## 3 Results
### Vertical Direction  
Figure 9 shows strain versus stress results in the vertical direction for all SVEs of sizes \(L_{SVE}=56~{}\upmu\)m and \(L_{SVE}=140~{}\upmu\)m, respectively. The elastic property corresponds to the slope of the linear part of the response and is the elastic modulus of SLM material in the vertical direction (\(E_{v}\)). Two key fracture properties correspond to the maximum stress (\(\sigma_{v,\text{max}}\)) and fracture energy (the area under the strain-stress curve). A few trends can be observed. First, for both sizes, there are more variations in fracture (maximum and post-maximum stress) than there are in elastic response. Second, the variations in both elastic and fracture properties are more noticeable for the smaller SVEs in Figure 9a. Third, the captured response is a relatively brittle pre-maximum stress level. In reference [46], several brittleness indicators are defined based on the ratios of  
Figure 8: (a) Domain and boundary condition for vertical loading for parameter calibration; (b) vertical stress versus strain for the domain in (a) with calibrated parameters. Figure 9: Vertical stress vs. strain overlay for (a) \(L_{SVE}=56~{}\upmu\)m; (b) \(L_{SVE}=140~{}\upmu\)m.  
initiation (i.e., where the stress-strain response before the maximum stress significantly deviates from the linear response), maximum (where the stress is the maximum), and final (the first instance when the stress reaches zero) strains or energies. For example, in Figure 9, the ratio of strain at the initiation stage to that at the maximum stage is very close to one, reflecting the little softening response before the elastic branches reach the maximum stress level, and indicating a brittle failure response. Figure 10a shows the probability distribution function (PDF) plots of \(E_{v}\) for all relative SVE sizes. As can be seen, the variation in \(E_{v}\) decreases as the SVE size increases. This is in agreement with the second observation above for the two relative SVE sizes of \(L_{SVE}=56\) and \(L_{SVE}=140\) um. Interestingly, the mean and mode of \(E_{v}\) are relatively unaffected as a function of SVE size. Another important vertical elastic material property is C\({}_{22}\), the vertical normal component of the elasticity stiffness matrix. Due to Poisson effects, C\({}_{22}\) is not equal to \(E_{v}\). For brevity, we do not present the results for C\({}_{22}\), but note that similar trends to \(E_{v}\) are observed for \(C_{22}\). The red dot corresponds to the experimental value reported in Table 1 from [38] and is provided as a reference point for other sample size results. The experimental sample, called RVE in this paper, is about six times larger than the largest SVE size of 140 um. Thus, if we had more experimental images for this material, we would have expected the PDF of for this size to be a narrower bell-shaped curve than that of \(L_{SVE}=\)140 um, around the red dot. Size effect relation of vertical stiffness is shown in Figure 10b. The minimum and maximum values of homogenized stiffness for the population of SVEs at any given size are shown by the dashed and dot-dashed lines. The mean value of the homogenized property is shown by the solid line inside this region. As implied by the PDFs in Figure 10a, we observe that the mean value of \(E_{v}\) is quite stable among all SVE sizes. Moreover, there exists a relative difference of only 0.0027 across means values for each SVE size, highlighting that \(E_{v}\) is not size-sensitive. In [26], a mean-based RVE size limit criterion is proposed, wherein the RVE size for a property is defined as the size at which the mean value of the given property is close enough to its terminal value (in the limit of infinite VE size). We observe that from this perspective, even the smallest relative SBE, \(L_{SVE}=56\) um, can be considered a RVE size for \(E_{v}\) whose mean value closely aligns with that of the published value in Table 1. The variations in \(E_{v}\) gradually decrease from about 10% to 2.5% of the mean value as the SVE size increases.