1. Introduction and Motivation
The analysis of the Acoustic Emissions (AE) in materials under deformation in laboratory experiments, is a very important technique for understanding the damage accumulation and failure modes [
1,
2,
3,
4,
5,
6,
7].
AE are high frequency elastic waves due to micromechanical damage induced by the microcracking. These emissions therefore represent indicators of deformation and fracturing processes occurring within a tested specimen and allow to monitor cracking formation, growth and propagation. The detection of the AE can therefore allow both to understand how the damage accumulates and develops within the material, and to monitor the final rupture. This information could be very useful in geophysics, for the interpretation of field scale seismic signals and the understanding of earthquakes precursors. Furthermore, a deep insight in the cracking propagation in the considered rocks can also be very important in civil engineering, for monitoring the integrity of bridges and buildings and develop strategies for safe design of tunnels.
From the seismological point of view AE obey, as earthquakes, to the GutenbergRichter relationship between frequency and magnitude. Rock fracture and earthquake rupture are processes obeying similar statistics for source dimensions over more than eight orders of magnitude [
8,
9]. In order to link experimental detailed studies to geophysical signatures recorded at the field scale, controlled laboratory rock deformation experiments equipped with dense microseismic arrays for AE detection have become a routinely used tool [
10,
11,
12]. Here, fault growth may be considered analogous to the field scale development of earthquake rupture generating acoustic emissions, which is a wellused analog to tectonic earthquakes due to the scale invariance of these processes [
13,
14,
15]. Source mechanisms evolution with respect varying effective pressure can also be inferred from AE, allowing to determine the effect of the increasing lithostatic pressure on the fracturing mechanisms and failure mode [
16].
In a civil engineering context, any project for the seismic design or monitoring of large structures needs to consider soilstructure interaction. The dynamic response of buildings or bridges, for example, is closely related to the mechanical behavior of the foundation soil. Other structures of great importance in civil engineering, whose behavior under seismic excitations is closely related to the nature of the surrounding rocks, are the tunnels. A reliable geological information is very important in seismic design of tunnels [
17,
18]. Excavation methods, dimensions and design parameters of a tunnel strictly depend on the type of rocks along its alignment. The mechanical information regarding soil and rocks are obtained by means of appropriate tests performed either in situ or from samples. In the case of tunnels, the surrounding rocks are subjected to distributed pressures which may vary in direction and intensity. When samples of rocks are extracted, it is therefore important to reproduce in 3D experimental tests the effects of the distributed confinement pressures. A reliable 3D laboratory test on soils and rocks, is the triaxial test which allows to evaluate the effect of confinement pressures on the failure load of the specimens [
19,
20,
21,
22].
In this paper we present a new analysis of the AE recorded during conventional triaxial deformation tests. In particular, our experimental setup allows recording AE generated during the loading by the crack initiation, propagation and growth leading eventually to macrofractures formation and sample failure. In this context, the use of nonextensive qstatistics has recently proved to be particularly effective in capturing some universal features which emerge during the cracks propagation under loading [
1,
7]. In a previous paper, some of the authors of this paper investigated, by means of qstatistics, AE in uniaxial compression experiments on samples of Basalt and Concrete subjected to cyclic loading [
5]. The aim of the present study is to apply a similar statistical analysis to AE obtained in triaxial compression tests in order to take into account the effect of the confining pressure on the AE release and assess the influence of increasing confining pressure, different deformation and failure mode on the AE statistical properties.
2. Experimental Setup and Data Analysis
The AE used in this work have been recorded during conventional triaxial deformation experiments on cylindrical 40mm x 100mm samples with an array of twelve 1MHz singlecomponent PiezoElectric Transducers for AE detection (REF). Triaxial compression tests have been performed at 5,10,20 and 40MPa on geologically and physically (i.e., fabric, porosity, grain size, and cementation) different lithologies such the Darley Dale Sandstone (DDS, porosity 14%) and Alzo Granite (AG, porosity < 1%). The AE data sets have been analysed in terms of source mechanisms time and spatial evolution before the rupture from some of the authors [
16]. AE source mechanisms analysis has evidenced single fracture nucleation for AG and multiple fracture nucleation for DDS due to single or multiple competing dilatant and compactant regions. Fracture growth and propagation appears controlled by the different confinements, with increasing pressure controlling the time evolution and size of dilatant and compactant regions, eventually controlling cracks coalescence into macroscopic fractures [
16].
In order to give representative insights on the AE time and amplitude distribution prior failure, we plot in
Figure 1 the AE amplitude (a) and the AE Interevent time (b) as function of time for two samples: AG at 40 MPa (top panel) and DDS at 20 MPa (bottom panel). The interevent time
$\delta \tau \left(n\right)$ is the time interval (in seconds) between two consecutive recordings AE(n) and AE(n − 1) and can be defined as:
where t
_{AE}(n) is the time at which the nth AE event does occur and t
_{AE}(n−1) the time of the previous event.
The coupled analysis of AE amplitude and interevent times reveals that AG shows an abrupt increase at the end of the elastic phase, between 500 and 1000 seconds, then AE amplitude remains quite constant until failure (which happens at about 2800s) and, finally, it suddenly decreases during the stickslip AE events occurring in the postfailure phase, with occasional AE clusters of higher amplitude driven by stress at the fault asperities; on the other hand, DDS presents a much more gradual AE amplitude increase from 500 to 1500 seconds, reaching steady values before failure (which happens at about 2400s), then a decrease driven by the postfailure stickslip processes. This difference can be explained by the different deformation mechanisms acting on the different lithologies. In fact, at the end of each sequence, macroscopic structure forms. For AG, a single damage cycle of crack nucleation and growth is sufficient to propagate fractures and develop the planar localization leading to dynamic failure. Whilst in DDS it can take multiple cycles of nucleation for coalescence to take place due to interacting mechanisms induced by multiple fracture nucleation sites [
16].
Thus, the DDS shows a much clear premonitory phase, or foreshock, before a critical damage threshold which would allow the coalescence into a larger scale deformation structure. For both the samples, failure is characterized by a peak in the AE interevent time, which suddenly appears after a sequence of very low values corresponding to the sequence of high amplitude events before rupture, corresponding to the transition from mm scale propagating microfractures to a fully developed cm scale fault zone.
In
Table 1 we report, for each sample, the time t
_{B} at which the breakdown occurs, the total time t
_{TOT} of the experiment, the number N* of AE events before breakdown and the total number N of AE events present in the corresponding time series. In the following we will investigate if qstatistics can help in revealing different structures in these data sets for increasing levels of confinement.
AE Amplitudes
To start, let us look at the amplitude probability distributions of the acoustic emissions for our eight considered samples. Firstly, we explore these distributions by dividing each time series in four parts (time intervals):
from the beginning to 30% t_{B}
from 30% t_{B} to 70% t_{B}
from 70% t_{B} to breakdown
after breakdown.
Of course, the number of data included in each time interval could be even quite small, however such a procedure could bring out otherwise hidden details of the process leading to the rupture.
In
Figure 2, the amplitude PDFs for AG (a) and DDS (b) samples, within each time interval and for each level of confinement, are plotted; the number of events included in each time interval is also reported in legenda.
We will not show distributions corresponding to time intervals with less than 50 AE events, due, of course, to the too poor statistics. We also anticipate that amplitude data for DDS at 5 MPa are not reliable, since they have been affected by a problem with the preamplifier gain during the experiment.
Comparing the various panels for the two types of lithologies, one immediately notices several features which reveal some kind of universal behavior:
Events in the first time interval (within 30% t_{B}) are always too few to give consistent distributions, regardless of the material;
Distributions before and after failure are quite similar, again regardless of the material, with an initial sudden increase, a peak and a slow decrease for high amplitudes;
Distributions after failure are more peaked for both AG and DDS.
Our approach identifies the background general behaviour driving the main mechanical phases of nucleation, growth, and coalescence of micro fractures into a macroscale fault zone, regardless the prevalence of a specific phase in a specific deformation stage driven by the different lithologies and effective pressures.
In
Figure 3 the complete probability distributions for the whole time series of the different samples are reported both in LinLin (ab) and in LogLin (cd) scale. It immediately appears that, for both the materials and all the confinements, all the curves collapse one onto the others (with the exception of DDS 5Mpa, which has been excluded for the reasons explained before) and in all cases show powerlaw tails which overlap one over the others. The shape of the LinLin curves is similar to that found in
Figure 2, but the plots in LogLin scale tell us that we are in presence of fat tails. This result confirms that the amplitude of the AE events is scaleinvariant and does not depend on lithology and confinement. Both the distributions (c) and (d) can be well fitted with the following function, given by the product of a quadratic term and a
qexponential one (which, for
q > 1, is a power law):
where
${A}_{0}$ = 7000,
${\beta}_{A}=40$ and
${q}_{A}$ = 1.19. The powerlaw which is introduced as a prefactor of the
qexponential function is similar to the density of states which is present in the Planck law for the blackbody radiation. Its origin here is possibly related to threedimensional nearly isotropic stresses. Similar powerlaw prefactors turn out to be necessary in diverse complex situations, such as the volume distributions in stock exchanges [
23], distributions of transverse momenta of hadronic jets produced in protonproton highenergy collisions at CERN/LHC [
24], and COVID19 peaks in the recent pandemics [
25], among others.
AE InterEvent Times
Let us now shift our attention to the AE interevent times. In particular, we will study the complementary cumulative (decumulative) probability distributions of AE interevent times for both the AG and DDS samples, adopting only data before breakdown.
To build the decumulative distribution P(> $\delta \tau $) of the interevent time series, one has to report, for each value of $\delta \tau $ in the interval [0,1000], the fraction of interevent times which are greater of that value. Therefore, in LogLog scale, one could expect a curve starting from the unit for small values of $\delta \tau $, then, after a certain inflection point, gradually decreasing with some kind of peculiar behavior for high values of $\delta \tau $.
In similar experiments [
1,
5], these decumulative distributions exhibited clear powerlaw tails which can be framed in the context of the
qgeneralized thermostatistics: actually, in these cases simple
qexponential functions were able to well fit the obtained curves, thus unveiling the fractal or multifractal nature of the breakdown process. In the results presented here, as shown in
Figure 4 for AG and in
Figure 5 for DDS, we found something more complex than the expected powerlaw tails: in fact, the decumulative PDFs for both the type of analyzed materials seem to further change slope in correspondence of a second inflection point, whose time position decreases with increasing the confinement for DDS while seems to oscillate for AG. This new kind of behavior could be still described in the framework of
qthermostatistics, but adopting the following more general fitting function [
26]:
It is composed by the sum of a first standard
qexponential function (with normalization factor
$A$, inverse temperature
${\beta}_{1}$ and entropic index
${q}_{1}$) and a second function containing (in addition to a normalization factor
$(1A)$, an inverse temperature
${\beta}_{2}$ and an entropic index
${q}_{2}$) a further parameter
$\lambda >0$ which ensures that the total function monotonically vanishes for increasing intertimes with an appropriate behavior. See [
27] for a similar crossover in the areapreserving dynamics of the standard map for intermediate values of the control parameter.
Looking to
Figure 4 and
Figure 5, it can be appreciated that function (2) is able to well fit all the PDFs, regardless of the material, provided that the following parameters are chosen:
In the previous
Table 2 one can notice that, for any material and confinement, typically
${\beta}_{1}\gg {\beta}_{2}$ and
${q}_{1}<{q}_{2}$. Moreover, in correspondence of the same amount of confinement, values of
$A$ coincide for AG and DDS. A double qexponential behavior, although kind of rare, occasionally emerges in complex systems. Such is the case for the nucleotide interdistances in DNA sequences of Homo Sapiens [
28].
AE Positions and Interevent Distances
Finally, let us investigate about if the AE events are clustered in space. In order to do this, we first explore the behavior of their subsequent positions (expressed in meters), ordered in time before breakdown (with a blue scale of decreasing intensity) and projected on the three planes XY, XZ and YZ, for both AG (
Figure 6) and DDS (
Figure 7) samples with the usual confinements.
Spatial distributions of AE show a higher clustering of events in AG, where fracturing occurs throughout localized planar fractures, while more scattered nucleation centers related to dilatant patches occur prior failure in DDS.
It is also useful to study the probability distribution of the AE interevent distances (also expressed in meters), defined as the metric distance between the 3D spatial positions of two subsequent recorded AE events inside a given sample during an experiment:
In
Figure 8 we plot the distributions of the interevent distances before rupture obtained for both AG (top panel) and DDS (bottom panel) at the different levels of confinement. What we observe is a Plancklike distribution, with a maximum and an asymmetric tail, which can be well fitted by the following function:
where the values of the two fitting parameters (A = 4.8 10
^{7} and B = 132) are independent of the type of materials and of the confinement, thus revealing again some kind of universal behavior. On the other hand, it is well visible a single narrow peak around zero for some AG samples, in particular those with intermedium levels of confinement, a peak which is completely absent for DDS samples. This might be explained by the single fracture nucleation mechanisms observed for AG, implying high spatial clustering with respect to the multiple fracture nucleation mechanisms observed for DDS, due to multiple fracturing regions [
16] and implying more diffused seismicity and a lower clustering.
To close this analysis, let us finally look at the decumulative distributions P(>
$d$) of the interevent distances before rupture, which are shown in
Figure 9 for AG samples and in
Figure 10 for DDS ones.
As for the interevent times decumulative PDFs, also in this case it is possible to fit all the distributions with
qexponential functions, but in this case are enough single standard
qexponentials with inverse temperature
$\beta $ and an entropic index
q. Notice that, in analogy to what has been found in [
29], values of the entropic indexes are all below 1, indicating not a power law behavior but revealing the presence of a cutoff in the distributions. Moreover, as shown in
Table 3, in agreement with a further finding of [
29], calling
${q}_{\tau}$ the first entropic index
${q}_{1}$ obtained for the interevent time PDFs (see
Table 2) and
${q}_{d}$ the entropic index just found for the interevent distance, it can be noticed that the sum
${q}_{\tau}+{q}_{d}$ oscillates around 2 for all the samples, regardless of both material and confinement. This results is also similar to what has been observed in regional seismicity data from Japan and California and numerically verified using the twodimensional BurridgeKnoppoff model [
29].
In
Figure 11 the entropic index
${q}_{\tau}$ is reported as function of
${q}_{d}$. The figure clearly shows that the linear behavior holds quite well inside an error of 10% for almost all the samples.
The results of this study could be applied in different research fields, such as the collapse prediction of building materials and the interpretation of deformation mechanisms preceding and accompanying earthquake ruptures. Actually, regardless the deformation mechanisms, whether planar localization or dilatant patches developing at different effective pressures and lithologies, and driving the precursory phase before a critical damage threshold is reached, the macroscopic coalescence into a larger scale deformation structure and its seismic output would be controlled by the transition from mm scale fractures into a cm scale fault zone.
Figure 1.
AE Amplitude (a) and interevent time (b) as function of time (s) for AG 40MPa (top) and DDS 20MPa (bottom) samples.
Figure 1.
AE Amplitude (a) and interevent time (b) as function of time (s) for AG 40MPa (top) and DDS 20MPa (bottom) samples.
Figure 2.
Amplitude probability distributions for both AG (a) and DDS (b) and for the four increasing levels of confinement. Each data set has been divided in four time intervals, three before breakdown (030%, 30%70%, 70%100%) and the last one after failure. In the legend, for each interval and each level of confinement, the number of AE events is reported in parentheses. Only distributions extracted from more than 50 events are shown in the panels.
Figure 2.
Amplitude probability distributions for both AG (a) and DDS (b) and for the four increasing levels of confinement. Each data set has been divided in four time intervals, three before breakdown (030%, 30%70%, 70%100%) and the last one after failure. In the legend, for each interval and each level of confinement, the number of AE events is reported in parentheses. Only distributions extracted from more than 50 events are shown in the panels.
Figure 3.
Amplitude probability distributions for the entire time series of AG (a,c) and DDS (b,d) at the four levels of confinement are reported, both in LinLin (a,b) and in LogLin (c,d) scale. The LogLin curves have been also fitted with a nonlinear function.
Figure 3.
Amplitude probability distributions for the entire time series of AG (a,c) and DDS (b,d) at the four levels of confinement are reported, both in LinLin (a,b) and in LogLin (c,d) scale. The LogLin curves have been also fitted with a nonlinear function.
Figure 4.
AG samples: double qexp fits for the decumulative PDFs of interevent times.
Figure 4.
AG samples: double qexp fits for the decumulative PDFs of interevent times.
Figure 5.
DDS samples: double qexp fits for the decumulative PDFs of interevent times.
Figure 5.
DDS samples: double qexp fits for the decumulative PDFs of interevent times.
Figure 6.
Scatter plots of the time ordered positions before rupture (blue scale of decreasing intensity), projected on the three coordinate planes XY, XZ and YZ, for AE events in the four AG samples.
Figure 6.
Scatter plots of the time ordered positions before rupture (blue scale of decreasing intensity), projected on the three coordinate planes XY, XZ and YZ, for AE events in the four AG samples.
Figure 7.
Scatter plots of the time ordered positions before rupture (blue scale of decreasing intensity), projected on the three coordinate planes XY, XZ and YZ, for AE events in the four DDS samples.
Figure 7.
Scatter plots of the time ordered positions before rupture (blue scale of decreasing intensity), projected on the three coordinate planes XY, XZ and YZ, for AE events in the four DDS samples.
Figure 8.
PDFs of the AE interevent distance for AG (top panel) and DDS (bottom panel) at the various levels of confinement.
Figure 8.
PDFs of the AE interevent distance for AG (top panel) and DDS (bottom panel) at the various levels of confinement.
Figure 9.
AG samples: single qexp fits for the decumulative PDFs of interevent distances.
Figure 9.
AG samples: single qexp fits for the decumulative PDFs of interevent distances.
Figure 10.
DDS samples: single qexp fits for the decumulative PDFs of interevent distances.
Figure 10.
DDS samples: single qexp fits for the decumulative PDFs of interevent distances.
Figure 11.
The entropic index ${q}_{\tau}$ is reported as function of ${q}_{d}$ for the eight considered samples (colored circles) and it is compared with the line ${q}_{\tau}=2{q}_{d}\text{}$. The $\pm $10% area around the line is colored in brown.
Figure 11.
The entropic index ${q}_{\tau}$ is reported as function of ${q}_{d}$ for the eight considered samples (colored circles) and it is compared with the line ${q}_{\tau}=2{q}_{d}\text{}$. The $\pm $10% area around the line is colored in brown.
Table 1.
Details of the time series for the different samples.
Table 1.
Details of the time series for the different samples.
Sample 
t_{B}

t_{TOT}

N* events before breakdown 
Tot. N events 
AG5 MPa 
2398 s 
3395 s 
2367 
2751 
AG10 MPa 
1767 s 
2445 s 
1577 
1956 
AG20 MPa 
2160 s 
2973 s 
1874 
2367 
AG40 MPa 
2857 s 
4566 s 
4419 
5533 
DDS5 MPa 
2400 s 
2540 s 
1067 
1100 
DDS10 MPa 
5808 s 
6248 s 
4802 
5334 
DDS20 MPa 
2409 s 
3348 s 
5760 
6714 
DDS40 MPa 
9710 s 
11736 s 
10659 
11696 
Table 2.
Details of the fitting parameters of Equation (2) for the eight samples considered.
Table 2.
Details of the fitting parameters of Equation (2) for the eight samples considered.
Sample 
$\mathit{A}$ 
${\mathit{\beta}}_{1}$ 
${\mathit{q}}_{1}$ 
${\mathit{\beta}}_{2}$ 
${\mathit{q}}_{2}$ 
$\mathit{\lambda}$ 
AG5 MPa 
0.975 
3.5 
1.48 
0.01 
1.7 
0.09 
AG10 MPa 
0.98 
2.0 
1.32 
0.02 
2.5 
0.19 
AG20 MPa 
0.98 
1.3 
1.2 
0.04 
2.0 
0.13 
AG40 MPa 
0.983 
2.6 
1.16 
0.017 
1.4 
0.12 
DDS5 MPa 
0.975 
1.5 
1.4 
0.0001 
1.7 
0.055 
DDS10 MPa 
0.98 
4.0 
1.32 
0.02 
1.7 
0.19 
DDS20 MPa 
0.98 
4.1 
1.24 
0.017 
1.5 
0.20 
DDS40 MPa 
0.983 
4.6 
1.16 
0.015 
2.4 
0.19 
Table 3.
The sum of the entropic indices ${q}_{\tau}$ and ${q}_{d}$ for the 8 considered samples oscillates around 2.
Table 3.
The sum of the entropic indices ${q}_{\tau}$ and ${q}_{d}$ for the 8 considered samples oscillates around 2.
Sample 
${\mathit{q}}_{\mathit{\tau}}$ 
${\mathit{q}}_{\mathit{d}}$ 
${\mathit{q}}_{\mathit{\tau}}+{\mathit{q}}_{\mathit{d}}$ 
AG5 MPa 
1.48 
0.76 
2.24 
AG10 MPa 
1.32 
0.75 
2.07 
AG20 MPa 
1.2 
0.69 
1.89 
AG40 MPa 
1.16 
0.75 
1.91 
DDS5 MPa 
1.4 
0.78 
2.18 
DDS10 MPa 
1.32 
0.75 
2.07 
DDS20 MPa 
1.24 
0.69 
1.93 
DDS40 MPa 
1.16 
0.79 
1.95 