Systematic advances in the resolution and analytical interpretation of acoustic emission (AE) spectroscopy have, over the last decade, allowed for extensions into novel fields. The same dynamic failure patterns, which have been identified in earthquakes, magnetism, and switching of ferroelastic and ferroelectric materials, are shown, in this paper, to be equally important in medicine, and minerals, in the geological context, to give just two examples. In the first application, we show that biological samples, i.e., kidney stones, can be analyzed with acoustic emission and related to the progression of mechanical avalanches. Discrepancies between strong and weak AE signals are shown to have separate avalanche exponents for a urate kidney stone, with evidence of slight multi-branching. It is proposed that investigations of this nature can be adopted to the field of medicine, and in the case of kidney stones, can provide a blueprint for selecting ideal combinations of energy and frequency to instigate their destruction. In a second example, porous geological material failure is shown to proceed equally in avalanches, and precursors to catastrophic failure can be detected via AE. Warning signs of impeding macroscopic collapse, e.g., in mining activities, show systematic evolution of energy exponents. Ultimately, this behavior is a result of geological processes, man-made bio-mineralization, or the burning of carbon inclusions, creating pores and holes, causing cracks, and accelerating their interactions.

Acoustic emissions (AEs), are the elastic waves produced within a material, resulting from irreversible structural changes, stepwise phase transitions, cracks, domain wall motion, etc.1 Fundamentally, they are traveling elastic waves hitting the surface of the material. When a material emits the elastic waves, the waves are received by piezoelectric transducers mounted on the surface of the sample and stored as electrical signals.2,3 Often, local structural changes follow avalanche behavior; they generate new singularities, proliferate, split, and lead to complex geometrical patterns. The behavior is well known in snow avalanches, the pathway of lightning flashes during storms, and the domain patterns formed in metals upon quenching from high temperatures. The emitted signals during such singular events, when they occur over a shared time window, are considered “jerks.” The AE spectrum, therefore, consists of a series of jerks. When jerks arise from avalanches, they follow the predicted behavior of avalanche dynamics and crackling noise emitters.2 In AE spectroscopy, the time window that the emitted signals share varies drastically, depending on the material system. Their amplitudes, energies, and duration are typically power law distributed. This power law dependence implies that the dynamic behavior is scale invariant and ubiquitous.

The same dynamics are observed across a range of disciplines. For example, neuronal avalanches in zebrafish larvae follow the same dynamics as global earthquakes, compressed sugar lumps, pico-scale earthquakes, and fluctuations in the stock market before a collapse.2,4,5 Studying the “crackling noise” dynamics in porous materials such as kidney stones can lead to new insights into non-invasive kidney stone removal procedures.6 Similarly, the study of crackling noise dynamics performed on minerals in the geological context can provide new insights into the early warning signs of impending macroscopic collapse, e.g., in mining activities, which show systematic evolution of energy exponents.

Breaking an arm or cracking a tooth can lead to irreversible damage of the arm or the tooth. The arm bone and the tooth may split. Similarly, minerals in a geological setting may crumble under stress. While small elastic deformations are well understood and elegantly described by appropriate elastic moduli, the bigger anharmonic effects are much less understood. How does a bone break and how does a tooth split? How do minerals under geological conditions transform? What happens to the large class of ferroelastic materials and minerals, where substantial stresses lead to changes in their microstructure without further destruction? How does porosity “soften” such materials?

This paper is written in honor of Cheetham, who investigated such effects in metal–organic materials. Early research in this field set the stage for developments in metalorganic frameworks (MOFs), and hybrid organic–inorganic perovskites (HOIPs), which show, again, that material scientists occasionally mimic nature’s best devices.7,8

Organic biological samples are often harder to characterize than inorganic materials, and, in general, their failure mechanisms are not well understood. One of the most well-studied biological materials is human teeth, which consist of an outer layer, commonly known as enamel. Enamel is composed of mostly hydroxyapatite crystals linked in prisms, the hardest mineral in the body, which protect the teeth. Below the enamel is dentin, which is composed of a mixture of hydroxyapatite crystals and organic proteins, which protect the pulp chamber.9 Teeth crack first in small segments, then cracks bifurcate, or trifurcate, proceeding along intricate pathways, which compromises the mechanical integrity.10 Such a crack forms one or many avalanches. Initial cracks are very common, and most people live with partially broken teeth, without being aware of the damage. Dental implants gained popularity in the 1960s when the biocompatibility of titanium alloys was realized. While titanium implant failure was relatively rare, the main failure mechanism was attributed to metal fatigue from high cyclic loading. Deformation at the screw interface generates dislocations and stress concentrations, resulting in cracks. The cracks typically follow the thread, and when the crack encircles, the entirety of the implant typically fails in a tear-like fashion.11 In the early 2000s yttrium-stabilized tetragonal zirconia polycrystalline (Y-TZP) implants began to replace their titanium predecessor. The high temperature tetragonal phase of zirconia can be stabilized down to room temperature by changing the concentration of yttria. When cracks form in Y-TZP, they trigger a ferroelastic tetragonal–monoclinic phase transition, accompanied by a volume increase of 4.5%, which closes the crack tip, preventing (or at least delaying) further propagation and avalanche effects.11 The study of failure mechanisms and avalanche dynamics in human teeth has greatly advanced dental implant technologies and has proven that in-depth studies of material failure in biological systems can greatly impact the medical field, as well as improve our understanding of the underlying physical failure mechanisms.10 

Neural activities are also known to proceed in avalanches,12 and neuromorphic computation makes use of these avalanches for fast switches in neural computers.13,14 Fracking, the extraction of hydrocarbons by small explosions, and breaking shales next to bore holes, equally produces local avalanches, with very well-defined elastic properties.15 The collapse of porous materials leads to the collapse of holes in a highly correlated way, again following the mechanism of avalanche propagation.16–24 

Many of these developments were first stimulated independently by the discovery of Barkhausen noise and by systematic investigations of earthquakes, mainly in Japan.25–27 This research has permeated the field of material sciences, particularly in metallurgy, where collapse processes often happen on a very short time scale.28–30 Geotectonic processes, on the other hand, are perceived as slow; an idea encapsulated in the colloquial expression of “long geological time scales” and hence thought to be not suitable for an avalanche analysis, unless they occur during earthquakes. This view has dramatically changed because of improvements in experimental facilities, allowing closer identification of geological time dependences, which occur simultaneously on many different time scales. Long-term time evolution is typically constituted by a long sequence of short singular events, and research has partially shifted from the continuous description of geological processes to the analysis of short singular events. The investigation of “rare events” became prominent in mineralogy31,32 and medical research33–35 after it became a key research field in metallurgy,28,36 material sciences,20,37,38 and solid-state physics.5,39,40 It is the purpose of this paper to review some of the key elements in this research field, and to propose its application in two new systems—kidney stones and porous materials—in the geological context.

Experimental approaches to observe the dynamics of avalanches range from magnetic measurements to electrical depolarization currents in ferroelectrics and optical observations of crack patterns and the determination of fractal dimensions.41–43 Some experiments are conducted under in vivo conditions, where the actual changes of domain structures are determined; others are post-mortem analyses where, for example, the fractal dimensions of cracks are measured.16 All methods have advantages and disadvantages and evolved for the purpose of a specific research activity.

Over the past decade came the desire to measure the main avalanche parameters with the highest possible accuracy. The current method of choice is acoustic emission (AE) spectroscopy. When a material releases elastic waves, the waves are received by piezoelectric transducers on the surface and stored as electrical signals.2,3,32 The typical experimental arrangement for acoustic emission spectroscopy is shown in Fig. 1. The acoustic waves emitted by a sample during changing fields are detected by piezoelectric receivers and analyzed using straightforward amplification devices. A simple summary is found in Refs. 24 and 28.

FIG. 1.

Schematic representation of the composition of an avalanche signal in acoustic emission experiments reproduced with permission from Salje et al., Appl. Sci. 11, 8801 (2021). Copyright 2021 Author(s), licensed under a Creative Commons Attribution 4.0 License. An example of an emitted elastic wave is shown below the setup with a breakdown of the various parameters i.e., duration, rise-time, amplitude, and threshold.

FIG. 1.

Schematic representation of the composition of an avalanche signal in acoustic emission experiments reproduced with permission from Salje et al., Appl. Sci. 11, 8801 (2021). Copyright 2021 Author(s), licensed under a Creative Commons Attribution 4.0 License. An example of an emitted elastic wave is shown below the setup with a breakdown of the various parameters i.e., duration, rise-time, amplitude, and threshold.

Close modal

Figure 1 shows a schematic representation of the composition of an avalanche signal in AE experiments. The diameter of the detector is 10 mm, which often covers a large part of the sample. During local switching, an avalanche emits a strain signal, which propagates through the sample and is eventually measured by the detector. During propagation, the sample signal is modified by elastic wave reflections, which are propagated by surfaces, and scattering on lattice imperfections, before being reconstructed into a digital waveform. The profile of a source function is changed by T(t), the so-called transfer function. The measured AE amplitude profile A(t) of the energy profile E(t) is the convolution of the source function with the transfer function.16,44,45 A catalog of avalanche parameters and characteristics, such as the energy of an emitted signal, its amplitude, duration, and correlation with other AEs, and the probability of pre-shocks and aftershocks, mimic those found in recordings of earthquakes. This supports the claim that many of these characteristics are scale invariant.

The frequency range of AE is very large and extends to several MHz, while seismometers are typically limited to frequencies below 100 Hz. The time resolution is hence very good in “lab-quakes” (25 ns) and rather poor in earthquakes (∼1 h).46 Here, we focus on the energy E of avalanche movements (typically in the attojoule range), which follows the power-law distributed probability p(E) of the form

p(E)EEεEmin1ε;EEmin.
(1)

The power law exponent, ε, is expected to range between 1.3 and 3, overlapping with typical mean-field exponents of 43 and 53 (Ref. 6). The estimator for the exponent ε is given by the maximum likelihood method (ML), which also allows for the analysis of mixed processes with different exponents.47 Analysis of crackling noise using the maximum-likelihood method gives an estimated probability, to trigger a jerk at any possible energy minimum. The mathematical description is given by Eq. (2):

11ε=1Nj=1,NlnEjE0.
(2)

Here, ε is the energy exponent, N′ is the size of the Ej summation, Ej is the jerk signal energy, and E0 is the minimum energy of the spectrum. Crackling noise under compression and external electric or magnetic fields has been observed in many physical systems. Typical examples include the compression of SiO2 based materials,18 domain walls, and topological defects in ferroelastic and ferroelectric materials,48 the core–mantle boundary,49 dislocation motion in single crystals,50–52 detwinning/twinning in porous Ti–Ni and Mg–Ho alloys and steel,22,28,53,54 debris flow,55–57 plasticity,58 and fracture.59 Avalanche dynamics can even be observed 10 parsecs away on the young M dwarf AU microscopii, whose dust clouds form from avalanches of submicron dust being repelled by the host star’s wind.60 

Power law exponents are often linked to the underlying failure mechanisms taking place. For example, dislocation motion will typically have very high energy exponents (ε = 1.9; Refs. 28 and 61) and are typically received as “weak” AE signals, while pore collapse and intergranular motion are seen as “strong” signals, with exponents ranging from ε = 1.3 to ε = 1.6.

Urolithiasis, i.e., having kidney stones, has a strong correlation with the region of residence, diet, race, season, and sex. The age distribution is bimodal for women, with peaks at 35 and 55 years of age, and unimodal for men, with a peak at 30.62 Many correlations of this type, however, are not well characterized. There are obvious interplays between race, region of residence, diet, socioeconomic class etc., making it difficult to isolate which of these factors are really contributing to the incidence rate. Dietary habits have the strongest impact on incidence rate, specifically high protein and salt intake, coupled with high consumption of carbonated beverages with high fructose corn syrup, leads to a significant increase in calcium oxalate renal stone formation, which is among the most common stone types.63–65 

There are several kidney stone removal procedures and techniques, most of which rely on the mechanical breakdown of the kidney stone in vivo. Surprisingly, not much is known about the composition or the physical breakdown mechanics of kidney stones. In this section, we show an example of how to analyze such materials using acoustic emission.

A kidney stone provided by Addenbrookes Hospital Urology Department (Addenbrookes Hospital, Hills Rd, Cambridge CB2 0QQ) was systematically loaded using a piezoelectric transducer with a small amount of silicon grease (sensor coupling agent, Vallen-Systeme GmbH) in direct contact with the sample surface. The kidney stone was irregular in shape and had a complex microstructure, as shown in Fig. 8. The kidney stone was roughly 5 × 3 × 3 mm3 in shape and was reduced to a fine powder by the end of the experiment. Progressive loading was applied to the sample using a continuous water drip hanging from a lever that was attached to the transducer. Before the start of the experiment, a noise measurement was performed for 10 min to determine the baseline noise (Fig. 9). The baseline noise was measured to be 100 aJ and 100 µV. The emitted acoustic emissions were transferred to the AMSY-6 AE- measurement system (Vallen-Systeme GmbH). The selected kidney stone is of type IIIc according to the Daudon scale66 and was measured in the frequency range 100–1800 kHz, with a sampling rate of 25 ns. This type of kidney stone is classified as a urate salt, which forms as a consequence of low urine pH, which facilitates the precipitation of urate from solution.66–68 

The Daudon classification system is a useful tool in identifying and diagnosing the type of kidney stone morphology present. However, chemically, the kidney stones are not always accurately described by their assigned class because they are typically a combination of stone classes.67,68 The chemical analysis of the kidney stone used in the present study confirmed that it is a mixture of 34% oxalate and 66% CaP.

Figure 2 shows a standard statistical analysis highlighting five power law exponents. Figure 2(a) is a jerk spectrum showing amplitude (μV) vs time (s). Alternatively, this could be energy (aJ) vs time (s). While the sample is being continuously loaded, elastic waves are emitted and received by the transducers, which are converted into waveforms with associated energies and amplitudes. Sharp bundles of AE signals are referred to as jerks in this context and are shaded in gray in Fig. 2(a). Each point in Fig. 2(a) above a certain energy level corresponds to an individual AE signal with an associated amplitude, duration, and rise-time of the kind shown in Fig. 3. Noise signals are also common and marked by noticeably low values of energy and amplitude. Most noise signals can be filtered out by selecting the appropriate threshold, which is determined by performing a noise experiment. The noise experiment measures the sample with zero load for a minimum of 10 min, and establishes the baseline energy and amplitude, which varies greatly, depending on the sample and laboratory conditions.

FIG. 2.

(a) Jerk spectrum where each point is a single AE of the form shown in Fig. 3. The y-axis represents the peak amplitude for each AE and the x-axis is the elapsed time from the start of loading. Each individually shaded region represents a jerk. (b) Maximum likelihood analysis which analyses energy values from each AE. The y-intercept of the horizontal fit in red provides the energy power-law exponent ε. (c) Amplitude probability distribution function, where the slope of the fit provides the amplitude power-law exponent τ. (d) Duration probability distribution function where the slope of the fit provides the duration power-law exponent α. (e) Amplitude – duration scaling where points below the noise threshold (100 µV) are shown in blue. The slope of the red line provides the exponent χ = 1.5 which is the expected value from theoretical calculations. (f) Energy – amplitude scaling where, again, points below the noise threshold (100 µV) are shown in blue and the slope of the fit provides the power-law exponent x.

FIG. 2.

(a) Jerk spectrum where each point is a single AE of the form shown in Fig. 3. The y-axis represents the peak amplitude for each AE and the x-axis is the elapsed time from the start of loading. Each individually shaded region represents a jerk. (b) Maximum likelihood analysis which analyses energy values from each AE. The y-intercept of the horizontal fit in red provides the energy power-law exponent ε. (c) Amplitude probability distribution function, where the slope of the fit provides the amplitude power-law exponent τ. (d) Duration probability distribution function where the slope of the fit provides the duration power-law exponent α. (e) Amplitude – duration scaling where points below the noise threshold (100 µV) are shown in blue. The slope of the red line provides the exponent χ = 1.5 which is the expected value from theoretical calculations. (f) Energy – amplitude scaling where, again, points below the noise threshold (100 µV) are shown in blue and the slope of the fit provides the power-law exponent x.

Close modal
FIG. 3.

Individual waveforms showing amplitude (μV) vs burst signal duration (μs) from the kidney stone sample with burst signal energy (a) 749, (b) 567, and (c) 363 aJ. Each individual waveform corresponds to the low energy region of Fig. 2(b) (E ≤ 1000 aJ) i.e., where ε = 1.8.

FIG. 3.

Individual waveforms showing amplitude (μV) vs burst signal duration (μs) from the kidney stone sample with burst signal energy (a) 749, (b) 567, and (c) 363 aJ. Each individual waveform corresponds to the low energy region of Fig. 2(b) (E ≤ 1000 aJ) i.e., where ε = 1.8.

Close modal

Figure 2(b) shows a maximum likelihood analysis, which takes points from Fig. 2(a), or rather, its energy analog, and describes the likelihood of observing an avalanche of a certain energy (given by the x–axis). Figure 2(b) provides the power-law exponent, ε, and for this experiment, there are two distinct exponents: ε = 1.8 and ε = 1.6. The extremely low energy regime (100 aJ) in Fig. 2(b) shows an increase in the ML exponent as the energy approaches zero, rather than a decrease, which would be expected from theoretical calculations.47 The increase in the ML exponent indicates that there is a high probability of finding noise signals in this region that are usually suppressed, by choosing a higher energy threshold. In contrast, a less noisy system would not have a higher probability of finding a signal in this region, and, therefore, the ML exponent would tend toward zero as the energy approached zero.

Similar to Ref. 28, there are some discrepancies between “strong” and “weak” AE signals. These discrepancies are commonly referred to as “wild” and “mild” signals in the realm of ferroelectrics13 and are marked by a sharp increase or decrease in ε. This behavior is indicative of power-law mixing, in which material systems will show two distinct exponents.47Figure 2(b) shows classic power–law mixing with distinct energy exponents of ε = 1.6, and ε = 1.8. The way to discriminate between these two types of signals is to examine specific waveforms and establish the noise level. The noise for this experiment was measured to be 100 aJ, as shown in the  Appendix (Fig. 9). Figure 3 shows individual waveforms, i.e., points in Fig. 2(a) between 100 and 1000 aJ.

Some signals arise from the noise just above 100 aJ, but the majority of signals have energies between 100 and 1000 aJ and are “weak” AE signals for this experiment. This fact is supported by the waveforms shown in Fig. 3, with the individual AE signals in the low-energy regime. Therefore, the low-energy exponent represents low-energy jerks, lending support to the fact that this kidney stone has two real exponents—ε = 1.8 for mild signals and ε = 1.6 for wild signals (similar to Ref. 50).

Figure 2(c) shows the probability distribution function (PDF) of AE burst peak amplitudes. Each point in Fig. 2(a) has an associated waveform of the kind shown in Figs. 3(a)3(c). Each wave has a peak amplitude, and Fig. 2(c) shows the probability distribution of those emitted waves. Aside from looking at individual waveforms, the way to distinguish between noise signals and real signals is to examine the PDF. If the probability is much higher than the bulk of higher amplitude signals, it is a good indicator that the region in question represents noise signals. For example, Fig. 2(c) shows an abrupt change in the slope around peak amplitudes of 95μV. The slope that represents the exponent τ is artificially inflated below this value by noise signals that are not related to avalanches. Markers such as this one are important when studying extremely low energy (1000 aJ AEs) because they can serve as a tool to discriminate between noise and very low energy signals.

Duration is the total time between the first threshold crossing and the last threshold crossing in an emitted acoustic wave, i.e., the x-axis of Fig. 3. The waves shown in Fig. 3 are examples of classic singular acoustic emissions. However, these singular waves can cause aftershocks, which means that multiple acoustic waves will not cross the minimum threshold, resulting in the inflated duration shown in Fig. 4.

FIG. 4.

AE signals from the kidney stone sample showing amplitude in μV vs time in μs with peak amplitude and duration: (a) A = 1190.4 µV, D = 62 631.7 µs, (b) A = 339.5 µV, D = 57 051 µs, and (c) A = 138.5 µV, D = 61 038.2 µs.

FIG. 4.

AE signals from the kidney stone sample showing amplitude in μV vs time in μs with peak amplitude and duration: (a) A = 1190.4 µV, D = 62 631.7 µs, (b) A = 339.5 µV, D = 57 051 µs, and (c) A = 138.5 µV, D = 61 038.2 µs.

Close modal

While Figs. 4(a)4(c) are each individual points in the energy spectra, they should be stored as multiple points—each with a shorter duration. However, because there is no minimum threshold crossing, multiple avalanches are coupled together, resulting in very high values of duration. Figure 2(d) shows the duration PDF for this experiment, and only the non-linear regime accurately reflects single events, i.e., where α = 2.8. The regime with α = 1.5 represents bundled events, as shown in Fig. 4, which artificially inflate the duration. Excessively long durations, i.e., where α approaches 1.5, can, however, provide some useful information about aftershocks. Tangent lines can be drawn across the curve for D ≤ 20 000 µs, and the slopes would vary from 1.5 to 2, until D ≤ 10 000 µs, where the slope would continuously change from 2 to 3. This artifact can be avoided by increasing the threshold. However, the trade-off is that the mild signals are often filtered out as a result of increasing the threshold.

Figure 2(e) shows the amplitude–duration scaling and corresponds to the power-law exponent χ. It is important to note that χ is constrained to 1.5, the predicted value from mean-field theory, not as a data fit. Nevertheless, this line fits the data for large amplitudes at any given duration. Low- amplitude signals correspond to bundled events, which is an artifact of inflated duration. Upon decreasing the proximity to the fit, more signals correspond to bundled events. Points above and slightly below the χ line are representative of singular events. The singular signals are likely attributed to pore–collapse and typically occur on shorter time scales compared to intergranular sliding, which is likely to cause aftershocks and bundled events.

Figure 2(f) shows the energy–amplitude scaling for this experiment, and it is apparent that there is some slight multibranching, given by two parallel lines with x = 2. This is indicative of multiple failure processes intertwining. For urate salts, this is likely to be pore collapse vs intergranular motion.

A summary of the various exponents and the statistical analyses from this experiment is provided in Table I. The key message for advancements in instigating kidney stone failure at lower energies lies in the maximum likelihood estimation shown in Fig. 2(b). It is shown in Fig. 3 that the regime between 300 and 1000 aJ has real signals, and Fig. 2(b) shows that there is a high probability of observing a jerk with an energy in this range. This low-energy regime is the focal point for future work because if many AEs can be generated in this low-energy regime and cause failure, then lower energy inputs can be used to instigate kidney stone destruction. A more detailed study considering more kidney stone types and input parameters from a medical perspective could prove to be useful in forming a procedure to break down stones at lower power settings in vivo to reduce the risk of injury and prolonged recovery time.

TABLE I.

Summary of exponents for the various power-law distributions.

Parameter/sample:Urate kidney stone
Energy (ε1.6 for E ≥ 103 aJ; 
1.8 for E < 103 aJ 
Amplitude (τ3.0 
Duration (α1.5 for D ≥ 104µs; 
2.8 for D < 104µ
Amplitude–duration (χ1.5 
Energy–amplitude (x
Parameter/sample:Urate kidney stone
Energy (ε1.6 for E ≥ 103 aJ; 
1.8 for E < 103 aJ 
Amplitude (τ3.0 
Duration (α1.5 for D ≥ 104µs; 
2.8 for D < 104µ
Amplitude–duration (χ1.5 
Energy–amplitude (x

Ceramics are often lighter than their chemical composition would suggest. The reason is that many are porous. They contain holes as structural elements, which are widely used as filters, fillers, low thermal conduction materials, and so on. Porous materials are particularly important, due to their relevance in the collapse forecast of both natural and artificial structures such as mines,31 and buildings.69 Typical sandstones used as building material, or for sculpturing, have porosities between 20% and 60%.

Figure 5 shows the energy PDF and the ML analysis of synthetic Vycor, which is a SiO2--based ceramic with nano-pores. Baró et al. (Ref. 17) demonstrated that the acoustic emission events produced during the compression of Vycor follow the Gutenberg–Richter law, the modified Omori’s law, and the law of aftershock productivity, for a minimum of 5 decades. They are independent of the compression rate and remain stationary for the duration of the experiment. The waiting-time distribution fulfills a unified scaling law, with a power-law exponent close to 2.45 for long times, which is explained in terms of the temporal variations of the activity rate. While the energy exponents are not exactly the same as during most earthquakes, the similarity between avalanches in ceramics and large-scale geophysical scenarios is stunning. This shows that avalanches follow a high degree of universality between these two extreme length scales.

FIG. 5.

Distribution of avalanche energies in Vycor during seven different subperiods reproduced with permission from Baró et al., Phys. Rev. Lett. 110, 088702 (2013). Copyright 2013 American Physical Society. The line shows the behavior corresponding to ε = 1.39. The inset shows the ML-fitted exponent ε as a function of a lower threshold Emin for the three experiments with different strain rates.

FIG. 5.

Distribution of avalanche energies in Vycor during seven different subperiods reproduced with permission from Baró et al., Phys. Rev. Lett. 110, 088702 (2013). Copyright 2013 American Physical Society. The line shows the behavior corresponding to ε = 1.39. The inset shows the ML-fitted exponent ε as a function of a lower threshold Emin for the three experiments with different strain rates.

Close modal

Another application of avalanche research relates to the use of bio-mineralization to generate large amounts of material, to stabilize parts of ocean islands. Microbially induced calcite precipitation (MICP) is used for bio-cementation of calcareous sand using a microbial metabolism to generate calcite precipitation. A sketch of MICP is shown in Figs. 6(a)6(d). This technology has also been proposed for sealing damage in geological reservoirs, repair of cracks in stone buildings, and strengthening of foundations in coastal engineering. Reference 70 showed that all characteristic avalanche parameters (energy, amplitude, inter-event time, etc.) are power-law distributed during plastic deformation. The energy exponents are ε = 1.35–1.6, depending on the degree of collapse, as shown in Fig. 7. Sands without cementation show avalanches with energy exponents near ε = 1.7 and very low strength. In contrast, compressed sand grains have high strength and energy exponents near ε = 1.4. Correlations between AE spectra show that compressed bio-cementation samples immediately re-convert to sand, with some larger grains resisting. The distribution of low AE energies in porous bio-cementation samples has the same avalanche exponents as that of sand. This demonstrates that some bio-cementation samples behave exactly as sand, and some consolidating bio-cemented structures in between, as shown in Fig. 7. When these structures are destroyed, the local grains collapse, under further compression.

FIG. 6.

Reproduced with permission from Wang et al., Eng. Fract. Mech. 247, 107675 (2021). Copyright 2021, Elsevier.(a)–(d) Sketch of the MICP (microbial induced calcite precipitation) process; (e) SEM photograph of bio-cemented sand sample treated by MICP method; and (f) local enlargement shows the microbial induced calcite on the surface of calcareous sand grains forming calcite bridges connecting sand grains.

FIG. 6.

Reproduced with permission from Wang et al., Eng. Fract. Mech. 247, 107675 (2021). Copyright 2021, Elsevier.(a)–(d) Sketch of the MICP (microbial induced calcite precipitation) process; (e) SEM photograph of bio-cemented sand sample treated by MICP method; and (f) local enlargement shows the microbial induced calcite on the surface of calcareous sand grains forming calcite bridges connecting sand grains.

Close modal
FIG. 7.

(a) Distribution of AE energies, (b) shows the ML-fitted exponent as a function of the lower threshold Emin for three experiments during the full duration of the experiment reproduced with permission from Wang et al., Eng. Fract. Mech. 247, 107675 (2021). Copyright 2021 Elsevier.

FIG. 7.

(a) Distribution of AE energies, (b) shows the ML-fitted exponent as a function of the lower threshold Emin for three experiments during the full duration of the experiment reproduced with permission from Wang et al., Eng. Fract. Mech. 247, 107675 (2021). Copyright 2021 Elsevier.

Close modal

Similarly, when mining materials such as sandstone and coal are subjected to a compressive stress, failure can be heralded by a significant precursor activity.20,31 In the precursor regime, the response of the system to the applied compressive stress is not smooth as classically expected for elastoplastic materials, but, instead, occurs as a sequence of avalanches. Typical examples where pores were induced by burning carbon inclusions include shales,71 sandstones,18 ceramic berlinite,19 ceramic alumina,21 and goethite ore.72 Avalanche behavior was found in all these materials, and the energy exponents varied between 1.33 and 2, with another common value near 1.66. This value coincides with the predictions of field integrated mean field theory.6 

The thing that cracks and hole–hole interactions have in common is that they do not form simple microstructures. Simillar to cracks in scattered window glass, they form complex patterns where the crack propagation does not follow linear trajectories, but progresses by junctions, bifurcations, spirals, and specific patterns, such as Turing patterns. Similarly, strain fields of a multitude of cavities form patterns of great complexity. These patterns are virtually always fractal, even when the holes are man-made in a periodic fashion.73,74 Any simple description of such patterns for cracks and clouds of holes75,76 remains on a rather coarse length scale because the knowledge of any finer details still exceeds our current ability to understand patterns. Local configurations matter greatly for the macroscopic properties of the material. One particularly impressive property is that such disordered patterns are always piezoelectric and are often polar even when the crystal structure is centrosymmetric.14,77,78 This will have massive implications for device manufacturing, particularly in the computing hardware industry, because previously excluded material systems can be brought back into the discussion. Centrosymmetric materials can have anharmonic effects, which can be switched and tailored to desired specifications.

Concepts that were initially derived for the (rather unsuccessful) prediction of local earthquakes have penetrated other fields of science. The first success in ferromagnetism was enhanced by the analysis of switching behavior in ferroelectric and ferroelastic materials. This development took some ∼100 years, whereas the applications in ferroic materials and neuromorphic computation are less than 5 years old, and are still in full acceleration. In this paper, we argue that two other fields, which were initially seen as not suitable for such approaches, show great promise, to constitute the next successful step forward for avalanche research. These areas are the investigation of minerals and rocks in the geological context and applications in clinical medicine. In both cases, existing nano- and microstructures greatly influence the stability of hard materials and determine how materials collapse under external forces. In the case of geology, this helps understand the geological history of the material and to generate “geological” materials such as in the case of bio-mineralization. Breaking geological materials is also at the core of human exploration, including fracking, extraction of building materials, sculpturing, and reservoir construction for nuclear waste.

In the field of clinical medicine, kidney stones, teeth, and bones are the first important materials to be investigated. They show with great sensitivity the mechanical breakdown with respect to a multitude of material parameters. The diversity of properties makes it compulsory to understand the breakdown mechanism before clinical intervention is used to destroy kidney stones and to tackle bone diseases. For kidney stones, this research has distinguished between several avalanche dynamics and failure mechanisms, identifying five power-law exponents. Future work can adapt the results into the medical field, and highlight ideal combinations of energy and frequency, to cause avalanches in vivo, at lower power levels, to prevent internal burns and speed up recovery time. This research is still in its infancy, and large research initiatives are planned to investigate the breaking of biological materials further.

The project has received funding from the EU’s Horizon 2020 program, under the Marie Sklodowska-Curie Grant Agreement No. 861153. This work was supported by the Engineering and Physical Sciences Research Council, under Grant No. EP/P024904/1 to E.K.H.S. and M.A.C.

The authors have no conflicts to disclose.

Jack T. Eckstein: Methodology (equal); Writing – original draft (lead); Writing – review & editing (lead). Michael A. Carpenter: Funding acquisition (equal); Project administration (equal); Supervision (equal); Writing – original draft (supporting); Writing – review & editing (supporting). Ekhard K. H. Salje: Funding acquisition (equal); Investigation (equal); Supervision (lead); Writing – original draft (equal); Writing – review & editing (supporting).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

FIG. 8.

Image of kidney stone of type IIIc, according to the Daudon scale.66–68 The kidney stone is ∼5 × 3 × 3 mm3 and irregular in shape.

FIG. 8.

Image of kidney stone of type IIIc, according to the Daudon scale.66–68 The kidney stone is ∼5 × 3 × 3 mm3 and irregular in shape.

Close modal
FIG. 9.

Noise measurement from the kidney stone AE experiment analyzed in Sec. III showing energy (aJ) vs time (s). The noise measurement was started just before placing the transducer in contact with the kidney stone. The jerk at t ≈ 10 s is when the detector was placed onto the sample. The baseline noise is shown to be 100 aJ.

FIG. 9.

Noise measurement from the kidney stone AE experiment analyzed in Sec. III showing energy (aJ) vs time (s). The noise measurement was started just before placing the transducer in contact with the kidney stone. The jerk at t ≈ 10 s is when the detector was placed onto the sample. The baseline noise is shown to be 100 aJ.

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