## 2 The Rate of Generation of Acoustic Hits (F-Function) in the Natural Time Domain  
Assume that, during the loading procedure, i.e., from the onset of loading until the fracture of the specimen, N acoustic hits are recorded at the time instants t\({}_{\mathrm{i}}\), t\({}_{\mathrm{2}}\),..., t\({}_{\mathrm{i}}\),..., t\({}_{\mathrm{N}}\). The respective time series, expressed in terms of the Natural Time, \(\chi\), is denoted as: \(\chi_{1}=\frac{1}{\mathrm{N}}\), \(\chi_{2}=\frac{2}{\mathrm{N}}\),..., \(\chi_{i}=\frac{1}{\mathrm{N}}\),..., \(\chi_{\mathrm{N}}=\frac{\mathrm{N}}{\mathrm{N}}=1\). Along the same lines, the respective N values of the load applied at each time instant t\({}_{\mathrm{i}}\) are recorded and denoted as \(\ell_{1}\), \(\ell_{2}\),..., \(\ell_{i}\),..., \(\ell_{N}\). In order to calculate the values of the F-function, the interevent time intervals \(\delta\tau_{i}=t_{i}-t_{i-1}\), are used, where \(t_{i}\) and \(t_{i-1}\) are the instants at which any two successive AE hits were recorded. Denoting, as already mentioned, by \(\delta\tau_{i}=t_{i}-t_{i-1}\) the interevent time, i.e., the time interval between two successive acoustic hits, then the quantity \(f_{i}=\frac{1}{\delta\tau_{i}}=\frac{1}{t_{i-1}-t_{i-1}}\) represents the instantaneous frequency of generation of acoustic hits when the \(i_{th}\) hit is recorded. Taking advantage of the instantaneous frequencies \(f_{i}\) of n successive acoustic hits, the respective value of the F-function is determined as follows:  
\[F_{i}=\frac{\sum\limits_{j=i-n+1}^{j=i}f_{j}}{n} \tag{2}\]  
Sliding by one acoustic hit the \"window\" of the n successive acoustic hits the next value of the F-function is obtained and so on. Any specific value of the F-function is paired to the average value (\(\tau\)) of the time instants at which the n successive acoustic hits of the specific \"window\" were recorded. It is thus evident that the F-function is in fact a time series of \(m=N-n+1\) discrete values of F(\(\tau\)) and can be plotted graphically along the time parameter \(\tau\). As a next step, each value of the time parameter \(\tau_{i}\) is paired to a Natural Time \"instant\" \(X_{i}\) and a load value \(L_{i}\), according to the following scheme:  
\[\tau_{i}=\frac{\sum\limits_{j=i-n+1}^{j=i}t_{j}}{n}\!\rightarrow\!X_{i}=\frac {\sum\limits_{j=i-n+1}^{j=i}\chi_{j}}{n}\rightarrow L_{i}=\frac{\sum\limits_{j =i-n+1}^{j=i}\ell_{j}}{n} \tag{3}\]  
Therefore, instead of the time series F(\(\tau\)), with \(m=N-n+1\) discrete values, the corresponding time series F(X) and L(X) are obtained in the Natural Time domain and can be plotted in juxtaposition to each other in the same graph. In case the load is not recorded simultaneously with the acoustic hit, but rather it is recorded from an \"independent\" sampling technique, using a different hardware, then it is possible to apply the above-described procedure with the aid of the table of values \(X_{i}(\tau_{i})\) by means of interpolation. The above procedure is schematically described in Table 1.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{} & & & & & & & & & \\ \hline Time & \(t_{1}\) & \(t_{2}\) &... & \(t_{n}\) &... & \(t_{i-n+1}\) &... & \(t_{i}\) &... & \(t_{N}\) \\ \hline Load & \(\ell_{1}\) & \(\ell  
It is concluded that the procedure for converting the time series from the conventional time domain to the respective ones in the Natural Time domain adopted here, is not identical to that adopted in ref. [13], in which the first value of the Natural Time, \(\chi_{1}\), was paired to the first value of the F-function and the last value, \(\chi_{\text{N}}\), to the last value of the F-function. In this study, the Natural Time starts counting simultaneously with the generation of the first acoustic hit.