakdown and fracture processes

Hybrid universality classes of systemic cascades

Cascades are self-reinforcing processes underlying the systemic risk of many complex systems. Understanding the universal aspects of these phenomena is of fundamental interest, yet typically bound to numerical observations in ad-hoc models and limited insights. Here, we develop a unifying approach that reveals two distinct universality classes of cascades determined by the global symmetry of the cascading process. We provide hyperscaling arguments predicting hybrid critical phenomena characterized by a combination of both mean-field spinodal exponents and d-dimensional corrections, and show how parity invariance influences the geometry and lifetime of critical avalanches. Our theory applies to a wide range of networked systems in arbitrary dimensions, as we demonstrate by simulations encompassing classic and novel cascade models, revealing universal principles of cascade critical phenomena amenable to experimental validation.

Physiological mechanical forces accelerate the degradation of bovine lung collagen fibers by bacterial collagenase

Collagen fibers, one of the key load-bearing components of the extracellular matrix, contribute significantly to tissue integrity through their mechanical properties of strain-dependent stiffening. This study investigated the effects of bacterial collagenase on the mechanical behavior of individual bovine lung collagen fibers in the presence or absence of mechanical forces, with a focus on potential implications for emphysema, a condition associated with collagen degradation and alveolar wall rupture. Tensile tests were conducted on individual collagen fibers isolated from bovine lung tissue. The rate of degradation was characterized by the change in fiber Young's modulus during 60 min of digestion under various mechanical conditions mimicking the mechanical stresses on the fibers during breathing. Compared to digestion without mechanical forces, a significantly larger drop of fiber modulus was observed in the presence of static or intermittent mechanical forces. Fiber yield stress was also reduced after digestion indicating compromised fiber failure. By incorporating fibril waviness obtained by scanning electron microscopic images, an analytic model allowed estimation of fibril modulus. A computational model that incorporated waviness and the results of tensile tests was also developed to simulate and interpret the data. The simulation results provided insights into the mechanical consequences of bacterial collagenase and mechanical forces on collagen fibers, revealing both fibril softening and rupture during digestion. These findings shed light on the microscale changes in collagen fiber structure and mechanics under enzymatic digestion and breathing-like mechanical stresses with implications for diseases that are impacted by collagen degradation such as emphysema.

Ductile and brittle yielding of athermal amorphous solids: A mean-field paradigm beyond the random-field Ising model

Amorphous solids can yield in either a ductile or brittle manner under strain: plastic deformation can set in gradually, or abruptly through a macroscopic stress drop. Developing a unified theory describing both ductile and brittle yielding constitutes a fundamental challenge of nonequilibrium statistical physics. Recently, it has been proposed that, in the absence of thermal effects, the nature of the yielding transition is controlled by physics akin to that of the quasistatically driven random field Ising model (RFIM), which has served as the paradigm for understanding the effect of quenched disorder in slowly driven systems with short-ranged interactions. However, this theoretical picture neglects both the dynamics of, and the elasticity-induced long-ranged interactions between, the mesoscopic material constituents. Here, we address these two aspects and provide a unified theory building on the Hébraud-Lequeux elastoplastic description. The first aspect is crucial to understanding the competition between the imposed deformation rate and the finite timescale of plastic rearrangements: We provide a dynamical description of the macroscopic stress drop, as well as predictions for the shifting of the brittle yield strain and the scaling of the peak susceptibility with inverse shear rate. The second is essential to capture properly the behavior in the limit of quasistatic driving, where avalanches of plasticity diverge with system size at any value of the strain. We fully characterise the avalanche behavior, which is radically different to that of the RFIM. In the quasistatic, infinite-size limit, we find that both models have mean-field Landau exponents, obscuring the effect of the interactions. We show, however, that the latter profoundly affect the behavior of finite systems approaching the spinodal-like brittle yield point, where we recover qualitatively the finite-size trends found in particle simulations. The interactions also modify the nature of the random critical point separating ductile and brittle yielding, where we predict critical behavior on top of the marginality present at any value of the strain. We finally discuss how all our predictions can be directly tested against particle simulations and eventually experiments, and make first steps in this direction.

Record statistics of fracture in the random spring network model

We study the role of record statistics of damage avalanches in predicting the fracture of a heterogeneous material under tensile loading. The material is modeled using a two-dimensional random spring network where disorder is introduced through randomness in the breakage threshold strains of the springs. It is shown that the waiting strain interval between successive records of avalanches has a maximum for moderate disorder, thus showing an acceleration in occurrence of records when approaching final fracture. Such a signature is absent for low disorder when the fracture is nucleation-dominated, as well as for high disorder when the fracture is percolation type. We examine the correlation between the record with the maximum waiting strain interval and the crossover record at which the avalanche statistics change from off-critical to critical. Compared to the avalanche exponent crossover based prediction for failure, we show that the record statistics have the advantage of both being real-time as well as being a precursor significantly prior to final fracture. We also find that in the avalanche-dominated regime, the failure strain is at best weakly correlated with the strain at the maximum waiting strain interval. A stronger correlation is observed between the index of the largest record and the index of the record at the maximum waiting strain interval.

Percolation versus depinning transition: The inherent role of damage hardening during quasibrittle failure

The intermittent damage evolution preceding the failure of heterogeneous brittle solids is well described by scaling laws. In deciphering its origins, failure is routinely interpreted as a critical transition. However at odds with expectations of universality, a large scatter in the value of the scaling exponents is reported during acoustic emission experiments. Here we numerically examine the precursory damage activity to reconcile the experimental observations with critical phenomena framework. Along with the strength of disorder, we consider an additional parameter that describes the progressive damageability of material elements at mesoscopic scale. This hardening behavior encapsulates the microfracturing processes taking place at lower length scales. We find that damage hardening can not only delay the final failure but also affect the preceding damage accumulation. When hardening is low, the precursory activity is strongly influenced by the strength of the disorder and is reminiscent of damage percolation. On the contrary, for large hardening, long-range elastic interactions prevail over disorder, ensuring a rather homogeneous evolution of the damage field in the material. The power-law statistics of the damage bursts is robust to the strength of the disorder and is reminiscent of the collective avalanche dynamics of elastic interfaces near the depinning transition. The existence of these two distinct universality classes also manifests as different values of the scaling exponent characterizing the divergence of the precursor size on approaching failure. Our finding sheds new light on the connection between the level of quasibrittleness of materials and the statistical features of the failure precursors. Finally, it also provides a more complete description of the acoustic precursors and thus paves the way for quantitative techniques of damage monitoring of structures-in-service.

Criticality in the fracture of silica glass: Insights from molecular mechanics

The universality of avalanches characterizing the inelastic response of disordered materials has the potential to bridge the gap from micro to macroscale. In this study, we explore the statistics and the scaling behavior of avalanches occurring during the fracture process in silica glass using molecular mechanics. We introduce a robust method for capturing and quantifying these avalanches, allowing us to perform rigorous statistical analyses, revealing universal power laws associated with critical phenomena. The influence of an initial crack is explored, observing deviations from mean-field predictions while maintaining the property of criticality. However, the avalanche exponents in the unnotched samples are predicted correctly by the mean-field depinning model. Furthermore, we investigate the strain-dependent probability density function, its cutoff function, and the interrelation between the critical exponents. Finally, we unveil distinct scaling behavior for small and large avalanches of the crack growth, shedding light on the underlying fracture mechanisms in silica glass.

Analysis of Acoustic Emission Energy Distribution and Avalanche Dynamics of Sandstone with Different Particle Sizes

The mechanical properties of sandstone have an important impact on the stability of the coal mine roof and floor, sandstone gas mining, and underground engineering safety. In order to study the critical characteristics on the failure process of sandstone with different particle sizes under uniaxial compression conditions, avalanche dynamics theory and a critical model are used to analyze the distribution of acoustic emission (AE) parameters, and the maximum likelihood estimation is used to accurately estimate the critical parameters. The results showed that the AE phenomenon of sandstone can be divided into four stages: initial compaction period, quiet period, crack stable growth period and outbreak period. During the process of compression failure, the larger the particle size is, the more seriously the sandstone is damaged. The AE energy probability density distribution follows single power-law distribution, and the AE energy critical exponent is 1.20 and follows the characteristics of scale-free regarding the power-law distribution on the particle sizes. When the stress runs up to 90% of peak stress, the bifurcation ratio increases sharply and shows the characteristics of the critical state. The waiting time and the avalanche size distribution follow double power-law distribution, and the inflection points are 0.03 and 37. Before and after the inflection point, the waiting time critical exponent and the avalanche size critical exponent are 1.90, 0.40 and 2.40, 1.60. This shows that the dynamic evolution process of sandstone under uniaxial compression condition can be characterized well by the fiber bundle model.

Role of composition in fracture behavior of two-phase solids

In a two-phase solid, we examine the growth of a preexisting macroscopic crack based on simulations of a random spring network model. We find that the enhancement in toughness, as well as strength, is strongly dependent on the ratio of elastic moduli as well as on the relative proportion of the phases. We find that the mechanism that leads to enhancement in toughness is not the same as that for enhancement in strength; however, the overall enhancement is similar in mode I and mixed-mode loading. Based on the crack paths, and the spread of the fracture process zone, we identify the type of fracture to transition from nucleation type, for close to single-phase compositions, whether hard or soft, to avalanche type for more mixed compositions. We also show that the associated avalanche distributions exhibit power-law statistics with different exponents for each phase. The significance of variations in the avalanche exponents with the relative proportion of phases and possible connections to the fracture types are discussed in detail.

Scale-free behavioral cascades and effective leadership in schooling fish

Behavioral contagion and the presence of behavioral cascades are natural features in groups of animals showing collective motion, such as schooling fish or grazing herbivores. Here we study empirical behavioral cascades observed in fish schools defined as avalanches of consecutive large changes in the heading direction of the trajectory of fish. In terms of a minimum turning angle introduced to define a large change, avalanches are characterized by distributions of size and duration showing scale-free signatures, reminiscent of self-organized critical behavior. We observe that avalanches are generally triggered by a small number of fish, which act as effective leaders that induce large rearrangements of the group's trajectory. This observation motivates the proposal of a simple model, based in the classical Vicsek model of collective motion, in which a given individual acts as a leader subject to random heading reorientations. The model reproduces qualitatively the empirical avalanche behavior observed in real schools, and hints towards a connection between effective leadership, long range interactions and avalanche behavior in collective movement.

Statistical physics perspective of fracture in brittle and quasi-brittle materials

We discuss the physics of fracture in terms of the statistical physics associated with the failure of elastic media under applied stresses in presence of quenched disorder. We show that the development and the propagation of fracture are largely determined by the strength of the disorder and the stress field around them. Disorder acts as nucleation centres for fracture. We discuss Griffith's law for a single crack-like defect as a source for fracture nucleation and subsequently consider two situations: (i) low disorder concentration of the defects, where the failure is determined by the extreme value statistics of the most vulnerable defect (nucleation regime) and (ii) high disorder concentration of the defects, where the scaling theory near percolation transition is applicable. In this regime, the development of fracture takes place through avalanches of a large number of tiny microfractures with universal statistical features. We discuss the transition from brittle to quasi-brittle behaviour of fracture with the strength of disorder in the mean-field fibre bundle model. We also discuss how the nucleation or percolation mode of growth of fracture depends on the stress distribution range around a defect. We discuss the corresponding numerical simulation results on random resistor and spring networks.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

Avalanche precursors of failure in hierarchical fuse networks

We study precursors of failure in hierarchical random fuse network models which can be considered as idealizations of hierarchical (bio)materials where fibrous assemblies are held together by multi-level (hierarchical) cross-links. When such structures are loaded towards failure, the patterns of precursory avalanche activity exhibit generic scale invariance: irrespective of load, precursor activity is characterized by power-law avalanche size distributions without apparent cut-off, with power-law exponents that decrease continuously with increasing load. This failure behavior and the ensuing super-rough crack morphology differ significantly from the findings in non-hierarchical structures.

Anisotropic particles strengthen granular pillars under compression

We probe the effects of particle shape on the global and local behavior of a two-dimensional granular pillar, acting as a proxy for a disordered solid, under uniaxial compression. This geometry allows for direct measurement of global material response, as well as tracking of all individual particle trajectories. In general, drawing connections between local structure and local dynamics can be challenging in amorphous materials due to lower precision of atomic positions, so this study aims to elucidate such connections. We vary local interactions by using three different particle shapes: discrete circular grains (monomers), pairs of grains bonded together (dimers), and groups of three bonded in a triangle (trimers). We find that dimers substantially strengthen the pillar and the degree of this effect is determined by orientational order in the initial condition. In addition, while the three particle shapes form void regions at distinct rates, we find that anisotropies in the local amorphous structure remain robust through the definition of a metric that quantifies packing anisotropy. Finally, we highlight connections between local deformation rates and local structure.

Conducting-insulating transition in adiabatic memristive networks

The development of neuromorphic systems based on memristive elements-resistors with memory-requires a fundamental understanding of their collective dynamics when organized in networks. Here, we study an experimentally inspired model of two-dimensional disordered memristive networks subject to a slowly ramped voltage and show that they undergo a discontinuous transition in the conductivity for sufficiently high values of memory, as quantified by the memristive ON-OFF ratio. We investigate the consequences of this transition for the memristive current-voltage characteristics both through simulation and theory, and demonstrate the role of current-voltage duality in relating forward and reverse switching processes. Our work sheds considerable light on the statistical properties of memristive networks that are presently studied both for unconventional computing and as models of neural networks.

Critical phenomena at a first-order phase transition in a lattice of glow lamps: Experimental findings and analogy to neural activity

Networks of non-linear electronic oscillators have shown potential as physical models of neural dynamics. However, two properties of brain activity, namely, criticality and metastability, remain under-investigated with this approach. Here, we present a simple circuit that exhibits both phenomena. The apparatus consists of a two-dimensional square lattice of capacitively coupled glow (neon) lamps. The dynamics of lamp breakdown (flash) events are controlled by a DC voltage globally connected to all nodes via fixed resistors. Depending on this parameter, two phases having distinct event rate and degree of spatiotemporal order are observed. The transition between them is hysteretic, thus a first-order one, and it is possible to enter a metastability region, wherein, approaching a spinodal point, critical phenomena emerge. Avalanches of events occur according to power-law distributions having exponents ≈3/2 for size and ≈2 for duration, and fractal structure is evident as power-law scaling of the Fano factor. These critical exponents overlap observations in biological neural networks; hence, this circuit may have value as building block to realize corresponding physical models.

Avalanches in dry and saturated disordered media at fracture

This paper analyzes fracturing in inhomogeneous media under dry and fully saturated conditions. We adopt a central force model with continuous damage to study avalanche behavior in a two-dimensional truss lattice undergoing dilation. Multiple fractures can develop at once and a power-law distribution of the avalanche size is observed. The values for the power-law exponent are compared with the ones found in the literature and scale-free behavior is suggested. The fracture evolves intermittently in time because only some avalanches correspond to fracture advancement. A fully saturated model with continuous damage based on the extended Biot's theory is developed and avalanche behavior is studied in the presence of fluid, varying the fluid boundary conditions. We show that power-law behavior is destroyed when the fluid flux governs the problem. Fluid pressure behavior during intermittent crack tip advancement is studied for the continuous-damage fully saturated model. It is found that when mechanical loading prevails, the pressure rises when the crack advances, while when fluid loading prevails, the pressure drops when the crack advances.

Acoustic Emission from Breaking a Bamboo Chopstick

The acoustic emission from breaking a bamboo chopstick or a bundle of spaghetti is found to exhibit similar behavior as the famous seismic laws of Gutenberg and Richter, Omori, and Båth. By the use of a force-sensing detector, we establish a positive correlation between the statistics of sound intensity and the magnitude of a tremor. We also manage to derive these laws analytically without invoking the concept of a phase transition, self-organized criticality, or fractal. Our model is deterministic and relies on the existence of a structured cross section, either fibrous or layered. This success at explaining the power-law behavior supports the proposal that geometry is sometimes more important than mechanics.

Complexity and emergent phenomena

Complex biological systems operate under non-equilibrium conditions and exhibit emergent properties associated with correlated spatial and temporal structures. These properties may be individually unpredictable, but tend to be governed by power-law probability distributions and/or correlation. This article reviews the concepts that are invoked in the treatment of complex systems through a wide range of respiratory-related examples. Following a brief historical overview, some of the tools to characterize structural variabilities and temporal fluctuations associated with complex systems are introduced. By invoking the concept of percolation, the notion of multiscale behavior and related modeling issues are discussed. Spatial complexity is then examined in the airway and parenchymal structures with implications for gas exchange followed by a short glimpse of complexity at the cellular and subcellular network levels. Variability and complexity in the time domain are then reviewed in relation to temporal fluctuations in airway function. Next, an attempt is given to link spatial and temporal complexities through examples of airway opening and lung tissue viscoelasticity. Specific examples of possible and more direct clinical implications are also offered through examples of optimal future treatment of fibrosis, exacerbation risk prediction in asthma, and a novel method in mechanical ventilation. Finally, the potential role of the science of complexity in the future of physiology, biology, and medicine is discussed.

Brittle-to-ductile transition in a fiber bundle with strong heterogeneity

We analyze the failure process of a two-component system with widely different fracture strength in the framework of a fiber bundle model with localized load sharing. A fraction 0≤α≤1 of the bundle is strong and it is represented by unbreakable fibers, while fibers of the weak component have randomly distributed failure strength. Computer simulations revealed that there exists a critical composition α(c) which separates two qualitatively different behaviors: Below the critical point, the failure of the bundle is brittle, characterized by an abrupt damage growth within the breakable part of the system. Above α(c), however, the macroscopic response becomes ductile, providing stability during the entire breaking process. The transition occurs at an astonishingly low fraction of strong fibers which can have importance for applications. We show that in the ductile phase, the size distribution of breaking bursts has a power law functional form with an exponent μ=2 followed by an exponential cutoff. In the brittle phase, the power law also prevails but with a higher exponent μ=9/2. The transition between the two phases shows analogies to continuous phase transitions. Analyzing the microstructure of the damage, it was found that at the beginning of the fracture process cracks nucleate randomly, while later on growth and coalescence of cracks dominate, which give rise to power law distributed crack sizes.

Long-time damage under creep experiments in disordered materials: transition from exponential to logarithmic fracture dynamics

Some materials, and in particular some polymer materials, can display an important range of stress levels for which slow and progressive damage can be observed before they finally break. In creep or fatigue experiments, final rupture can happen after very long times, during which the mechanical properties have progressively decayed. We model here some generic features of the long-time damage evolution of disordered elastic materials under constant load, characterized by a progressive decrease of the elastic modulus. We do it by studying a two-dimensional electric random fuse network with quenched disorder and thermal noise. The time evolution of global quantities (conductivity or, equivalently, elastic modulus) is characterized by different regimes ranging from faster than exponential to slower than logarithmic, which are governed by the stress level and the relative magnitude of disorder with respect to temperature. A region of widely distributed rupture times exists where the modulus decays (more slowly than) logarithmically for not too small values of the disorder and for not too large values of the load. A detailed analysis of the dynamical regimes is performed and presented through a phase diagram.

Avalanche process of the fiber-bundle model with stick-slip dynamics and a variable Young modulus

In order to more accurately describe the fracture process of extensive biological fibers, a fiber-bundle model with stick-slip dynamics and a variable Young modulus is constructed. In this model, the Young modulus of a fiber is assumed to increase or decrease by multiplying with a changing ratio after local sliding events. So, the maximum number of stick-slip events of a single fiber and the changing ratio of the Young modulus are the two key parameters of the model. By means of analytical theory and numerical simulation, the constitutive law, the critical stress, the average size of the largest avalanche, and the avalanche size distribution are shown against the two parameters of the model. From a macroscopic viewpoint, the constitutive curves show different morphologies varying from a local plastic state to a unimodal parabola, while from a microscopic viewpoint, the avalanche size distributions can be well fitted into a power law relationship, which is in accord with the classical fiber-bundle model.

Universal fluctuations and extreme statistics of avalanches near the depinning transition

We derive exact predictions for universal scaling exponents and scaling functions associated with the statistics of maximum velocities v(m) during avalanches described by the mean-field theory of the interface depinning transition. In particular, we find a robust power-law regime in the statistics of maximum events that can explain the observed distribution of the peak amplitudes in acoustic emission experiments of crystal plasticity. Our results are expected to be broadly applicable to a broad range of systems in the mean-field interface depinning universality class, ranging from magnets to earthquakes.

Distribution of maximum velocities in avalanches near the depinning transition

We report exact predictions for universal scaling exponents and scaling functions associated with the distribution of the maximum collective avalanche propagation velocities v(m) in the mean field theory of the interface depinning transition. We derive the extreme value distribution P(v(m)|T) for the maximum velocities in avalanches of fixed duration T and verify the results by numerical simulation near the critical point. We find that the tail of the distribution of maximum velocity for an arbitrary avalanche duration, v(m), scales as P(v(m))~v(m)(-2) for large v(m). These results account for the observed power-law distribution of the maximum amplitudes in acoustic emission experiments of crystal plasticity and are also broadly applicable to other systems in the mean-field interface depinning universality class, ranging from magnets to earthquakes.

Damage-cluster distributions and size effect on strength in compressive failure

We investigate compressive failure of heterogeneous materials on the basis of a continuous progressive-damage model. The model explicitly accounts for tensile and shear local damage and reproduces the main features of compressive failure of brittle materials like rocks or ice. We show that the size distribution of damage clusters, as well as the evolution of an order parameter--the size of the largest damage cluster--argue for a critical interpretation of fracture. The compressive failure strength follows a normal distribution with a very small size effect on the mean strength, in good agreement with experiments.

Statistical model with two order parameters for ductile and soft fiber bundles in nanoscience and biomaterials

Traditional fiber bundles models (FBMs) have been an effective tool to understand brittle heterogeneous systems. However, fiber bundles in modern nano- and bioapplications demand a new generation of FBM capturing more complex deformation processes in addition to damage. In the context of loose bundle systems and with reference to time-independent plasticity and soft biomaterials, we formulate a generalized statistical model for ductile fracture and nonlinear elastic problems capable of handling more simultaneous deformation mechanisms by means of two order parameters (as opposed to one). As the first rational FBM for coupled damage problems, it may be the cornerstone for advanced statistical models of heterogeneous systems in nanoscience and materials design, especially to explore hierarchical and bio-inspired concepts in the arena of nanobiotechnology. Applicative examples are provided for illustrative purposes at last, discussing issues in inverse analysis (i.e., nonlinear elastic polymer fiber and ductile Cu submicron bars arrays) and direct design (i.e., strength prediction).

Energy dissipation statistics in the random fuse model

We study the statistics of the dissipated energy in the two-dimensional random fuse model for fracture under different imposed strain conditions. By means of extensive numerical simulations we compare different ways to compute the dissipated energy. In the case of an infinitely slow driving rate (quasistatic model), we find that the probability distribution of the released energy shows two different scaling regions separated by a sharp energy crossover. At low energies, the probability of having an event of energy E decays as approximately E(-1/2), which is robust and independent of the energy quantifier used (or lattice type). At high energies, fluctuations dominate the energy distribution, leading to a crossover to a different scaling regime, approximately E(-2.75), whenever the released energy is computed over the whole system. On the contrary, strong finite-size effects are observed if we consider only the energy dissipated at microfractures. In a different numerical experiment, the quasistatic dynamics condition is relaxed, so that the system is driven at finite strain load rates, and we find that the energy distribution decays as P(E) approximately E(-1) for all the energy range.

Critical behavior of random fibers with mixed Weibull distribution

A random fiber bundle model with a mixed Weibull distribution is studied under the global load sharing scheme. The mixed model consists of two sets of fibers. The threshold strength of one set of fibers is randomly chosen from a Weibull distribution with a particular Weibull index, and another set of fibers with a different index. The mixing tunes the critical stress of the bundle and the variation of critical stress with the amount of mixing is determined using a probabilistic method where the external load is increased quasistatically. In a special case which we illustrate, the critical stress is found to vary linearly with the mixing parameter. The critical exponents and power-law behavior of burst avalanche size distribution is found to remain unaltered due to mixing.

Crack surface roughness in three-dimensional random fuse networks

Using large system sizes with extensive statistical sampling, we analyze the scaling properties of crack roughness and damage profiles in the three-dimensional random fuse model. The analysis of damage profiles indicates that damage accumulates in a diffusive manner up to the peak load, and localization sets in abruptly at the peak load, starting from a uniform damage landscape. The global crack width scales as W approximately L(0.5) and is consistent with the scaling of localization length xi approximately L(0.5) used in the data collapse of damage profiles in the postpeak regime. This consistency between the global crack roughness exponent and the postpeak damage profile localization length supports the idea that the postpeak damage profile is predominantly due to the localization produced by the catastrophic failure, which at the same time results in the formation of the final crack. Finally, the crack width distributions can be collapsed for different system sizes and follow a log-normal distribution.

Failure process of a bundle of plastic fibers

We present an extension of fiber bundle models considering that failed fibers still carry a fraction 0 < or = alpha < or = 1 of their failure load. The value of alpha interpolates between the perfectly brittle failure (alpha = 0) and perfectly plastic behavior (alpha = 1) of fibers. We show that the finite load bearing capacity of broken fibers has a substantial effect on the failure process of the bundle. In the case of global load sharing it is found that for alpha --> 1 the macroscopic response of the bundle becomes perfectly plastic with a yield stress equal to the average fiber strength. On the microlevel, the size distribution of avalanches has a crossover from a power law of exponent approximately 2.5 to a faster exponential decay. For localized load sharing, computer simulations revealed a sharp transition at a well-defined value alpha(c) from a phase where macroscopic failure occurs due to localization as a consequence of local stress enhancements, to another one where the disordered fiber strength dominates the damage process. Analyzing the microstructure of damage, the transition proved to be analogous to percolation. At the critical point alpha(c), the spanning cluster of damage is found to be compact with a fractal boundary. The distribution of bursts of fiber breakings shows a power-law behavior with a universal exponent approximately 1.5 equal to the mean-field exponent of fiber bundles of critical strength distributions. The model can be relevant to understand the shear failure of glued interfaces where failed regions can still transmit load by remaining in contact.

Universal mean moment rate profiles of earthquake ruptures

Earthquake phenomenology exhibits a number of power law distributions including the Gutenberg-Richter frequency-size statistics and the Omori law for aftershock decay rates. In search for a basic model that renders correct predictions on long spatiotemporal scales, we discuss results associated with a heterogeneous fault with long-range stress-transfer interactions. To better understand earthquake dynamics we focus on faults with Gutenberg-Richter-like earthquake statistics and develop two universal scaling functions as a stronger test of the theory against observations than mere scaling exponents that have large error bars. Universal shape profiles contain crucial information on the underlying dynamics in a variety of systems. As in magnetic systems, we find that our analysis for earthquakes provides a good overall agreement between theory and observations, but with a potential discrepancy in one particular universal scaling function for moment rates. We primarily use mean field theory for the theoretical analysis, since it has been shown to be in the same universality class as the full three-dimensional version of the model (up to logarithmic corrections). The results point to the existence of deep connections between the physics of avalanches in different systems.

Experimental realization of the fuse model of crack formation

In this work, we present an experimental investigation of the fuse model. Our main goal was to study the influence of the disorder on the fracture process. The experimental apparatus used consisted of an L x L square lattice with fuses placed on each bond of the lattice. Two types of materials were used as fuses: copper and steel wool wires. The lattice composed only of copper wires varied from a weakly disordered system to a strongly disordered one. The lattice formed only by steel wool wires corresponded to a strongly disordered one. The experimental procedure consisted of applying a potential difference V to the lattice and measuring the respective current I. The characteristic function I(V) obtained was investigated in order to find the scaling law dependence of the voltage and the current on the system size L when the disorder was changed. Our results show that the scaling laws are only verified for the disordered regime.

Statistical physics models for nacre fracture simulation

Natural biological materials such as nacre (or mother-of-pearl), exhibit phenomenal fracture strength and toughness properties despite the brittle nature of their constituents. For example, nacre's work of fracture is three orders of magnitude greater than that of a single crystal of its constituent mineral. This study investigates the fracture properties of nacre using a simple discrete lattice model based on continuous damage random thresholds fuse network. The discrete lattice topology of the proposed model is based on nacre's unique brick and mortar microarchitecture, and the mechanical behavior of each of the bonds in the discrete lattice model is governed by the characteristic modular damage evolution of the organic matrix that includes the mineral bridges between the aragonite platelets. The analysis indicates that the excellent fracture properties of nacre are a result of their unique microarchitecture, repeated unfolding of protein molecules (modular damage evolution) in the organic polymer, and the presence of fiber bundle of mineral bridges between the aragonite platelets. The numerical results obtained using this simple discrete lattice model are in excellent agreement with the previously obtained experimental results, such as nacre's stiffness, tensile strength, and work of fracture.

Scale-free fuse network and its robustness

The robustness and reliability of scale-free networks are tested as a fuse network. The idea is to examine the robustness of a scale-free network when links are irreversibly removed after failing. Due to inherent characteristics of the fuse network model, the sequence of links removal is deterministic and conditioned to fuse tolerance and connectivity of its ends. It is a different situation from classical robustness analysis of complex networks, when they are usually tested under random fails and deliberate attacks of nodes. The use of this system to study the fracture of elastic material brought some interesting results.

A continuous damage random thresholds model for simulating the fracture behavior of nacre

This study investigates the fracture properties of nacre using a discrete lattice model based on continuous damage random threshold fuse network. The discrete lattice topology of the model is based on nacre's unique brick and mortar microarchitecture. The mechanical behavior of each of the bonds in the discrete lattice model is governed by the characteristic modular damage evolution of the organic matrix and the mineral bridges between the aragonite platelets. The numerical results obtained using this simple discrete lattice model are in very good agreement with the previously obtained experimental results, such as nacre's stiffness, tensile strength, and work of fracture. The analysis indicates that nacre's superior toughness is a direct consequence of ductility (maximum shear strain) of the organic matrix in terms of repeated unfolding of protein molecules, and its fracture strength is a result of its ordered brick and mortar architecture with significant overlap of the platelets, and shear strength of the organic matrix.

Breakdown of disordered media by surface loads

We model an interface layer connecting two parts of a solid body by N parallel elastic springs connecting two rigid blocks. We load the system by a shear force acting on the top side. The springs have equal stiffness but are ruptured randomly when the load reaches a critical value. For the considered system, we calculate the shear modulus G as a function of the order parameter phi describing the state of damage, and also the "spalled"; material (burst) size distribution. In particular, we evaluate the relation between the damage parameter and the applied force and explore the behavior in the vicinity of material breakdown. Using this simple model for material breakdown, we show that damage, caused by applied shear forces, is analogous to a first-order phase transition. The scaling behavior of G with phi is explored analytically and numerically, close to phi=0 and phi=1 and in the vicinity of phi(c), when the shear load is close to but below the threshold force that causes material breakdown. Our model calculation represents a first approximation of a system subject to wear induced loads.

Statistical properties of fracture in a random spring model

Using large-scale numerical simulations, we analyze the statistical properties of fracture in the two-dimensional random spring model and compare it with its scalar counterpart: the random fuse model. We first consider the process of crack localization measuring the evolution of damage as the external load is raised. We find that, as in the fuse model, damage is initially uniform and localizes at peak load. Scaling laws for the damage density, fracture strength, and avalanche distributions follow with slight variations the behavior observed in the random fuse model. We thus conclude that scalar models provide a faithful representation of the fracture properties of disordered systems.

Interacting damage models mapped onto Ising and percolation models

We introduce a class of damage models on regular lattices with isotropic interactions between the broken cells of the lattice. Quasi-static fiber bundles are an example. The interactions are assumed to be weak, in the sense that the stress perturbation from a broken cell is much smaller than the mean stress in the system. The system starts intact with a surface-energy threshold required to break any cell sampled from an uncorrelated quenched-disorder distribution. The evolution of this heterogeneous system is ruled by Griffith's principle which states that a cell breaks when the release in potential (elastic) energy in the system exceeds the surface-energy barrier necessary to break the cell. By direct integration over all possible realizations of the quenched disorder, we obtain the probability distribution of each damage configuration at any level of the imposed external deformation. We demonstrate an isomorphism between the distributions so obtained and standard generalized Ising models, in which the coupling constants and effective temperature in the Ising model are functions of the nature of the quenched-disorder distribution and the extent of accumulated damage. In particular, we show that damage models with global load sharing are isomorphic to standard percolation theory and that damage models with a local load sharing rule are isomorphic to the standard Ising model, and draw consequences thereof for the universality class and behavior of the autocorrelation length of the breakdown transitions corresponding to these models. We also treat damage models having more general power-law interactions, and classify the breakdown process as a function of the power-law interaction exponent. Last, we also show that the probability distribution over configurations is a maximum of Shannon's entropy under some specific constraints related to the energetic balance of the fracture process, which firmly relates this type of quenched-disorder based damage model to standard statistical mechanics.

Creep ruptures in heterogeneous materials

We present creep experiments on fiber composite materials with different controlled heterogeneity. All samples exhibit a power-law relaxation of the strain rate in the primary creep regime (Andrade's law) followed by a power-law acceleration up to rupture. We discover that the rupture time is proportional to the duration of the primary creep regime, showing the interplay between the two regimes and offering a method of rupture prediction. These experimental results are rationalized by a mean-field model of representative elements with nonlinear viscoelastic rheology and with a large heterogeneity of strengths.

Crack roughness and avalanche precursors in the random fuse model

We analyze the scaling of the crack roughness and of avalanche precursors in the two-dimensional random fuse model by numerical simulations, employing large system sizes and extensive sample averaging. We find that the crack roughness exhibits anomalous scaling, as recently observed in experiments. The roughness exponents (zeta, zeta(loc) ) and the global width distributions are found to be universal with respect to the lattice geometry. Failure is preceded by avalanche precursors whose distribution follows a power law up to a cutoff size. While the characteristic avalanche size scales as s(0) approximately L(D) , with a universal fractal dimension D , the distribution exponent tau differs slightly for triangular and diamond lattices and, in both cases, it is larger than the mean-field (fiber bundle) value tau=5/2 .

Universality class of the fiber bundle model on complex networks

We investigate the failure characteristics of complex networks within the framework of the fiber bundle model subject to the local load sharing rule in which the load of the broken fiber is transferred only to its neighbor fibers. Although the load sharing is strictly local, it is found that the critical behavior belongs to the universality class of global load sharing where the load is transferred equally to all fibers in the system. From the numerical simulations and the analytical approach applied to the microscopic behavior, it is revealed that the emergence of a single dominant hub cluster of broken fibers causes the global load sharing effect in the failure process.

Two-threshold model for scaling laws of noninteracting snow avalanches

The sizes of snow slab failure that trigger snow avalanches are power-law distributed. Such a power-law probability distribution function has also been proposed to characterize different landslide types. In order to understand this scaling for gravity-driven systems, we introduce a two-threshold 2D cellular automaton, in which failure occurs irreversibly. Taking snow slab avalanches as a model system, we find that the sizes of the largest avalanches just preceding the lattice system breakdown are power-law distributed. By tuning the maximum value of the ratio of the two failure thresholds our model reproduces the range of power-law exponents observed for land, rock, or snow avalanches. We suggest this control parameter represents the material cohesion anisotropy.

Three-dimensional mapping of dislocation avalanches: clustering and space/time coupling

There is growing evidence for the complex, intermittent, and heterogeneous character of plastic flow. Here we report a three-dimensional mapping of dislocation avalanches during creep deformation of an ice crystal, from a multiple-transducers acoustic emission analysis. Correlation analysis shows that dislocation avalanches are spatially clustered according to a fractal pattern and that the closer in time two avalanches are, the larger the probability is that they will be closer in space. Such a space/time coupling may contribute to the self-organization of the avalanches into a clustered pattern.

Acoustic emission from paper fracture

We report tensile failure experiments on paper sheets. The acoustic emission energy and the waiting times between acoustic events follow power-law distributions. This remains true while the strain rate is varied by more than 2 orders of magnitude. The energy statistics has the exponent beta approximately 1.25+/-0.10 and the waiting times the exponent tau approximately 1.0+/-0.1, in particular, for the energy roughly independent of the strain rate. These results do not compare well with fracture models, for (brittle) disordered media, which as such exhibit criticality. One reason may be residual stresses, neglected in most theories.

Fracture of disordered solids in compression as a critical phenomenon. I. Statistical mechanics formalism

This is the first of a series of three articles that treats fracture localization as a critical phenomenon. This first article establishes a statistical mechanics based on ensemble averages when fluctuations through time play no role in defining the ensemble. Ensembles are obtained by dividing a huge rock sample into many mesoscopic volumes. Because rocks are a disordered collection of grains in cohesive contact, we expect that once shear strain is applied and cracks begin to arrive in the system, the mesoscopic volumes will have a wide distribution of different crack states. These mesoscopic volumes are the members of our ensembles. We determine the probability of observing a mesoscopic volume to be in a given crack state by maximizing Shannon's measure of the emergent-crack disorder subject to constraints coming from the energy balance of brittle fracture. The laws of thermodynamics, the partition function, and the quantification of temperature are obtained for such cracking systems.

Heterogeneous interfacial failure between two elastic blocks

We investigate numerically the failure process when two elastic media, one hard and one soft that have been glued together thus forming a common interface, are pulled apart. We present three main results: (1) The area distribution of simultaneously failing glue (bursts) follows a power law consistent with the theoretically expected exponent 2.5, (2) the maximum load and displacement before catastrophic failure scale as L(2) and L(0), respectively, where L is the linear size of the system, and (3) the area distribution of failed glue regions (clusters) is a power law with exponent -1.6 when the system fails catastrophically.

Bursts in a fiber bundle model with continuous damage

We study the constitutive behavior, the damage process, and the properties of bursts in the continuous damage fiber bundle model introduced recently. Depending on its two parameters, the model provides various types of constitutive behaviors including macroscopic plasticity. Analytical results are obtained to characterize the damage process along the plastic plateau under strain controlled loading; furthermore, for stress controlled experiments we develop a simulation technique, and numerically explore the distribution of bursts of fiber breaks assuming an infinite range of interaction. Simulations revealed that under certain conditions power law distribution of bursts arises with an exponent significantly different from the mean field exponent 5/2. A phase diagram of the model characterizing the possible burst distributions is constructed.

Modeling relaxation and jamming in granular media

We introduce a stochastic microscopic model to investigate the jamming and reorganization of grains induced by an object moving through a granular medium. The model reproduces the experimentally observed periodic sawtooth fluctuations in the jamming force and predicts the period and the power spectrum in terms of the controllable physical parameters. It also predicts that the avalanche sizes, defined as the number of displaced grains during a single advance of the object, follow a power law P(s) approximately s(-tau), where the exponent is independent of the physical parameters.

Dynamic scaling for avalanches in disordered systems

Dynamic scaling for fracture or breakdown process in disordered systems is investigated in a two-dimensional random field Ising model (RFIM). We find two evolving stages in the avalanche process in the RFIM. At the short-time regime, a power-law growth of the avalanche size Deltas is observed; and at late times, the conventional nucleation and growth process is found. At the critical point of the RFIM, the avalanche size is found to obey the dynamic scaling law Delta(s) approximately equal t((d-beta/nu)/z). From this dynamic scaling relation, the critical strength of the random field D(c) and the critical exponents, beta, nu, and z, are determined. The observed dynamics is explained by a simple nucleation theory of first-order phase transformations.