## 5 Results and Discussion
### Particle Size Distribution Analysis: Analysis Two  
Figure 13 shows the stress-strain curves of the PS DEMs compared with the experimental stress-strain curve. It is observed that by increasing the particle size distribution resolution (i.e., increasing the number of particle sizes from one to four), the DEM's stress-strain shows a better fit with the experimental data. This effect is also validated from the optimization's objective function residuals of 0.33, 0.24, 0.17, and 0.09, corresponding to specimens PSD1_S2, PSD2_S2, PSD3_S2, and PSD4_S2, respectively. Figure 12: Contact force magnitude at a measured strain \(\epsilon_{yy}\) equal to 0.001 obtained from PSD4 DEMs with different particle size scaling. Figure 13: Stress-strain curves from the DEMs with different PSDs. Also, a reduction in simulation time was observed when there was a reduction in the number of particles constituting the specimens (Figure 14). The reduction of particles was due to a reduction in the PSD resolution decrease. The micro-parameter values obtained from the optimization process for each specimen are listed in Table 5. Additionally, Table 5 introduces two additional DEM characteristics: The \(Drat\), which constitutes the ratio between the maximum and the minimum particle diameter (\(D_{max}/D_{min}\)), and the \(L/D_{nt}\) ratio, where \(L\) is the largest dimension of the specimen, which in this case corresponds to the height (0.30 m). The \(L/D_{nt}\) and \(D_{nt}\) have been used previously [14; 22] to measure the homogeneity of the particle diameter sizes conforming to a DEM. \(E_{LC}\) and \(\overline{c}\) show decreases of 40% and 67%, respectively, and \(\overline{\phi}\) and \(\mu\) shows about a two-times increase when the PSD is also increased. The \(\overline{\sigma}_{c}\) and \(E_{F\!f}\) micro-parameters do not show a significant variation. Figure 15 includes plots showing the trends of the four DEMs' micro-parameters obtained through the optimization process. The variation of the \(\overline{\phi}\), \(\overline{c}\), and \(\mu\) is analyzed considering different specimen characteristics. Particular trends for each parameter can be identified with respect to \(D_{nt}\). Lower values of \(D_{nt}\) produce particle size homogenization, which effect generates lower particle interlocking, producing compensation from the \(\overline{\phi}\), \(\overline{c}\), and \(\mu\) micro-parameters values in order to reproduce the mortar's reference experimental stress-strain curve. Additionally, Figure 16 shows a representation of the specimen's cross-section showing the contact forces concentration for the four DEMs. From this figure, it is observed that the contact forces concentrations are significantly more localized. This is due to the FJA per 1000 contacts (FJA/CN/1000) increasing, denoting larger contact interfaces. Therefore, the stresses generated by the axial loading are distributed among a higher quantity of contacts when the PSD is increased. Hence, the effect of heterogeneity on the DEM impacts the micro-parameters values and the mechanical behavior of the  
Figure 14: Particle quantity vs. Simulation time obtained from DEMs with different PSDs. \begin{table}
\begin{tabular}{c c c c c c} \hline  & & **PSD1\_S2** & **PSD2\_S2** & **PSD3\_S2** & **PSD4\_S2** \\ \hline \multirow{8}{*}{Micro parameters} & \(E_{LC}\) [GPa] & 5.28 & 6.94 & 7.17 & 7.34 \\  & \(E_{F\!f}\) [GPa] & 9.68 & 9.84 & 10.1 & 10.38 \\  & \(\overline{c}\) [MPa] & 0.28 & 0.28 & 0.27 & 0.29 \\  & \(\overline{c}\) [MPa] & 0.65 & 1.35 & 1.8 & 1.98 \\  & \(\mu\) [Coeff.] & 0.41 & 0.28 & 0.21 & 0.21 \\  & \(\overline{\phi}\) [\({}^{\circ}\)] & 42.52 & 24.83 & 21.48 & 20.70 \\  & Number of particles & 18,270 & 28,326 & 57,709 & 55,138 \\  & CN & 51,906 & 76,522 & 152,812 & 147,235 \\  & FJA [m\({}^{2}\)] & 43.6 & 40.6 & 49.9 & 47.3 \\  & ST [h] & 0.24 & 0.6 & 1.9 & 1.9 \\  & \(D_{nt}\) [m/m] & 1.0 & 3.2 & 5.86 & 8.47 \\ \hline \end{tabular}
\end{table} 
Table 5: Optimized micro-parameters for DEMs with PSD variations. DEMs.