## 3 Results and Discussion
### Plastic Properties  
The size dependency of apparent plastic properties is now investigated based on the MBC form of BCs (explained in Section 2.3). The applied displacements on the facets of the SVE gradually increase until the macroscopic stress responses pass a maximum value and tend to zero beyond this stage. Figure 7 shows homogenized \(\sigma_{xx}\) versus \(\varepsilon_{xx}\) for each SVE size, when the SVEs are loaded along the \(x\)-direction (load case 1). The yield strength, \(Y_{x}\), and plastic hardening modulus, \(H_{x}\), are calculated from the homogenized curves for different loading cases using the process explained in Section 2.4. The statistical variation in the overall strain stress response and yield point location decreases as the SVE size increases.  
Figure 6: Convergence of SVE size to \(\Delta^{c_{f}}\) for (**a**) bulk modulus, \(\kappa\), and (**b**) shear modulus, \(G\). 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 \(\epsilon=0.01\).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & \(\mathbf{f}\) & \(\mathbf{\alpha_{f}}\) & \(\mathbf{b_{f}}\) & \(\Delta^{c_{f}}\) \\ \hline \multirow{2}{*}{M1} & \(\kappa\) & 1.268 & 10.057 & 233.35 \\  & \(G\) & 1.042 & 3.822 & 300.44 \\ \hline \multirow{2}{*}{M2} & \(\kappa\) & 1.414 & 25.072 & 253.05 \\  & \(G\) & 1.117 & 7.459 & 373.45 \\ \hline \multirow{2}{*}{M3} & \(\kappa\) & 1.312 & 7.210 & 150.87 \\  & \(G\) & 0.912 & 2.175 & 366.65 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Convergence and power law fit data for elastic properties and \(\epsilon=0.01\).  
Figure 7: _Cont._Figure 8 shows the mean, minimum and maximum values of the yield strength, \(Y\), and the plastic hardening modulus, \(H\), obtained by averaging the corresponding \(x\) and \(y\) load case values; cf. Equation (4a). Similar to elastic properties, the variation in the initial yield strength decreases as the SVE size increases. Similarly, mean properties over the SVE sizes exhibit minimal variations even for the smaller sizes of the same material. However, one interesting observation is the ordering of \(Y\) and \(H\) values across the three materials; the yield strength ordering in Figure 7c is the same as those of \(\kappa\) and \(G\) in Figure 5, as the martensite phase is not only stiffer but also has a higher saturation slip strength; cf. Table 1. The ordering, however, is reversed for \(H\) as M1 has the highest homogenized hardening modulus.  
The convergence plots for the RVE limit of plastic properties are shown in Figure 9. Table 4 summarizes the corresponding data. In terms of convergence rate \(\alpha_{f}\), there is no consistent trend when comparing the elastic (Table 3) and plastic properties. That is, the convergence rate for elastic properties is higher for only some properties and materials. However, in terms of RVE size, the elastic properties consistently have a smaller RVE size. This observation is consistent with [16] where it is stated that more complex phenomena have larger RVE sizes, based on thermal conductivity and linear elastic stiffnesses results. In our case, we deem plastic properties to stem from the more complex constitutive equations, cf. (Equation (1)), and the larger RVE size mainly stems from larger proportionality factors, \(b_{f}\) (Table 4).  
Figure 8: Minimum, mean, and maximum of (**a**) yield strength, \(Y\) and (**b**) plastic hardening modulus, \(H\), versus \(L_{\text{SVE}}\) for the three materials.