**4. Results**

Although results for both the ferrite and martensite phase were processed and analyzed in this work, in this section, the results for the ferrite phase are presented. Similar

numerical results (Tables A1 and A2) of the martensite phase are provided in Appendix A of this article for interested readers.

#### *4.1. Local Stress/Strain Distribution in 2D and 3D*

Initially, local stress/strain distributions for case 1 to case 9 are shown in Figure 3 to address how the local results vary between 2D and 3D separately sourced from 1- and 50-layers only for ferrite matrix. Note that case 1 to case 3 indicates different total grain numbers; case 4 to case 6 indicate different strain levels; case 7 to case 9 indicates different volume fractions. The detailed microstructure information is demonstrated in Table 1. To point out the difference between 2D and 3D-RVEs, the local stress and strains are arithmetically subtracted (3D−2D) as *fer*0*i* = *f*50*i* − *f*1*i*, and the difference map with the true stress or strain scales −700 to 700 MPa −60 to 60%, respectively, is shown in Figure 3.

**Figure 3.** Local von Mises stress and strain distribution for ferrite phase from case 1 to case 9. Note

that case 1 to case 3 indicate different total grain numbers; case 4 to case 6 indicate different strain levels; case 7 to case 9 indicate different volume fractions. The extreme left and right column indicates the 2D and 3D local stress and strain distribution differences, respectively.

For 2D conditions, high contrast in the stress/strain concentration is observed. It can be noted that there is a significant difference between the 3D and 2D-RVEs local attribute distribution. In the 3D RVE case, the stress and strain are relatively homogeneously distributed and display lower contrast. Meanwhile, the higher stress/strain transfer to the martensite/ferrite interface. It can be concluded that the local stress/strain distribution is different from 3D and 2D-RVEs, regardless of the total grain numbers, strain levels, and volume fractions. Therefore, a straightforward transformation method can hardly be adapted to modify 2D results to 3D distribution for a specific element. The results from the proposed numerical statistical analysis are given in the next section.

#### *4.2. Step by Step for Transformation from 2D to 3D*

In Section 2, the proposed iterative method was derived for this specific problem. Figure 4 shows a scatter diagram of the ferrite phase with two extreme "total grain number" cases (case 1 (a, c) and case 3 (b, d)) via different iterative steps both for stress (a, b) and strain (c, d) value. Since all the cases are similar, this article only presents cases 1 and case 3 for discussion. Regarding the x-axis, the repaired stress/stain is indicative of the corresponding different revised stress/strain of 2D. It assumes 2D stress/strain ( *f*1*i*) for the zero iteration, but for the first iteration and onwards, it is derived to *f*er0i and Equation (6) for the first and second iteration, respectively. For the y-axis, the different stress/strain comes from the general form in Equation (14), where the 2D revised stress/strain compared with 3D stress/strain is computed for each solution point. This term is now clearly used and mentioned in the explanation of *f*er0i for the zero iteration. Then, Equations (6) and (10) indicate the first and second iteration, respectively.

**Figure 4.** Scatter diagram showing the numerical tendency of stress in (**a**) case 1 (**b**) case 3 and strain in (**c**) case 1 (**d**) case 3 by the proposed iterative method for ferrite phase at 25% global true strain and 0.1 volume fraction.

For the same iterative step, the numerical distribution reveals a similar tendency regardless of different total grain numbers for stress cases, as shown in Figure 4a,b, or for strain cases, as shown in Figure 4c,d, respectively. For the initial condition, the stress/strain on the 2D layer result ( *f*1*i*) with respect to the difference of non-revised stress/strain between 2D and 3D layer results ( *fer*0*i*) are defined as black scatter, as shown in Figure 4. It can be observed that the stress/strain results in 2D have inverse behavior to the difference in stress/strain between the 3D and 2D layer results. Due to inverse behavior, it can be speculated that the larger value of stress/strain is overestimated, and the lower value of stress/strain is underestimated for the 2D layer result. Transferring the local stress/strain of a 2D layer to match the 3D layer better, a larger value of stress/strain should be forcibly applied to a negative value of stress/strain, and a positive value of stress/strain should be forcibly applied to a smaller one.

Based on the least square method, the linear equation for the original condition can be calculated in Equations (3) and (4). Tables 3 and 4 demonstrate the constants *am*,*bm* and *R*<sup>2</sup> *m* (m = 1, 2, 3, 4) of the linear equation by the proposed iterative method for stress/strain conditions during different iterative steps. Using *a*1 and *b*1, the value of stress/strain in 2D layer result will be forcibly modified by a corresponding value and become the first revised stress/strain in Equation (5). Meanwhile, the different first revised stress/strain can be calculated by Equation (6).

After applying the approach mentioned above, the first repaired stress/strain ( *fer*0*i*) concerning different first revised stress/strain ( *f*er1i) are shown in purple scatters in Figure 4. It can be observed that different first revised stress/strain results have proportional behavior to the result of the first repaired stress/strain.

Repeatedly, through the least square method, the linear equation for the second iteration condition can be calculated by Equations (7) and (8). Through *a*2 and *b*2, the first revised stress/strain result ( *f*r1i) will forcibly be modified with a corresponding value of a linear equation and become the second revised stress/strain result ( *f*r2i) in Equation (9). Additionally, the different second revised stress/strain ( *f*er2i) can be calculated by Equation (10). Followed by a similar iterative procedure, the second repaired stress strain ( *f*er1i) concerning the different second revised stress/strain ( *f*er2i) is defined as blue scatters, as shown in Figure 4. Similarly, third repaired stress/strain ( *f*er2i) concerning the different third revised stress/strain ( *f*er3i) is defined as green scatters.


**Table 3.** Constants *am*, *bm*, and *R*<sup>2</sup> *m* (m = 1, 2, 3, 4) for the ferrite phase of linear equation with stress condition by the proposed iterative method.


**Table 4.** Constants *am*, *bm*, and *<sup>R</sup>*2*m* (m = 1, 2, 3, 4) for the ferrite phase of linear equation with strain condition by the proposed iterative method.

#### *4.3. Convergence and Statistical Analysis*

A quantitative analysis method determines the convergen<sup>t</sup> behavior and confirms the error analysis via different iterative steps. The function of average difference is defined as follows:

$$\left| cr\_{\text{av}\_{\text{SV}}}(\mathbf{x}) = \sum\_{i=1}^{n} \frac{|f\_{\text{err}}|}{n} \mathbf{m} = 0, 1, 2 \dots, \mathbf{n} = 10,000 \right. \tag{15}$$

where *f*ermi is the difference in stress/strain between 3D and revised 2D layer results as derived in Equation (14), which can also be obtained by the magnitude of the y-axis in Figure 4, "m" is an iterative step from 0 to 5, and n is element number in 100 × 100 voxels RVEs. Figure 5 shows the average difference in stress/strain with different iterative steps. To avoid confusion, only case 1 to case 3 are shown in the convergence analysis. As can be seen, the convergen<sup>t</sup> behavior with a small error is observed in the fourth iterative step. Therefore, it can be concluded that the fourth iterative step can reach the desired 3D local stress/strain distribution after almost 70% average difference in stress/strain conditions.

**Figure 5.** Convergence analysis for the ferrite phase at 25% global true strain. (**a**) Stress and (**b**) strain by the proposed iterative method with different iterative steps.

Previously presented qualitative comparisons of transformation are mainly based on individual visual perceptions. An appropriate statistical quantitative analysis is carried out here to compare the transformation results more accurately. For the statistical method, the probability distribution function is adopted. The probability distribution function can be derived in terms of *μ* and *σ* as follows:

$$p(\mathbf{x}) = \frac{1}{\sigma\sqrt{2\pi}}\exp(-\frac{(\mathbf{x}-\boldsymbol{\mu})^2}{2\sigma^2}), \ -\infty < \mathbf{x} < \infty \tag{16}$$

where *μ* is mean value of the probability distribution function *σ* is the standard deviation value. The standard normal distribution-based probability distribution function of the difference in stress/strain distribution for case 1 with different iterative steps is shown in Figure 6. Note that five different stress/strains from zero to the fourth iterative step have been considered in the probability distribution function. Again, the different stress/strain comes from the general form in Equation (14), where the 2D revised stress/strain compared with 3D stress/strain is computed for each solution point. It can be seen that the peak considerably protrudes with the increase in iterative steps in the probability diagram. Therefore, it can be concluded that the error is convergen<sup>t</sup> as iterative steps increases, which represents the derived 3D stress and strain distribution, as effectively calculated by the proposed iterative method.

**Figure 6.** Probability distribution function of different (**a**) stress and (**b**) strain distributions between 3D result and iteratively revised 2D result for the ferrite phase of case 1 with different iterative steps.

#### *4.4. Derivation of 3D Stress and Strain Distribution from the 2D Result*

The graphic of derived 3D local stress and strain distribution with different iterative steps using the identified iterative constants will be demonstrated in this section. Due to the convergence and statistical analysis in Figures 5 and 6, the transformation is carried out up to the fourth iterative step. Local stress/strain distribution for the revised 2D layer result, 3D layer result, and differences between 2D and 3D layer results for the ferrite phase of case 1 with different iterative steps is as displayed in Figure 7. In the first iteration, the stress/strain distribution displays the virtually perfectly averaged arrangemen<sup>t</sup> when forcibly applying a negative value on a larger stress/strain value and a positive value on a smaller stress/strain value by modification from corresponding *a*1 and *b*1 in Tables 3 and 4. The high stress/strain zones disappear from the actual results. However, the difference in stress/strain between the first revised 2D and 3D layers can still be easily identified. The second revised stress/strain is obtained for the second iteration after modifying the first revised result by corresponding *a*2 and *b*2 in Tables 3 and 4. It can be observed that

the stress/strain concentration is generated along with the ferrite/martensite interfaces. The second iteration result displays a dramatic difference compared with the first iteration result. Subsequently, the third and fourth revised stress/strain are obtained during the modification of the second and third results revised by the corresponding *a*3 and *b*3 as well as *a*4 and *b*4 in Tables 3 and 4. It can be seen that stress/strain concentration along the interface of the fourth revised result is more enhanced than the second revised result. It can be concluded that the derived 3D stress/strain distribution closely resembles the 3D result at the fourth iterative step.

**Figure 7.** Local von Mises stress and strain distribution for the ferrite phase of case 1 with different iterative steps. The extreme left and right column indicate the 2D and 3D local stress and strain distribution differences, respectively.
