## 3 Results and Discussion
### Parametric Analysis with Inter-Granular Cracks (Debonding)  
The presence of inter-granular cracks is a prevalent phenomenon in poly-crystalline materials because they can lead to debonding and subsequent material embrittlement, thus increasing the chance of significant risks such as structural failure [50; 51; 52; 53; 54]. Therefore, it is crucial to develop models that can accurately capture these failure mechanisms and the interaction with UGWs. To address this, a parametric analysis involving inter-granular cracks was conducted. In this analysis, aluminum micro-structures with dimensions of \(100\,\times\,100\) [mm\({}^{2}\)] were employed. Simulations were carried out by introducing cracks into the micro-structure while considering the following effects:  
* Position/orientation: the cracks were located at the same distance from the upper edge with equal length but were positioned at different locations (left, center, and right);
* Crack length: the cracks were situated at the same distance from the upper edge, but varied in length;
* Distance: the cracks had the same length but were positioned at different distances from the upper edge.  
Figures 21-23 present illustrative examples for each effect, thus visually demonstrating the configurations of the cracks within the micro-structure.  
The excitation signal was applied at the upper edge of the structure, in which a prescribed traction with a 3-cycle tone burst with a Hanning window and center frequency of 1 [MHz] was utilized. The simulation duration was 40 [us], with a time step of 10 [ns] and an excitation time interval of 3 [us]. The ToA was calculated from the beginning of the simulation until the starting point of the wave packet, which was scattered by the crack itself for all three analyzed effects. The starting point of the wave packet that was scattered by the crack itself was evaluated as the value of the wave amplitude above a threshold. The simulated response corresponded to the averaged displacement over the entire upper edge against time. Figure 21 investigates the effect of the crack position and orientation. The cracks were randomly distributed in the simulated domain, and it could be observed that the numerical model successfully detected the presence of cracks based on the amplitude and ToA of the signal scattered by the cracks. Figure 22 evaluates the influence of the crack distance from the upper edge. As expected, it was observed that the ToA of the backscattered signal from the crack increased as the crack became more distant from the upper edge.  
Figure 20: Averaged Y displacement at the upper edge for a 100-grain micro-structure for pristine vs. damaged conditions.  
In Figure 23, the impact of crack length is examined. The signal amplitude that originated from the shortest crack (represented by the black signal) was comparable to the previously analyzed cases. However, as the crack length increased, the signal amplitude from the crack also increased. Consequently, the amplitude of the back-wall signal from the upper edge decreased for the green and purple signals, while the crack signal amplitude became more prominent. This illustrated that longer cracks have a greater influence on the overall signal amplitude. A summary of these findings can be found in Table 5.  
Figure 21: Effect of the position/orientation.  
Figure 22: Effect of the distance.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Signal** & **Crack Signal Amplitude [m]** & **Back-Wall Signal Amplitude [m]** \\ \hline Black color & 4.723 \(\times\) 10\({}^{-14}\) & 1.201 \(\times\) 10\({}^{-13}\) \\ Green color & 7.587 \(\times\) 10\({}^{-14}\) & 1.073 \(\times\) 10\({}^{-14}\) \\ Purple color & 1.377 \(\times\) 10\({}^{-13}\) & 3.882 \(\times\) 10\({}^{-15}\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Crack and back-wall signal amplitudes.