Abstract

Acoustic emissions exhibit complex correlations between space, time, and magnitude and as such they present a unique example for a complex time series. We apply the recently introduced method of natural time analysis, which enables the detection of long-range temporal correlations even in the presence of heavy tails and find that the acoustic emissions exhibits features similar to that of other equilibrium or nonequilibrium critical systems such as the worldwide seismicity as presented in the Centennial earthquake catalogue which includes global seismicity event with magnitude Mw>7.0. It is recognized that earthquake is the failure of the focal earth matterial accompanied by a rapid release of moment. Similarly, the acoustic emissions (AEs) in a rock experiment, are elastic waves generated in conjunction with energy release during crack onset, propagation and internal deformations in rock’s body. Experiments of rock deformation are considered as a tool for understanding the occurrence of natural earthquakes [1]. Acoustic emission studies can give us an insight into the fracture network evolution processes that take place and provide us with the opportunity to develop laws suitable for testing at larger scales [1]. The latter could be useful in understanding earthquake mechanisms and may contribute to solving the problem of earthquake prediction [2]. Fracturing is one of the most important examples of a complex process in heterogeneous materials involving a wide range of time and length scales, from the microto the structural scale. This process is governed by the nucleation, growth and coalescence of microcracks, eventually leading to failure. In this context, fracture can be seen as the outcome of the irreversible dynamics of a long-range interacting, disordered system. [3] During rock deformation, energy released as high-frequency AE from microfractures within the sample. These emissions provide a passive indicator of the progression of inelastic damage, during the approach to failure. Characterisation of the sources that produce AE can provide us with an insight into the microscopic processes that are involved in the initiation and coalescence of damage within a loaded rock sample. Laboratory AE exhibit some remarkable similarities with large scale seismological events and earthquake physics, such as power law, frequency–magnitude distributions and Omori law aftershock behaviour [4-6]. Monitoring and characterisation of AE during experiments can improve our understanding of a wide range of processes, including fault asperity rupture and volcano–seismic events [7-9]. Recently this spatio-temporal similarity has been views in the frame of non-extensive statistical physics [10] in addition to the views where brittle fracture has been associated with a first –order transition [11-13] or to a critical point phenomenon [14]. We note that both the aforementioned approaches lead to power-law distributions since second-order transitions present scaling close to the critical point, while the first-order transitions follow scaling laws when the range of interactions is large [15] The main motivation of our work is to investigate fracture in a heterogeneous brittle material (Etna basalt) under triaxial deformation, analyzing the temporal correlation of moment release of AE from microfractures that occur before the final fracture. We focus on the analysis of acoustic emissions in natural time [16-20]. We apply natural time analysis because it has been shown [18] that the analysis of time series of complex systems in this time domain reduces uncertainty and extracts signal information as much as possible. Natural time analysis enables [18-19] among others, the identification of long-range correlations even in the presence of ―heavy tails‖ [21]. In addition, since the applications of this new type of analysis with interesting results have been presented in a variety of cases including seismicity [see 18 and the references therein, 21-28] and self-organized criticality, [29-32] the question whether fracture (i.e., AEs) is described by natural time parameters, even at the phenomenological level presents a challenge, possibly leading to a universal principle from rocks crack up to geodynamic scale. High-speed multi-channel waveform recording technology enables us to monitor the spatio-temporal evolution of fracturing processes using AEs activity in triaxially deformed rock samples with high precision. [7-9]. Here we study acoustic emissions catalogue collected in laboratory experiments on highly fractured samples of Etna basalt, a porphyritic, alkali, lava-flow basalt from Mount Etna, Italy, which comprises millimetre-sized phenocrysts of pyroxene, olivine and feldspar in a fine-grained groundmass [7-9] deformed at a constant axial strain rate of 5 × 10 s and at an effective confining pressure of 40 MPa . Previous studies [8] have shown that the Etna basalt used in this study contains a ubiquitous network of pre-existing microcracks, which are distributed relatively isotropically with the opening of new, dilatant microcracks to be present with their long axes parallel to the σ1 direction. Figure 1a shows the AEs magnitude MAE similar defined in earthquakes, versus time. We observe that AE are mainly observed in the final stage of deformation in consistency with that stated in [7-9] that rapid acceleration to failure often observed in the final phase of triaxial deformation of brittle rocks is accompanied by a fast increase in AE activity. The record between the arrows A and B (figure 1a), which spans the period of crack growth and dynamic failure, has used to analyse AE in natural time. As reported in [7-9] within the period A-B microcracks appear to nucleate in the lower right-hand part of the sample and then propagate to the upper left-hand part of the sample with the bulk of the total acoustic emission activity from the whole experiment to contained within this period. As presented in figure 1c, during the period A-B the cluster of AE propagates diagonally across the whole sample. The natural time analysis of a complex system presented for first in [16] and in detailed in [18]. Here we recapitulate the concept of natural time analysis as applied in acoustic emission data. In a time series consisting of N acoustic emissions, the natural time χ serves as an index for the occurrence of the k event and is defined as χk = k/N. For the analysis of AE the pair (χk, Mk) is considered, where Mk is the seismic moment released during the k event. Considering the evolution of (χk, Mk), the continuous function F(ω) is defined as: F ω = Mk N k=1 exp iω k N (1), where ω = 2πφ and φ stands for the natural frequency. We normalize F(ω) dividing it by F(0), Φ ω = Mk N k=1 exp iω k N Mn N n=1 = pk N k=1 exp iω k N (2), where pk = Mk Mn N n=1 . The quantities pk describe a probability to observe the acoustic event at natural time χk. From (2) a normalized power spectrum can be obtained: Π ω = |Φ(ω)|. For natural frequencies φ less than 0.5, Π(ω) or Φ(ω) reduces to a characteristic function for the probability distribution pk in the context of probability theory. It has been shown [18] that the following relation holds: Π ω = 18 5ω2 − 6 cos ω 5ω2 − 12 sin ω 5ω3 (3) According to the probability theory, the moments of a distribution and hence the distribution itself can be approximately determined once the behavior of the characteristic function of the distribution is known around zero. For ω→0, (3) leads to: Π ω ≈ 1 − κ1ω 2 (4), where κ1 is the variance in natural time given as κ1 = χ 2 − χ 2 = pk N k=1 χk 2 − pk N k=1 χk 2 (5). The quantity κ1 has been proposed [18] as an order parameter for seismicity, based on three important findings: (a) The quantity κ1 abruptly changes acquiring values very close to zero, once a final fracture (i.e a strong earthquake in case of seismicity) takes place, (b) when studying the fluctuations of κ1 through an events catalogue using its scaled probability distribution function a universal curve appears and (c) the resulting universal curve exhibits fluctuations similar with those observed for other equilibrium and nonequilibrium critical systems. When studying an acoustic emissions catalogue comprising W events by using a sliding natural time window of length l and examine the window starting at k = k0, the quantities pj = Mk0−j−1/ Mk0+m−1 l m=1 are well defined and give rise to an average value μj equal to: μj = 1 W−l+1 Mk0+j−1 Mk0+m−1 l m =1 W−l+1 k0=1 (6) The second order moments of pj, as the variance is given as : Var pj = 1 W−l+1 Mk0+j−1 Mk0+m−1 l m=1 − μj 2 W−l+1 k0=1 , while the covariance [18, 21] is calculated by the expression : Cov pj , pi = 1 W − l + 1 Mk0+j−1 Mk0+m−1 l m=1 − μj W−l+1 k0=1 × Mk0+i−1 Mk0+m−1 l m=1 − μi Then the expectation value of κ1 [18, 21] is expressed as : E κ1 = 1 W−l+1 j l 2 Mk0+j−1 Mk0+m−1 l m=1 − j l Mk0+j−1 Mk0+m−1 l m =1 l j=1 2 l j=1 W−l+1 k0=1 (7) obtained from the W − l + 1 windows of the acoustic emissions catalogue and is given by: E κ1 = κ1,M + . l i=j+1 l−1 j=1 j−1 2 l2 Cov pj , pi (8), whereκ1,M is the value obtained from equation (5) when substituting μk for pk, [21]. Natural time analysis enables the identification and quantification of magnitude correlation in a catalogue of acoustic emissions in matter analogue of that of seismicity [21], by comparing the value of E κ1 of the original AE series with the distribution obtained for E(κ1,shuf h) when many randomly shuffled copies of the original AE catalogue used. Following the latter approach we consider a randomly shuffled copy of the original AEs catalogue, expecting that all pj to be equivalent independent of j and thus μj= 1/l . It has been shown [18] that the expectation value for κ1, denoted by E(κ1,shuf ) in the ensemble of randomly shuffled copy case, is given by: E(κ1,shuf ) = κu 1 − 1 l2 − κu l + 1 Var p (8), where κu = 1/12 corresponding to a uniform distribution and Var(p) the expectation value for(p