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.

Figure 1

(a) Sequence of events in a single simulation, i.e., the size of events $\mathrm{\Delta }$ as a function of the strain $\varepsilon$ where they occurred. Record-breaking events are highlighted by red bars, while subsequences between records have randomly assigned colors. The bold continuous line represents the moving average of event sizes using 30 consecutive events. For clarity, a relatively small system of 20000 particles was considered, where $n_{\max }=1832$ event occurred with 10 records. (b) Record events in a time series of a real fracture experiment performed under conditions similar to the simulations [27].

Figure 2

Average number of records $〈N_n〉$ occurred until the $n\mathrm{th}$ event as a function of $n$ on a double logarithmic plot. Power-law behavior is obtained over a broad range of $n$. The straight line represents a power law of exponent $0.33$. The inset presents the same quantity $〈N_n〉$ on a semilogarithmic plot. For comparison the shuffled data set is also included where a logarithmic dependence is found.

Figure 3

Standard deviation of the number of records $\mathrm{Var}\left(N_n\right)$ occurred until the $n\mathrm{th}$ event divided by the average record number $〈N_n〉$ on a double logarithmic plot. The red (dashed) and the blue (dashed-dotted) lines represent the corresponding result of the shuffled data set and a power law of exponent 1/3, respectively.

Figure 4

Probability distribution of the total number of records $p\left(N_n^{\mathrm{tot}}\right)$ that occurred up to failure for the original and surrogate data sets. The continuous lines represent the Gaussian distributions obtained with the mean and standard deviation calculated directly from the data.

Figure 5

Probability distribution $p\left(\mathrm{\Delta }\right)$ of the event size $\mathrm{\Delta }$ considering all events up to failure. The size distribution of records $p\left({\mathrm{\Delta }}_r\right)$ is presented both for the original and surrogate data sets. In all cases the bold lines represent best fits with Eq. (7) obtained with the parameter values: $\xi =2.15, c=1.0$ and ${\mathrm{\Delta }}^{*}=1810$ for $p\left(\mathrm{\Delta }\right), {\xi }_r=1.0, c_r=0.9$ and ${\mathrm{\Delta }}_r^{*}=1052$ for $p\left({\mathrm{\Delta }}_r\right)$, and ${\xi }_r^s=0.8, c_s=0.9$ and ${\mathrm{\Delta }}_s^{*}=1270$ for for shuffled series.

Figure 6

Average value of the characteristic quantities of records as a function of the record rank $k$: (a) size and increment, (b) relative increment, (c) waiting time, and (d) strain of records normalized by the critical strain of failure. The gray stripe highlights the strain range between the records of rank $k=7$ and $k=8$. In (a), (b), (c) the open and filled symbols represent the original and the surrogate data sets, respectively.

Figure 7

Distribution $p\left(m\right)$ of waiting times $m$ between consecutive records for fracture simulations and for the surrogate data. The bold lines represent best fit, which was obtained using the generic functional form of Eq. (7). The value of the power-law exponent is $z=0.62$ and $z=0.98$ for the simulated and shuffled data, respectively.

Figure 8

Probability distribution $p\left({\mathrm{\Delta }}_r^k\right)$ of record sizes ${\mathrm{\Delta }}_r^k$ for fixed ranks $k$ rescaled with the average record size presented in Fig. 6. Good quality collapse is obtained.

Figure 9

Probability distribution $p\left({\mathrm{\Delta }}^k\right)$ of event sizes in subsequences of events between consecutive records. For demonstration three curves were fitted with the functional form Eq. (10). For the highest ranks $k\ge 20$ the distributions practically fall on top of each other.

Figure 10

The value of the exponent ${\tau }_k$ of the size distribution of subsequences of events between consecutive records as a function of the record rank $k$, together with the error bars.

Figure 11

Probability distribution $p\left({\mathrm{\Delta }}^k\right)$ of event sizes in subsequences of the shuffled event series between consecutive records. Except for the lowest ranks the curves fall on the top of each other. The white curve was obtained by fitting the functional form Eq. (10) with an exponent $\tau =2.1$ nearly equal to the one of the size distribution including all the events.

Figure 12

Average size of events $〈{\mathrm{\Delta }}_n〉$ in the vicinity of records as a function of their integer index relative to records $n-n_k$. High rank records are preceded by events of increasing size and they are followed by a relaxation process where the event size gradually decreases. The bold line represents fit with Eq. (11).

Figure 13

Average value of the physical waiting time $〈T_n〉$ between consecutive events as a function of the event index relative to records $n-n_k$. The low and high rank records are presented separately in (a) and (b), respectively. The red bold continuous lines in (b) represent fit with Eq. (11).