## 3 Results and Discussion
### Missing Grains: Porosity  
To investigate the impact of porosity on the macroscopic material properties, a parametric analysis was conducted by varying the number (#) of grains and the volume fraction of porosity [47; 48; 49]. The analysis was performed under the assumption of a plain strain condition, and the material considered was aluminum. The dimensions of the micro-structure were 100 \(\times\) 100 [mm\({}^{2}\)]. The specific details of the porosity model employed in this study can be found in the corresponding section of this paper. To assess the macroscopic properties, such as the elastic modulus (E) and shear modulus (G), the homogenization approach based on the RVE was utilized. The procedure for evaluating these properties is explained in detail in the dedicated section of this paper. The volume fraction of porosity was calculated with Equation (19). Figures 16 and 17 show the Young and Shear moduli, respectively, against the volume fraction of porosity, thus sweeping the number of grains from 30 up to 300. As expected, the higher the volume fraction of porosity, the lower the Young and Shear moduli. To investigate the impact of porosity on grain morphology, three different materials with distinct grain shapes were tested: gold-silver (Ag-Si) with a rhombic crystal morphology, yttria (Y\({}_{2}\)O\({}_{3}\)) with a cubic crystal morphology, and alumina (Al\({}_{2}\)O\({}_{3}\)) with a truncated octahedral crystal morphology. The material properties for each of these materials are provided in Table 1. Figure 18 illustrates the variation of both the Young and Shear moduli against the volume fraction of porosity for alumina (189 grains), yttria (125 grains), and Ag-Si (63 grains). As expected, an increase in the volume fraction of porosity lead to a decrease in both the Young and Shear moduli for all three materials. Figure 14: Back wall comparison of the BEM vs. the FEM. Figure 15: Aluminum and Copper. Figure 16: Young’s modulus at different numbers of grains and porosity. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & **Density [\(\frac{\text{kg}}{\text{m}^{2}}\)]** & **Young’s Modulus [GPa]** & **Poisson’s Ration** \\ \hline Gold –Silver (Ag-Si) & 12,000–15,000 & 90 & 0.286 \\ Yttria (Y\({}_{2}\)O\({}_{3}\)) & 5010 & 174 & 0.308 \\ Alumina (Al\({}_{2}\)O\({}_{3}\)) & 3600 & 403 & 0.232 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties. Figure 17: Shear modulus at different numbers of grains and porosity. Furthermore, by analyzing the BEM results, it was possible to quantitatively evaluate the influence of the volume fraction of porosity on the macroscopic material properties. Tables 2-4 present the volume fraction of porosity, Young's modulus, Shear modulus, and the percentage difference of E and G compared to the pristine condition for all three materials. The analysis revealed that the volume fraction of porosity had a greater effect on E compared to G, particularly for the rhombic crystal morphology (Ag-Si), while the effect was less pronounced for both the cubic crystal morphology (yttria) and the truncated octahedral crystal morphology (alumina). To conclude the porosity analysis, simulations were performed to detect the presence of voids. The detection of voids was analyzed based on the changes in the amplitude and ToA of the backscattered signal. The simulations were performed with voids in different positions for an aluminum micro-structure. The input signal was defined at the upper edge of the micro-structure, where a prescribed traction 3-cycle tone burst with a Hanning window and center frequency of 1 [MHz] was enforced. The time interval of the simulation was 40 [ns] with a time step of 10 [ns] and a time interval of the excitation of 3 [ns]. The ToA was calculated from 0 to the starting point of the wave packet backscattered by the void itself. The starting point of the wave packet backscattered by the void itself was evaluated as the value of the wave above a threshold.