## 3 Results and Discussion
### Anisotropy of Elastic and Plastic Properties  
The mean, maximum, and minimum of anisotropy measures are presented in Figure 10. A few observations can be discerned regarding the anisotropy of the elasticity stiffness tensor \(\mathbf{C}\), based on the Ranganathan-Ostoja anisotropy index \(A_{\text{RO}}\). First, a semi-analytical model [57] shows that for a given \(L_{\text{SVE}}\) and a composite formed by two isotropic phases, the anisotropy index is zero for \(V_{f}=0\) and \(1\) and becomes the most anisotropic for \(V_{f}\) around half (the exact value depends on the bulk and shear moduli of the two phases). In this case, we observe higher anisotropy values (especially maximum values) for M2 which is consistent with the model in [57]. Second, we observe that some SVEs have negative values for \(A_{\text{RO}}\). We attribute this to the nature of MBC wherein the Hill-Mandel condition is not exactly satisfied. As expected, this discrepancy diminishes as the SVE size increases. We note that this form of MBC [52] is popular as it provides an elasticity tensor that is between the upper KUBC and lower SUBC limits. Moreover, the roller nature of the BCs allows the formation of through cracks (or failure zones) along the width of the SVE in nonlinear simulations. Given that the latter is important in our CP simulations, we chose this BC for linear analysis as well. We refer the reader to [53] for the class of materials, e.g., orthotropic materials, for which the Hill-Mandel condition is exactly satisfied for MBC.  
The anisotropy indices for plastic yield strength and hardening modulus are shown in Figures 10 and 10, respectively. The elastic anisotropy and yield strength anisotropy exhibit similar trends and converge to the isotropic limit. Because the hardening modulus has a complex behavior, its anisotropy index decreases more slowly versus \(L_{\text{SVE}}\). The convergence of the anisotropy index to zero for all properties is expected, because at the macroscale the two-phase materials are isotropic. As mentioned in Section 2.6, the isotropy limit of a property is taken as the maximum of mean-based and variation-based RVE sizes for the corresponding anisotropy index. The power relations for this variation-based condition for \(A_{\text{RO}}\), \(A_{\text{Y}}\), and \(A_{H}\) are shown in Figures 11, 11, and 11, respectively. The convergence of mean values to zero follows a similar power relation. Hence, for brevity,  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & \(\mathbf{f}\) & \(\mathbf{\kappa}_{f}\) & \(\mathbf{b}_{f}\) & \(\Delta^{\varepsilon/}\) \\ \hline \multirow{2}{*}{M1} & \(\gamma\) & 1.390 & 33.859 & 345.93 \\  & \(H\) & 1.114 & 6.320 & 326.37 \\ \hline \multirow{2}{*}{M2} & \(\gamma\) & 1.252 & 20.381 & 439.38 \\  & \(H\) & 1.146 & 16.244 & 634.75 \\ \hline \multirow{2}{*}{M3} & \(\gamma\) & 1.209 & 7.864 & 248.11 \\  & \(H\) & 1.072 & 22.094 & 1312.96 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Convergence and power law fit data for plastic properties and \(\epsilon=0.01\).  
Figure 9: Convergence of SVE size to \(\Delta^{\varepsilon/}\) for the (**a**) yield strength, \(Y\) and (**b**) plastic hardening modulus, \(H\) versus \(L_{\text{SVE}}\) for the three materials. Individual markers represent numerical results, and dashed lines represent the linear regression fit. The green line at the ordinate of -2 corresponds to the RVE limit for \(\epsilon=0.01\).