## 4 Case Study
### Results
Simulation results are presented in Figures 9 and 10 and summarized in Table 3. Although the damage process begins at normalized time = 0.25, its effect is negligible until \(t\) = 0.77. Figure 9 depicts the system's energy balance with respect to normalized time. In that same figure, the peaks in the kinetic energy evolution coincide with the region of significant AE activity (see Figure 10), and the elastic energy maximum occurred at the same normalized time as the maximum load was reached. Figure 10 illustrates the simulated applied load as a function of normalized time, with the corresponding AE signal from the virtual sensors. The \(b\)-value and DPH parameters are calculated analogously to the experimental data presented in Figure 8. However, two different metrics of elastic energy are used to extract the DPH parameter: the first is the elastic energy computed within the specimen (i.e., the \"actual\" energy involved in the process); the second relies on the product between the reaction support and the prescribed displacement, i.e., the same method used for estimating the elastic energy behavior for the analysis presented in Figure 8. The results indicate significantly fewer AE events throughout the simulated process in comparison with the experiment. The two last graphs in Figure 10 concern the DPH parameter's evolution, which was computed by two methods: with the \"actual\" elastic energy computed during the simulation (\(U\)) and with its estimated counterpart \(\tilde{U}\), given by the product between the excitation displacement and the corresponding load applied by the mobile support to induce it. These results are summarized in Table 3. Since the simulated results did not indicate clear local failures before the total collapse, the usefulness of the DPH index as their predictor could not be confirmed through simulations. However, the indication of total collapse was confirmed with the indexes computed from both \(U\) and \(\tilde{U}\) (points \(a^{\dagger}\) and \begin{table}
\begin{tabular}{c c c c c c} \hline  & \((a^{\star})\) & \((b^{\star})\) & \((c^{\star})\) & \((d^{\star})\) & \((e^{\star})\) \\ \hline \(t_{n}\) & 0.340 & 0.418 & 0.707 & 0.955 & 0.987 \\ \(t_{i}\) & 0.348 & 0.434 & 0.718 & 0.970 & 1.000 \\ \(t_{n}/t_{i}\) [\%] & 97.626 & 96.356 & 98.360 & 98.466 & 98.674 \\ \hline \end{tabular}
\end{table} 
Table 2: Collapse prediction time (\((\cdot)^{\star}\)) relationship with DPH characteristic values—experimental test (see Figure 8). Figure 9: Time evolution of energy of the LDEM model. The dissipated energy is zero up to the normalized time = 0.25; then, it begins to increase in sensible form after time = 0.77. \(a^{*}\) in Figure 10), which suggests the validity of the proposed estimation method for the elastic energy in an experimental setup. Concluding this section, the experimental and simulated data observations can be summed up in three main points. First, the predictions from the DPH index were consistent with the developing damage pattern and the indication carried out for the \(b\)-value evolution during the experimental test. Since the \(b\)-value is a widely employed failure precursor with several successful application examples, this result alone cannot mean that the DPH is a consolidated precursor of local and global failures. On the other hand, since its supporting data are entirely independent from those used for calculating the \(b\)-value, the DPH can be treated as a complementary metric, leading to a more complete analysis of the damage process. Second, although the simulated results were not conclusive with respect to predicting the damage pattern from either the DPH or the \(b\)-value, they did confirm that the behavior of the actual elastic energy within the system (which is necessary to calculate the DPH index in its original form) can be successfully approximated through the product  
Figure 10: Results in terms of Acoustic Emission signal vs, time, load vs. time, and the time evolution of \(b\)-value and DPH index vs. the normalize time.