## 3 First Application: AE Events in a Simulated Fracture Process
### Results  
The first results referred to the overall damage. As shown in Figure 2(a), the final simulated configuration showed a main diagonal fissure throughout the plate, which was typical for this test. That information was complemented by the distribution of bars that surpassed their limit strain \(\epsilon_{r}\), from the start of the process to immediately before the main fracture occurred (Figure 2(b)). As shown in Figure 2(b), a distinctly dense cluster of broken bars (a crack) appeared where the main fissure began. This region's dimensions were nearly 0.022 m, which was consistent with the value of \(d_{eq}\) in Table 1 and its definition in Section 2, i.e., the critical crack size that would generate unstable propagation throughout the specimen. Figure 3(a) illustrates the same test regarding the shear stresses along each plate edge as a function of the distortion angle. Due to the non-homogeneous nature of the material, the homogeneous stress state was perturbed once the first micro-fracture had appeared. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\mu(G_{f})\) & \(CV(G_{f})\) & \(CV_{p}\) & \(E\) & \(d_{eq}\) & \(\rho\) & \(\
u\) & \(L_{n}\) & \(l_{c}\) \\ \hline
70 N m\({}^{-1}\) & 100\% & 2.50\% & 32 GPa & 0.02 m & 2400 kg m\({}^{-3}\) & 0.25 & 4 mm & 4 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensions and material properties of the plate. Figure 3: (a) Final configuration obtained with DEM model subjected to pure-shear stress. Broken elements are in red. (b) Spatial distribution of bars where the strain exceeded \(\epsilon_{r}\) from the beginning to immediately before the main fracture occurred. The ellipse indicates where the main fissure started. Afterwards, it remained approximately homogeneous until the peak load was reached. Then, unstable fracture propagation took place, leading to large changes in the stress distribution. We noted that the test concentrated on the damage process leading to collapse, so the post-peak behavior was not considered. The corresponding energy balance is shown in Figure 4b. The sudden increases in kinetic and dissipated energies after the normalized time of 0.80 were clear indicators of unstable growth in the main fissure that had nucleated during the damage process. We then analyzed the same results in terms of the AE signals captured by the virtual sensors \(S_{1}\) and \(S_{2}\). As shown in Figure 5, the analysis was performed according to the \(b\)-value parameter, which has been widely used in seismology in connection with the Gutenberg-Richter (G-R) law [57]. As damage progressed, the acoustic waves appeared within the structure as the fissures were formed. These wave amplitudes were a measure of the fissure sizes and, therefore, of the damage extension. Mathematically, each wave connected with an event, and the G-R law determined that the cumulative number \(N\) of events with amplitudes larger than or equal to a given cutoff amplitude \(A\), according to the following:  
\[N(\geq A)=cA^{-b}. \tag{11}\]  
Otherwise, in the bi-log domain:  
\[N(\geq A)=\log(c)-b\log(A), \tag{12}\]  
where \(c\) and \(b\) reflect both the intrinsic material properties and the boundary conditions applied to the structure, respectively. This relation has been widely investigated in several works, such as [19; 21; 22]. In particular, ref. [29] interpreted the \(b\)-value as one-half of the fractal dimension of an AE event. According to this interpretation, during the damage process, the \(b\)-value was expected to change from an ideal initial value around \(b=1.5\) (\(D=3\)) to a final result close to \(b=1.0\) (\(D=2\)), because the individual (punctual) micro-fissures tended to combine with the main crack. Figure 4: (a) Global shear stress vs. global distortion obtained with DEM. (b) Global energy balance. As shown in Figure 5, the calculation of the \(b\)-value was superposed to the AE amplitude distribution during the simulated process.