Avalanches and extreme value statistics in interfacial crackling dynamics

Abstract

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 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’.

1. Introduction

The mechanical response of heterogeneous materials submitted to simple loading can present a complex behaviour, in the form of collective excitations, bursts, with very broad size distributions usually following power laws, separated by periods of rest, also broadly distributed [1]. This complexity culminates for instance when considering faults dynamics during earthquakes, where slowly accumulated energy during the very slow motion of tectonic plates is suddenly released and dissipated by failure and frictional processes. The corresponding intermittent ‘crackling noises’ [2] recorded by seismologists have indeed specific statistical characteristics, with for instance the famous scale-free Gutenberg–Richter law [3], relating the magnitude and total number of earthquakes, as well as temporal correlations with aftershocks sequences following Omori's scaling law [4].

Such physical behaviour is a common feature of numerous systems covering a broad range of scales, when compressing or elongating laboratory heterogeneous samples like paper [5–7], wood [8], rocks [9], porous materials [10] and artificial rocks made of sintered polystyrene beads [11], or weakly sintered Plexiglas plates [12–16]. Nevertheless, this phenomenology is not limited to the physical processes of deformation and rupture of disordered materials. Indeed, such intermittent avalanche dynamics can be observed for very different heterogeneous systems, slowly driven in out-of-equilibrium conditions, with, for instance, the magnetic response of a disordered ferromagnet, also known as ‘Barkhausen noise’ [17,18], the motion of vortex lines in superconductors [19], the fluid imbibition in porous and fractured media [20–23], the motion of contact lines over substrates with wetting heterogeneities [24], but also the yielding of amorphous materials [25–27] and single crystals [28–30] or even in the biological activity of neuronal networks [31,32] and cell migration [33].

Understanding the morphology and dynamics of interfaces slowly driven in disordered media has thus become a real challenge over the last 30 years. The common point of the so-called disordered elastic systems [34] is the competition between the medium heterogeneities, which deform the fronts, while the elasticity of the system tends to maintain an orderly structure and smooth the interfaces. Those competing forces act at different spatial and temporal scales, leading indeed to complex structural and dynamical properties of the interfaces, with self-affine roughening morphologies and intermittent scale-free burst dynamics. Since the very first theoretical descriptions of the propagation of interfaces in random media [35,36], much progress has been achieved in recent years, both theoretically and experimentally.

From a theoretical perspective, the detailed description of these systems is difficult and requires sophisticated approaches from nonlinear statistical physics [37,38]. Nevertheless, their critical-like scaling behaviour could be interpreted in terms of non-equilibrium phase transitions [39], separating quiescent and active phases. Close to such dynamical phase transitions, the temporal fluctuations of an activity signal (as, for instance, an interfacial velocity) acting as an order parameter display power-law scaling avalanches [1]. Statistical analyses and functional renormalization group calculations have succeeded to predict avalanche scaling exponents, within the mean field approximation and even beyond [38,40–42], and suggest classifying those systems displaying avalanche dynamics into ‘universality classes’, characterized by their critical scaling exponents, directly related to the spatial dimension and the interaction range of the system, but a priori independent of the microscopic material properties.

Thanks to experimental efforts to design model systems [12,21,43,44], where the structure and dynamics of interfaces could be directly observed and tracked very accurately both in space and time at the microscopic scale of the disorder, recent studies have been able to reconcile theoretical predictions and experimental observations. Specifically, it has been shown that both the structure and dynamics of planar cracks could be described quantitatively, at scales larger than the disorder, within the framework of a ‘critical depinning transition’ [15,16,45–48].

In this work, we describe novel experimental results obtained for such a model system that allows one to access the structure and the dynamics of an interfacial crack, propagating along a weak disordered interface, locally at the scale of the heterogeneities. Usually, such spatial resolution is difficult to obtain, and the avalanche dynamics is typically studied only through the temporal evolution of a global, spatially averaged quantity—also called ‘crackling noise’. By contrast, our experimental set-up allows this gap to be bridged, from local to global dynamics [49].

We present here an analysis of the avalanche and extreme statistics of the global velocity of a planar crack front, propagating slowly along a weak heterogeneous interface within a polymethyl methacrylate (PMMA) block. This global velocity is obtained from the spatial average of the front velocity measured at local microscopic scales, below the scale of heterogeneities. The originality of our study is indeed to analyse the crackling dynamics at a global scale, characterizing the statistics of global avalanches, extracted from the spatially averaged crack front speed, while most of our previous studies on crack front avalanches have focused on local bursts, e.g. clusters of high local crack front velocity [12–14]. Moreover, in contrast with the more recent works reported in [15,16] (that focused on the avalanche shape and on the waiting time between subsequent avalanches, respectively), different loading conditions were used here. We thus could obtain a very high-resolution large dataset, with in particular a broad range of average crack front velocities, spanning close to five decades, allowing us to characterize in detail the crackling dynamics. Indeed, we quantify the avalanche crack front dynamics by measuring the size and duration of global bursts in this crackling noise signal, e.g. the global crack front velocity. We show that our results are in quantitative agreement with previous theoretical and numerical results [15,46,48], with power-law exponents characterizing the size S, as P(S)∼S^−τ, with τ = 1.15 ± 0.15, and duration D, as P(D)∼D^−α, with α = 1.32 ± 0.15 of those global avalanches, as well as their scaling 〈D | S〉∼S^ξ with ξ = 0.60 ± 0.15. We thus confirm that the planar crack front avalanche dynamics at large scales can be well described by the critical depinning of a long-range elastic string in a two-dimensional random medium.

Furthermore, we have computed the extreme statistics of the velocity fluctuations corresponding to global avalanches. We find that the maximum avalanche velocity follows a power-law tailed distribution with an exponent ϵ = 1.34 ± 0.15. The average maximum velocity also scales with the avalanche duration as , with the exponent β = 0.90 ± 0.20 that is also related by the scaling relation to the exponents of the avalanche duration and maximum amplitude distribution, namely ϵ = 1 + (α − 1)/β. We note that the values of these exponents are different from those predicted by the mean field theory of interface depinning [50,51], but the scaling relation between them is still valid. We observe nevertheless a quantitative agreement between the experimental values and those obtained from numerical simulations of a long-range elastic interface slowly driven along a disordered plane, beyond the mean field approximation. However, there are also fundamental differences when comparing the scaling functions for the distribution of maximal velocities within avalanches of fixed duration. Albeit the mean values are in agreement with simulations, we find that maximal velocities are broadly varying with a tail distribution that is similar to that of the local crack front velocity distribution and in particular its fat-tail P(v)∼v^−η with η∼2.8. This is not captured by the interfacial model which gives an exponential tail. Therefore, it can question the validity of the elastic line approach to describe the crack front dynamics at small scales close to the disorder, for which a percolation-like approach [52,53] might be more relevant.

2. Experiments and methods

We will first recall briefly the experimental set-up together with the samples preparation and methods of analysis, described in detail in [14,54].

Our samples consist of two sand-blasted Plexiglas plates that are annealed together to create a single block with a weak heterogeneous interface. Those plates have the following dimensions (30, 14, 1) cm for the top (thicker and wider) one, and (30, 12, 0.4) cm for the bottom one. Before annealing at a controlled temperature of 205°C for 30–50 min, both plates are blasted on one side with 50 μm steel particles or 100 μm glass beads. Such procedure roughens the Plexiglas plate surfaces with a random topography which induces random toughness fluctuations at microscopic scales (few tenths of microns) during the annealing procedure [12]. A cylindrical press bar mounted on a motorized vertical translation stage imposes a normal displacement to the thin Plexiglas bottom plate, while the upper one is fixed to a rigid aluminium frame. Subsequently, a fracture propagates along the weak heterogenous plane between the sintered plates, in mixed mode I/II conditions, due to the unequal thickness of the two plates and the loading asymmetry conditions (see appendix in [54] for more details). Different loading conditions have been used, by either pushing the bottom plate at a constant velocity, constant velocity boundary conditions (F), or by imposing a constant deflection d to the bottom plate, creep boundary conditions (R).

A high-resolution and fast digital camera mounted on a microscope images a small central region of the crack front at the millimetre scale, as shown in figure 1. The large width of the bent PMMA plate (10 cm) ensures that this central region of interest is not influenced by finite size effects. The sandblasting procedure—crucial to introduce toughness fluctuations along the weak plane—leads moreover to a very good contrast between the cracked and uncracked part of the sample, allowing the fracture front to be detected and extracted rather easily, as shown in the bottom panel of figure 1. For a more detailed description of the front extraction and image treatment, see [13]. The interfacial crack front propagation is captured with a high frame rate (from 1 to 2000 frames per second (fps)) compared to the average crack front velocity (typically two orders of magnitude faster), recording between 12 000 and 30 000 frames, using either a Photron Fastcam-Ultima APX (512 × 1024 pixels) or a Pixelink Industrial Vision PL-A781 (2200 × 3000 pixels), at a spatial resolution approximately 1 − 10 μm pixel⁻¹. We have performed various experiments with different loading conditions, leading to average crack front velocity spanning a very broad range (five orders of magnitude). Indeed, our experiments could last a few seconds to several hours, whereas the average distance of front propagation is approximately 500 μm in all cases.

Figure 1.

(a) Sketch of the experimental set-up. Two initially sand-blasted and sintered PMMA plates create a weak heterogeneous interface, where a fracture—initiated by lowering a cylindrical press bar onto the lower plate—propagates. The fracture front is imaged from above by a high-resolution fast camera mounted on a microscope. (b) Typical fracture front h(x, t′) extracted at some time t′ superimposed on the corresponding raw image, propagating from top to bottom. The framed raw image corresponds to a tiny central part of the full sample. (Online version in colour.)

The spatially random toughness fluctuations (induced by the sandblasting procedure) along the weak interface of the Plexiglas block leads to a rough fracture front with complex self-affine scaling properties [45,47] and an intermittent avalanche dynamics [12]. In order to characterize this jerky front propagation, we have developed an original procedure [12], based on the measurement of the local waiting time fluctuations of the crack front during its propagation. Such method has been successfully applied to quantify the avalanche dynamics of various systems, such as fluid imbibition fronts [55,56] or biological cell migration [33]. Specifically, we first computed the local front velocities v(x, y) from measurements of the local waiting times wt(x, y) elapsed while the crack front was located at a given position (x, y), and defining v(x, y) = r/wt(x, y), where r is the pixel resolution. Using the detected crack front position, h(x, t), it is straightforward to obtain the spatio-temporal local velocity map v(x, t) and the total average front velocity 〈v〉 = 〈v(x, t)〉_x,t of the fracture. Finally, from this local velocity field v(x, t), we compute the spatially averaged front velocity V_l(t) over a variable window size l∈(r, L), namely as V_l(t) = 〈v(x, t)〉_x0, where the brackets correspond to an average of the local velocity over the abscissa x₀ of a window of size l. We have systematically varied the scale l = L/N at which we measure the global velocity V_l(t), obtaining finally N different crackling signals from one experiment with a crack length L. This window size l is always larger than the correlation length of the local velocities along the crack front, measured around 100 μm for our samples.

Figure 2a shows the global velocity V_L(t) for one typical creep experiment. This signal is obtained from the spatio-temporal local velocity map v(x, t), spatially averaged at the scale L of the recorded images. This grey scale map illustrates the intermittent crack front dynamics. In particular, it has been shown previously that the front dynamics is governed by local avalanches—connected clusters of high local velocity—with very large size and velocity fluctuations following a fat-tail distribution P(v)∼v^−η, decaying with a power-law exponent η∼2.7. Consequently, the global crackling noise V_l(t) signal follows non-Gaussian statistics and, by the generalized central limit theorem, converges to a Levy-distribution [49] at large measuring scales l.

Figure 2.

(b) A typical spatio-temporal local velocity field v(x, t) of the interfacial fracture during a slow creep experiment, which reflects the intermittent dynamics over a broad range of scales. In (a), the corresponding global velocity of the front V_L(t) = 〈v(x, t)〉_L, obtained from the local velocity field spatially averaged, here, at the scale L of the system size (e.g. image size). The dashed line indicates the threshold V_c used to define the global avalanches, of size S, duration D, and of maximum velocity , as explained in inset. Here, the threshold is equal to the total average front velocity. The size equals the area under connected points in V_L(t) above the threshold line. For a given avalanche, the duration is the time difference between the two subsequent intersections with the threshold line. The maximum velocity is the peak value inside an avalanche minus the threshold velocity V_c.

Here, we go further by studying the avalanche and extreme statistics of those global velocity fluctuations V_l(t). Similar to earlier works [17,18,21,46], we introduce the concept of global avalanches in the crack velocity signal. Each avalanche is characterized by its size S, duration D′ and maximum amplitude for a given threshold level V_c = 〈v〉 + C · σ, where C is a constant and σ corresponds to the standard deviation of the crack velocity signal V_l(t):

2.1

2.2

2.3

The dimension of the global avalanche size S is length, and it corresponds to the extra displacement of the fracture front in comparison to the global advancement of the crack at the velocity equal to the threshold value V_c, during the avalanche duration D′. In order to compare the duration statistics from experiments with very different average velocity 〈v〉, we compute the normalized duration D as D = D′/D₀, where D₀ = r/〈v〉.

3. Avalanches and extreme values statistics

Figure 3 shows the size and duration distribution P(S) and P(D) of the global avalanche for one given fracture experiment. The avalanches were computed from crackling signals V_l(t) computed at various scales l = L/N, with N = [1, 2, 4, 6, 8, 10, 12, 16, 20], while the threshold value was fixed using C =  − 0.1. We observe power-law distributions with exponential cut-offs, both for the size S and duration D. Interestingly, the distributions do not depend on the scale l at which the global quantity V_l(t) is measured (as soon as l is larger than the lateral correlation length scale of the local crack front velocities around 100 μm [14]). Nevertheless, obviously, the statistics drastically improves as l decreases, since the number of crackling signals and thus the number of avalanches detected increases. We have previously reported that the local dynamics of the fracture front does not depend on the average front velocity, for either creep or constant velocity loading conditions [14]. Therefore, we expect a similar behaviour for the global crackling noise signal V_l(t).

Figure 3.

Probability distributions functions of the size S and normalized duration D of global avalanches computed from crackling noise signals measured at various length scales l = L/N, with N = [1, 2, 4, 6, 8, 10, 12, 16, 20], from a single fracture experiment. The global avalanches are detected with a threshold level C = − 0.1. (Online version in colour.)

In figure 4, we represent the probability distribution functions of the size S and normalized duration D of global avalanches detected for the global velocity of the crack front, measured at a length scale l = L/N, with N = 16 for a threshold level C =  − 0.1 for five experiments with very different average velocities, from around 0.02 μm s⁻¹ to around 150 μm s⁻¹ (obtained in different loading conditions). Indeed, we can observe that those distributions—power law up to an exponential cut-off—do not depend on the average crack front velocity. Moreover, we have also checked that we can obtain similar collapses for other values of N and C.

Figure 4.

Probability distributions functions of the size S and normalized duration D of global avalanches detected for five different experiments, performed in different conditions (constant velocity (F) or creep relaxation (R)) leading to very different average crack front velocity, from around 0.02 μm s⁻¹ to around 150 μm s⁻¹. The global velocity is measured at a length scale l = L/16 and for a threshold level with C = − 0.1. (Online version in colour.)

Therefore, we are then justified to accumulate statistics from all our experiments, with very different average front velocities. Moreover, we can also increase the statistics by computing, for each experiment, N = 16 different crackling noise signals, with averaged velocity V_l(t) measured at length scales l = L/16, as done previously in figure 4. This is valuable in order to find reliable scaling exponents and to characterize precisely the distribution cut-offs and their dependence on the threshold value, which is exactly the goal of the analysis shown in the following figures. Indeed, we display the size and duration distributions of the global avalanches, changing systematically the threshold values C≡ [ − 0.3, − 0.1, 0, 0.2, 0.5, 0.8, 1.2], in figure 5. In figure 5a, it is clear that the size distributions of the global avalanches P(S) can be well fitted by a power law with an exponential cut-off S* (represented by dashed black lines). For the range of thresholds used, we observe that the power-law exponent does not evolve, while the cut-off S*decreases systematically as the threshold increases: S* = 128c^−1.9_r with the normalized threshold velocity c_r = V_c/〈v〉 = 1 + Cσ/〈v〉. A similar scaling behaviour is observed for the avalanche duration distributions (top-right) with the cut-off D* = 68c^−3.0_r. Therefore, we could use the values of those cut-offs to collapse the various distributions P(S) and P(D), allowing reliable power-law exponents to be extracted as shown in figure 5c,d,

3.1

and

3.2

Moreover, we also show (inset, figure 5d) that the average avalanche duration of the global avalanches scales with the avalanche size as

3.3

Those scaling exponents for the avalanche size and duration distributions are in remarkable agreement (within the rather large experimental error bars) with theoretical (using functional renormalization group calculations) and numerical predictions considering an interface with a long-range elasticity propagating in a random medium, giving τ = 1.25, α = 1.43 and ξ = 0.58 (see [15–17,46,48,57,58] and references therein). Moreover, it is important to note that the values of those exponents characterizing the scaling behaviour of the global avalanches are different when considering local crack front bursts [12,48]. Our new experimental results clearly confirm that high local velocity clusters and global avalanches measured on crackling time series are different observables with different statistical properties, as already reported in numerical simulations [46,48], and in imbibition experiments [23,55,56]. Finally, we can also remark that those scaling exponents are also different from those reported for a three-dimensional crack front propagating in a bulk artificial rock [59]. This could suggest that the fracture of three-dimensional solids may not be simply reduced to a two-dimensional depinning problem. However, the scaling exponents reported in [59] evolve with the driving velocity, in strong contrast to our measurements, leading to robust scaling exponent values, for different loading conditions, with very different average crack front velocities. Therefore, this discrepancy might simply arise from the fact that the quasi-static assumption of the depinning model is not satisfied in those experiments.

Figure 5.

Distributions of avalanche sizes S (a) and normalized durations D (b), for various threshold values C≡ [ − 0.3, − 0.1, 0, 0.2, 0.5, 0.8, 1.2], where the crackling noise signals have been measured at a scale l = L/16 (where L is the system size), for various experiments with a broad range of average crack front velocities (thanks to different loading conditions). Extracting the evolution of the exponential cut-off of each power-law distributions, S* and D*, for sizes and durations, respectively (as shown in the top insets), we can collapse those distributions P(S) and P(D), when plotting P(S)S^*τ or P(D)D^*α as a function of rescaled observables S/S* or D/D*. The collapses displayed on (c,d) allow us to extract reliable scaling exponents τ = 1.15 ± 0.15 and α = 1.32 ± 0.15. In the inset of (d), we moreover show how the average avalanche duration scales with their average size, 〈D|S〉∼S^ξ with ξ = 0.60 ± 0.15. (Online version in colour.)

To further characterize the statistics of these avalanches, we also examine their maximum amplitudes measured by , where we have renormalized the crackling noise signal by the average crack front velocity value, i.e. V_l(t)/〈v〉, to compare experiments performed at very different average velocities. Figure 6 shows the distributions of the maximum amplitudes of avalanches measured for the same dataset analysed previously (figure 5), e.g. five different experiments with very different average crack front velocities, and for each experiments N = 16 different crackling noise signals (averaged velocity V_l(t), measured at length scales l = L/16) were computed changing systematically the threshold values C≡ [ − 0.3, − 0.1, 0, 0.2, 0.5, 0.8, 1.2]. Strikingly, we observe that the distributions follow a power-law behaviour with an exponential cut-off, independent of the threshold C,

3.4

Moreover, in the inset of figure 6, we also show that the avalanche maximum velocity scales in average with the avalanche duration as

3.5

Finally, using a simple change of variables, we can relate those different scaling exponents, characterizing the statistics of the maximum amplitude of the global avalanche and their duration, as, ϵ = 1 + (α − 1)/β. Based on our previous measurements of α and β, we find ϵ≃1.35, which is consistent with our direct experimental measured value.

Figure 6.

Distributions of the avalanche maximum velocities for various threshold values C≡ [ − 0.3, − 0.1, 0, 0.2, 0.5, 0.8, 1.2], for the same velocities time series V_L/16(t) studied in figure 5. The inset gives the scaling of the average avalanche maximum velocities with the avalanche duration, , with β∼0.90. (Online version in colour.)

It is interesting that the mean field theory of interface depinning predicted a different exponent for the [50]. This is consistent with the other avalanche exponents that also deviate from the mean field predictions. Additionally, the distribution of maximum amplitudes in avalanches of fixed durations was found to follow a scaling form , where the scaling function is determined analytically in terms of Bessel functions [51]. We find that a similar scaling form also holds beyond the mean field approximation, but with different expression for the scaling function . Even though, it is difficult to extract clearly this scaling function from experimental measurements, we were able to compute from our large data set, obtained for different fracture experiments, driven at slow velocities of a few tenths of μm s⁻¹.

Figure 7 shows in inset the distributions P(V_max | D) of maximum velocities of avalanches of various fixed durations (chosen within the scaling range of the 〈V_max〉(D) relation shown in the inset of figure 6). Those distributions can be collapsed by plotting as a function of the maximum amplitude of the global avalanches renormalized by their average amplitude , as shown in the main panel. The resulting scaling function exhibits a power-law tail describing the largest maximum velocities , which—as expected—is very different from the mean field prediction [50]. Interestingly, this scaling function, and in particular its fat-tail, is similar to the distributions of the local front velocities P(v/〈v〉), plotted with black circles in the main panel of figure 7, at the origin of the non-Gaussian Levy statistics of the global front velocity [49].

Figure 7.

Distributions of maximum amplitudes of global avalanches (detected with a clip level C = 0) P(V_max|D) for various fixed durations D-ranges (inset). The main panel demonstrates that these distributions can be collapsed as . The scaling function is well described by the distribution of the corresponding local fracture front velocities P(v/〈v〉), displaying, in particular, a fat-tail at large velocities P(v)∼v^−2.8. (Online version in colour.)

4. Numerical modelling of elastic interface depinning

Since the large-scale avalanche dynamics of our planar crack seems to be well-described by the behaviour of a long-range elastic string in a two-dimensional random medium [15,48], we propose now to study the extremal statistics of the global avalanches, within the framework of such numerical model. So, here, we consider a discretized version of this line model, using periodic boundary conditions, with integer interface heights h_i(t), i = 1…L, where L is the system size and the lateral coordinates x_i of the interface are given by x_i = i. The total force acting on the interface element i is , where the long-range elastic interactions exhibit a 1/x² decay along the interface, F_ext is the external driving force and η is quenched spatially uncorrelated disorder describing toughness fluctuations of the disordered weak plane.

The crackling noise signal is given by , where v_i = θ(F_i), with θ the Heaviside step function; this implies that the local velocity at a given instant in time is either one (the interface is advancing locally) or zero (the interface is locally pinned). The interface is driven with a finite constant spatially averaged velocity 〈V 〉, by imposing F_ext = K(〈V 〉t − 〈h〉), where K describes the stiffness of the specimen-machine system and controls the cut-off avalanche size S* and duration D*, and 〈h〉 is the average interface height. We show here results for L = 32 768 and a slow but finite driving velocity 〈V 〉 = 4/32 768. We have checked that other small 〈V 〉 values yield similar results. To define global avalanches and their maximum amplitude, we set the threshold value to V_c = 〈V 〉; in what follows, we denote by V_max the global maximum velocity of avalanches from which the threshold value V_c has been subtracted.

The main panel of figure 8 shows the scaling of the distributions P(V_max) for all avalanches detected for different values of the stiffness parameter K. We observe a clear power-law behaviour terminated at a K-dependent cut-off, with a power-law exponent ϵ = 1.4 ± 0.1, in quantitative agreement with our experimental measurements. The inset of figure 8 shows the scaling of 〈V_max〉 with D, displaying again a power-law relation similar to the experimental one with β = 0.8 ± 0.1. This scaling exponent is slightly smaller but still in very good agreement within the large error bars with the one measured in our fracture experiments.

Figure 8.

Main panel: Distributions P(V_max) for various K, displaying a power law with ϵ = 1.4 ± 0.1, and a K-dependent cut-off. Inset: Scaling of 〈V_max〉 with D, displaying a power-law with β = 0.8 ± 0.1. (Online version in colour.)

We have also computed the distributions P(V_max|D) of the maximum amplitudes of avalanches for various fixed duration ranges D (taken within the scaling range of the 〈V_max〉(D) relation), as shown in the inset of figure 9. The main panel of figure 9 demonstrates that those distributions can be collapsed as for our experimental data by representing as a function of , in a semi-logarithmic plot. The resulting scaling function exhibits a roughly exponential tail describing the largest maximum velocities. As expected, this scaling function has a different shape than the analytical prediction obtained within the mean field approach [51]. Nevertheless, it is important to note that this scaling function is also different from the experimental one which displays a power-law tail instead of an exponential one.

Figure 9.

The inset shows the distributions P(V_max | D) at various fixed durations D-ranges. The main panel demonstrates that these distribution can be collapsed, when plotting as a function of in a semi-log scale. The scaling function has a clear exponential tail different from the power-law behaviour of the experimental data.

Finally, considering the scaling form observed for the conditional distribution of maximal velocities within avalanches of duration D, and the power-law distributions for the avalanche durations P(D)∼D^−α, we can show that the distribution of the maximum amplitudes of avalanches, , using the change of integration variables , can be written as

4.1

Therefore, this formula shows that the scaling of does not depend on the conditional distribution P(V_max|D) as long as the integral in equation (4.1) converges; thus, it explains why we could observe the same ‘universal’ power-law distributions for the maximum velocities considering all avalanches, , between our experiments and numerical simulations (with a quantitative agreement), despite different underlying conditional distributions P(V_max|D).

5. Conclusion

We have presented an experimental study of the global, spatially averaged velocity of an interfacial crack propagating along a weak disorder plane in a transparent PMMA block. This crackling noise signal is highly intermittent, with power-law distributed avalanches in size, duration and maximum amplitudes. We confirm that the critical depinning of a long-range elastic string propagating in a two-dimensional disordered medium describes quantitatively the scaling behaviour of most of our experimental observables. However, we could also show that such a model fails to reproduce the power-law distribution of the maximum amplitude of avalanches at fixed duration, which may be the signature of the fat-tail power-law statistics of the local crack front velocity. A different approach describing material failure as a correlated percolation process considering crack coalescence [52,53] could lead to such largely power-law distributed local velocity of the crack front [12,14,49]. Therefore, our results suggest that different approaches may be needed to fully describe the intermittent crack front dynamics, as for its complex self-affine morphology [45,47,60]. While at large scales, the long-range elastic line model appears very successful, different approaches such as a percolation-like model [52,53] or thermally activated processes [61–63] could be more relevant to describe material failure at the local scale of the heterogeneities.

Data accessibility

Experimental and numerical data are accessible upon reasonable request to the authors.

Authors' contributions

All authors significantly contributed to this work as a team effort. S.S., K.J.M., R.T. and L.A. designed the study. S.S. and K.T.T. performed the experiments and its analysis. L.L. performed the numerical modelling and its analysis. All authors participated in the analysis and interpretation of the data. S.S. wrote the manuscript first drafted by K.T.T. and L.A. All authors read, revised critically and approved the manuscript.

Competing interests

We declare we have no competing interests.

Funding

This work was supported by the Russian Government with grant no. 14.W03.31.0002, the CNRS with the French/Norwegian International Associated Laboratory (LIA, ‘D-FFRACT’) and by the Academy of Finland through an Academy Research Fellowship (L.L., project no. 268302). We also acknowledge the support of the University of Oslo and the support by the Research Council of Norway through its Centres of Excellence funding scheme, project number 262644, and through the Frinat grant no. 205486.