## 4 Case Study
### Results
Figure 8a also features slashed lines marking the highest instability points within each region, i.e., when a sudden loading reduction signals the appearance of a significant crack. Completing the information provided in the figure, the most relevant points identified through the two parameters of interest appear as (\(a\)), (\(b\)), (\(c\)), and (\(d\)), where the \"diamond\" (\(\cdot\)) marker indicates a \(b\)-value local minimum and the \"asterisk\" (\((\cdot\))) refers to a sharp variation in the DPH index. Figure 7: Evolution of damage in the experimental trial in three regions localized in Figure 8 in normalized time \(t\). (**a**) Region (I), \(t\): 0.290. (**b**) Region (II), \(t\): 0.529. (**c**) Region (III), \(t\): 0.818. The evolution of the \(b\)-value appears in Figure 8a's middle graph. This parameter acts as a failure precursor when a local minimum follows a \"plateau\" where its value is relatively stable. Notice that one such \"plateau\" occurs before each of the four main regions in the specimen behavior and that each of those \"plateaus\" is followed by one of the local minima marked as (\(a^{\circ}\)), (\(b^{\circ}\)), (\(c^{\circ}\)), and (\(d^{\circ}\)). However, none of those minima occur near the major instability points. Thus, the \(b\)-value appears as a reliable indicator of changes in the specimen's overall behavior regions but it is not as effective for predicting the actual instability points. The bottom graph in Figure 8a illustrates the evolution of the DPH index. This parameter was introduced in ref. [38], where its maximum overall value indicated a structure's total collapse. Here, we investigate its usefulness to identify not only the final collapse but also the intermediate critical points as the damage process evolves. Notice that the DPH index presents a local maximum ((\(a^{*}\)), (\(b^{*}\)), (\(c^{*}\)), (\(d^{*}\)), and (\(e^{*}\))) near each main instability point, including even the relatively minor one associated with point (\(d^{*}\)). Also, the final collapse is predicted by the overall maximum (\(e^{*}\)), which confirms the results presented in ref. [38]. To the best of the authors' knowledge, this is the first instance of applying the DPH index as a local-failure predictor and the first time it is applied to experimental data. Notice that, close to the final collapse, the DPH index shows a maximum value; however, in this situation, there is no high activity in the AE signals, perhaps due to the shielding effect produced by the damage advance in the specimen region making it difficult for the AE signals activity to arrive to the sensor. This is a very good example where the DPH index could be a useful complement to the AE analysis. A different tendency between the two precursors could indicate the presence of a local effect that prevents the AE wave from arriving at the sensors used in the analysis. Notice also that in the same process in other local instabilities, the maxima of the DPH index and the \(b\)-value decreasing during the process both indicate the eminence of local critical regions. The information in Figure 8a's bottom graph is complemented by Table 2, relating the moments of critical instabilities (\(t_{i}\)) with the times when the DPH index presented a local  
Figure 8: (**a**) AE experimental results: top—the results of the temporal evolution of Load, Signal AE; middle—\(b\)-value evolution; bottom—DPH parameter evolution during the test. \((\cdot)^{\circ}\) and \((\cdot)^{*}\) instants of particular interest for the predictors \(b\)-value and DPH, respectively. (**b**) Visual representation of the damage process throughout the test. maximum (\(t_{n}\)), including the ratio between those two values. Notice that this ratio presents small variability [0.96-0.99] and is close to 1.0, reinforcing the DPH index's consistency as a failure predictor. This ratio's stability is discussed in further detail in [38].