## 4 Discussion  
In this section, we summarize the results from the presented numerical results in the prior sections. First, we examine whether or not one can claim any of the four homogenized properties has reached its RVE size for \(L_{SVE}=140~{}\upmu\)m. In [26], we discussed mean-based,  
Figure 14: Horizontal results for \(\sigma_{h,max}\) (**a**) PDF of \(\sigma_{h,max}\) for all SVE sizes. The red dot serves as a reference to the published value from [38]; (**b**) Size effect plot for \(\sigma_{h,max}\). Figure 13: Horizontal results for \(E_{h}\): (**a**) PDF of \(E_{h}\) for all SVE sizes. The red dot serves as a reference to the published value from [38]; (**b**) the size effect plot of \(E_{h}\) for all SVE sizes. variation-based and boundary condition (BC)-based criteria to examine whether or not a homogenized properties has reached its RVE size limit. In the mean-based criterion, the RVE size is defined as the \(L_{SVE}\), such that for all SVEs larger than that, the relative variations in the mean value (e.g., the middle line in Figures 10b, 11b, 13b and 14b) stay below a certain user-defined threshold, \(\epsilon\), across all larger SVE sizes. In the variation-based criterion, the RVE size corresponds to the size beyond with the coefficient of variation (COV) staying below the threshold, \(\epsilon\). The COV is defined as the ratio of the standard deviation to the mean value of the quantity. Often, the arbitrary threshold of \(\epsilon=0.01\) is used for both criteria [22; 25]. This variation-based criterion is the most common choice to determine the RVE size in the literature; see for example [25; 47]. We refer the reader to [26] for a further discussion of these criteria and the BC-based criterion not used in this work. In [26], we first obtain best fits to the means and COVs of a homogenized property versus \(L_{SVE}\) before determining when the normalized mean and COV fall below \(\epsilon\) for the definition of mean- and variation-based RVEs. In this work, we adopt a simpler approach given the lower variation in homogenized properties discussed in Section 3. For the mean-based criterion, we define the error in the mean values, e(\(\upmu\)), as the largest difference between the mean values of all sizes divided by the largest mean value. That is, it is a relative error among all the mean values of the given property. These values are reported in the first data row of Table 3. As evident, for all properties, e(\(\upmu\)) is below or very close to \(\epsilon=0.01\). Thus, from this perspective, all homogenized properties can be claimed to have reached the mean-based RVE size below or around \(L_{SVE}=140\)\(\upmu\)m.  
The results for the variation-based criterion are shown in the second data line of Table 3. Herein, the COVs of the largest SVE size of \(L_{SVE}=140\)\(\upmu\)m are provided. For elastic properties and vertical strength, the COVs are below \(\epsilon=0.01\). Even for the homogenized strength, the COV of 0.023 is close to the often-used threshold of 0.01. Thus, even for the variation-based criterion, the properties are deemed to have reached their RVE size below or around \(L_{SVE}=140\)\(\upmu\)m. This is an anticipated result, as [48] also obtained a similar finding pertaining to elastic properties. Next, we examine the differences in the homogenized properties from two perspectives. First, for both criteria, elastic properties take smaller values, implying that they reach their RVE limit at a smaller size. This is consistent with our results in [26; 41] for different two-phase composites. In [41], stress contours demonstrate that minute local variations in the microstructure result in highly varied levels of stress concentration and thus the strength value assigned to an SVE. In contrast, the elastic properties reflect the average response of the SVE (as opposed to the weakest point-type response for fracture properties) and are less sensitive to microstructural variations.