Local waiting time fluctuations along a randomly pinned crack front

Fracture process of composite materials in a spring network model

We analyze a two-dimensional spring network model comprising breakable and unbreakable springs. Computer simulations showed this system to exhibit intermittent stress drops in a larger strain regime, and these stress drops resulted in ductilelike behavior. The scaling analysis reveals that the avalanche size distribution demonstrates a cutoff, depending on its internal structure. This study also investigates the relationship between cluster growth and stress drop, and we show that the amount of stress drop increases in terms of power law, corresponding to crack growth. The crack length distribution also demonstrates a cutoff depending on its internal structure. The results show that both the cluster growth-stress drop relationship and the crack size distribution are scaled by the quantity related to the internal structure, and the relevance of the exponent that scales the cluster growth-stress drop relationship to the exponent that scales crack size distribution is verified.

Loss of memory of an elastic line on its way to limit cycles

Oscillatory-driven amorphous materials forget their initial configuration and converge to limit cycles. Here we investigate this memory loss under a nonquasistatic drive in a minimal model system, with quenched disorder and memory encoded in a spatial pattern, where oscillating protocols are formally replaced by a positive-velocity drive. We consider an elastic line driven athermally in a quenched disorder with biperiodic boundary conditions and tunable system size, thus controlling the area swept by the line per cycle as would the oscillation amplitude. The convergence to disorder-dependent limit cycle is strongly coupled to the nature of its velocity dynamics depending on system size. Based on the corresponding phase diagram, we propose a generic scenario for memory formation in disordered systems under finite driving rate.

Scale-free bursting activity in shrinkage induced cracking

Based on computer simulations of a realistic discrete element model we demonstrate that shrinkage induced cracking of thin layers of heterogeneous materials, generating spectacular crack patterns, proceeds in bursts. These crackling pulses are characterized by scale free distributions of size and duration, however, with non-universal exponents depending on the system size and shrinking rate. On the contrary, local avalanches composed of micro-cracking events with temporal and spatial correlation are found to obey a universal power law statistics. Most notably, we demonstrate that the observed non-universality of the integrated signal is the consequence of the temporal superposition of the underlying local avalanches, which pop up in an uncorrelated way in homogeneous systems. Our results provide an explanation of recent acoustic emission measurements on drying induced shrinkage cracking and may have relevance for the acoustic monitoring of the electro-mechanical degradation of battery electrodes.

Mechanical excitation and marginal triggering during avalanches in sheared amorphous solids

We study plastic strain during individual avalanches in overdamped particle-scale molecular dynamics (MD) and mesoscale elastoplastic models (EPM) for amorphous solids sheared in the athermal quasistatic limit. We show that the spatial correlations in plastic activity exhibit a short length scale that grows as t^{3/4} in MD and ballistically in EPM, which is generated by mechanical excitation of nearby sites not necessarily close to their stability thresholds, and a longer lengthscale that grows diffusively for both models and is associated with remote marginally stable sites. These similarities in spatial correlations explain why simple EPMs accurately capture the size distribution of avalanches observed in MD, though the temporal profiles and dynamical critical exponents are quite different.

Clusters in an Epidemic Model with Long-Range Dispersal

In the presence of long-range dispersal, epidemics spread in spatially disconnected regions known as clusters. Here, we characterize exactly their statistical properties in a solvable model, in both the supercritical (outbreak) and critical regimes. We identify two diverging length scales, corresponding to the bulk and the outskirt of the epidemic. We reveal a nontrivial critical exponent that governs the cluster number and the distribution of their sizes and of the distances between them. We also discuss applications to depinning avalanches with long-range elasticity.

Theory and experiments for disordered elastic manifolds, depinning, avalanches, and sandpiles

Domain walls in magnets, vortex lattices in superconductors, contact lines at depinning, and many other systems can be modeled as an elastic system subject to quenched disorder. The ensuing field theory possesses a well-controlled perturbative expansion around its upper critical dimension. Contrary to standard field theory, the renormalization group (RG) flow involves a function, the disorder correlator Δ(w), and is therefore termed the functional RG. Δ(w) is a physical observable, the auto-correlation function of the center of mass of the elastic manifold. In this review, we give a pedagogical introduction into its phenomenology and techniques. This allows us to treat both equilibrium (statics), and depinning (dynamics). Building on these techniques, avalanche observables are accessible: distributions of size, duration, and velocity, as well as the spatial and temporal shape. Various equivalences between disordered elastic manifolds, and sandpile models exist: an elastic string driven at a point and the Oslo model; disordered elastic manifolds and Manna sandpiles; charge density waves and Abelian sandpiles or loop-erased random walks. Each of the mappings between these systems requires specific techniques, which we develop, including modeling of discrete stochastic systems via coarse-grained stochastic equations of motion, super-symmetry techniques, and cellular automata. Stronger than quadratic nearest-neighbor interactions lead to directed percolation, and non-linear surface growth with additional Kardar-Parisi-Zhang (KPZ) terms. On the other hand, KPZ without disorder can be mapped back to disordered elastic manifolds, either on the directed polymer for its steady state, or a single particle for its decay. Other topics covered are the relation between functional RG and replica symmetry breaking, and random-field magnets. Emphasis is given to numerical and experimental tests of the theory.

Preventing a global transition to thermoacoustic instability by targeting local dynamics

The burning of fossil fuels to generate power produces harmful emissions. Lowering such emissions in gas turbine engines is possible by operating them at fuel-lean conditions. However, such strategies often fail because, under fuel-lean conditions, the combustors are prone to catastrophic high-amplitude oscillations known as thermoacoustic instability. We reveal that, as an operating parameter is varied in time, the transition to thermoacoustic instability is initiated at specific spatial regions before it is observed in larger regions of the combustor. We use two indicators to discover such inceptive regions: the growth of variance of fluctuations in spatially resolved heat release rate and its spatiotemporal evolution. In this study, we report experimental evidence of suppression of the global transition to thermoacoustic instability through targeted modification of local dynamics at the inceptive regions. We strategically arrange slots on the flame anchor, which, in turn, reduce the local heat release rate fluctuations at the inceptive regions and thus suppress the global transition to thermoacoustic instability. Our results open new perspectives for combustors that are more environmental-friendly.

Energy exponents of avalanches and Hausdorff dimensions of collapse patterns

A simple numerical model to simulate athermal avalanches is presented. The model is inspired by the "porous collapse" process where the compression of porous materials generates collapse cascades, leading to power law distributed avalanches. The energy (E), amplitude (A_{max}), and size (S) exponents are derived by computer simulation in two approximations. Time-dependent "jerk" spectra are calculated in a single avalanche model where each avalanche is simulated separately from other avalanches. The average avalanche profile is parabolic, the scaling between energy and amplitude follows E∼A_{max}^{2}, and the energy exponent is ε = 1.33. Adding a general noise term in a continuous event model generates infinite avalanche sequences which allow the evaluation of waiting time distributions and pattern formation. We find the validity of the Omori law and the same exponents as in the single avalanche model. We then add spatial correlations by stipulating the ratio G/N between growth processes G (linked to a previous event location) and nucleation processes N (with new, randomly chosen nucleation sites). We found, in good approximation, a power law correlation between the energy exponent ε and the Hausdorff dimension H_{D} of the resulting collapse pattern H_{D}-1∼ɛ^{-3}. The evolving patterns depend strongly on G/N with the distribution of collapse sites equally power law distributed. Its exponent ɛ_{topo} would be linked to the dynamical exponent ε if each collapse carried an energy equivalent to the size of the collapse. A complex correlation between ɛ,ɛ_{topo}, and H_{D} emerges, depending strongly on the relative occupancy of the collapse sites in the simulation box.

Thermally activated intermittent dynamics of creeping crack fronts along disordered interfaces

We present a subcritical fracture growth model, coupled with the elastic redistribution of the acting mechanical stress along rugous rupture fronts. We show the ability of this model to quantitatively reproduce the intermittent dynamics of cracks propagating along weak disordered interfaces. To this end, we assume that the fracture energy of such interfaces (in the sense of a critical energy release rate) follows a spatially correlated normal distribution. We compare various statistical features from the obtained fracture dynamics to that from cracks propagating in sintered polymethylmethacrylate (PMMA) interfaces. In previous works, it has been demonstrated that such an approach could reproduce the mean advance of fractures and their local front velocity distribution. Here, we go further by showing that the proposed model also quantitatively accounts for the complex self-affine scaling morphology of crack fronts and their temporal evolution, for the spatial and temporal correlations of the local velocity fields and for the avalanches size distribution of the intermittent growth dynamics. We thus provide new evidence that an Arrhenius-like subcritical growth is particularly suitable for the description of creeping cracks.

Controlling crackling dynamics by triggering low-intensity avalanches

We examine the effect of small, spatially localized excitations applied periodically in different manners, on the crackling dynamics of a brittle crack driven slowly in a heterogeneous solid. When properly adjusted, these excitations are observed to radically modify avalanche statistics and considerably limit the magnitude of the largest events. Surprisingly, this does not require information on the front loading state at the time of excitation; applying it either at a random location or at the most loaded point gives the same results. Subsequently, we unravel how the excitation amplitude, spatial extent, and frequency govern the effect. We find that the excitation efficiency is ruled by a single reduced parameter, namely the injected power per unit front length; the suppression of extreme avalanches is maximum at a well-defined optimal value of this control parameter. analysis opens another way to control the largest events in crackling dynamics. Beyond fracture problems, it may be relevant for crackling systems described by models of the same universality class, such as the wetting of heterogeneous substrates or magnetic walls in amorphous magnets.

Criticality in sheared, disordered solids. I. Rate effects in stress and diffusion

Rate effects in sheared disordered solids are studied using molecular dynamics simulations of binary Lennard-Jones glasses in two and three dimensions. In the quasistatic (QS) regime, systems exhibit critical behavior: the magnitudes of avalanches are power-law distributed with a maximum cutoff that diverges with increasing system size L. With increasing rate, systems move away from the critical yielding point and the average flow stress rises as a power of the strain rate with exponent 1/β, the Herschel-Bulkley exponent. Finite-size scaling collapses of the stress are used to measure β as well as the exponent ν which characterizes the divergence of the correlation length. The stress and kinetic energy per particle experience fluctuations with strain that scale as L^{-d/2}. As the largest avalanche in a system scales as L^{α}, this implies α<d/2. The diffusion rate of particles diverges as a power of decreasing rate before saturating in the QS regime. A scaling theory for the diffusion is derived using the QS avalanche rate distribution and generalized to the finite strain rate regime. This theory is used to collapse curves for different system sizes and confirm β/ν.

Criticality in sheared, disordered solids. II. Correlations in avalanche dynamics

Disordered solids respond to quasistatic shear with intermittent avalanches of plastic activity, an example of the crackling noise observed in many nonequilibrium critical systems. The temporal power spectrum of activity within disordered solids consists of three distinct domains: a novel power-law rise with frequency at low frequencies indicating anticorrelation, white-noise at intermediate frequencies, and a power-law decay at high frequencies. As the strain rate increases, the white-noise regime shrinks and ultimately disappears as the finite strain rate restricts the maximum size of an avalanche. A new strain-rate- and system-size-dependent theory is derived for power spectra in both the quasistatic and finite-strain-rate regimes. This theory is validated using data from overdamped two- and three-dimensional molecular dynamics simulations. We identify important exponents in the yielding transition including the dynamic exponent z which relates the size of an avalanche to its duration, the fractal dimension of avalanches, and the exponent characterizing the divergence in correlations with strain rate. Results are related to temporal correlations within a single avalanche and between multiple avalanches.

Universality class for loopless invasion percolation models and a percolation avalanche burst model for hydraulic fracturing

Invasion percolation is a model that was originally proposed to describe growing networks of fractures. Here we describe a loopless algorithm on random lattices, coupled with an avalanche-based model for bursts. The model reproduces the characteristic b-value seismicity and spatial distribution of bursts consistent with earthquakes resulting from hydraulic fracturing ("fracking"). We test models for both site invasion percolation and bond invasion percolation. These have differences on the scale of site and bond lengths l. But since the networks are characterized by their large-scale behavior, l≪L, we find small differences between scaling exponents. Though data may not differentiate between models, our results suggest that both models belong to different universality classes.

Dynamic crack growth along heterogeneous planar interfaces: Interaction with unidimensional strips

We examine theoretically and numerically fast propagation of a tensile crack along unidimensional strips with periodically evolving toughness. In such dynamic fracture regimes, crack front waves form and transport front disturbances along the crack edge at speed less than the Rayleigh wave speed and depending on the crack speed. In this configuration, standing front waves dictate the spatiotemporal evolution of the local crack front speed, which takes a specific scaling form. Analytical examination of both the short-time and long-time limits of the problem reveals the parameter dependency with strip wavelength, toughness contrast and overall fracture speed. Implications and generalization to unidimensional strips of arbitrary shape are lastly discussed.

Spatial Clustering of Depinning Avalanches in Presence of Long-Range Interactions

Disordered elastic interfaces display avalanche dynamics at the depinning transition. For short-range interactions, avalanches correspond to compact reorganizations of the interface well described by the depinning theory. For long-range elasticity, an avalanche is a collection of spatially disconnected clusters. In this Letter we determine the scaling properties of the clusters and relate them to the roughness exponent of the interface. The key observation of our analysis is the identification of a Bienaymé-Galton-Watson process describing the statistics of the number of clusters. Our work has concrete importance for experimental applications where the cluster statistics is a key probe of avalanche dynamics.

How heat controls fracture: the thermodynamics of creeping and avalanching cracks

While of paramount importance in material science, the dynamics of cracks still lacks a complete physical explanation. The transition from their slow creep behavior to a fast propagation regime is a notable key, as it leads to full material failure if the size of a fast avalanche reaches that of the system. We here show that a simple thermodynamics approach can actually account for such complex crack dynamics, and in particular for the non-monotonic force-velocity curves commonly observed in mechanical tests on various materials. We consider a thermally activated failure process that is coupled with the production and the diffusion of heat at the fracture tip. In this framework, the rise in temperature only affects the sub-critical crack dynamics and not the mechanical properties of the material. We show that this description can quantitatively reproduce the rupture of two different polymeric materials (namely, the mode I opening of polymethylmethacrylate (PMMA) plates, and the peeling of pressure sensitive adhesive (PSA) tapes), from the very slow to the very fast fracturing regimes, over seven to nine decades of crack propagation velocities. In particular, the fastest regime is obtained with an increase of temperature of thousands of Kelvins, on the molecular scale around the crack tip. Although surprising, such an extreme temperature is actually consistent with different experimental observations that accompany the fast propagation of cracks, namely, fractoluminescence (i.e., the emission of visible light during rupture) and a complex morphology of post-mortem fracture surfaces, which could be due to the sublimation of bubbles.

Universal Scaling of the Velocity Field in Crack Front Propagation

The propagation of a crack front in disordered materials is jerky and characterized by bursts of activity, called avalanches. These phenomena are the manifestation of an out-of-equilibrium phase transition originated by the disorder. As a result avalanches display universal scalings which are, however, difficult to characterize in experiments at a finite drive. Here, we show that the correlation functions of the velocity field along the front allow us to extract the critical exponents of the transition and to identify the universality class of the system. We employ these correlations to characterize the universal behavior of the transition in simulations and in an experiment of crack propagation. This analysis is robust, efficient, and can be extended to all systems displaying avalanche dynamics.

Anisotropic avalanches and critical depinning of three-dimensional magnetic domain walls

Simulations with more than 10^{12} spins are used to study the motion of a domain wall driven through a three-dimensional random-field Ising magnet (RFIM) by an external field H. The interface advances in a series of avalanches whose size diverges at a critical external field H_{c}. Finite-size scaling is applied to determine critical exponents and test scaling relations. Growth is intrinsically anisotropic with the height of an avalanche normal to the interface ℓ_{⊥} scaling as the width along the interface ℓ_{∥} to a power χ=0.85±0.01. The total interface roughness is consistent with self-affine scaling with a roughness exponent ζ≈χ that is much larger than values found previously for the RFIM and related models that explicitly break orientational symmetry by requiring the interface to be single-valued. Because the RFIM maintains orientational symmetry, the interface develops overhangs that may surround unfavorable regions to create uninvaded bubbles. Overhangs complicate measures of the roughness exponent but decrease in importance with increasing system size.

Seismiclike organization of avalanches in a driven long-range elastic string as a paradigm of brittle cracks

Crack growth in heterogeneous materials sometimes exhibits crackling dynamics, made of successive impulselike events with specific scale-invariant time and size organization reminiscent of earthquakes. Here, we examine this dynamics in a model which identifies the crack front with a long-range elastic line driven in a random potential. We demonstrate that, under some circumstances, fracture grows intermittently, via scale-free impulse organized into aftershock sequences obeying the fundamental laws of statistical seismology. We examine the effects of the driving rate and system overall stiffness (unloading factor) onto the scaling exponents and cutoffs associated with the time and size organization. We unravel the specific conditions required to observe a seismiclike organization in the crack propagation problem. Beyond failure problems, implications of these results to other crackling systems are finally discussed.

Continuously Sheared Granular Matter Reproduces in Detail Seismicity Laws

We introduce a shear experiment that quantitatively reproduces the main laws of seismicity. By continuously and slowly shearing a compressed monolayer of disks in a ringlike geometry, our system delivers events of frictional failures with energies following a Gutenberg-Richter law. Moreover, foreshocks and aftershocks are described by Omori laws and interevent times also follow exactly the same distribution as real earthquakes, showing the existence of memory of past events. Other features of real earthquakes qualitatively reproduced in our system are both the existence of a quiescence preceding some main shocks, as well as magnitude correlations linked to large quakes. The key ingredient of the dynamics is the nature of the force network, governing the distribution of frictional thresholds.

Depinning Dynamics of Crack Fronts

We investigate experimentally and theoretically the dynamics of a crack front during the microinstabilities taking place in heterogeneous materials between two successive equilibrium positions. We focus specifically on the spatiotemporal evolution of the front, as it relaxes to a straight configuration, after depinning from a single obstacle of controlled strength and size. We show that this depinning dynamics is not controlled by inertia, but instead by the rate dependency of the dissipative mechanisms taking place within the fracture process zone. This implies that the crack speed fluctuations around its average value v_{m} can be predicted from an overdamped equation of motion (v-v_{m})/v_{0}=[G-G_{c}(v_{m})]/G_{c}(v_{m}) involving the characteristic material speed v_{0}=G_{c}(v_{m})/G_{c}^{'}(v_{m}) that emerges from the variation of fracture energy with crack speed. Our findings pave the way to a quantitative description of the critical depinning dynamics of cracks in disordered solids and open up new perspectives for the prediction of the effective failure properties of heterogeneous materials.

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.

Avalanches and extreme value statistics in interfacial crackling dynamics

We study the avalanche and extreme statistics of the global velocity of a crack front, propagating slowly along a weak heterogeneous interface of a transparent polymethyl methacrylate block. The different loading conditions used (imposed constant velocity or creep relaxation) lead to a broad range of average crack front velocities. Our high-resolution and large dataset allows one to characterize in detail the observed intermittent crackling dynamics. We specifically measure the size S, the duration D, as well as the maximum amplitude [Formula: see text] of the global avalanches, defined as bursts in the interfacial crack global velocity time series. Those quantities characterizing the crackling dynamics follow robust power-law distributions, with scaling exponents in agreement with the values predicted and obtained in numerical simulations of the critical depinning of a long-range elastic string, slowly driven in a random medium. Nevertheless, our experimental results also set the limit of such model which cannot reproduce the power-law distribution of the maximum amplitudes of avalanches of a given duration reminiscent of the underlying fat-tail statistics of the local crack front velocities.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

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'.

Crack growth in heterogeneous brittle solids: intermittency, crackling and induced seismicity

Crack growth is the basic mechanism leading to the failure of brittle materials. Engineering addresses this problem within the framework of continuum mechanics, which links deterministically the crack motion to the applied loading. Such an idealization, however, fails in several situations and in particular cannot capture the highly erratic (earthquake-like) dynamics sometimes observed in slowly fracturing heterogeneous solids. Here, we examine this problem by means of innovative experiments of crack growth in artificial rocks of controlled microstructure. The dynamical events are analysed at both global and local scales, from the time fluctuation of the spatially averaged crack speed and the induced acoustic emission, respectively. Their statistics are characterized and compared with the predictions of a recent approach mapping fracture onset to the depinning of an elastic interface. Finally, the overall time-size organization of the events is characterized to shed light on the mechanisms underlying the scaling laws observed in seismology.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

Thermally activated crack fronts propagating in pinning disorder: simultaneous brittle/creep behaviour depending on scale

We study theoretically the propagation of a crack front in mode I along an interface in a disordered elastic medium, with a numerical model considering a thermally activated rheology, toughness disorder and long-range elastic interactions. This model reproduces not only the large-scale dynamics of the crack front position in fast or creep loading regimes, but also the small-scale self-affine behaviour of the front. Two different scaling laws are predicted for the front morphology, with a Hurst exponent of 0.5 at small scales and a logarithmic scaling law at large scales, consistently with experiments. The prefactor of these scaling laws is expressed as a function of the temperature, and of the quenched disorder characteristics. The cross-over between these regimes is expressed as a function of the quenched disorder amplitude, and is proportional to the average energy release rate, and to the inverse of temperature. This model captures as well the experimentally observed local velocity fluctuation probability distribution, with a high-velocity tail P(v)∼v -2.6 This feature is shown to arise when the quenched disorder is sufficiently large, whereas smaller toughness fluctuations lead to a lognormal-like velocity distribution. Overall, the system is shown to obey a scaling determined by two distinct mechanisms as a function of scale: namely, the large scales display fluctuations similar to an elastic line in an annealed noise excited as the average front travels through the pinning landscape, while small scales display a balance between thresholds in possible elastic forces and quenched disorder.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

Spatiotemporal Organization of Correlated Local Activity within Global Avalanches in Slowly Driven Interfaces

We study the jerky response of slowly driven fronts in disordered media, just above the depinning transition. We focus on how spatially disconnected clusters of internally correlated activity lead to large-scale velocity fluctuations in the form of global avalanches and identify three different ways in which local activity clusters may organize within a global avalanche, depending on the distance to criticality. Our analysis provides new scaling relations between the power-law exponents of the statistical distributions of sizes and durations of local bursts and global avalanches. Fluid fronts of imbibition in heterogeneous media are taken as a case study to validate these scaling relations.

Mode-III interfacial crack propagation in heterogeneous media

We monitor optically the propagation of a slow interfacial mode III crack along a heterogeneous weak interface and compare it to mode I loading. Pinning and depinning of the front on local toughness asperities within the process zone are the main mechanisms for fracture roughening. Geometrical properties of the fracture fronts are derived in the framework of self-affine scale invariance and Family-Vicsek scaling. We characterize the small and large scale roughness exponents ζ_{-}=0.6 and ζ_{+}=0.35, the growth exponent at large scale β_{+}=0.58, and the power-law exponent of the local velocity distribution of the fracture fronts, η=2.55. All these analyzed properties are similar to those previously observed for mode I interfacial fractures. We also observe a common power-law decay of the probability distribution function of avalanche area. We finally observe that amplitude of front fluctuations, local rupture velocity correlation in time, and larger size of events highlight more dynamically unstable behavior of mode III crack ruptures.

Aftershock sequences and seismic-like organization of acoustic events produced by a single propagating crack

Brittle fractures of inhomogeneous materials like rocks, concrete, or ceramics are of two types: Nominally brittle and driven by the propagation of a single dominant crack or quasi-brittle and resulting from the accumulation of many microcracks. The latter goes along with acoustic noise, whose analysis has revealed that events form aftershock sequences obeying characteristic laws reminiscent of those in seismology. Yet, their origin lacks explanation. Here we show that such a statistical organization is not only specific to the multi-cracking situations of quasi-brittle failure and seismology, but also rules the acoustic events produced by a propagating crack. This simpler situation has permitted us to relate these laws to the overall scale-free distribution of inter-event time and energy and to uncover their selection by the crack speed. These results provide a comprehensive picture of how acoustic events are organized upon material failure in the most fundamental of fracture states: single propagating cracks.

The multiple microcracking events underlying damage in inhomogeneous brittle materials form characteristic aftershocks sequences obeying similar laws to those in seismology. Here, Barés et al. evidence and explain the same organization in the acoustic noise produced by a single propagating crack.

Localization of soft modes at the depinning transition

We characterize the soft modes of the dynamical matrix at the depinning transition, and compare the matrix with the properties of the Anderson model (and long-range generalizations). The density of states at the edge of the spectrum displays a universal linear tail, different from the Lifshitz tails. The eigenvectors are instead very similar in the two matrix ensembles. We focus on the ground state (soft mode), which represents the epicenter of avalanche instabilities. We expect it to be localized in all finite dimensions, and make a clear connection between its localization length and the Larkin length of the depinning model. In the fully connected model, we show that the weak-strong pinning transition coincides with a peculiar localization transition of the ground state.

Verification of a Dynamic Scaling for the Pair Correlation Function during the Slow Drainage of a Porous Medium

In this Letter we give experimental grounding for the remarkable observation made by Furuberg et al. [Phys. Rev. Lett. 61, 2117 (1988)PRLTAO0031-900710.1103/PhysRevLett.61.2117] of an unusual dynamic scaling for the pair correlation function N(r,t) during the slow drainage of a porous medium. Those authors use an invasion percolation algorithm to show numerically that the probability of invasion of a pore at a distance r away and after a time t from the invasion of another pore scales as N(r,t)∝r^{-1}f(r^{D}/t), where D is the fractal dimension of the invading cluster and the function f(u)∝u^{1.4}, for u≪1 and f(u)∝u^{-0.6}, for u≫1. Our experimental setup allows us to have full access to the spatiotemporal evolution of the invasion, which is used to directly verify this scaling. Additionally, we connect two important theoretical contributions from the literature to explain the functional dependency of N(r,t) and the scaling exponent for the short-time regime (t≪r^{D}). A new theoretical argument is developed to explain the long-time regime exponent (t≫r^{D}).

Crack propagation through disordered materials as a depinning transition: A critical test of the theory

The dynamics of a planar crack propagating within a brittle disordered material is investigated numerically. The fracture front evolution is described as the depinning of an elastic line in a random field of toughness. The relevance of this approach is critically tested through the comparison of the roughness front properties, the statistics of avalanches, and the local crack velocity distribution with experimental results. Our simulations capture the main features of the fracture front evolution as measured experimentally. However, some experimental observations such as the velocity distribution are not consistent with the behavior of an elastic line close to the depinning transition. This discrepancy suggests the presence of another failure mechanism not included in our model of brittle failure.

Avalanches dynamics in reaction fronts in disordered flows

We report on numerical studies of avalanches of an autocatalytic reaction front in a porous medium. The front propagation is controlled by an adverse flow resulting in upstream, static, or downstream regimes. In an earlier study focusing on front shape, we identified three different universality classes associated with this system by following the front dynamics experimentally and numerically. Here, using numerical simulations in the vicinity of the second-order transition, we identify an avalanche dynamics characterized by power-law distributions of avalanche sizes, durations, and lateral extensions. The related exponents agree well with the quenched-Kardar-Parisi-Zhang theory, which describes the front dynamics. However, the geometry of the propagating front differs slightly from that of the theoretical one. We show that this discrepancy can be understood in terms of the nonquasistatic correction induced by the finite front velocity.

Interevent Correlations from Avalanches Hiding Below the Detection Threshold

Numerous systems ranging from deformation of materials to earthquakes exhibit bursty dynamics, which consist of a sequence of events with a broad event size distribution. Very often these events are observed to be temporally correlated or clustered, evidenced by power-law-distributed waiting times separating two consecutive activity bursts. We show how such interevent correlations arise simply because of a finite detection threshold, created by the limited sensitivity of the measurement apparatus, or used to subtract background activity or noise from the activity signal. Data from crack-propagation experiments and numerical simulations of a nonequilibrium crack-line model demonstrate how thresholding leads to correlated bursts of activity by separating the avalanche events into subavalanches. The resulting temporal subavalanche correlations are well described by our general scaling description of thresholding-induced correlations in crackling noise.

Bursts of activity in collective cell migration

Dense monolayers of living cells display intriguing relaxation dynamics, reminiscent of soft and glassy materials close to the jamming transition, and migrate collectively when space is available, as in wound healing or in cancer invasion. Here we show that collective cell migration occurs in bursts that are similar to those recorded in the propagation of cracks, fluid fronts in porous media, and ferromagnetic domain walls. In analogy with these systems, the distribution of activity bursts displays scaling laws that are universal in different cell types and for cells moving on different substrates. The main features of the invasion dynamics are quantitatively captured by a model of interacting active particles moving in a disordered landscape. Our results illustrate that collective motion of living cells is analogous to the corresponding dynamics in driven, but inanimate, systems.

How cracks are hot and cool: a burning issue for paper

Material failure is accompanied by important heat exchange, with extremely high temperature - thousands of degrees - reached at crack tips. Such a temperature may subsequently alter the mechanical properties of stressed solids, and finally facilitate their rupture. Thermal runaway weakening processes could indeed explain stick-slip motions and even be responsible for deep earthquakes. Therefore, to better understand catastrophic rupture events, it appears crucial to establish an accurate energy budget of fracture propagation from a clear measure of various energy dissipation sources. In this work, combining analytical calculations and numerical simulations, we directly relate the temperature field around a moving crack tip to the part α of mechanical energy converted into heat. By monitoring the slow crack growth in paper sheets using an infrared camera, we measure a significant fraction α = 12% ± 4%. Besides, we show that (self-generated) heat accumulation could weaken our samples by microfiber combustion, and lead to a fast crack/dynamic failure/regime.

Experimental study of stable imbibition displacements in a model open fracture. I. Local avalanche dynamics

We report the results of an experimental investigation of the spatiotemporal dynamics of stable imbibition fronts in a disordered medium, in the regime of capillary disorder, for a wide range of experimental conditions. We have used silicone oils of various viscosities μ and nearly identical oil-air surface tension and forced them to slowly invade a model open fracture at different constant flow rates v. In this first part of the study we have focused on the local dynamics at a scale below the size of the quenched disorder. Changing μ and v independently, we have found that the dynamics is not simply controlled by the capillary number Ca∼μv. Specifically, we have found that the wide statistical distributions of local front velocities, and their large spatial correlations along the front, are indeed controlled by the capillary number Ca. However, local velocities exhibit also very large temporal correlations, and these correlations depend more strongly on the mean imposed velocity v than on the viscosity μ of the invading fluid. Correlations between local velocities lead to a burstlike dynamics. Avalanches, defined as clusters of large local velocities, follow power-law distributions-both in size and duration-with exponential cutoffs that diverge as Ca→0, the pinning-depinning transition of stable imbibition displacements. Large data sets have led to reliable statistics, from which we have derived accurate values of critical exponents of the relevant power-law distributions. We have investigated also the dependence of their cutoffs on μ and v and related them to the autocorrelations of local velocities in space and time.

Fractal frontiers of bursts and cracks in a fiber bundle model of creep rupture

We investigate the geometrical structure of breaking bursts generated during the creep rupture of heterogeneous materials. Using a fiber bundle model with localized load sharing we show that bursts are compact geometrical objects; however, their external frontiers have a fractal structure which reflects their growth dynamics. The perimeter fractal dimension of bursts proved to have the universal value 1.25 independent of the external load and of the amount of disorder in the system. We conjecture that according to their geometrical features, breaking bursts fall in the universality class of loop-erased self-avoiding random walks with perimeter fractal dimension 5/4 similar to the avalanches of Abelian sand pile models. The fractal dimension of the growing crack front along which bursts occur proved to increase from 1 to 1.25 as bursts gradually cover the entire front.

Moving line model and avalanche statistics of Bingham fluid flow in porous media

In this article, we propose a simple model to understand the critical behavior of path opening during flow of a yield stress fluid in porous media as numerically observed by Chevalier and Talon (2015). This model can be mapped to the problem of a contact line moving in an heterogeneous field. Close to the critical point, this line presents an avalanche dynamic where the front advances by a succession of waiting time and large burst events. These burst events are then related to the non-flowing (i.e. unyielded) areas. Remarkably, the statistics of these areas reproduce the same properties as in the direct numerical simulations. Furthermore, even if our exponents seem to be close to the mean field universal exponents, we report an unusual bump in the distribution which depends on the disorder. Finally, we identify a scaling invariance of the cluster spatial shape that is well fit, to first order, by a self-affine parabola.

Fluctuations of global energy release and crackling in nominally brittle heterogeneous fracture

The temporal evolution of mechanical energy and spatially averaged crack speed are both monitored in slowly fracturing artificial rocks. Both signals display an irregular burstlike dynamics, with power-law distributed fluctuations spanning a broad range of scales. Yet, the elastic power released at each time step is proportional to the global velocity all along the process, which enables defining a material-constant fracture energy. We characterize the intermittent dynamics by computing the burst statistics. This latter displays the scale-free features signature of crackling dynamics, in qualitative but not quantitative agreement with the depinning interface models derived for fracture problems. The possible sources of discrepancies are pointed out and discussed.

Statistical properties of Barkhausen noise in amorphous ferromagnetic films

We investigate the statistical properties of the Barkhausen noise in amorphous ferromagnetic films with thicknesses in the range between 100 and 1000 nm. From Barkhausen noise time series measured with the traditional inductive technique, we perform a wide statistical analysis and establish the scaling exponents τ,α,1/σνz, and ϑ. We also focus on the average shape of the avalanches, which gives further indications on the domain-wall dynamics. Based on experimental results, we group the amorphous films in a single universality class, characterized by scaling exponents τ=1.28±0.02,α=1.52±0.3, and 1/σνz=ϑ=1.83±0.03, values compatible with that obtained for several bulk amorphous magnetic materials. Besides, we verify that the avalanche shape depends on the universality class. By considering the theoretical models for the dynamics of a ferromagnetic domain wall driven by an external magnetic field through a disordered medium found in literature, we interpret the results and identify an experimental evidence that these amorphous films, within this thickness range, present a typical three-dimensional magnetic behavior with predominant short-range elastic interactions governing the domain-wall dynamics. Moreover, we provide experimental support for the validity of a general scaling form for the average avalanche shape for non-mean-field systems.

Disorder-induced capillary bursts control intermittency in slow imbibition

A multiscale analysis of the spatially averaged velocity of an imbibition front V_{ℓ}(t) measured at scale ℓ reveals that the slow front dynamics is intermittent: the distributions of ΔV_{ℓ}(τ)=V_{ℓ}(t+τ)-V_{ℓ}(t) evolve continuously through time scales τ, from heavy-tailed to Gaussian-reached at a time lag τ_{c} set by the extent of the medium heterogeneities. Intermittency results from capillary bursts triggered from the smallest scale of the disorder up to the scale ℓ_{c} at which viscous dissipation becomes dominant. The effective number of degrees of freedom of the front ℓ/ℓ_{c} controls its intensity.

High frequency monitoring reveals aftershocks in subcritical crack growth

By combining direct imaging and acoustic emission measurements, the subcritical propagation of a crack in a heterogeneous material is analyzed. Both methods show that the fracture proceeds through a succession of discrete events. However, the macroscopic opening of the fracture captured by the images results from the accumulation of more-elementary events detected by the acoustics. When the acoustic energy is cumulated over large time scales corresponding to the image acquisition rate, a similar statistics is recovered. High frequency acoustic monitoring reveals aftershocks responsible for a time scale dependent exponent of the power law energy distributions. On the contrary, direct imaging, which is unable to resolve these aftershocks, delivers a misleading exponent value.

Stability and roughness of tensile cracks in disordered materials

We study the stability and roughness of propagating cracks in heterogeneous brittle two-dimensional elastic materials. We begin by deriving an equation of motion describing the dynamics of such a crack in the framework of linear elastic fracture mechanics, based on the Griffith criterion and the principle of local symmetry. This result allows us to extend the stability analysis of Cotterell and Rice [B. Cotterell and J. R. Rice, Int. J. Fract. 16, 155 (1980)] to disordered materials. In the stable regime we find stochastic crack paths. Using tools of statistical physics, we obtain the power spectrum of these paths and their probability distribution function and conclude that they do not exhibit self-affinity. We show that a real-space fractal analysis of these paths can lead to the wrong conclusion that the paths are self-affine. To complete the picture, we unravel a systematic bias in such real-space methods and thus contribute to the general discussion of reliability of self-affine measurements.

Role of the sample thickness in planar crack propagation

We study the effect of the sample thickness in planar crack front propagation in a disordered elastic medium using the random fuse model. We employ different loading conditions and we test their stability with respect to crack growth. We show that the thickness induces characteristic lengths in the stress enhancement factor in front of the crack and in the stress transfer function parallel to the crack. This is reflected by a thickness-dependent crossover scale in the crack front morphology that goes from from multiscaling to self-affine with exponents, in agreement with line depinning models and experiments. Finally, we compute the distribution of crack avalanches, which is shown to depend on the thickness and the loading mode.

Adhesion failure at 180,000 frames per second: direct observation of the detachment process of a mushroom-shaped adhesive

Nature has successfully evolved the mushroom-shaped contact geometry in many organisms in order to solve the attachment problem. We studied the detachment process of individual bioinspired artificial mushroom-shaped adhesive microstructures (MSAMSs) resolving the failure dynamics at high spatiotemporal resolution. The experimental data provide strong evidence for a homogeneous stress distribution in MSAMS, which was recently proposed. Our results allow us to explain the advantage of such contact geometry and provide a suggestion for the widely observed mushroom-shaped contact geometry.

Statistics of avalanches with relaxation and Barkhausen noise: a solvable model

We study a generalization of the Alessandro-Beatrice-Bertotti-Montorsi (ABBM) model of a particle in a Brownian force landscape, including retardation effects. We show that under monotonous driving the particle moves forward at all times, as it does in absence of retardation (Middleton's theorem). This remarkable property allows us to develop an analytical treatment. The model with an exponentially decaying memory kernel is realized in Barkhausen experiments with eddy-current relaxation and has previously been shown numerically to account for the experimentally observed asymmetry of Barkhausen pulse shapes. We elucidate another qualitatively new feature: the breakup of each avalanche of the standard ABBM model into a cluster of subavalanches, sharply delimited for slow relaxation under quasistatic driving. These conditions are typical for earthquake dynamics. With relaxation and aftershock clustering, the present model includes important ingredients for an effective description of earthquakes. We analyze quantitatively the limits of slow and fast relaxation for stationary driving with velocity v>0. The v-dependent power-law exponent for small velocities, and the critical driving velocity at which the particle velocity never vanishes, are modified. We also analyze nonstationary avalanches following a step in the driving magnetic field. Analytically, we obtain the mean avalanche shape at fixed size, the duration distribution of the first subavalanche, and the time dependence of the mean velocity. We propose to study these observables in experiments, allowing a direct measurement of the shape of the memory kernel and tracing eddy current relaxation in Barkhausen noise.

Temporal and spacial evolution of bursts in creep rupture

We investigate the temporal and spacial evolution of single bursts and their statistics emerging in heterogeneous materials under a constant external load. Based on a fiber bundle model we demonstrate that when the load redistribution is localized along a propagating crack front, the average temporal shape of pulses has a right-handed asymmetry; however, for long range interaction a symmetric shape with parabolic functional form is obtained. The pulse shape and spatial evolution of bursts proved to be correlated, which can be exploited in materials' testing. The probability distribution of the size and duration of bursts have power law behavior with a crossover to higher exponents as the load is lowered. The crossover emerges due to the competition of the slow and fast modes of local breaking being dominant at low and high loads, respectively.

Crackling versus continuumlike dynamics in brittle failure

We study how the loading rate, specimen geometry, and microstructural texture select the dynamics of a crack moving through an heterogeneous elastic material in the quasistatic approximation. We find a transition, fully controlled by two dimensionless variables, between dynamics ruled by continuum fracture mechanics and crackling dynamics. Selection of the latter by the loading, microstructure, and specimen parameters is formulated in terms of scaling laws on the power spectrum of crack velocity. This analysis defines the experimental conditions required to observe crackling in fracture. Beyond failure problems, the results extend to a variety of situations described by models of the same universality class, e.g., the dynamics in wetting or of domain walls in amorphous ferromagnets.