Encyclopedia of Continuum Mechanics

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| Editors: Holm Altenbach, Andreas Öchsner

Avalanches in Solids

  • Karin A. DahmenEmail author
  • Wendelin J. WrightEmail author
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DOI: https://doi.org/10.1007/978-3-662-53605-6_73-2



Avalanches are domino-like processes where one event triggers another.

Bulk metallic glasses (BMGs) are noncrystalline metals, typically produced by rapid quenching and comprising three or more elements. The need for rapid quenching limits the thickness of metallic glasses in one dimension.

The complementary cumulative distribution C(S) of avalanche sizes S gives the probability of finding an avalanche with size greater than S.

A low-pass filter is an electronic circuit that passes signals with frequencies lower than a specified cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency.

Mean-field theory is an approximation to a model for which the physical interactions are replaced by infinite-range interactions in order to solve the model analytically.

A piezoelectric load cell is a sensor that generates a potential when a force is applied; the output is calibrated so that the sensor can be used to measure load.

The renormalization group is a mathematical method to coarse grain a model in order to compute the behavior on length scales that are large compared to the microscopic details.

A serration is a sudden stress drop in a stress-strain curve, commonly observed during plastic deformation of metallic glasses; it is an avalanche or slip, specifically in a metallic glass.

Avalanches in Solids

Slowly sheared bulk metallic glasses, densely packed granular materials, and many other materials deform with slip avalanches that have a broad size distribution. Recent experiments, analytic models, and simulations show that the avalanche size distribution often follows a power law over a broad range of sizes, similar to the Gutenberg Richter law of earthquakes. Here a series of statistical and dynamical quantities are reviewed that are useful for the systematic quantitative comparison of experiments to theory and simulations in a wide range of systems. As an example, results from a simple mean-field model are discussed in comparison to experiments on bulk metallic glasses. An important issue to consider when comparing theory to experiments is that inadequate data acquisition rates can distort the experimental results. The effects of temporal resolution on the data analysis and techniques for circumventing common resolution problems are described. The methods discussed here are broadly applicable to many systems with avalanches.

Many solids respond with sudden slips to slow deformation (Zaiser 2006; Sethna et al. 2017). Examples include slowly compressed nanocrystals, microcrystals, bulk metallic glasses (BMGs), and other alloys, as well as slowly sheared granular materials and earthquakes. The slips are visible as sudden steps in the stress-strain curves (see Fig. 1) or as “crackling” observed in the acoustic/seismic emission. A simple model (Dahmen et al. 2009, 2011) suggests that these slips are avalanches of slipping weak spots, i.e., a slipping spot in the material can trigger other weak spots to also slip, which triggers others to slip and so on, creating a slip avalanche. The model assumes that each of the weak spots has a failure stress for slipping. If the local stress reaches that failure stress, the weak spot slips. When it slips, the associated stress drop leads to a stress change for the other weak spots in the material. (Those resulting stress changes can trigger other weak spots to slip also in a slip avalanche.) In many cases, the slips are mostly confined to a slip-plane, such as a shear band in glassy materials, a glide plane in crystals, or a narrow fault zone in earthquakes. It can be shown (Dahmen et al. 2009, 2011) that in this case a local slip leads to an increase in the local stress everywhere else in the slip-plane, i.e., the interactions are ferroelastic. The stress evolution can easily be expressed in formulas in a discrete version of the mean-field theory: Dividing the slip plane into N lattice points or cells on the slip plane, the local stress τl at a lattice point l = 1,…,N, is given by τl = J/N Σm (um – ul+ F. Here J is the mean-field coupling constant, F is the applied stress (or the stress added to the slip plane by the slow motion of a far away sample boundary), and ul and um (m = 1,,N) are the local displacement discontinuities (total accumulated slip) across the slip plane at their respective sites, which are indicated by the subscripts. When a cell slips, its local accumulated slip increases by a finite amount, and its local stress is reduced to a randomly chosen local arrest stress, and the cell resticks (Dahmen et al. 2009, 2011). This is the simplest version of the model for ductile materials. For brittle materials, like most BMGs, the local failure threshold for each weak spot can weaken by a finite amount after a slip (e.g., due to dilation caused by local free volume generation). For slow driving, when the times between avalanches are sufficiently long, it may be assumed that the local cells reheal to their original strength between avalanches. For loading at a slow strain rate, the weakening can create almost periodically recurring large avalanches, with smaller avalanches during the intervening time intervals (McFaul et al. 2018). For hardening materials, such as microcrystals, the failure threshold strengthens, e.g., due to the interaction and multiplication of dislocations. More details on the model can be found elsewhere (Dahmen et al. 2009, 2011).
Fig. 1

Engineering stress as a function of time for a uniaxial compression test on Zr45Hf12Nb5Cu15.4Ni12.6Al10 BMG at a nominal strain rate of 10–4 s–1 (constant displacement rate). The right inset shows most of the test. The main figure shows only the portion that is included in the blue box in the right inset. The left inset shows a magnified view of a single serration. (Figure reprinted from Antonaglia et al. (2014a))

Tools from the theory of phase transitions, such as the renormalization group, predict that mean-field theory correctly captures the scaling behavior of the slip avalanche statistics on length scales that are large compared to the microscopic details (Fisher et al. 1997). (In the mean-field approximation, the physical long-range elastic interactions along the slip plane are replaced with infinite-range interactions.) The prediction of the renormalization group means that the analytic mean-field model can be used to compute many scaling properties of the avalanche statistics and their dynamics exactly, especially in cases where the slips are localized in a slip plane or shear band (Dahmen et al. 2009, 2011). In the following some of the predictions of the mean-field model are briefly reviewed and discussed in comparison to experiments.

The slip avalanche size S can be measured in various ways. For example, for deformation at a slow imposed strain rate, the avalanche size is the size of the stress drop measured during each slip. For loading by a slowly increasing stress, the avalanche size equals the displacement during a slip.

The distribution D(S) of the slip avalanche sizes S is typically broad, following a power law over several decades, D(S∼S–τ with a predicted mean-field exponent τ = 3/2 [for nonhardening materials near a critical stress (which coincides with the failure stress for a ductile material), and for hardening materials for a broad range of stresses], or τ = 2 in a nonhardening material, if the data are integrated over a wide interval of applied stresses. More precisely, the exponent τ = 2 can typically be seen if the stress-strain curve has negative curvature over the range of the stress integration interval. The exponent τ is predicted to be universal, i.e., independent of the microscopic details of the material. In many experiments, its value is indeed close to the mean-field value τ = 3/2 (Antonaglia et al. 2014a, b; Sun et al. 2010; Friedman et al. 2012; Uhl et al. 2015; Denisov et al. 2016, 2017; Maaß et al. 2015). Several simulations based on various assumptions give similar exponent values, although not always exactly the same as seen in mean-field theory and experiments (Salerno et al. 2012; Lin et al. 2014; Zaiser 2006; Tsekenis et al. 2013; Zhang et al. 2017; Csikor et al. 2007). Other approaches to materials with randomly distributed properties (also called “random media”) give related scaling behavior of the slip statistics (Kale and Ostoja-Starzewski 2014).

All materials have a critical stress at which discrete deformation transforms to continuous flow (similar to a phase transition). Whether a real material reaches this critical stress depends on specific features of the material. The materials properties depend on the proximity of the experimental conditions to this critical point, e.g., avalanches become larger near the critical point.

Some predictions of this model are universal: e.g., the exponents of the statistical distributions are predicted to be the same for all materials (Uhl et al. 2015); however, there are system-specific features as well. These are the tuning parameters of the model. The tuning parameters include the strain rate, temperature, applied stress, extent of weakening, specimen size, and machine stiffness. Changing the tuning parameters can change the exponents, but in predictable ways (e.g., the change from τ = 3/2 to τ = 2 for the case of negative curvature of the stress-strain curve over the range of the stress integration interval mentioned previously). Regarding the distribution of avalanche sizes, the tuning parameters only affect the cutoff of the scaling behavior. The cutoff is where the power-law scaling breaks down at large avalanche sizes. The large avalanches occur only for materials with weakening. They prevent the critical point from being reached and are indicative of system-specific behavior. The cutoff shifts to larger sizes when the strain rate decreases, or the applied stress increases, or the weakening decreases, or the specimen size is larger, and/or the machine is less stiff.

Comparing the slip statistics and slip dynamics to the model predictions can be a powerful approach to confirm or rule out certain assumptions made in models for materials deformation. The slip statistics at lower stresses can also be used to predict the failure stress. This is especially true for materials for which the size of the largest slip avalanche grows with the applied stress in the way predicted by mean-field theory (Friedman et al. 2012; Uhl et al. 2015).

Several open questions are listed in the following section; statistical quantities that can be extracted from the slip statistics in experiments and simulations are then outlined. Measuring these statistical quantities in experiments can be used to test the model assumptions for the deformation dynamics of solid materials. This method has been successfully applied to experimental results on slowly compressed BMGs as will be shown here; similar studies have been performed on granular and other materials (see Baldassari et al. 2006; Daerr and Douady 1999; Hartley 2005; Hayman et al. 2011; Majmudar and Behringer 2005; Palassini and Goethe 2012; Petri et al. 2008; Sen et al. 1994; Thomas et al. 1994).

Some Open Questions in the Field

Below a few open research questions are presented, for which the detailed study of avalanche statistics and dynamics will likely provide answers (Kavli 2014).
  • What are the microscopic processes that control the intermittent material response in different materials?

  • What can be stated in general about the competition between disorder and elasticity? (Elasticity is the source of avalanching behavior, while disorder generally tends to break up avalanches.)

  • Which scales are crucial for understanding large-scale events?

  • Which features control the timescales of microstructural processes, and how do they enter a description of material response under time-dependent driving?

  • Which features of collective phenomena are general and which are system specific?

  • What theoretical predictions can be tested from experimental data and what experiments need to be done to improve our understanding of these questions?

  • What kind of statistical descriptions are needed for a minimal representation of intermittent response and large fluctuations in materials?

In order to answer some of these questions, a thorough understanding of the slip avalanche statistics and dynamics is needed. Our goal here is to compile a list of statistical quantities that hopefully are useful for organizing experimental data and testing model predictions.

In the following, mean-field theory predictions for experiments at fixed slow strain rate Ω (typically in the range 10−5/s ≤ Ω ≤ 10−3/s) are listed. Analogous predictions apply to experiments at slowly increasing stress (Dahmen et al. 2009, 2011). A table summarizing many of the scaling exponents and functions predicted by the model is given in Dahmen (2017), Salje and Dahmen (2014), and LeBlanc et al. (2013).
  1. 1.

    The complementary cumulative distribution \( C(S)=\underset{S}{\overset{\infty }{\int }}D\left({S}^{\prime}\right)d{S}^{\prime } \) of stress-drop sizes S is given by the probability of the occurrence of avalanches larger than size S. For steady-state deformation, and for hardening materials, the mean-field model predicts C(S∼S(τ–1) for a certain scaling regime range of (small) avalanche sizes Smin < S < Smax, with τ = 3/2. Experimental data on crystals and BMGs give values for τ that are consistent with mean-field theory τ = 1.51 ± 0.03 (Friedman et al. 2012; Antonaglia et al. 2014a). For brittle materials, (e.g., for BMGs), the model predicts additional large (i.e., system spanning) avalanches that recur almost periodically, with the power-law-distributed smaller avalanches occurring during the time intervals between the large avalanches (Dahmen et al. 2009, 2011). Again, consistent with the mean-field predictions, both small and large avalanches have recently been observed in high data acquisition rate experiments on BMGs (Antonaglia et al. 2014a; Wright et al. 2016).

  2. 2.

    The power spectrum P(ω) of the stress-drop rate dσ/dt is predicted by mean-field theory to scale as P(ω∼ω1/(σ’υz) with 1/(σ’υz) = 2 (Dahmen et al. 2009, 2011). Here P(ω) is the absolute square of the Fourier transform of the stress-drop rate – time series dσ/dt (t), where σ(t) is the applied stress, and ω is the Fourier frequency.

  3. 3.

    The distribution D(T) of the durations T of the stress drops is predicted to scale as D(T∼T–α with α = 2 in mean-field theory in a certain scaling range of durations Tmin < T < Tmax. Or analogously, the complementary cumulative stress-drop duration distribution C(T) is predicted by mean-field theory to scale over a certain duration range as C(T∼T(α–1) with α = (τ – 1)/(σ’υz+ 1 (White and Dahmen 2003), where 1/(σ’υz) is again the exponent with which the power spectrum P(ω) decays at frequencies ω in the range 2π/Tmax < ω < 2π/Tmin. Mean-field theory predicts α = 2 and 1/(σ’υz) = 2 (Dahmen et al. 2009, 2011).

  4. 4.

    For the typical duration <T(S)> for a given stress-drop size S, the mean-field model predicts the following scaling relation: <T(S)> ∼Sσ’υz with σ’υz = 1/2 (Dahmen et al. 2009, 2011) in the scaling regime of the avalanche sizes and durations.

  5. 5.

    The average maximum stress-drop rate <(−dσ/dt)max | S> during an avalanche of size S is predicted by the mean-field model to scale as <(−dσ/dt)max | S> ∼Sσ’ρ (LeBlanc et al. 2013) with σ’ρ = 1/2 (Dahmen et al. 2009, 2011; LeBlanc et al. 2012, 2013).

  6. 6.

    The average temporal profile <−dσ/dt (t) | T> for the average stress-drop rate –dσ/dt (t) for avalanches of total duration T as a function of time t is obtained by averaging the time profile of the stress-drop rate –dσ/dt (t) over all avalanches of the same duration T. The mean-field model predicts that it follows a parabola. Machine stiffness effects lead to a flattened shape. An asymmetric profile can result either from the effects of inertia (typically tilted to the right) or delay effects (typically tilted to the left). It is possible for different systems to share the same exponent values and yet differ in this particular stress-drop rate profile, i.e., this average temporal profile can be used to differentiate between different slip dynamics (Dahmen et al. 2009, 2011; LeBlanc et al. 2013; Ramanathan and Fisher 1998; Schwarz and Fisher 2001, 2003; Fisher et al. 1997; Zapperi et al. 2005).

  7. 7.

    The time profile for the average stress-drop rate for avalanches of the same total size is denoted by <−dσ/dt (t) | S>. It is computed by averaging –dσ/dt (t) over all avalanches of the same total size S. It is predicted by mean-field theory to follow a function of the form < −dσ/dt (t) | S> ∼Sσ’ρ G(t/Sσ’υz) with σ’ρ = 1/2 σ’υz = 1/2 and G(x= A x exp(–Bx2), where A and B are constants that depend on the microscopic details and are expected to be different for different materials (Dahmen et al. 2009, 2011).

  8. 8.

    The model predicts small avalanches and large avalanches with different temporal stress-drop rate profiles for each (Dahmen et al. 2009, 2011). This agrees with the shapes obtained from experimental data (Antonaglia et al. 2014a).

  9. 9.

    In many systems, the distributions of avalanche sizes and durations have cutoffs that change with the applied stress, or the applied strain-rate, or other tuning parameters (Dahmen et al. 2009, 2011). In this case Widom scaling-collapses can be used to test the scaling forms predicted by the models (Antonaglia et al. 2014a). For example, for ductile materials such as crystals, the mean-field model predicts that for slowly increasing applied stress, the probability density function D(S,F) of the slip avalanche sizes S measured in a small stress bin around the applied stress F scales as D(S,F∼S–τ exp(–S (Fc – F)1/σ’), with τ = 3/2 and σ’ = 1/2 in mean-field theory (Dahmen et al. 2009, 2011). Here Fc > F is the flow stress (or failure stress) of the ductile material. In a scaling collapse, plotting Sτ D(S,F) versus S (Fc – F)1/σ’ should then collapse the distributions D(S,F) for all stress bins around stresses F near the failure stress Fc onto the exponentially decaying cutoff scaling function exp(−x). This prediction has been confirmed in recent experiments on nanocrystals (Friedman et al. 2012). A related analysis has been performed for BMGs (Antonaglia et al. 2014b).


Comparison of Model Predictions for the Avalanche Statistics and Dynamics to Recent High-Resolution Experiments on Bulk Metallic Glasses

Recently these quantities were determined from experimental stress-time data from quasi-static uniaxial compression of a BMG (Antonaglia et al. 2014a, b; Sun et al. 2010). The following focuses especially on the work in Antonaglia et al. (2014a) because it clarifies not only how to extract avalanche statistics but also how to extract and analyze avalanche dynamics. (The experiments in Antonaglia et al. (2014a) represent the first time that adequate high-resolution data have been obtained for a structural material to demonstrate agreement with not only the predicted scaling exponents of the mean-field avalanche statistics but also the predicted dynamical scaling functions of the avalanche time profiles.)

For the high-resolution experiments in Antonaglia et al. (2014a), rectangular specimens of Zr45Hf12Nb5Cu15.4Ni12.6Al10 (6 mm in the direction of the loading axis × 1.5 mm × 2 mm) were compressed at a strain rate of 10−4 s−1 (Fig. 1). The data were acquired at 100 kHz (Antonaglia et al. 2014a; Wright et al. 2013). Successful comparison of the above predictions of the mean-field model with experiments requires that the data have high-temporal resolution and low noise. Wiener filtering was employed to reduce the noise and minimize the influence of the system electronics (Antonaglia et al. 2014a). For more details on the experimental methods, see Antonaglia et al. (2014a) and Wright et al. (2013).

Figure 2a shows the distribution of stress-drop sizes S for the total of 3744 avalanches recorded from testing of two specimens. The expected mean-field exponent of –(τ – 1) = −1/2 is observed over the range of 0.26–3.3 MPa (i.e., the scaling regime) for the complementary cumulative distribution C(S). This plot satisfies the first of the above predictions. Notice that the sharp drop-off of C(S) at the largest avalanche sizes indicates that there are many large avalanches, which have different dynamics than the avalanches in the power-law scaling regime of the smaller avalanches in the distribution (Dahmen et al. 2009, 2011; Antonaglia et al. 2014a; Wright et al. 2016). This observation in the experiments is also consistent with the model predictions (Dahmen et al. 2009, 2011; Antonaglia et al. 2014a).
Fig. 2

Avalanche statistics for quasi-static uniaxial compression of two specimens of Zr45Hf12Nb5Cu15.4Ni12.6Al10. For comparison, the predictions of the mean-field model are indicated by the dashed red lines. The data in the scaling regime are highlighted in red. (a) The complementary cumulative distribution of stress drop sizes for the 3744 avalanches; the data in the scaling regime are consistent with the expected mean-field exponent of –½. (b) The avalanche duration as a function of avalanche size. The data in the scaling regime are consistent with the expected mean-field exponent of ½. (Figures reprinted from Antonaglia et al. (2014a))

Figure 2b shows the avalanche duration as a function of avalanche size, and the expected mean-field exponent of 1/2 is measured for the same scaling regime as shown in Fig. 2a. This plot confirms prediction (4) in the above list.

Figure 3a shows the stress-drop rate profiles divided by duration T averaged over all avalanches in the scaling regime. This plot confirms a specific version of prediction (6) in the above list. A more detailed comparison with that prediction is given in Antonaglia et al. (2014a). The inset of Fig. 3b shows the unscaled average stress-drop rate profiles for different stress-drop sizes S. The main plot of Fig. 3b is the same data as the inset but scaled by a factor of S–1/2 on both axes. The prediction of the mean-field model is also shown. This plot demonstrates the agreement between the collapse of the experimental data and the predicted mean-field collapse scaling function. This plot confirms prediction (7) of the above list. Confirmation of the remaining predictions of the above list are shown in Antonaglia et al. (2014a, b).
Fig. 3

(a) The average avalanche profile obtained from all avalanches in the scaling regime. The stress drop rate is rescaled by the maximum average stress drop rate for each avalanche on the vertical axis and by the avalanche duration on the horizontal time axis. The mean-field prediction is indicated by the black line. (b) The unscaled average profiles for several small avalanches of various sizes are displayed in the inset. When both axes are rescaled by S–1/2, the shapes collapse onto the universal scaling function predicted by the mean-field model (shown by the black dashed line). (Figures reprinted from Antonaglia et al. (2014a))

The data in Fig. 2 span approximately one decade of avalanche size and duration. A larger range of data is desirable; however, mechanical testing of metallic glasses poses several experimental challenges. It is well documented that as specimen size increases, the ductility of metallic glasses decreases for a fixed test frame stiffness (Han et al. 2009). Thus, the number of avalanches sustained during a single test decreases as specimen size increases, limiting the statistics that can be obtained. The size of specimens is also restricted by the cooling rates that are required to achieve fully amorphous specimens. The low end of the scaling regime is obscured by the noise and finite stress resolution of the system for macroscopic samples.

This work demonstrates that a simple mean-field model accurately captures the exponents and dynamics of the avalanche events in metallic glasses. The interpretation of the mean-field model produces important insights about deformation of metallic glasses. Since the mean-field model holds that slipping weak spots give rise to avalanche behavior, the agreement between the model and experiment provides experimental evidence for shear transformation zones as the mechanism of deformation in metallic glasses (Antonaglia et al. 2014a). These data also establish the existence of two modes of shear banding in metallic glasses. The small avalanches that fall within the scaling regime of the mean-field model correspond to shear bands that initiate at stress concentrations and then propagate incrementally into regions of lower stress. The mean-field model predicts that for these small avalanches in the scaling regime S~∼~A where A is the total area that slipped in a single event, and the maximum stress-drop rate (which is expected to be proportional to the maximum slip rate) \( {<}(-d\sigma/dt)_{max} | S{>}\,{\sim}\, {S}^{1/2} \). Those avalanches that have stress-drop sizes and durations that are larger than the scaling regime scale as S ∼ A3/2, much like a Mode II crack. For these large avalanches, \( {<}(-d\sigma/dt)_{max} | S{>}\,{\sim}\, {S}^{2/3} \). These shear bands propagate in a simultaneous fashion across the specimen, meaning that all points in the shear band slip at the same rate and at the same time. The simultaneous mode of propagation has been verified by direct imaging (Song et al. 2010; Wright et al. 2013, 2016).

In summary, experiments on BMGs and a simple mean-field model together show that the dynamics of the large avalanches are very different from those of the small avalanches. In both model and experiments, the small avalanches propagate progressively, i.e., in a pulse-like fashion, while the large avalanches propagate in a crack-like fashion (Antonaglia et al. 2014a; Wright et al. 2016; Dahmen et al. 2009, 2011). Distinguishing between these dynamics is expected to be important to future experiments and modeling efforts on a wide range of systems. The application of the mean-field model predictions to extract exponents and slip dynamics for other materials will yield useful information as to the similarities and differences between different systems. The work described here can serve as a model for designing experiments and performing data analysis on a wide range of systems with avalanches.

On the Proper Determination of Power-Law Exponents for Slip Statistics Using Experimental Data from Bulk Metallic Glasses

Now guidelines are presented for the experimental conditions required to accurately record slip avalanches and compute power-law exponents. Proper determination of the power-law exponents and slip dynamics requires careful attention to the details of the instrumentation including parameters of low-pass filters in the signal conditioning electronics as well as filtering of experimental noise and selection of a sufficient data acquisition rate. Inadequate attention to these details may lead to unresolvable merging of avalanches, loss of small avalanche data, calculation of incorrect exponents, and possibly failure to recognize a power-law regime in the data even if one truly exists. Here the critical characteristics of sensor electronics are discussed, and a method for extracting the true materials behavior from data with a suboptimal acquisition rate is demonstrated. Improperly filtered and purposely downsampled data from compression testing of metallic glasses are presented to illustrate the importance of these issues.

Figure 1 is a plot of engineering stress as a function of engineering strain for serrated flow in a Zr45Hf12Nb5Cu15.4Ni12.6Al10 BMG subjected to quasi-static constant displacement rate compression (Antonaglia et al. 2014a; Wright et al. 2016). Each one of the serrations or “avalanches” in the plastic region corresponds to the propagation of typically one, but in some instances perhaps as many as a few, shear bands. In these experiments, the strain rate is 10−4 s−1, a typical quasi-static strain rate. The timescales for these serration events are short – on the order of fractions of a millisecond to several milliseconds (Wright et al. 2009, 2013).

In a typical test, a load cell generates a signal through the transformation of mechanical force to a time-varying voltage, which is then digitized and stored. This voltage signal is typically amplified and filtered, by an external signal conditioner or by a data acquisition card. The signal conditioners of conventional systems designed for quasi-static testing anticipate relatively slowly changing signals and are intended to provide low-noise data. This is achieved by incorporating low-pass filters that have low cutoff frequencies. For example, with modern systems, in software, the user often chooses between low-pass filters with 100 Hz, 10 Hz, or 1 Hz cutoff frequencies, but these frequencies are too low for rapid events to be accurately measured. Figure 4 shows the output of an often used low-pass filter known as a Butterworth filter. The plot shows signal gain as a function of frequency. The goal is to provide a low-noise signal. This works well if the high frequency components are simply noise, but if the high frequency components are real and contain valuable information, they will be distorted and possibly lost. An ideal low-pass filter passes all signal components below the cutoff frequency and rejects all signal components above the cutoff frequency, but no filter is perfect. The “order” of a real filter represents how steep the attenuation of high frequency components is above the cutoff frequency.
Fig. 4

Gain as a function of frequency for a Butterworth filter with a cutoff frequency of 1 rad/s. The ideal response and the responses for filter orders n = 1, 2, 4, and 8 are shown

The data acquisition rate also requires special consideration. According to the Nyquist-Shannon sampling theorem, in order to accurately reproduce a sampled analog signal, the analog signal must be sampled at a rate that is at least two times greater than the highest frequency component of the signal (Nyquist 2002; Shannon 1998). Thus, as a rule of thumb, the data acquisition rate should be at least two times greater than the cutoff frequency of the low-pass filter in the sensor electronics.

Figure 5 illustrates the effects on the data of low-pass filters with cutoff frequencies that are too low. These measurements for serrated flow were acquired using a conventional load cell and linear variable differential transformers (LVDTs) to measure the displacement at a data acquisition rate of 100 Hz (Wright et al. 2001). The load cell was conditioned by an amplifier with a 3-Hz low-pass filter. The LVDTs had an external signal conditioner with a 400-Hz low-pass filter. The two data sets of Fig. 5a, b are synchronized in time. The load data indicate that the serration that occurred at approximately 22.5 s lasted 100 ms; the displacement data indicate that the same serration lasted 20 ms. This discrepancy is due to the filters. The displacement data show shorter elapsed times because the filter has a higher cutoff frequency although it likewise is insufficient.
Fig. 5

(a) Load as a function of time for a portion of serrated flow in a Pd40Ni40P20 bulk metallic glass during quasi-static uniaxial compression. (b) Total displacement as a function of time for the same data as shown in (a). The two data sets indicate different elapsed times for each serration due to the different cutoff frequencies of the low-pass filters in the different sensor electronics. For example, the serration that occured at approximately 22.5 s appears to have lasted approximately 100 ms according to the load data and only 20 ms according to the displacement data. The data points are separated by 10 ms in both plots. (Figures reprinted from Wright et al. (2001))

Use of a piezoelectric load cell is now the standard when the goal is to measure load-time signals with high temporal resolution during compression testing of BMGs (Wright et al. 2009; Dalla Torre et al. 2010). Piezoelectric load cells generate an electric potential when subjected to an applied force. They typically can be placed closer to the test specimen than a conventional load cell, minimizing the distance over which elastic waves must propagate and therefore also minimizing the time until which the elastic waves are registered. Piezoelectric load cells are much stiffer than conventional cells, enabling a higher frequency response; stiffer test systems also lead to enhanced ductility when testing BMGs (Han et al. 2009).

For the remainder of this entry, it is assumed that the experimental data are acquired with the proper sensors so that the signals are passed undistorted by the instrumentation to a data acquisition unit. If the signals are distorted, the following analysis cannot restore them to their proper form. After signal acquisition, Wiener filtering most likely needs to be performed. For guidance on how to perform Wiener filtering, see Antonaglia et al. (2014a). The discussion that follows is based on the work in LeBlanc et al. (2016), but is presented here in such a way as to provide a more streamlined approach to make implementation of this method simpler for the reader.

Quantities that are affected by inadequate time resolution include the number of observed avalanches, avalanche size, and avalanche duration. Errors in these quantities affect the power-law exponents of their distributions. Avalanches are typically identified as drops in stress or as events that begin when the slope of the stress versus time plot becomes negative and end when the slope becomes positive (assuming that increasing compressive stress is plotted as positive as in Figs. 1 and 5). As shown in Fig. 6, if the signal is noisy, a small negative threshold may be used to mark a slope value to indicate the beginning of the avalanche. The avalanche ends when the slope crosses the threshold again.
Fig. 6

Stress as a function of time from a simulation of the mean-field model with two avalanches (top) and its time derivative (bottom) (Dahmen et al. 2009). The avalanches are detected when the time derivative drops below a threshold (the dashed line in the bottom figure). The circles and squares indicate the starts and ends of the avalanches, respectively. The avalanche size is the total decrease in stress, as indicated in the top figure. (Figure reprinted from LeBlanc et al. (2016))

Figure 7 shows results from a simulation of the discrete version of the mean-field model described at the beginning of this entry (Dahmen et al. 2009, 2011). It is a plot of stress versus time for a portion of serrated flow. The simulation results are shown as a solid line. The circles indicate data points sampled a factor of 20 slower than the simulated data. In this plot, the interevent time between avalanches is much longer than the avalanche duration. Large avalanches can be readily identified by inspection even with low-time-resolution data. For the indicated large serration, the avalanche size is underestimated and the duration is overestimated, but the occurrence of the large event is correctly identified. The small serration, on the other hand, is not even observed at low resolution, using the conventional method of identification, because the slope does not decrease between the two data points bounding it.
Fig. 7

Stress as a function of time for a portion of serrated flow from a simulation of the mean-field model (Dahmen et al. 2009). The circles indicate data points sampled a factor of 20 times slower than the simulated data. For the indicated large serration, the avalanche size is underestimated and the duration is overestimated, but the occurrence of the large event is correctly identified. As the inset shows, a small stress drop does not have a corresponding drop in the low-resolution signal. The slope of the stress-time series when the stress is increasing is denoted as r. (Figure reprinted from LeBlanc et al. (2016))

Figure 8 is a plot of the complementary cumulative distribution C(S) for avalanche size using experimental data acquired at 100 kHz (Antonaglia et al. 2014a; LeBlanc et al. 2016). Data that were downsampled to 10 kHz are also shown and exhibit essentially the same behavior except at the smallest avalanche sizes where the Wiener filtering cannot effectively filter all of the noise in either case and microscopic details come into play. As the time resolution decreases, the power-law regime becomes narrower until it is indiscernible. A simple solution to remedy the effect of the downsampled data is to “tilt the curve” as described below.
Fig. 8

Avalanche size complementary cumulative distributions from quasi-static compression experiments on Zr45Hf12Nb5Cu15.4Ni12.6Al10 (Antonaglia et al. 2014a; Wright et al. 2016) originally acquired at a data acquisition rate of 100 kHz. The data have been downsampled to demonstrate the effect of time resolution on the distributions. Data that were downsampled to 10 kHz exhibit essentially the same behavior as the 100 kHz data. As the time resolution decreases further, the power-law regime narrows until it is indiscernible below 50 Hz. The trend lines are power laws with exponent 1 − τ =  − 1/2 as predicted by mean-field theory. (Figure reprinted from LeBlanc et al. (2016))

Because there are periods of elastic loading between avalanche events with a nominally constant slope denoted as r, the elastic portions can be subtracted from the stress(or load) signal F(t) using Ftilted(t) = F(t) – rt, where t is time; doing so tilts the curve so that the elastic portions are horizontal. After tilting, even small events in Ftilted(t) show a stress drop that is registered as an avalanche. Figure 9 (LeBlanc et al. 2016) shows tilted data with numerous small avalanches that would have been missed using the conventional analysis with low-time-resolution data. Figure 10 (LeBlanc et al. 2016) demonstrates that the low-resolution data obscure the power-law behavior on, for example, a log-log plot of the complementary cumulative distribution for avalanche size C(S), but by tilting the curve, even data that have 1% of the temporal resolution show nearly identical behavior to the full-resolution data.
Fig. 9

Simulated data for stress as a function of time (Dahmen et al. 2009), tilted according to Ftilted(t) = F(t) – rt. Subtracting the elastic stress component makes small avalanches (that would have been otherwise missed) visible as drops in the signal. The inset shows the stress-time curve before tilting. (Figure reprinted from LeBlanc et al. (2016))

Fig. 10

Avalanche size complementary cumulative distributions at full resolution (ts = 1) and 1/100th resolution (ts = 100) for both the untilted and tilted stress-time series. The tilted data give a distribution that is close to the one obtained at full resolution, whereas the untilted signal gives a distribution that appears rounded. (Figure reprinted from LeBlanc et al. (2016))

As shown in Fig. 7, when the time resolution is insufficient, the avalanche size depends on the sampling time: as sampling time increases, the avalanche size decreases to the point that small avalanches become undetectable. Figure 11 highlights the fact that the number of detected avalanches Ndetected shows a power-law dependence on the sampling time according to Ndetected∼(ts)−(τ – 1), where the exponent τ is the avalanche size exponent in the mean-field model and ts is the sampling time (LeBlanc et al. 2016). This equation is a useful tool to determine if the temporal resolution of an experiment is adequate. One can simply make a log-log plot of the number of detected avalanches as a function of sampling time by downsampling the full-resolution data. If the plot shows a power-law dependence with no plateau at short sampling times, then the time resolution is insufficient. Additionally, this method can be used to find the size exponent τ without determining the size of each avalanche.
Fig. 11

Number of measured avalanches as a function of sampling time from a simulation of the mean-field model (Dahmen et al. 2009). As the sampling time increases, the number of detected avalanches decreases according to Ndetected ∼  ts−1/2. This relationship permits a test of the adequacy of the data acquisition rate. If the data are purposely downsampled and the number of detected avalanches immediately begins to decrease with downsampling according to this relationship, then the data acquisition rate is too low. (Figure reprinted from LeBlanc et al. (2016))

Thus far, the discussion has focused on the case when the interevent time is much longer than the avalanche duration. Additional considerations are required when the time between avalanches is similar to the avalanche duration. Figure 12 (LeBlanc et al. 2016) shows the full-resolution simulation data with circles indicating data that have been downsampled by a factor of 2000. In this case, each time interval will contain typically at least one or possibly several avalanches. If several avalanches occur within a single time step as shown in the inset, it is impossible to distinguish one avalanche from another because they simply merge into a single larger and longer avalanche. In this case, the number of avalanches will be underestimated from the low-resolution data. It is possible, however, to recover the proper power-law distribution of avalanche size by taking each time interval with a stress drop to be the size of a single avalanche regardless of how many avalanches occur. For the complementary cumulative distribution of avalanche size C(S), the distribution of the low-resolution data analyzed in this way is shifted up since the number of measured avalanches is fewer, but importantly, the power-law behavior is the same even when the full-resolution data are downsampled by three orders of magnitude as shown in Figure 13 (LeBlanc et al. 2016). To reiterate, if the data have low-temporal resolution, and this method is not used, the wrong exponent will be obtained or the power-law behavior will be obscured entirely. This method is successful because the combined avalanches also show power-law behavior.
Fig. 12

Stress as a function of time from a simulation of the mean-field model (Dahmen et al. 2009) tilted according to Ftilted(t) = F(t) – rt. The circles mark every 2000th data point and thus correspond to a very low-resolution signal. This sampling time is about twice the inverse avalanche nucleation rate, so most increments contain at least one avalanche. As a result, measuring avalanches as successive drops in the tilted stress will give stress drops of very long duration, corresponding to many successive underlying avalanches. The stress drops measured between successive sample points will be close in size to the underlying avalanches, although sometimes several avalanches may be merged, as is the case for the first and last time step of size ts of the inset plot. (Figure reprinted from LeBlanc et al. (2016))

Fig. 13

Avalanche size complementary cumulative distributions from simulations of the mean-field model (Dahmen et al. 2009) at full resolution and from tilted stress-time curves at 1/1000th resolution using the improved method for very low-resolution data that defines the sample stress drops after tilting. The distribution from the improved method is much closer to the full-resolution distribution, and the same rough power-law behavior can be seen. (Figure reprinted from LeBlanc et al. (2016))

The mean-field model predicts that a scaling collapse should be achieved for data that were acquired using mechanical test systems with different stiffnesses K as shown in Fig. 14a. Figure 14 (LeBlanc et al. 2016) is a log-log plot of the complementary cumulative distribution of avalanche size C(S), where the vertical axis is scaled by S(τ – 1) on the vertical axis and the horizontal axis is scaled by the machine stiffness to the power of α. Using the full-resolution simulation data, the mean-field exponents of τ = 3/2 and α = 2 are obtained as expected. When the data are downsampled by a factor of 100 and tilted, the correct exponents are obtained as shown in Fig. 14c. Figure 14d demonstrates that when the data are downsampled by a factor of 1000, tilted, and each sample is taken to represent a single avalanche when a stress drop occurs, the correct exponents are still obtained. In contrast, Fig. 14b demonstrates that when the data are downsampled to every 500th data point, but the curve is not tilted, the power-law exponents are different from what is known from the full-resolution data.
Fig. 14

(a) Data collapse of the avalanche size complementary cumulative distribution for different machine stiffness values (K) from a simulation of the mean-field model (Dahmen et al. 2009) at full resolution. The collapse uses mean-field exponents τ = 3/2 and α = 2. (b) An apparent collapse for 1/500th of the resolution in (a) without using any of the techniques discussed here for circumventing resolution issues. The collapse yields different exponents τ = 1 and α = 1.5 because the time resolution is insufficient. (c) A collapse with mean-field exponents τ = 3/2 and α = 2 for 1/100th of the resolution in (a) using a tilted signal. (d) A successful collapse with mean-field exponents τ = 3/2 and α = 2 for 1/1000th of the resolution in (a) using the techniques described herein. Insets are the original distributions before rescaling. (Figure reprinted from LeBlanc et al. (2016))

In summary, tools to organize and analyze experimental data on (slip-) avalanches were reviewed. Methods to circumvent problems created by low data acquisition rates also were discussed. Model predictions show strong agreement with experiments on BMGs, as an important paradigm for slip avalanches in materials with disorder. It is expected that the reviewed methods and results are applicable to many other systems with avalanches.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2020

Authors and Affiliations

  1. 1.Department of Physics and Institute for Condensed Matter TheoryUniversity of Illinois at Urbana ChampaignUrbanaUSA
  2. 2.Kavli Institute for Theoretical Physics, Kohn HallUniversity of California at Santa BarbaraSanta BarbaraUSA
  3. 3.Department of Mechanical Engineering, One Dent DriveBucknell UniversityLewisburgUSA
  4. 4.Department of Chemical Engineering, One Dent DriveBucknell UniversityLewisburgUSA

Section editors and affiliations

  • Martin Ostoja-Starzewski
    • 1
  1. 1.Department of Mechanical Science & Engineering, Institute for Condensed Matter Theory and Beckman InstituteUniversity of Illinois at Urbana–ChampaignUrbanaUSA