## 3 Results and Discussion
### The Form of Homogenized Properties  
The derivation of apparent elastic and plastic properties not only has merit on its own in terms of understanding the size dependency of properties but can also provide _mesoscopic_ mechanical properties in a numerical multiscale framework. There are two options for the values of these mesoscopic properties in a macroscopic domain. In option one, the macroscopic domain is covered by moving windows and at the center of each window (SVE) its apparent homogenized properties are used as local material properties. In option two, the statistics of the apparent homogenized properties are computed first. The advantage of this approach is that the properties of new macroscopic domains can be assigned by generating random fields that are consistent with such statistics, e.g., one- and two-point statistics of the apparent properties. We refer the reader to [10; 65; 66] for some examples of these _direct_ and _random field_ options. We collectively refer to all attributes such as homogeneity, isotropy, and other aspects such as size-dependency (discussed in [13]) as the form of the material properties. For either of the options above, the form of the homogenized field has a direct effect on the accuracy and complexity of the multiscale approach. For large enough SVEs, the homogenized fields become homogeneous and isotropic depending on a material's microstructural design. The multiscale modeling of such RVE-based properties is straightforward but the lack of heterogeneity and randomness (sample-to-sample variation) are some of the drawbacks of using large VEs for homogenization. On the other hand, for very small SVEs the homogenized fields are genuinely heterogeneous and anisotropic. Computational solutions for such material models are difficult for both multiscale approaches. Specifically, the generation of anisotropic tensorial random fields is challenging [67; 68], using the random field multiscale approach. Moreover, as the SVE size decreases the choice of BCs on the SVE has a noticeable effect on homogenized apparent properties. As elaborated in [13; 15], the SVE size analysis can provide suitable intermediate SVE sizes that strike a balance between these limiting cases. Figure 12 summarizes the form of the apparent properties versus the SVE size for the three materials. The homogeneity limits correspond to the size at which the coefficient of variation in the homogenized properties falls below the tolerance \(\epsilon=0.01\). The values for volume fraction, taken from Table 2, are shown by green circles. The homogeneity limits for elastic and plastic properties are taken from Table 3 and Table 4, respectively, and are shown by red lines. The elastic homogeneity limit is taken as the maximum of \(\Delta^{c_{x}}\) and \(\Delta^{c_{G}}\) and the corresponding value for plastic properties is taken as the maximum of \(\Delta^{c_{Y}}\) and \(\Delta^{c_{H}}\). The isotropy limits for elastic and plastic properties, shown by magenta lines, are determined from the process discussed in Section 3.4. For elastic properties, the spatial and sample-to-sample variations of the elasticity tensor can be neglected beyond the homogeneity limit. That is, a constant representation of \(\mathbf{C}\) suffices. In addition, if the observation size (\(L_{\text{SVE}}\)) is larger than the isotropy limit, this matrix is uniquely determined by the RVE limit \(\kappa\) and \(G\). A similar logic applies to the homogenized plastic properties. The results can be summarized as follows. The elastic field can be considered homogeneous and isotropic for M1 if \(L_{\text{SVE}}\gtrsim 300\). For the plastic fields, the yield strength \(Y\) tends to its homogeneity and isotropy limits faster than the hardening modulus \(H\). The plastic properties can not be characterized as isotropic and homogeneous until \(L_{\text{SVE}}\gtrsim 576\) for M1. While similar trends are observed for M2 and M3, we note the following. First, the \(L_{\text{SVE}}\) of plastic properties is significantly larger for M3 than for M1 and M2.