## 4 Conclusions  
We used SVE homogenization to study the size dependency of elastic and plastic properties in a two-phase metallic polycrystalline material by varying the volume fractions of ferrite and martensite phases with random grain orientations. The SVEs were generated by partitioning a large domain using the moving window method and were simulated by CPFEM. Referring to Table 6 which, summarizes the effective mechanical properties and RVE sizes, and considering other results we draw the following conclusions,  
1. For both geometry (\(v_{f}\)) and elastic properties, the largest RVE is required for M2. This ordering is justified as M2 has the highest point-wise variation in volume fraction. In contrast, for plastic properties, the RVE size monotonically increases as \(V_{f}\) increases from 10% to 90% for M1 to M3. Figure 12: Summary of the homogeneous and isotropic limits of geometric (volume fraction), elastic and plastic properties. Green dots denote the RVE limit for volume fraction. Red and magenta represent the homogenous and anisotropic RVE limits, respectively. The abbreviations Iso., Ani., Hom., and Het. denote isotropic, anisotropic, homogenous, and heterogenous, respectively. 2. The smallest RVE size corresponds to the geometry property \(v_{f}\), followed by elastic properties (\(L^{\epsilon}_{\text{RVE}}\) about equal to twice larger than the geometry RVE size), and plastic properties. The combined elastoplastic RVE size has the highest variation across the three materials with \(L^{\epsilon p}_{\text{RVE}}\approx 1.5\) to \(9L^{\epsilon}_{\text{RVE}}\). The corresponding RVE size ranges (geometry, elastic, and elastoplastic) are approximately 250 to 330, 300 to 640, and 600 to 4400 \(\upmu\)m (50 to 66, 60 to 128, and 120 to 880 for \(L_{\text{RVE}}/\bar{d}\), grains along the RVE edge) for the three materials considered. Given that \(\bar{d}\) is the key inherent length scale of the material, \(L_{\text{RVE}}/\bar{d}\) can be considered as the scale factor of the RVE size. 3. The homogenized hardening modulus \(H\) shows a higher variability and anisotropy than the yield strength \(Y\). This parameter is the main driver of large plastic RVE sizes, especially for M3 where the inclusion of \(H\) increases this size by approximately threefold. In fact, except for the response of \(H\) for M3, all the RVEs are smaller than 1 mm (\(L_{\text{RVE}}<200\bar{d}\)). 4. The isotropic and homogeneous elastic and plastic effective properties shown in Table 6 can be used for a bulk plasticity model. We believe that the macroscopic domain should be in the cm range or larger (at least for M3) to justify the use of these bulk-effective properties. We conclude by mentioning potential extensions of this work. First, the effect of other microstructural features such as grain size, shape, and orientation distributions can be considered. Second, CPFEM and SVE-based analysis using the moving window method lend themselves to fatigue crack nucleation and propagation and as well as other problems that exhibit high statistical variability and size dependency. Third, CP models with an internal length scale, e.g., [34; 38; 40], can be used to provide a more accurate size-dependency of material properties, especially for smaller SVEs. Finally, the CPFEM of SVEs can provide random field material properties and constitutive equations for a stochastic continuum plasticity model. Conceptualization, A.D.A., R.A. and S.T. ; methodology, A.D.A., R.A. and S.T. ; software, A.D.A. and R.F. ; formal analysis, A.D.A. ; resources, R.A. and S.T. ; writing--original draft preparation, A.D.A. ; writing--review and editing, R.F., R.A. and S.T. ; visualization, A.D.A. ; supervision, R.A. and S.T. ; funding acquisition, R.A and S.T. All authors have read and agreed to the published version of the manuscript. This research was funded by Air Force Office of Scientific Research (AFOSR) under grant number FA9550-22-1-0359. Not applicable. The data presented in this study are available on request from the corresponding author.