# Detecting Criticality by Exploring the Acoustic Activity in Terms of the \"Natural-Time\" Concept
## 2 Theoretical Preliminaries  
The concept of Natural Time, \(\chi\), was introduced in 2001 by [PERSON] et al. [24; 25] and it was initially used in the field of Seismology. Since then the analysis in the Natural Time domain has been proven quite advantageous when describing the dynamic evolution of complex systems: It reduces uncertainty, allowing optimal extraction of signal information, especially for signals originating from dynamic systems approaching the criticality [26]. In this context, the Natural Time concept became gradually a valuable alternative tool in the tedious effort of the scientific community to develop theories for the prediction of upcoming catastrophic events. Soon after the concept was introduced it was adopted for data analysis in an amazingly wide range of scientific disciplines, from seismology [26; 27; 28; 29; 30], to medicine [31; 32], condensed matter physics [33] and mechanics of materials [34; 35; 36; 37; 38; 39; 40]. As mentioned above in a fracture process, the function \(F(\tau)\) is created by assuming that is a time series consisting of \(N\) values, namely: \(F_{1}=F(\tau_{1})\), \(F_{2}=F(\tau_{2})\),..., \(F_{k}=F(\tau_{k})\),..., \(F_{N}=F(\tau_{N})\). Instead of the conventional time \(\tau\), a value \(k\) of order (1, 2, 3... \(N\)) will be characterized by the evolution of the pair \((\chi_{k},F_{k})\) where \(\chi_{k}=k/N\) and \(F_{k}\) is the value of \(F(\tau_{k})\). This creates the time series \(F(\chi)\) where \(\chi\) is the Natural Time (see Figure 1). In order to further clarify the above-described procedure, a detailed example will be discussed in this section, indicating the way of representing the acoustic activity (expressed here as the rate of occurrence of the acoustic hits) in terms of the \(F\)-function, both in the conventional (\(\tau\)) and the Natural Time (\(\chi\)) domains: a uniaxial tensile test with a Double Edge Notched (DENT) marble specimen (Figure 2a) with notch length equal to \(\alpha=8\) cm is considered. Details of the experimental protocol concerning the specimen and the experimental set-up have been reported in previous works [41; 42]. During the specific test, a total of 137 acoustic hits of amplitude \(A\geq 40\) dB were recorded, the vast majority of which were detected at the very last loading stages, i.e., when the externally applied load was approaching the tensile strength of the specimen. Figure 1: Schematic interpretation of the \(F\)-function in terms of conventional time (**up**) and in the Natural Time domain (**down**). For the calculation of the \(F\)-function \(n=10\) consecutive inter-event time intervals were used and its evolution in conventional time \(\tau\) is presented in Figure 3a. The sharp increase of the \(F\)-function starts at the instant \(\tau\approx 1245\) s when the externally applied tensile force attained a value equal to about 2889 N, corresponding to 98.4% of the maximum force attained during the specific test. It is clearly seen that densely packed information is hidden in a very narrow time interval (just before fracture) and thus it is quite possible that critical details are lost. Therefore, the need for an alternative representation of the acoustic activity becomes a pressing demand. In this direction, the evolution of the \(F\)-function is considered in the Natural Time domain (\(\chi\)) as it is shown in Figure 3b. It is highlighted that, for the whole time interval for which the applied force varies around its maximum value (i.e., the catastrophic fracture is impending), the function \(F(\chi)\) is governed by a power law of the form:  
\[F(\chi)=A\cdot\chi^{m} \tag{2}\]  
where \(A\) is a constant [\(A=F(\chi=1)\)] and \(m\) is an exponent related to how strongly the acoustic activity is intensified at the time instants just before macroscopic fracture. For the specific experiment analyzed in this section and for the time interval corresponding to the shaded rectangular in Figure 3b, proper curve fitting yields for \(m\) a value equal to 4.65.