## 3 Results and Discussion
### Geometric Property  
The martensitic volume fractions in the generated 400 \(\upmu\)m \(\times\) 400 \(\upmu\)m domains of M1, M2, and M3 are \(V_{f}=\) 10%, 45%, and 90%, respectively (Figure 1). However, the volume fraction of martensitic phase in an SVE, \(v_{f}\), can have a different value. For example, in smaller SVEs, \(v_{f}\) can vary significantly due to local heterogeneity. As the size of an SVE increases, the distribution of \(v_{f}\) narrows and approaches a delta function with the center at \(V_{f}\). Figure 4a shows the mean, maximum, and minimum volume fractions observed for all the considered SVE sizes. The solid, dot-dashed, and dashed lines denote the mean, max, and minimum values, respectively. The variation in volume fractions is the largest in small SVEs and the variations in \(v_{f}\) gradually decrease as the SVE size increases. The mean values are equal to \(V_{f}\) (M1: = 0.10, M2: 0.45, and M3: 0.90) for any SVE size of a specific material, as any SVE size of the same material covers the entire domain. Figure 4b represents the power relations for convergence of the variation in volume fraction. The plot is given in the log-log form with a base of 10 and the horizontal green line represents the RVE limit with tolerance \(\epsilon=0.01\). Linear regression lines are fitted to obtain the parameters of Equation (5). The data describing the variation-based convergence of RVE limits for different materials are provided in Table 2.  
A few conclusions can be drawn from the size dependency of \(D_{v_{f}}\). First, we observe that the \(D_{v_{f}}\) of M2 is larger than that of M1 and M3, while the values for M1 and M3 are 246.57 and 237.76, respectively. This relationship is expected from the theoretical results on additive properties such as volume fraction in [63; 64], where it is shown that the standard deviation of volume fraction for a volume element is proportional to its point-wise  
Figure 4: (a) Minimum, mean, and maximum of \(v_{f}\) versus \(L_{\text{SVE}}\) for the three materials. (b) Convergence of SVE size to \(\Delta^{Df}\) for volume fraction, \(v_{f}\). 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 \(\epsilon=0.01\).  
variation, that is \(\sqrt{V_{f}(1-V_{f})}\). This expression takes its largest value of 0.5 for \(V_{f}=0.5\), 0.4975 for \(V_{f}=0.45\) (M2), and the values for \(V_{f}=0.1\) (M1) and \(V_{f}=0.9\) (M3) are 0.3. This relationship explains the observed trends. The same theoretical models [63; 64] also describe the slope of convergence, \(a_{v_{f}}\) as \(d/2\), where \(d\) is the spatial dimension. In [16; 63; 64], the variance of a property is related to the SVE volume and the convergence rate for additive properties is 1. In [13] the relation of such an equation to (5) and the power of \(d/2\) for additive properties is explained. For the spatial dimension of 3 in this study, a convergence rate of 1.5 is predicted. The size of SVEs in the thickness direction was not increased; therefore, it is expected that their response fall between 2D (convergence rate of 1) and 3D (convergence rate of 1.5), settling at convergence rates ranging from 1.3 to 1.4 (Table 2).