# Detecting Criticality by Exploring the Acoustic Activity in Terms of the \"Natural-Time\" Concept
## 1 Introduction  
The irreversible processes (for example, microcracking, propagation and coalescence of microcracks, various changes of the microstructure) taking place due to the activation of damage mechanisms at the interior of loaded structural elements are responsible for the generation of acoustic emissions (AEs). The respective sensing technique that detects and records these emissions, namely the AE technique, was introduced long ago [1; 2; 3] and is still under further development. Gradually it became an extremely useful tool in the hands of engineers from various disciplines since it allows the real-time monitoring of the damage evolution under either quasi-static or dynamic loads, as well as the characterization/classification of the failure modes taking place [4]. Nowadays, the AE technique is considered the most mature and reliable tool in the attempt to understand the mechanisms activated before the loaded structure enters into its critical stage (i.e., that of impending fracture) and during this critical stage. Moreover, it is widely used for structural health monitoring purposes. Quite a few characteristics of the AEs (such as, for example, their rate of occurrence, their cumulative energy, the temporal evolution of b-values, the cumulative number of counts, etc.) are properly elaborated statistically enlightening interesting aspects of the overall response of the loaded structures [4; 5; 6; 7; 8; 9]. The number of hits recorded per second is an often-used index for monitoring the acoustic activity during a loading/fracture process. Recently an alternative procedure was introduced, which offers monitoring the acoustic activity with increased resolution by taking advantage of the inter-event timeintervals between two successive hits. The procedure is based on the \(F\)-function [10], which is determined by utilizing the interevent time intervals of a sufficient number of \(n\) (\(n>10\)) consecutive AE hits/events. In fact, the \(F\)-function represents the average frequency of appearance of AE hits/events in a sliding time window including \(n\) consecutive AE hits/events. Each value of the \(F\)-function is paired to an average time instant, \(\tau\), determined as the average of the time instants at which each one of the \(n\) successive hits of the group contained in the specific time window appeared. Contrary to the traditional way of representing the AE activity through the \"number-of-hits-per-second\", the temporal evolution of the \(F\)-function is advantageous since it highlights according to a clearer manner the variability of the acoustic activity, especially during the last phase of the loading process just before the catastrophic fracture. Taking into account that the vast majority of acoustic events are recorded during the very last time instants prior to the macroscopically observed catastrophic fracture, the time evolution of the F-function is usually described in terms of the \"time-to-failure\" parameter, (\(t_{f}-\tau\)), where \(t_{f}\) is the time instant when the last AE hit/event was recorded (i.e., the time instant of the catastrophic fracture, when usually an AE hit/event with an amplitude of 99 dB is detected). Adopting the specific description offers an enlightening view of what happens during the very last loading instants. The use of the \(F\)-function for the exploration of acoustic activity is nowadays adopted by many researchers worldwide [11; 12; 13; 14; 15; 16; 17]. The temporal evolution of the \(F\)-function in terms of the \"time-to-failure\" parameter, during the last loading stages before the catastrophic fracture, was found to obey a power law [10; 18] of the form:  
\[F=A\cdot\left(t_{f}-\tau\right)^{-m} \tag{1}\]  
where \(A\) is a constant and \(m\) is an exponent, the numerical value of which ranges around the unit. In general, the exponent \(m\) characterizes the intensity of the rate of production of acoustic hits/events just before fracture.