Predicting mining collapse: Superjerks and the appearance of record-breaking events in coal as collapse precursors

Record statistics of fracture in the random spring network model

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

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.

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.

Acoustic Emission Spectroscopy: Applications in Geomaterials and Related Materials

As a non-destructive testing technology with fast response and high resolution, acoustic emission is widely used in material monitoring. The material deforms under stress and releases elastic waves. The wave signals are received by piezoelectric sensors and converted into electrical signals for rapid storage and analysis. Although the acoustic emission signal is not the original stress signal inside the material, the typical statistical distributions of acoustic emission energy and waiting time between signals are not affected by signal conversion. In this review, we first introduce acoustic emission technology and its main parameters. Then, the relationship between the exponents of power law distributed AE signals and material failure state is reviewed. The change of distribution exponent reflects the transition of the material’s internal failure from a random and uncorrelated state to an interrelated state, and this change can act as an early warning of material failure. The failure process of materials is often not a single mechanism, and the interaction of multiple mechanisms can be reflected in the probability density distribution of the AE energy. A large number of examples, including acoustic emission analysis of biocemented geological materials, hydroxyapatite (human teeth), sandstone creep, granite, and sugar lumps are introduced. Finally, some supplementary discussions are made on the applicability of Båth’s law.

The duration-energy-size enigma for acoustic emission

Acoustic emission (AE) measurements of avalanches in different systems, such as domain movements in ferroics or the collapse of voids in porous materials, cannot be compared with model predictions without a detailed analysis of the AE process. In particular, most AE experiments scale the avalanche energy E, maximum amplitude Amax and duration D as E ~ A_max^x and A_max ~ D^χ with x = 2 and a poorly defined power law distribution for the duration. In contrast, simple mean field theory (MFT) predicts that x = 3 and χ = 2. The disagreement is due to details of the AE measurements: the initial acoustic strain signal of an avalanche is modified by the propagation of the acoustic wave, which is then measured by the detector. We demonstrate, by simple model simulations, that typical avalanches follow the observed AE results with x = 2 and ‘half-moon’ shapes for the cross-correlation. Furthermore, the size S of an avalanche does not always scale as the square of the maximum AE avalanche amplitude A_max as predicted by MFT but scales linearly S ~ A_max. We propose that the AE rise time reflects the atomistic avalanche time profile better than the duration of the AE signal.

Avalanche criticality during ferroelectric/ferroelastic switching

Field induced domain wall displacements define ferroelectric/ferroelastic hysteresis loops, which are at the core of piezoelectric, magnetoelectric and memristive devices. These collective displacements are scale invariant jumps with avalanche characteristics. Here, we analyse the spatial distribution of avalanches in ferroelectrics with different domain and transformation patterns: Pb(Mg_1/3Nb_2/3)O₃–PbTiO₃ contains complex domains with needles and junction patterns, while BaTiO₃ has parallel straight domains. Nevertheless, their avalanche characteristics are indistinguishable. The energies, areas and perimeters of the switched regions are power law distributed with exponents close to predicted mean field values. At the coercive field, the area exponent decreases, while the fractal dimension increases. This fine structure of the switching process has not been detected before and suggests that switching occurs via criticality at the coercive field with fundamentally different switching geometries at and near this critical point. We conjecture that the domain switching process in ferroelectrics is universal at the coercive field.

While classical approaches rely on the study of individual ferroelectric domain wall movement on long time scales, the authors consider collective movements of domain walls during short time scales, characterized by discrete jumps, as indicators of avalanches on a broad range of scales.

Impact-induced transition from damage to perforation

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

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

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

Avalanches in Compressed Sandstone: Crackling Noise under Confinement

The acoustic emission, AE, from avalanches of local cracks and microstructural changes of sandstone under confined compression have been reported. These avalanches soften the underlying minerals and play a key role as indicators for the prediction of geo-engineering disasters, such as mining collapses, rock outbursts caused by high ground stress, and man-made quakes by fracking. Compressed sandstone is a model material for the investigation of avalanches. The avalanche energies, amplitudes, and waiting times show the probability distributions that allow us to distinguish between three compression stages; namely, (I) pre-failure, (II) correlated failure, and (III) post-failure. The energy of stage I and stage II is power-law distributed and scale invariant, while post-failure experiments show power laws with high exponential damping (friction). The scaling behavior is close to the predictions of a mean-field (MF) model (stage II) and a force-integrated mean-field model (stage I). Confinement shifts the value of the energy exponent closer to the MF prediction. Omori’s law and waiting time distributions are independent of stress during the compression; their scaling exponents are very similar to those found in seismological studies.

Criticality in failure under compression: Acoustic emission study of coal and charcoal with different microstructures

A systematic study of acoustic emission avalanches in coal and charcoal samples under slow uniaxial compression is presented. The samples exhibit a range of organic composition in terms of chemical elements as well as different degrees of heterogeneity in the microstructure. The experimental analysis focuses on the energies E of the individual acoustic emission events as well as on the time correlations between successive events. The studied samples can be classified into three groups. The more homogeneous samples (group I) with pores in the micro and nanoscales, with signatures of hardening effects in the stress-strain curves, exhibit the cleanest critical power-law behavior for the energy distributions g(E)dE∼E^{-ε}dE with a critical exponent ε=1.4. The more heterogeneous samples with voids, macropores, and granular microstructures (group III), show signatures of weakening effects and a larger effective exponent close to the value ε=1.66, but in some cases truncated by exponential damping factors. The rest of the samples (group II) exhibit a mixed crossover behavior still compatible with an effective exponent ε=1.4 but clearly truncated by exponential factors. These results suggest the existence of two possible universality classes in the failure of porous materials under compression: one for homogeneous samples and another for highly heterogeneous samples. Concerning time correlations between avalanches, all samples exhibit very similar waiting time distributions although some differences for the Omori aftershock distributions cannot be discarded.

Acoustic Emission from Porous Collapse and Moving Dislocations in Granular Mg-Ho Alloys under Compression and Tension

We identified heterogeneous Mg-Ho alloys as an ideal material to measure the most extensive acoustic emission spectra available. Mg-Ho alloys are porous and show a high density of dislocations, which slide under external tension and compression. These dislocations nucleate near numerous heterogeneities. Two mechanisms compete under external forcing in the structural collapse, namely collapsing holes and the movements of dislocations. Their respective fingerprints in acoustic emission (AE) measurements are very different and relate to their individual signal strengths. Porous collapse generates very strong AE signals while dislocation movements create more but weaker AE signals. This allows the separation of the two processes even though they almost always coincide temporarily. The porous collapse follows approximately mean-field behavior (ε = 1.4, τ' = 1.82, α = 2.56, x = 1.93, χ = 1.95) with mean field scaling fulfilled. The exponents for dislocation movement are greater (ε = 1.92, τ' = 2.44, α = 3.0, x = 1.7, χ = 1.42) and follows approximately the force integrated mean-field predictions. The Omori scaling is similar for both mechanisms. The Bath's law is well fulfilled for the porous collapse but not for the dislocation movements. We suggest that such 'complex' mixing behavior is dominant in many other complex materials such as (multi-) ferroics, entropic alloys and porous ferroelastics, and, potentially, homogeneous materials with the simultaneous appearance of different collapse mechanisms.

Avalanche mixing and the simultaneous collapse of two media under uniaxial stress

Avalanches in coal and sandstone samples under common uniaxial stress serve as a model for mixing of avalanche exponents in ceramics, multiferroics, and alloys. The two media are sandwiched together and subjected to common uniaxial stress using high- and low-stress compression. Each medium collapses individually through avalanches that often coincide with secondary avalanches into the other medium. The total avalanche time sequence allows a detailed investigation of the mixing by superposition and delayed coincidence. Correlations can be described by an inter-media Båth's law.

Compressive Failure as a Critical Transition: Experimental Evidence and Mapping onto the Universality Class of Depinning

Acoustic emission (AE) measurements performed during the compressive loading of concrete samples with three different microstructures (aggregate sizes and porosity) and four sample sizes revealed that failure is preceded by an acceleration of the rate of fracturing events, power law distributions of AE energies and durations near failure, and a divergence of the fracturing correlation length and time towards failure. This argues for an interpretation of compressive failure of disordered materials as a critical transition between an intact and a failed state. The associated critical exponents were found to be independent of sample size and microstructural disorder and close to mean-field depinning values. Although compressive failure differs from classical depinning in several respects, including the nature of the elastic redistribution kernel, an analogy between the two processes allows deriving (finite-) sizing effects on strength that match our extensive data set. This critical interpretation of failure may have also important consequences in terms of natural hazards forecasting, such as volcanic eruptions, landslides, or cliff collapses.

Time-dependent fracture under unloading in a fiber bundle model

We investigate the fracture of heterogeneous materials occurring under unloading from an initial load. Based on a fiber bundle model of time-dependent fracture, we show that depending on the unloading rate the system has two phases: for rapid unloading the system suffers only partial failure and it has an infinite lifetime, while at slow unloading macroscopic failure occurs in a finite time. The transition between the two phases proved to be analogous to continuous phase transitions. Computer simulations revealed that during unloading the fracture proceeds in bursts of local breakings triggered by slowly accumulating damage. In both phases the time evolution starts with a relaxation of the bursting activity characterized by a universal power-law decay of the burst rate. In the phase of finite lifetime the initial slowdown is followed by an acceleration towards macroscopic failure where the increasing rate of bursts obeys the (inverse) Omori law of earthquakes. We pointed out a strong correlation between the time where the event rate reaches a minimum value and of the lifetime of the system which allows for forecasting of the imminent catastrophic failure.

Experimental Evidence of Accelerated Seismic Release without Critical Failure in Acoustic Emissions of Compressed Nanoporous Materials

The total energy of acoustic emission (AE) events in externally stressed materials diverges when approaching macroscopic failure. Numerical and conceptual models explain this accelerated seismic release (ASR) as the approach to a critical point that coincides with ultimate failure. Here, we report ASR during soft uniaxial compression of three silica-based (SiO_{2}) nanoporous materials. Instead of a singular critical point, the distribution of AE energies is stationary, and variations in the activity rate are sufficient to explain the presence of multiple periods of ASR leading to distinct brittle failure events. We propose that critical failure is suppressed in the AE statistics by mechanisms of transient hardening. Some of the critical exponents estimated from the experiments are compatible with mean field models, while others are still open to interpretation in terms of the solution of frictional and fracture avalanche models.