# Long-Range Correlations and Natural Time Series Analyses from Acoustic Emission Signals
## 3 Applications
### Three-Point Bending Test of a Glass Fiber-Reinforced Polymeric Plate
#### 3.1.2 Results  
Figure 7a depicts the AE activity during the TPB test in normalized time (time/timepeak, where timepeak = 72 s is the peak loading time). Approximately 5050 signals above a 0.25 mV threshold level were separated throughout the TPB test according to the methodology that was described in [26]. The number of instantaneous and accumulated signals are presented in Figure 7b, accompanied by the evolution of the load (F/Fmax) and the AE energy (EAE/EAEmax), which was computed as recommended in RILEM [60]. The results in the same figure indicate a considerable correlation between the AE energy and the number of accumulated signals. In addition, one identifies three distinct patterns throughout the test. The first occurred in the region 0-0.26, where a few sparse avalanches were identified. Their isolated nature suggests that they were probably caused by local effects, such as the specimen accommodation on its supports, without a significant change in the specimen stiffness. Within the 0.26-0.78 range, damage grew indistinctly throughout the structure, with low-amplitude AE signals from no particularly discernible source. Finally, cracking coalescence took place in the region 0.78-1.0. This was indicated by the increment in the signal avalanches, which can be considered a precursor to the imminent collapse in the structure. The \(b\)-value was the reference parameter to the analyses that was performed through all methods that were considered in this work, and its calculation is described in Figure 7a. The test period was divided into 14 intervals, and the \(b\)-value was calculated for each of them [26]. Instead of repeating the expected pattern that is shown in Figure 1, it decreased locally in several opportunities as large-amplitude AE signals occurred throughout the process. That suggests that the rupture happened in spasms that were uniformly distributed along the whole process instead of abruptly, i.e., the energy dissipates in several small events instead of being stored to a large amount and released at once in an explosive collapse. Figure 7c summarizes the corresponding AE signal long-range analysis according to the DFA and \(R\)/S\({}^{\star}\) methods. The DFA (blue points) and Hurst (pink points) exponents were evaluated for each event throughout the normalized test period, considering a 25-hit moving window, i.e., parameters were calculated at hit #25, based on a temporal series starting at hit #1, then from #2 to #26, #3 to #27 and so on. The dotted black transversal line separates the values of \(\alpha\) and \(H\) indicating correlated/persistent signals (>0.5) from noncorrelated/anti-persistent ones (>0.5). The responses from both methods were similar. The persistence they indicated at the beginning of the test is probably related to the specimen accommodation on its supports, as already pointed out in Figure 7a. Then, both exponents tended about 0.5 until the normalized time 0.75 was reached. At this instant, both exponents presented significant fluctuations, which indicates the spasmodic behavior of the structure. That agrees with the findings in [PERSON] et al., 2020 [61], where the DFA method yielded exponents of about 0.6 (persistent behavior) for seismic signals that were acquired when large earthquakes were about to occur, then tending to 0.5 (random) after they occurred. The evolution of natural-time parameters versus normalized time is plotted in Figure 7d. For this analysis, we follow the same reasoning as [47] and [62], taking the energy component (\(Q_{k}\)) as equal to \(A_{\text{max},k}\)[1, 5], where \(A_{\text{max},k}\) is the maximum amplitude of the signal, an approach commonly found also in seismology studies [63]. Calculation of \(A_{\text{max},k}\)[\(t\)] (with \(t\) denoting the conventional time) was performed hit by hit, i.e., every time a new signal comes up, a new signal is included in the NT analysis, leading to the rescaling of the (\(\chi_{k}\), \(Q_{k}\)) time series and the recalculation of \(\kappa_{1}\), \(S\), \(S_{\text{rev}}\) and \(\langle D\rangle\).