Role of disorder in the size scaling of material strength

Tuning load redistribution and damage near heterogeneous interfaces

We investigate interface failure of model materials representing architected thin films in contact with heterogeneous substrates. We find that, while systems with statistically isotropic distributions of impurities derive their fracture strength from the ability to develop rough detachment fronts, materials with hierarchical microstructures confine failure near a prescribed surface, where crack growth is arrested and crack surface correlations are suppressed. We develop a theory of network Green's functions for the systems at hand, and we find that the ability of hierarchical microstructures to control failure mode and locations comes at no performance cost in terms of peak stress and specific work of failure and derives from the quenched local anistotropy of the elastic interaction kernel.

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.

Effect of the loading condition on the statistics of crackling noise accompanying the failure of porous rocks

We test the hypothesis that loading conditions affect the statistical features of crackling noise accompanying the failure of porous rocks by performing discrete element simulations of the tensile failure of model rocks and comparing the results to those of compressive simulations of the same samples. Cylindrical samples are constructed by sedimenting randomly sized spherical particles connected by beam elements representing the cementation of granules. Under a slowly increasing external tensile load, the cohesive contacts between particles break in bursts whose size fluctuates over a broad range. Close to failure breaking avalanches are found to localize on a highly stressed region where the catastrophic avalanche is triggered and the specimen breaks apart along a spanning crack. The fracture plane has a random position and orientation falling most likely close to the centre of the specimen perpendicular to the load direction. In spite of the strongly different strengths, degrees of 'brittleness' and spatial structure of damage of tensile and compressive failure of model rocks, our calculations revealed that the size, energy and duration of avalanches, and the waiting time between consecutive events all obey scale-free statistics with power law exponents which agree within their error bars in the two loading cases.

Toughness of Network Materials: Structural Parameters Controlling Damage Accumulation

Many materials have a network of fibers as their main structural component and are referred to as network materials. Their strength and toughness are important in both engineering and biology. In this work we consider stochastic model fiber networks without pre-existing cracks and study their rupture mechanism. These materials soften as the crosslinks or fibers fail and exhibit either brittle failure immediately after the peak stress, or a more gradual, ductile rupture in the post peak regime. We observe that ductile failure takes place at constant energy release rate defined in the absence of pre-existing cracks as the strain derivative of the specific energy released. The network parameters controlling the energy release rate are identified and discussed in relation to the Lake-Thomas theory which applies to crack growth situations. We also observe a ductile to brittle failure transition as the network becomes more affine and relate the embrittlement to the reduction of mechanical heterogeneity of the network. Further, we confirm previous reports that the network strength scales linearly with the bond strength and with the crosslink density. The present results extend the Lake-Thomas theory to networks without pre-existing cracks which fail by the gradual accumulation of distributed damage and contribute to the development of a physical picture of failure in stochastic network materials.

Rifts in rafts

A particle raft floating on an expanding liquid substrate provides a macroscopic analog for studying material failure. The time scales in this system allow both particle-relaxation dynamics and rift formation to be resolved. In our experiments, a raft, an aggregate of particles, is stretched uniaxially by the expansion of the air-liquid interface on which it floats. Its failure morphology changes continuously with pulling velocity. This can be understood as a competition between two velocity scales: the speed of re-aggregation, in which particles relax towards a low-energy configuration determined by viscous and capillary forces, and the difference of velocity between neighboring particles caused by the expanding liquid surface area. This competition selects the cluster length, i.e., the distance between adjacent rifts. A model based on this competition is consistent with the experimental failure patterns.

Edge betweenness centrality as a failure predictor in network models of structurally disordered materials

Network theoretical measures such as geodesic edge betweenness centrality (GEBC) have been proposed as failure predictors in network models of load-driven materials failure. Edge betweenness centrality ranks which links are significant, based on the fraction of shortest paths that pass through the links between network nodes. We study GEBC as a failure predictor for two-dimensional fuse network models of load transmission in structurally disordered materials. We analyze the evolution of edge betweenness centrality in the run-up to failure and the correlation between GEBC and failure propensity for both hierarchical and non-hierarchical networks exhibiting various degrees of disorder. We observe a non trivial relationship between GEBC and failure propensity, which suggests that the idea of GEBC as a useful failure predictor needs to be strongly qualified.

Role of spatial patterns in fracture of disordered multiphase materials

Multiphase materials, such as composite materials, exhibit multiple competing failure mechanisms during the growth of a macroscopic defect. For the simulation of the overall fracture process in such materials, we develop a two-phase spring network model that accounts for the architecture between the different components as well as the respective disorders in their failure characteristics. In the specific case of a plain weave architecture, we show that any offset between the layers reduces the delocalization of the stresses at the crack tip and thereby substantially lowers the strength and fracture toughness of the overall laminate. The avalanche statistics of the broken springs do not show a distinguishable dependence on the offsets between layers. The power-law exponents are found to be much smaller than that of disordered spring network models in the absence of a crack. A discussion is developed on the possibility of the avalanche statistics being those near breakdown.

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.

Forecasting failure locations in 2-dimensional disordered lattices

Forecasting fracture locations in a progressively failing disordered structure is of paramount importance when considering structural materials. We explore this issue for gradual deterioration via beam breakage of 2-dimensional (2D) disordered lattices, which we represent as networks, for various values of mean degree. We study experimental samples with geometric structures that we construct based on observed contact networks in 2D granular media. We calculate geodesic edge betweenness centrality, which helps quantify which edges are on many shortest paths in a network, to forecast the failure locations. We demonstrate for the tested samples that, for a variety of failure behaviors, failures occur predominantly at locations that have larger geodesic edge betweenness values than the mean one in the structure. Because only a small fraction of edges have values above the mean, this is a relevant diagnostic to assess failure locations. Our results demonstrate that one can consider only specific parts of a system as likely failure locations and that, with reasonable success, one can assess possible failure locations of a structure without needing to study its detailed energetic states.

Crack growth and energy dissipation in paper

Here, we follow the stable propagation of a roughening crack using simultaneously Digital Image Correlation and Infra-Red imaging. In a quasi-two-dimensional paper sample, the crack tip and ahead of that the fracture process zone follow the slowly, diffusively moving “hot spot” ahead of the tip. This also holds when the crack starts to roughen during propagation. The well-established intermittency of the crack advancement and the roughening of the crack in paper are thus subject to the dissipation and decohesion in the hot spot zone. They are therefore not only a result of the depinning of the crack in a heterogeneous material.

Size scaling of failure strength with fat-tailed disorder in a fiber bundle model

We investigate the size scaling of the macroscopic fracture strength of heterogeneous materials when microscopic disorder is controlled by fat-tailed distributions. We consider a fiber bundle model where the strength of single fibers is described by a power law distribution over a finite range. Tuning the amount of disorder by varying the power law exponent and the upper cutoff of fibers' strength, in the limit of equal load sharing an astonishing size effect is revealed: For small system sizes the bundle strength increases with the number of fibers, and the usual decreasing size effect of heterogeneous materials is restored only beyond a characteristic size. We show analytically that the extreme order statistics of fibers' strength is responsible for this peculiar behavior. Analyzing the results of computer simulations we deduce a scaling form which describes the dependence of the macroscopic strength of fiber bundles on the parameters of microscopic disorder over the entire range of system sizes.

Rupture cascades in a discrete element model of a porous sedimentary rock

We investigate the scaling properties of the sources of crackling noise in a fully dynamic numerical model of sedimentary rocks subject to uniaxial compression. The model is initiated by filling a cylindrical container with randomly sized spherical particles that are then connected by breakable beams. Loading at a constant strain rate the cohesive elements fail, and the resulting stress transfer produces sudden bursts of correlated failures, directly analogous to the sources of acoustic emissions in real experiments. The source size, energy, and duration can all be quantified for an individual event, and the population can be analyzed for its scaling properties, including the distribution of waiting times between consecutive events. Despite the nonstationary loading, the results are all characterized by power-law distributions over a broad range of scales in agreement with experiments. As failure is approached, temporal correlation of events emerges accompanied by spatial clustering.

Approach to failure in porous granular materials under compression

We investigate the approach to catastrophic failure in a model porous granular material undergoing uniaxial compression. A discrete element computational model is used to simulate both the microstructure of the material and the complex dynamics and feedbacks involved in local fracturing and the production of crackling noise. Under strain-controlled loading, microcracks initially nucleate in an uncorrelated way all over the sample. As loading proceeds the damage localizes into a narrow damage band inclined at 30°-45° to the load direction. Inside the damage band the material is crushed into a poorly sorted mixture of mainly fine powder hosting some larger fragments. The mass probability density distribution of particles in the damage zone is a power law of exponent 2.1, similar to a value of 1.87 inferred from observations of the length distribution of wear products (gouge) in natural and laboratory faults. Dynamic bursts of radiated energy, analogous to acoustic emissions observed in laboratory experiments on porous sedimentary rocks, are identified as correlated trails or cascades of local ruptures that emerge from the stress redistribution process. As the system approaches macroscopic failure consecutive bursts become progressively more correlated. Their size distribution is also a power law, with an equivalent Gutenberg-Richter b value of 1.22 averaged over the whole test, ranging from 3 to 0.5 at the time of failure, all similar to those observed in laboratory tests on granular sandstone samples. The formation of the damage band itself is marked by a decrease in the average distance between consecutive bursts and an emergent power-law correlation integral of event locations with a correlation dimension of 2.55, also similar to those observed in the laboratory (between 2.75 and 2.25).

Emergent patterns of localized damage as a precursor to catastrophic failure in a random fuse network

We study the failure of disordered materials by numerical simulations of the random fuse model. We identify emergent patterns of localized damage prior to catastrophic failure by statistically averaging the density of damage around the eventual failure nucleation point. The resulting pattern depends on fracture density and obeys the same scaling relations as would be expected for the stress field generated by a critical crack nucleating in a finite, disorder-free effective medium of varying size. The growth of this critical crack absorbs preexisting clusters according to a well-defined scaling relation. Unfortunately, in single model runs such precursory signals are not obvious. Our results imply that reliable and accurate prediction of failure in time-independent, microscopically brittle random materials in a real case is inherently problematic, and degrades with system size.

Experimental study of the effect of disorder on subcritical crack growth dynamics

The growth dynamics of a single crack in a heterogeneous material under subcritical loading is an intermittent process, and many features of this dynamics have been shown to agree with simple models of thermally activated rupture. In order to better understand the role of material heterogeneities in this process, we study the subcritical propagation of a crack in a sheet of paper in the presence of a distribution of small defects such as holes. The experimental data obtained for two different distributions of holes are discussed in the light of models that predict the slowing down of crack growth when the disorder in the material is increased; however, in contradiction with these theoretical predictions, the experiments result in longer lasting cracks in a more ordered scenario. We argue that this effect is specific to subcritical crack dynamics and that the weakest zones between holes at close distance to each other are responsible for both the acceleration of the crack dynamics and the slightly different roughness of the crack path.

Microstructure of damage in thermally activated fracture of Lennard-Jones systems

We investigate the effect of thermal fluctuations on the critical stress and the microstructure of damage preceding macroscopic fracture of Lennard-Jones solids under a constant external load. Based on molecular dynamics simulations of notched specimens at finite temperatures, we show that the crystalline structure gets distorted ahead of the crack in the secondary creep regime. The damage profile characterizing the spatial distribution of lattice distortions is well described by an exponential form. The characteristic length of the exponential form provides the scale of damage, which is found to be an increasing function of the temperature: At low temperatures, damage is strongly localized to the crack tip, while at high temperatures, damage extends to a broader range, leading to more efficient relaxation of overloads. As a consequence, the stress intensity factor decreases with increasing temperature. The final macroscopic failure of the system occurs suddenly and is initiated by the creation of vacancies and voids. The creep strength exhibits inverse square root scaling with the notch size corrected by the extension of the process zone.

Crackling noise in three-point bending of heterogeneous materials

We study the crackling noise emerging during single crack propagation in a specimen under three-point bending conditions. Computer simulations are carried out in the framework of a discrete element model where the specimen is discretized in terms of convex polygons and cohesive elements are represented by beams. Computer simulations revealed that fracture proceeds in bursts whose size and waiting-time distributions have a power-law functional form with an exponential cutoff. Controlling the degree of brittleness of the sample by the amount of disorder, we obtain a scaling form for the characteristic quantities of crackling noise of quasibrittle materials. Analyzing the spatial structure of damage we show that ahead of the crack tip a process zone is formed as a random sequence of broken and intact mesoscopic elements. We characterize the statistics of the shrinking and expanding steps of the process zone and determine the damage profile in the vicinity of the crack tip.

Stress concentration and size effect in fracture of notched heterogeneous material

We study theoretically and experimentally the effect of long-range correlations in the material microstructure on the stress concentration in the vicinity of the notch tip. We find that while in a fractal continuum the notch-tip displacements obey the classic asymptotic for a linear elastic continuum, the power-law decay of notch-tip stresses is controlled by the long-range density correlations. The corresponding notch-size effect on fracture strength is in good agreement with the experimental tests performed on notched sheets of different kinds of paper. In particular, we find that there is no stress concentration if the fractal dimension of the fiber network is D≤d-0.5, where d is the topological dimension of the paper sheet.

Fracture toughness and maximum stress in a disordered lattice system

We model the disorder-induced increase in fracture strength by applying a two-dimensional elastic network with an initial crack. The model has all the nearest neighbors, except those on the initial crack, connected by springs, which are randomly chosen as either hard or soft. All the springs are cut at the same force threshold. Our calculations show that the maximum stress in the disordered system is increased if the initial crack length is sufficiently large, and we discuss the relationship between the crack length and the force distribution or the work done on the system. We also derive the enhancement of the stress intensity factor and fracture toughness by analyzing the generation of the fracture process zone.

Time-dependent fiber bundles with local load sharing. II. General Weibull fibers

Fiber bundle models (FBMs) are useful tools in understanding failure processes in a variety of material systems. While the fibers and load sharing assumptions are easily described, FBM analysis is typically difficult. Monte Carlo methods are also hampered by the severe computational demands of large bundle sizes, which overwhelm just as behavior relevant to real materials starts to emerge. For large size scales, interest continues in idealized FBMs that assume either equal load sharing (ELS) or local load sharing (LLS) among fibers, rules that reflect features of real load redistribution in elastic lattices. The present work focuses on a one-dimensional bundle of N fibers under LLS where life consumption in a fiber follows a power law in its load, with exponent rho , and integrated over time. This life consumption function is further embodied in a functional form resulting in a Weibull distribution for lifetime under constant fiber stress and with Weibull exponent, beta. Thus the failure rate of a fiber depends on its past load history, except for beta=1 . We develop asymptotic results validated by Monte Carlo simulation using a computational algorithm developed in our previous work [Phys. Rev. E 63, 021507 (2001)] that greatly increases the size, N , of treatable bundles (e.g., 10(6) fibers in 10(3) realizations). In particular, our algorithm is O(N ln N) in contrast with former algorithms which were O(N2) making this investigation possible. Regimes are found for (beta,rho) pairs that yield contrasting behavior for large N. For rho>1 and large N, brittle weakest volume behavior emerges in terms of characteristic elements (groupings of fibers) derived from critical cluster formation, and the lifetime eventually goes to zero as N-->infinity , unlike ELS, which yields a finite limiting mean. For 1/2<or=rho<or=1 , however, LLS has remarkably similar behavior to ELS (appearing to be virtually identical for rho=1 ) with an asymptotic Gaussian lifetime distribution and a finite limiting mean for large N. The coefficient of variation follows a power law in increasing N but, except for rho=1, the value of the negative exponent is clearly less than 1/2 unlike in ELS bundles where the exponent remains 1/2 for 1/2<rho<or=1. For sufficiently small values 0<rho1, a transition occurs, depending on beta , whereby LLS bundle lifetimes become dominated by a few long-lived fibers. Thus the bundle lifetime appears to approximately follow an extreme-value distribution for the longest lived of a parallel group of independent elements, which applies exactly to rho=0. The lower the value of beta , the higher the transition value of rho , below which such extreme-value behavior occurs. No evidence was found for limiting Gaussian behavior for rho>1 but with 0<beta(rho+1)<1, as might be conjectured from quasistatic bundle models where beta(rho+1) mimics the Weibull exponent for fiber strength.

Size scaling and bursting activity in thermally activated breakdown of fiber bundles

We study subcritical fracture driven by thermally activated damage accumulation in the framework of fiber bundle models. We show that in the presence of stress inhomogeneities, thermally activated cracking results in an anomalous size effect; i.e., the average lifetime t{f} decreases as a power law of the system size t{f} approximately L{-z}, where the exponent z depends on the external load sigma and on the temperature T in the form z approximately f(sigma/T{3/2}). We propose a modified form of the Arrhenius law which provides a comprehensive description of thermally activated breakdown. Thermal fluctuations trigger bursts of breakings which have a power law size distribution.