Issue 50

V. Saltas et alii, Frattura ed Integrità Strutturale, 50 (2019) 505-516; DOI: 10.3221/IGF-ESIS.50.42 512 has a sub-additive or super-additive behavior, respectively. I f → 1 , Tsallis entropy reduces to the classical Boltzman- Gibbs entropy. In the case of the recorded AE activity during the fracture of brittle materials (concrete and rocks), the continuous variable X corresponds to an AE parameter, such as the inter-event time or inter-event distance between two successive micro-cracks generated within the material, due to the applied mechanical stress [24,26,27]. The maximization of q -entropy under appropriate constraints generates probability distributions, the so-called q -distribu- tions, such as q -exponential, q -Gaussian, q -Weibull distributions, etc [32-34]. The q -exponential distribution has been extensively used to describe the cumulative distribution functions (CDF) of seismic parameters in global and regional scale, and the AE activity in laboratory-scale fracturing experiments of rocks [24-28]. It is defined as follows ሺ൐ ሻ ൌ exp ௤ ሺെ ሻ ൌ ሾ1 ൅ ሺ1 െ ሻ ሿ ଵ ሺଵି௤ሻ ⁄ (4) where the parameter β is related to the Lagrange multiplier [27]. In the limit → 1 , Eq.(4) leads to the ordinary exponential distribution. In this study, NESP is applied to analyze the series of the inter-event times ( IT , i.e. the time interval between successive micro-cracks) of the hits recorded during the SCDA-induced fracturing process of the concrete specimen (stage B). Specifi- cally, we chose to use the sequence of the hits recorded to sensor with the maximum number of recorded hits, i.e., sensor 6. The time history of the inter-event times ( τ ) of the recorded hits in channel 6 during the monitoring test, is shown in Fig.8a. The same data are also shown in a sequential order in Fig.8b, to increase the resolution of observation during the fracture (stage B), where most AE activity is recorded. We observe that IT values cover more than 6 orders of magnitude, spanning from hundreds of seconds down to the millisecond range. In range A, ITs have very high values, because of the rare occurrence of hits. Instead, the initiation of the fracturing process (range B) causes an abrupt decrease of IT of more than 4 orders of magnitude (from ~10 2 to less than 10 -2 sec), as it is clearly observed in Fig.8b. A subsequent characteristic decrease of IT is observed at recording time ൎ 43772 , i.e. after 154 sec of the cracks’ initiation (refer to Fig.8b)]. The prolonged recording of IT with low to intermediate values (~0.1-10 sec) suggests that a quasi-static fracture is taking place during the effect of the cracking agent to the concrete specimen. This is consistent with published reports stating that the quasi-static conditions should be taken into account in fracture modelling of rocks with SCDAs [5]. An alternative approach for presenting the AE activity in terms of the inter-event times has been recently proposed by Triantis and Kourkoulis [35]. According to this methodology, a time function F ( τ i ) is introduced, which represents the average frequency of occurrence of AE hits, in a sliding time window of N consecutive hits with a mean value of the inter-event times, ௜ ൌ ሺ ேା௜ିଵ െ ௜ିଵ ሻ/ . The time evolution of the function F of the recorded hits in channel 6 for N =10, during the stage B of the fracture process is illustrated in Fig.8c. The observed peak of F at 43618 sec is associated with the initiation of the fracture process which is significant during the first few minutes, but decreases afterwards. This behavior is quite similar to the observed fluctuations of the smoothed curve of IT, as it is shown in Fig. 8b. Depending on the low or higher values of N which adjusts the time resolution of the evolution of the recorded hits during the test, the details of the AE activity can be revealed or shadowed, respectively. In the present case, for the optimum selected value of N ( N =10), the rate of AE hits as expressed by the function F exhibit the same characteristics as those seen in Fig.3 and Fig.8b. Proceeding further with the analysis in the context of NESP, it should be noted that a single q -exponential function failed to fit the cumulative distribution of ITs in the entire region B. Thus, the aforementioned observed fluctuations of IT in region B have prompted us to separate this region into 2 different sub-regions. The 1 st sub-region (B1), where IT fluctuates considerably, includes hits from 71 to 470 and the 2 nd sub-region (B2) the remainder. In the latter case, the majority of ITs exhibit higher values on average. This separation of region B is also consistent with the approach mentioned above, regarding the values of the function F in each sub-region [35]. In sub-region B1, the function F maintains relatively high values, while in sub-region B2, F has low values, without any significant fluctuations. The normalized cumulative distributions (P> τ ) of IT associated to the recorded hits in channel 6 are depicted in Fig.9a,b, for the two distinct sub-regions of stage B, in a log-log representation. In Fig.9a, for low values of IT ( ൏ 1 ), the cumulative distribution was best fitted with a q -exponential function (red line), according to Eq.(4), while for higher values of IT, an ordinary exponential function (dashed green line) was sufficient to fit the data. Subsequently, the entire distribu- tion of ITs in region B2 (hits 471-1340) is best fitted with a single q -exponential function, as it is depicted in Fig. 9b. In the latter case, the calculated q -value which is close to unity ( q =1.12±0.03) indicates a rather weak q -exponential behavior, suggesting that long-range interactions should be also weak. On the contrary, the calculated high q -value ( q =2.58±0.05) in region B1 for low values of IT , suggests a highly sub-additive behavior, associated with a significantly well-organized fracturing process. However, the subsequent crossover to an expo- nential decay at higher IT values is indicative of randomness in the fracturing process. Actually, the behavior of the CDFs of