Acceleration and localization of subcritical crack growth in a natural composite material

Size effect of accelerated seismic release from failures under compression in naturally fractured media

It is crucial to understand the scaling behaviors of acoustic emissions (AEs) preceding damage localization in order to predict failures of brittle solids under compression. Yet, the effect of length scale on the complex interplay between the initiation, propagation, and coalescence of pre-existing fracture networks and corresponding AE behaviors is poorly understood. In this study, we perform laboratory compressional experiments on naturally fractured rocks at four sample sizes from 50 to 300 mm whose strength generally exhibits a finite-size effect. We analyze the time history of AE energy distribution and accelerated seismic release (ASR) until catastrophic failures of the specimens. We find their time evolution towards failure resembles the observations from specimens containing a single fault and highly microstructurally disordered materials. We observe clear evidence that a size effect exists at small AE magnitude, where larger specimens tend to produce a higher proportion of smaller microcracks. However, the AE energy distribution is scale-independent at high energies. Near to failure, the power-law component of the AE energy population is almost stationary. The temporal evolution of the AE activity rate is independent of sample size; instead, there exist fast and slow periods of the AE activity rate that could be attributed to the stress heterogeneities around the fracture network. ASRs are exclusively observed in the AE activity rate and are more general in the lack of criticality. Our interpretation of observed foreshock activities at varying sample sizes in fractured media may have significance for understanding and predicting failures from natural hazards and engineering instabilities.

Prediction of depinning transitions in interface models using Gini and Kolkata indices

The intermittent dynamics of driven interfaces through disordered media and its subsequent depinning for large enough driving force is a common feature for a myriad of diverse systems, starting from mode-I fracture, vortex lines in superconductors, and magnetic domain walls to invading fluid in a porous medium, to name a few. In this work, we outline a framework that can give a precursory signal of the imminent depinning transition by monitoring the variations in sizes or the inequality of the intermittent responses of a system that are seen prior to the depinning point. In particular, we use measures traditionally used to quantify economic inequality, i.e., the Gini index and the Kolkata index, for the case of the unequal responses of precritical systems. The crossing point of these two indices serves as a precursor to imminent depinning. Given a scale-free size distribution of the responses, we calculate the expressions for these indices, evaluate their crossing points, and give a recipe for forecasting depinning transitions. We apply this method to the Edwards-Wilkinson, Kardar-Parisi-Zhang, and fiber bundle model interface with variable interaction strengths and quenched disorder. The results are applicable for any interface dynamics undergoing a depinning transition. The results also explain previously observed near-universal values of Gini and Kolkata indices in self-organized critical systems.

Large-scale heterogeneities can alter the characteristics of compressive failure and accelerated seismic release

Externally stressed brittle rocks fail once the stress is sufficiently high. This failure is typically preceded by a pronounced increase in the total energy of acoustic emission (AE) events, the so-called accelerated seismic release. Yet, other characteristics of approaching the failure point such as the presence or absence of variations in the AE size distribution and, similarly, whether the failure point can be interpreted as a critical point in a statistical physics sense differs across experiments. Here, we show that large-scale stress heterogeneities induced by a notch fundamentally change the characteristics of the failure point in triaxial compression experiments under a constant displacement rate on Westerly granite samples. Specifically, we observe accelerated seismic release without a critical point and no change in power-law exponent ε of the AE size distribution. This is in contrast to intact samples, which exhibit a significant decrease in ε before failure. Our findings imply that the presence or absence of large-scale heterogeneities play a significant role in our ability to predict compressive failure in rock.

Hidden jerk in universal creep and aftershocks

Most materials exhibit creep memory under the action of a constant load. The memory behavior is governed by Andrade's creep law, which also has an inherent connection with the Omori-Utsu law of earthquake aftershocks. Both empirical laws lack a deterministic interpretation. Coincidentally, the Andrade law is similar to the time-varying part of the creep compliance of the fractional dashpot in anomalous viscoelastic modeling. Consequently, fractional derivatives are invoked, but since they lack a physical interpretation, the physical parameters of the two laws extracted from curve fit lack confidence. In this Letter, we establish an analogous linear physical mechanism that underlies both laws and relates its parameters with the material's macroscopic properties. Surprisingly, the explanation does not require the property of viscosity. Instead, it necessitates the existence of a rheological property that relates strain with the first order time derivative of stress, which involves jerk. Further, we justify the constant quality factor model of acoustic attenuation in complex media. The obtained results are validated in light of the established observations.

Seismic events miss important kinematically governed grain scale mechanisms during shear failure of porous rock

Catastrophic failure in brittle, porous materials initiates when smaller-scale fractures localise along an emergent fault zone in a transition from stable crack growth to dynamic rupture. Due to the rapid nature of this critical transition, the precise micro-mechanisms involved are poorly understood and difficult to image directly. Here, we observe these micro-mechanisms directly by controlling the microcracking rate to slow down the transition in a unique rock deformation experiment that combines acoustic monitoring (sound) with contemporaneous in-situ x-ray imaging (vision) of the microstructure. We find seismic amplitude is not always correlated with local imaged strain; large local strain often occurs with small acoustic emissions, and vice versa. Local strain is predominantly aseismic, explained in part by grain/crack rotation along an emergent shear zone, and the shear fracture energy calculated from local dilation and shear strain on the fault is half of that inferred from the bulk deformation.

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

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

A Terahertz Identification Method for Internal Interface Structures of Polymers Based on the Long Short-Term Memory Classification Network

Polymers are used widely in the power system as insulating materials and are essential to the power grid’s security and stability. However, various insulation defects may occur in the polymer., which can lead to severe insulation accidents. Terahertz (THz) detection is a novel non-destructive testing (NDT) method that is able to detect the interface structures inside polymers. The large quantity of information in the THz waveform has potential for the identification of interface types, and the long short-term memory (LSTM) network is one of the most popular artificial intelligence methods for time series data like THz waveform. In this paper, the LSTM classification network was used to identify the internal interfaces of the polymer with the reflected THz pulses of the internal interfaces. The experiment verified that it is feasible to identify and image the void interfaces and impurity interfaces in the polymer using the proposed method.

Sheared granular matter and the empirical relations of seismicity

The frictional instability associated with earthquake initiation and earthquake dynamics is believed to be mainly controlled by the dynamics of fragmented rocks within the fault gauge. Principal features of the emerging seismicity (e.g., intermittent dynamics and broad time and/or energy scales) have been replicated by simple experimental setups, which involve a slowly driven slider on top of granular matter, for example. Yet these setups are often physically limited and might not allow one to determine the underlying nature of specific features and, hence, the universality and generality of the experimental observations. Here, we address this challenge by a numerical study of a spring-slider experiment based on two-dimensional discrete element method simulations, which allows us to control the properties of the granular matter and of the surface of the slider, for example. Upon quasistatic loading, stick-slip-type behavior emerges which is contrasted by a stable sliding regime at finite driving rates, in agreement with experimental observations. Across large parameter ranges for damping, interparticle friction, particle polydispersity, etc., the earthquake-like dynamics associated with the former regime results in several robust scale-free statistical features also observed in experiments. At first sight, these closely resemble the main empirical relations of tectonic seismicity at geological scales. This includes the Gutenberg-Richter distribution of event sizes, the Omori-Utsu-type decay of aftershock rates, as well as the aftershock productivity relation and broad recurrence time distributions. Yet, we show that the correlations associated with tectonic aftershocks are absent such that the origin of the Omori-Utsu relation, the aftershock productivity relation, and Båth's relation in the simulations is fundamentally different from the case of tectonic seismicity. This, we believe, is mainly due to a lack of macroscale relaxation processes that are closely tied to the generation of real aftershocks. We argue that the same is true for previous laboratory experiments.

Quasistatic kinetic avalanches and self-organized criticality in deviatorically loaded granular media

The behavior of granular media under quasistatic loading has recently been shown to attain a stable evolution state corresponding to a manifold in the space of micromechanical variables. This state is characterized by sudden transitions between metastable jammed states, involving the partial micromechanical rearrangement of the granular medium. Using numerical simulations of two-dimensional granular media under quasistatic biaxial compression, we show that the dynamics in the stable evolution state is characterized by scale-free avalanches well before the macromechanical stationary flow regime traditionally linked to a self-organized critical state. This, together with the nonuniqueness and the nonmonotony of macroscopic deformation curves, suggests that the statistical avalanche properties and the susceptibilities of the system cannot be reduced to a function of the macromechanical state. The associated scaling exponents are nonuniversal and depend on the interactions between particles. For stiffer particles (or samples at low confining pressure) we find distributions of avalanche properties compatible with the predictions of mean-field theory. The scaling exponents decrease below the mean-field values for softer interactions between particles. These lower exponents are consistent with observations for amorphous solids at their critical point. We specifically discuss the relationship between microscopic and macroscopic variables, including the relation between the external stress drop and the internal potential energy released during kinetic avalanches.

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.

Prediction of creep failure time using machine learning

A subcritical load on a disordered material can induce creep damage. The creep rate in this case exhibits three temporal regimes viz. an initial decelerating regime followed by a steady-state regime and a stage of accelerating creep that ultimately leads to catastrophic breakdown. Due to the statistical regularities in the creep rate, the time evolution of creep rate has often been used to predict residual lifetime until catastrophic breakdown. However, in disordered samples, these efforts met with limited success. Nevertheless, it is clear that as the failure is approached, the damage become increasingly spatially correlated, and the spatio-temporal patterns of acoustic emission, which serve as a proxy for damage accumulation activity, are likely to mirror such correlations. However, due to the high dimensionality of the data and the complex nature of the correlations it is not straightforward to identify the said correlations and thereby the precursory signals of failure. Here we use supervised machine learning to estimate the remaining time to failure of samples of disordered materials. The machine learning algorithm uses as input the temporal signal provided by a mesoscale elastoplastic model for the evolution of creep damage in disordered solids. Machine learning algorithms are well-suited for assessing the proximity to failure from the time series of the acoustic emissions of sheared samples. We show that materials are relatively more predictable for higher disorder while are relatively less predictable for larger system sizes. We find that machine learning predictions, in the vast majority of cases, perform substantially better than other prediction approaches proposed in the literature.

Crossover from mean-field compression to collective phenomena in low-density foam-formed fiber material

We study the compression of low-weight foam-formed materials made out of wood fibers. Initially the stress-strain behavior follows mean-field like response, related to the buckling of fiber segments as dictated by the random three-dimensional geometry. Our Acoustic Emission (AE) measurements correlate with the predicted number of segment bucklings for increasing strain. However, the experiments reveal a transition to collective phenomena as the strain increases sufficiently. This is also seen in the gradual failure of the theory to account for the stress-strain curves. The collective avalanches exhibit scale-free features both as regards the AE energy distribution and the AE waiting time distributions with both exponents having values close to 2. In cyclic compression tests, significant increases in the accumulated acoustic energy are found only when the compression exceeds the displacement of the previous cycle, which further confirms other sources of acoustic events than fiber bending.

Avalanche Behavior in Creep Failure of Disordered Materials

We present a mesoscale elastoplastic model of creep in disordered materials, which considers temperature-dependent stochastic activation of localized deformation events that are coupled by internal stresses, leading to collective avalanche dynamics. We generalize this stochastic plasticity model by introducing damage in terms of a local strength that decreases, on statistical average, with increasing local plastic strain. The model captures failure in terms of strain localization in a catastrophic shear band concomitant with a finite-time singularity of the creep rate. The statistics of avalanches in the run-up to failure is characterized by a decreasing avalanche exponent τ that, at failure, approaches the value τ=1.5 typical of a critical branching process. The average avalanche rate exhibits an inverse Omori law as a function of time to failure. The distribution of interavalanche times turns out to be consistent with the epidemic-type aftershock sequences (ETAS) model of earthquake statistics.

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.

Universal avalanche statistics and triggering close to failure in a mean-field model of rheological fracture

The hypothesis of critical failure relates the presence of an ultimate stability point in the structural constitutive equation of materials to a divergence of characteristic scales in the microscopic dynamics responsible for deformation. Avalanche models involving critical failure have determined common universality classes for stick-slip processes and fracture. However, not all empirical failure processes exhibit the trademarks of criticality. The rheological properties of materials introduce dissipation, usually reproduced in conceptual models as a hardening of the coarse grained elements of the system. Here, we investigate the effects of transient hardening on (i) the activity rate and (ii) the statistical properties of avalanches. We find the explicit representation of transient hardening in the presence of generalized viscoelasticity and solve the corresponding mean-field model of fracture. In the quasistatic limit, the accelerated energy release is invariant with respect to rheology and the avalanche propagation can be reinterpreted in terms of a stochastic counting process. A single universality class can be defined from such analogy, and all statistical properties depend only on the distance to criticality. We also prove that interevent correlations emerge due to the hardening-even in the quasistatic limit-that can be interpreted as "aftershocks" and "foreshocks."

Triggering Processes in Rock Fracture

We study triggering processes in triaxial compression experiments under a constant displacement rate on sandstone and granite samples using spatially located acoustic emission events and their focal mechanisms. We present strong evidence that event-event triggering plays an important role in the presence of large-scale or macrocopic imperfections, while such triggering is basically absent if no significant imperfections are present. In the former case, we recover all established empirical relations of aftershock seismicity including the Gutenberg-Richter relation, a modified version of the Omori-Utsu relation and the productivity relation-despite the fact that the activity is dominated by compaction-type events and triggering cascades have a swarmlike topology. For the Gutenberg-Richter relations, we find that the b value is smaller for triggered events compared to background events. Moreover, we show that triggered acoustic emission events have a focal mechanism much more similar to their associated trigger than expected by chance.

Predicting sample lifetimes in creep fracture of heterogeneous materials

Materials flow-under creep or constant loads-and, finally, fail. The prediction of sample lifetimes is an important and highly challenging problem because of the inherently heterogeneous nature of most materials that results in large sample-to-sample lifetime fluctuations, even under the same conditions. We study creep deformation of paper sheets as one heterogeneous material and thus show how to predict lifetimes of individual samples by exploiting the "universal" features in the sample-inherent creep curves, particularly the passage to an accelerating creep rate. Using simulations of a viscoelastic fiber bundle model, we illustrate how deformation localization controls the shape of the creep curve and thus the degree of lifetime predictability.

Record-breaking events during the compressive failure of porous materials

An accurate understanding of the interplay between random and deterministic processes in generating extreme events is of critical importance in many fields, from forecasting extreme meteorological events to the catastrophic failure of materials and in the Earth. Here we investigate the statistics of record-breaking events in the time series of crackling noise generated by local rupture events during the compressive failure of porous materials. The events are generated by computer simulations of the uniaxial compression of cylindrical samples in a discrete element model of sedimentary rocks that closely resemble those of real experiments. The number of records grows initially as a decelerating power law of the number of events, followed by an acceleration immediately prior to failure. The distribution of the size and lifetime of records are power laws with relatively low exponents. We demonstrate the existence of a characteristic record rank k(*), which separates the two regimes of the time evolution. Up to this rank deceleration occurs due to the effect of random disorder. Record breaking then accelerates towards macroscopic failure, when physical interactions leading to spatial and temporal correlations dominate the location and timing of local ruptures. The size distribution of records of different ranks has a universal form independent of the record rank. Subsequences of events that occur between consecutive records are characterized by a power-law size distribution, with an exponent which decreases as failure is approached. High-rank records are preceded by smaller events of increasing size and waiting time between consecutive events and they are followed by a relaxation process. As a reference, surrogate time series are generated by reshuffling the event times. The record statistics of the uncorrelated surrogates agrees very well with the corresponding predictions of independent identically distributed random variables, which confirms that temporal and spatial correlation in the crackling noise is responsible for the observed unique behavior. In principle the results could be used to improve forecasting of catastrophic failure events, if they can be observed reliably in real time.