# Detecting Criticality by Exploring the Acoustic Activity in Terms of the \"Natural-Time\" Concept
## 4 Discussion and Conclusions  
The evolution of the \(F\)-function in the Natural Time domain was studied in a series of five experimental protocols, during which Dionysos marble specimens of various geometries were subjected to different loading schemes. Both elementary and structural protocols were considered. In spite of the wide variety of the protocols studied (in terms of the geometry of the specimens and the loading schemes imposed) the evolution of the \(F\)-function in the Natural Time domain exhibited a more or less self-consistent motive: The \(F(\chi)\) values increase progressively for the major portion of the loading process, with some fluctuations, the number and intensity of which depend on the specific protocol. During the very last loading stages, i.e., while macroscopic fracture is approaching, the evolution of the \(F(\chi)\) function is systematically governed by a power law of the form \(F(\chi)=A\chi^{m}\). The characteristics of the domain that is governed by the above power law are briefly summarized in Table 1.  
It is concluded, therefore, that in case the acoustic activity, expressed in terms of the \(F\)-function, is explored in the Natural Time domain, important information is extracted regarding the proximity of the loaded system into its critical stage: The onset of validity of the above-mentioned power law provides a clear signal indicating that the system entered into the stage of impending macroscopic fracture. The above conclusions are definitely supported by similar ones drawn by the analysis of the electric activity developed during all five protocols presented in this study. The temporal evolution of the electrical activity was here investigated by means of the pressure stimulated currents (PSC) technique [47], which is based on the detection and recording of weak electric currents, systematically produced during the mechanical loading of specific classes of brittle materials. This laboratory sensing technique is successfully applied worldwide, and it has been proven that it provides reliable pre-failure indicators [19; 42; 48; 49; 50; 51] in the form of a rapid increase of the intensity of the electric current while the specimens are about to enter into its critical stage, namely that of impending fracture. Therefore, it was considered that it would be useful to compare the findings concerning the pre-failure indicator provided by the evolution of the \(F\)-function in the Natural Time domain against the respective ones provided by the PSC technique. In this direction, PSC data are here considered as obtained from the protocol with Circular Semi Rings under diametral compression. The temporal evolution of the PSC recorded during the specific experiment of this protocol discussed in Section 3.3 is plotted in Figure 13. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Experiment** & **Natural Time** & **Load Region** & \
\begin{tabular}{c} **Maximum** \\ **Load** \\ \end{tabular} &
\begin{tabular}{c} **Exponent \(m\)** \\ \end{tabular} \\ \hline Direct tension & [0.36–0.85] & [2931 N–2928 N] & 2933 N & 4.65 \\ Three-point bending & [0.46–0.79] & [647 N–643 N] & 656.9 N & 8.80 \\ Uniaxial compression & [0.58–0.93] & [89.9 MPa–94.2 MPa] & 95.3 MPa & 5.26 \\ Diametral compression & [0.21–0.36] & [267 N–255 N] & 267 N & 6.62 \\ Shear & [0.91–0.97] & [27.25 kN–27.45 kN] & 27.5 kN & 21.5 \\ \hline \hline \end{tabular} Note: the blue color indicates a load value before it reaches its maximum value and the red color indicates the respective value after maximization of the load (stage of decreasing load).  
\end{table} 
Table 1: The characteristics of the interval during which the \(F\)-function expressed in terms of the Natural Time concept is governed by a power law of the form: \(F(\chi)=A\chi^{m}\). It is seen from this figure that for the major portion of the test's duration the PSC increases smoothly and almost linearly with some rather smooth fluctuations. However, at the instant \(t\approx 470\) s the intensity of the PSC exhibits a rapid increase.