## 4 Results and Discussion
### Correlation between \(q\) and \(\beta_{q}\)  
In the work described in Reference [30], a relationship between the entropic index \(q\) and the parameter \(1/\beta_{q}\) was investigated by analyzing the AE data recorded during cyclic compression tests on two different types of brittle materials, namely concrete and basalt. Results showed the emergence of a linear correlation between the quantities \(q\) and \(1/\beta_{q}\). The investigation of a possible correlation is considered in the context of this work in both loading protocols.  
Figure 8 shows the value of the index \(q\) as a function of the parameter \(1/\beta_{q}\) for the UC test. As can be seen, a linear relationship emerged, with a slope of \(-2.28\), that is found only in the stress region where the normalized stress \(\overline{S}_{n}\) ranges between \(69\%\) and \(96\%\) of the specimen's strength (see green circles in Figure 8). It is characteristic that the initial loading stages (blue circles), as well as those corresponding to the last loading stages during the collapse of the specimen (red circle), deviated from the linear fitting curve. More enlightening is the corresponding behavior that emerged in the UCSS protocol, in which the applied stress initiated at least \(84\%\) of the specimen's total strength (see Figure 9). The linear relationship between the index \(q\) and the parameter \(1/\beta_{q}\) was evident both when the values were plotted for the regions of stress increases (Figure 9a) and during the regions of constant stress (Figure 9b). It is remarkable that during the last stress step, a significant deviation from the linear relationship (red circles in Figure 9) emerged.  
Figure 7: The evolution of the \(q\) entropic indices of loading protocols UC (blue circles) and UCSS (green triangles and red squares).  
Figure 8: The values of the entropic index \(q\) as a function of the parameter \(1/\beta_{q}\) for the UC protocol.  
Figure 9: The values of the entropic index \(q\) as a function of the parameter \(1/\beta_{q}\) for the UC protocol, (**a**) for the stages of stress jumps; (**b**) for the stages when the stress was kept constant.  
A computational explanation of the linear relationship between \(q\) and \(1/\beta_{q}\), which emerged as a failure approach, was attempted based on Equation (5). Using the values of the entropic index \(q\) obtained from the UC loading protocol in the region where \(q\) vs. \(1/\beta_{q}\) presented a linear relationship, the values \(F(q)\) and \(Q(q)\) were calculated and subsequently plotted (see Figure 10a). As the specimen failure approached and the entropic index q value increased due to the organized system of microcracks, \(F(q)\) showed a relatively strong increase, while the function \(Q(q)\) showed a slight decrease. Then, as in Figure 10b, the resulting values were depicted based on the relationship \(1/\beta_{q}^{*}=Q(q)/F\), and thus, the linear relationship between \(q\) and \(1/\beta_{q}\) became apparent. Finally, in Figure 10c, the \(1/\beta_{q}^{*}\) values are presented together with the \(1/\beta_{q}\) values resulting from the fitting of Equation (5) which were presented in Figure 8. The small variations between the values of \(1/\beta_{q}^{*}\) and \(1/\beta_{q}\) were due to the deviations occurring in the long-time interval values (\(\delta\tau\)) with the fitting based on Equation (5). Indicatively, in Figure 10d, the CDFs and the corresponding fitting are presented; in this way, the \(q\) and \(\beta_{q}\) values for group 3 of the UC protocol were calculated. The deviations in the tail of the distribution, with values indicated by \(\delta\tau\), were smaller than those predicted by the fitting, and gave slightly smaller mean values of \(\overline{\delta\tau}\) and slightly larger values of \(F\) compared to the \(1/\beta_{q}\) value obtained through fitting. Based on the \(1/\beta_{q}^{*}=Q(q)/F\), \(1/\beta_{q}^{*}\) relationship, the values were slightly smaller than the results of the corresponding \(1/\beta_{q}\), which is evident in Figure 10d.