## 5 Results and Discussion  
### Particle Size Analysis: Analysis One  
Figure 9 shows the stress-strain curves of PSD4 DEMs with particle size scales 1.0, 1.5, and 2.0 and the micro-parameters obtained from the optimization procedure. It is observed that the stress-strain curve is well-fitted using the three scales considered in this work. The optimization objective function varies between 0.07 and 0.09. There is a decrease in computational effort when the particle size is up-scaled. As represented in Figure 10, there is an increment in simulation time when there is an increment in the number of particles constituting the specimen (the larger the scale factor, the smaller number of particles). Figure 10: Particle quantity vs. simulation time obtained from PD4 DEMs with different particle size scaling. Figure 9: Stress-strain curves obtained from PSD4 DEMs with different particle size scaling. Table 4 lists the calibrated micro-parameters for PSD4 DEMs with respect to their corresponding up-scale factor. All details about each parameterization are included in Table 4, where the DEM's characteristics are presented, the specimen scale (scale), number of particles, number of particle contacts (CN), flat joint contact area (FJA), and the simulation time (ST). The flat joint contact area is defined as the summation of the area generated by the interface plane (Figure 1) of the contact of two particles. It is observed in this table that the micro-parameters \(\mu\) and \(\overline{\phi}\) had a significant change when the particle size scale was increased, having an increment of 39% and 51% from scale 1 to 2, respectively. The remaining micro-parameters show minor variations across the differently-scaled DEM. Figure 11 presents the variation of the friction coefficient (\(\mu\)) and friction angle (\(\overline{\phi}\)) micro-parameters with respect to the flat joint area (FJA) for the three PSD DEMs described above. The FJA decreased as the number of particles was reduced and as the PS was up-scaled. The friction coefficient and the friction angle increment compensate for the reduction of particle interlocking, reduced by the increment of particle scale and the reduction of the number of particles, and hence the number of contacts within the sample. Figure 12 shows a representation of the mortar's specimen cross-section, illustrating the concentration of forces between particles for the three scaled PSD4 DEMs when the normal strain \(\varepsilon_{\text{yy}}\) is 0.001. Notice that the contact force's concentrations are more localized when the particle scale factor equals one; as the PS scale is increased, the concentration of forces cover larger regions within the specimen. This is due to the particle contact interface being larger, as presented in Table 4 (FJA/CN/1000), which measures the FJA per every 1000 contacts. The particle contact interface is larger due to larger particles, and it has to withstand larger forces than the PSD4_S1 DEM since there are fewer particle contacts, and thus, there is less FJA to distribute the forces within the sample. Consequently, in order to be able to reproduce the stress-strain behavior as the experimental observations, the friction coefficient and the friction angle increase by 61% and 107%, respectively, when the FJA decreases by approximately 100%.