Modes of failure in disordered solids

Scaling laws of failure dynamics on complex networks

The topology of the network of load transmitting connections plays an essential role in the cascading failure dynamics of complex systems driven by the redistribution of load after local breakdown events. In particular, as the network structure is gradually tuned from regular to completely random a transition occurs from the localized to mean field behavior of failure spreading. Based on finite size scaling in the fiber bundle model of failure phenomena, here we demonstrate that outside the localized regime, the load bearing capacity and damage tolerance on the macro-scale, and the statistics of clusters of failed nodes on the micro-scale obey scaling laws with exponents which depend on the topology of the load transmission network and on the degree of disorder of the strength of nodes. Most notably, we show that the spatial structure of damage governs the emergence of the localized to mean field transition: as the network gets gradually randomized failed clusters formed on locally regular patches merge through long range links generating a percolation like transition which reduces the load concentration on the network. The results may help to design network structures with an improved robustness against cascading failure.

Inequality of avalanche sizes in models of fracture

Prediction of an imminent catastrophic event in a driven disordered system is of paramount importance-from the laboratory scale controlled fracture experiment to the largest scale of mechanical failure, i.e., earthquakes. It has long been conjectured that the statistical regularities in the energy emission time series mirror the "health" of such driven systems and hence have the potential for forecasting imminent catastrophe. Among other statistical regularities, a measure of how unequal avalanche sizes are is potentially a crucial indicator of imminent failure. The inequalities of avalanche sizes are quantified using inequality indices traditionally used in socioeconomic systems: the Gini index g, the Hirsch index h, and the Kolkata index k. It is shown analytically (for the mean-field case) and numerically (for the non-mean-field case) with models of quasi-brittle materials that the indices show universal behavior near the breaking points in such models and hence could serve as indicators of imminent breakdown of stressed disordered systems.

Optimal Stopping Rules for Preventing Overloading of Multicomponent Systems

When random-strength components work as an interconnected parallel system, then its carrying capacity is random as well. In a case where such a multicomponent system is a subject of the stepwise-growing workload, some of its components fail and their loads are taken over by the ones that are intact. When the loading process is continued, the additional loads trigger consecutive failures that degrade the system, eventually leading to a complete failure. If the goal of the system is to carry as much load as possible, then the loading process should be continued, but no longer than until the loading capacity of the whole system is reached. On the other hand, with every additional load step, a failure of the system becomes more probable, as the carrying capacity is random and known solely through its probability distribution. In such cases, the decision on when to cease the loading process is not obvious. We introduce and analyse a minimal model of failure spreading in an array of progressively loaded pillars controlled by a decision-maker who stops the process when a required load is attained. We show how to construct an optimal stopping rule. Under some additional assumptions regarding the adopted loss function, it is argued that the optimal stopping rule is of the threshold type and it significantly depends on the shape of the load-step probability distribution.

Prediction of imminent failure using supervised learning in a fiber bundle model

Prediction of a breakdown in disordered solids under external loading is a question of paramount importance. Here we use a fiber bundle model for disordered solids and record the time series of the avalanche sizes and energy bursts. The time series contain statistical regularities that not only signify universality in the critical behavior of the process of fracture, but also reflect signals of proximity to a catastrophic failure. A systematic analysis of these series using supervised machine learning can predict the time to failure. Different features of the time series become important in different variants of training samples. We explain the reasons for such a switch over of importance among different features. We show that inequality measures for avalanche time series play a crucial role in imminent failure predictions, especially for imperfect training sets, i.e., when simulation parameters of training samples differ considerably from those of the testing samples. We also show the variation of predictability of the system as the interaction range and strengths of disorders are varied in the samples, varying the failure mode from brittle to quasibrittle (with interaction range) and from nucleation to percolation (with disorder strength). The effectiveness of the supervised learning is best when the samples just enter the quasibrittle mode of failure showing scale-free avalanche size distributions.

Modeling crack propagation in heterogeneous materials: Griffith's law, intrinsic crack resistance, and avalanches

Various kinds of heterogeneity in solids, including atomistic discreteness, affect the fracture strength as well as the failure dynamics remarkably. Here we study the effects of an initial crack in a discrete model for fracture in heterogeneous materials, known as the fiber bundle model. We find three distinct regimes for fracture dynamics depending on the initial crack size. If the initial crack is smaller than a certain value, it does not affect the rupture dynamics and the critical stress, while for a larger initial crack, the growth of the crack leads to breakdown of the entire system, and the critical stress depends on the crack size in a power-law manner with a nontrivial exponent. The exponent, as well as the limiting crack size, depend on the strength of heterogeneity and the range of stress relaxation in the system.

Stretchy and disordered: Toward understanding fracture in soft network materials via mesoscopic computer simulations

Soft network materials exist in numerous forms ranging from polymer networks, such as elastomers, to fiber networks, such as collagen. In addition, in colloidal gels, an underlying network structure can be identified, and several metamaterials and textiles can be considered network materials as well. Many of these materials share a highly disordered microstructure and can undergo large deformations before damage becomes visible at the macroscopic level. Despite their widespread presence, we still lack a clear picture of how the network structure controls the fracture processes of these soft materials. In this Perspective, we will focus on progress and open questions concerning fracture at the mesoscopic scale, in which the network architecture is clearly resolved, but neither the material-specific atomistic features nor the macroscopic sample geometries are considered. We will describe concepts regarding the network elastic response that have been established in recent years and turn out to be pre-requisites to understand the fracture response. We will mostly consider simulation studies, where the influence of specific network features on the material mechanics can be cleanly assessed. Rather than focusing on specific systems, we will discuss future challenges that should be addressed to gain new fundamental insights that would be relevant across several examples of soft network materials.

Survivability of Suddenly Loaded Arrays of Micropillars

When a multicomponent system is suddenly loaded, its capability of bearing the load depends not only on the strength of components but also on how a load released by a failed component is distributed among the remaining intact ones. Specifically, we consider an array of pillars which are located on a flat substrate and subjected to an impulsive and compressive load. Immediately after the loading, the pillars whose strengths are below the load magnitude crash. Then, loads released by these crashed pillars are transferred to and assimilated by the intact ones according to a load-sharing rule which reflects the mechanical properties of the pillars and the substrate. A sequence of bursts involving crashes and load transfers either destroys all the pillars or drives the array to a stable configuration when a smaller number of pillars sustain the applied load. By employing a fibre bundle model framework, we numerically study how the array integrity depends on sudden loading amplitudes, randomly distributed pillar strength thresholds and varying ranges of load transfer. Based on the simulation, we estimate the survivability of arrays of pillars defined as the probability of sustaining the applied load despite numerous damaged pillars. It is found that the resulting survival functions are accurately fitted by the family of complementary cumulative skew-normal distributions.

Impact-induced transition from damage to perforation

We investigate the impact-induced damage and fracture of a bar-shaped specimen of heterogeneous materials focusing on how the system approaches perforation as the impact energy is gradually increased. A simple model is constructed which represents the bar as two rigid blocks coupled by a breakable interface with disordered local strength. The bar is clamped at the two ends, and the fracture process is initiated by an impactor hitting the bar in the middle. Our calculations revealed that depending on the imparted energy, the system has two phases: at low impact energies the bar suffers damage but keeps its integrity, while at sufficiently high energies, complete perforation occurs. We demonstrate that the transition from damage to perforation occurs analogous to continuous phase transitions. Approaching the critical point from below, the intact fraction of the interface goes to zero, while the deformation rate of the bar diverges according to power laws as function of the distance from the critical energy. As the degree of disorder increases, farther from the transition point the critical exponents agree with their zero disorder counterparts; however, close to the critical point a crossover occurs to a higher exponent.

Tough-brittle transition in the planar fracture of unidirectional fiber composites

The transverse fracture of model unidirectional composite specimen, comprising up to 2^{20} fibers with random strengths, is studied using Monte Carlo simulations. The load sharing from broken to intact fibers is assumed to obey power-law scaling ∼r^{-γ} with distance r from the fiber break. Fiber breaks are assumed to interact in order to remain traction free. The pattern of fiber breaks that propagate catastrophically is interpreted through cluster analysis. The empirical strength distributions obtained from the simulations are interpreted using two probabilistic models of brittle fracture available in the literature. These point to a transition from the brittle to the tough fracture mode as γ↓2. The transitional γ is approximately equal to that reported in the literature for noninteracting fiber breaks.

Rigidity-Controlled Crossover: From Spinodal to Critical Failure

Failure in disordered solids is accompanied by intermittent fluctuations extending over a broad range of scales. The implied scaling has been previously associated with either spinodal or critical points. We use an analytically transparent mean-field model to show that both analogies are relevant near the brittle-to-ductile transition. Our study indicates that in addition to the strength of quenched disorder, an appropriately chosen global measure of rigidity (connectivity) can be also used to tune the system to criticality. By interpreting rigidity as a timelike variable we reveal an intriguing parallel between earthquake-type critical failure and Burgers turbulence.

Athermal Fracture of Elastic Networks: How Rigidity Challenges the Unavoidable Size-Induced Brittleness

By performing extensive simulations with unprecedentedly large system sizes, we unveil how rigidity influences the fracture of disordered materials. We observe the largest damage in networks with connectivity close to the isostatic point and when the rupture thresholds are small. However, irrespective of network and spring properties, a more brittle fracture is observed upon increasing system size. Differently from most of the fracture descriptors, the maximum stress drop, a proxy for brittleness, displays a universal nonmonotonic dependence on system size. Based on this uncommon trend it is possible to identify the characteristic system size L^{*} at which brittleness kicks in. The more the disorder in network connectivity or in spring thresholds, the larger L^{*}. Finally, we speculate how this size-induced brittleness is influenced by thermal fluctuations.

System-size-dependent avalanche statistics in the limit of high disorder

We investigate the effect of the amount of disorder on the statistics of breaking bursts during the quasistatic fracture of heterogeneous materials. We consider a fiber bundle model where the strength of single fibers is sampled from a power-law distribution over a finite range, so that the amount of materials' disorder can be controlled by varying the power-law exponent and the upper cutoff of fibers' strength. Analytical calculations and computer simulations, performed in the limit of equal load sharing, revealed that depending on the disorder parameters the mechanical response of the bundle is either perfectly brittle where the first fiber breaking triggers a catastrophic avalanche, or it is quasibrittle where macroscopic failure is preceded by a sequence of bursts. In the quasibrittle phase, the statistics of avalanche sizes is found to show a high degree of complexity. In particular, we demonstrate that the functional form of the size distribution of bursts depends on the system size: for large upper cutoffs of fibers' strength, in small systems the sequence of bursts has a high degree of stationarity characterized by a power-law size distribution with a universal exponent. However, for sufficiently large bundles the breaking process accelerates towards the critical point of failure, which gives rise to a crossover between two power laws. The transition between the two regimes occurs at a characteristic system size which depends on the disorder parameters.

Brittle to quasibrittle transition in a compound fiber bundle

The brittle to quasibrittle transition has been studied for a compound of two different kinds of fibrous materials, having distinct difference in their breaking strengths under the framework of the fiber bundle model. A random fiber bundle model has been devised with a bimodal distribution of the breaking strengths of the individual fibers. The bimodal distribution is assumed to consist of two symmetrically placed rectangular probability distributions of strengths p and 1-p, each of width d, and separated by a gap 2s. Different properties of the transition have been studied varying these three parameters and using the well-known equal load-sharing dynamics. Our study exhibits a brittle to quasibrittle transition at the critical width d_{c}(s,p)=p(1/2-s)/(1+p) confirmed by our numerical results.

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'.

Stability in a fiber bundle model: Existence of strong links and the effect of disorder

The present paper deals with a fiber bundle model which consists of a fraction α of infinitely strong fibers. The inclusion of such an unbreakable fraction has been proven to affect the failure process in early studies, especially around a critical value α_{c}. The present work has a twofold purpose: (i) a study of failure abruptness, mainly the brittle to quasibrittle transition point with varying α and (ii) variation of α_{c} as we change the strength of disorder introduced in the model. The brittle to quasibrittle transition is confirmed from the failure abruptness. On the other hand, the α_{c} is obtained from the knowledge of failure abruptness as well as the statistics of avalanches. It is observed that the brittle to quasibrittle transition point scales to lower values, suggesting more quasi-brittle-like continuous failure when α is increased. At the same time, the bundle becomes stronger as there are larger numbers of strong links to support the external stress. High α in a highly disordered bundle leads to an ideal situation where the bundle strength, as well as the predictability in failure process is very high. Also, the critical fraction α_{c}, required to make the model deviate from the conventional results, increases with decreasing strength of disorder. The analytical expression for α_{c} shows good agreement with the numerical results. Finally, the findings in the paper are compared with previous results and real-life applications of composite materials.