**1. Introduction**

Multiple-phase composites possess admirable mechanical properties and service life. Combining softer matrix with harder islands [1], multiple-phase materials demonstrate the strong structure and ductility, such as in dual-phase steel (DP steel) [2], metal matrix composites, and other advanced steel [3]. The material properties of the multiphase materials depend on the microstructural attributes, such as the size, shape, composition, and distribution of the second phase within the matrix [4]. The effect of these attributes is interdependent and plays a key role in defining the deformation and damage behavior under varying loading conditions [5,6], i.e., strain rate, temperature, and loading direction. Numerical simulation models provide an interesting outlook for targeted material development by avoiding the expensive and time-consuming experimentation of every iterative modification in the material microstructure [7–9].

There are different numerical simulation models, i.e., empirical, analytical, datadriven, and hybrid [10,11]. Although they are useful in the general modeling of material deformation and damage behavior, they lack the fundamental dependence on the local microstructural attributes [12,13]. Fast Fourier transformation-based crystal plasticity models provide a comprehensive and accurate solution for modeling the multiphase

**Citation:** Tseng, S.; Qayyum, F.; Guk, S.; Chao, C.; Prahl, U. Transformation of 2D RVE Local Stress and Strain Distributions to 3D Observations in Full Phase Crystal Plasticity Simulations of Dual-Phase Steels. *Crystals* **2022**, *12*, 955. https:// doi.org/10.3390/cryst12070955

Academic Editors: Pavel Lukác and ˇ Wojciech Polkowski

Received: 28 April 2022 Accepted: 6 July 2022 Published: 8 July 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

material's dependence on the microstructural attributes [14,15]. The RVEs in such models comprise all the necessary phase compositions, size, orientation, and distribution that represent the actual material [16]. Therefore, the model's accuracy and applicability largely depend on the constructed RVE for such simulations.

Different simple and sophisticated models for accurate RVE development have been proposed previously, i.e., single-step Voronoi tessellation, multi-step Voronoi tessellation, and artificial neural networks yielding accurate results that are usually 3D. According to the RVE [17] method, both 2D (2-dimension) [18] and 3D (3-dimension) [19] microstructures have been constructed by random grain size, orientation, phase, and texture or measured from electron backscatter diffraction (EBSD) data [20]. Based on EBSD patterns from the serial-sectioning experiments, 3D polycrystalline microstructures were constructed by Groeber et al. [21,22]. Recently, DREAM-3D [23] was used to construct realistic RVEs from virtual or real statistical grain size, orientation, and texture data. Applying DREAM-3D and DAMASK [24], the performance of multilayer composites can be evaluated under mechanical loading by the crystal plasticity material model.

Compared to the simulation and experimental data for DP steels, Ramazani et al. [25] discovered that the 2D model displayed underestimated behavior, while the 3D result demonstrated a quantitative description of the flow curve in comparison to the experimental data. Qayyum et al. [26,27] studied the local deformation and transformation behavior of transformation-induced plasticity (TRIP) steel. In these articles, the simulation results of global stress and strain behavior were compared to the experimental result. Nevertheless, the validation of local distribution in 3D has a challenge to discuss, owing to the inconsistency between experiment and simulation results. Diehl et al. [28] analyzed the effect of "columnarity" and studied the influence of the nearby environment on stress and strain in DP microstructures. It was discovered that the local stress and strain distributions are strongly influenced by both the nearby grain shape and grain orientation. Due to this effect, the 2D simulations of heterogeneous microstructures could be definitely misleading for the damage prediction [29] of crack initiation and propagation.

The CP models are also largely dependent on several physical and a few fitting parameters. These parameters are obtained by comparing averaged numerical simulation results with experimental stress and strain curves [26,30]. Although a certain set of fitted parameters seems to accurately represent the overall deformation behavior, a slightly different set of fitted parameters is also expected to yield similar global results with a significant change in local results. This reliability of such models on the globally calibrated parameters can lead to incorrect local results. In the recent past, researchers have developed algorithms for automated sensitivity analyses and parametric identifications from global stress and strain curves [31,32]. A more reliable way of calibrating and validating the CP model parameters can be through direct local comparison of experimental in situ and simulation results.

The local strain measurement is experimentally possible now due to the advancement of in situ measurement and data processing tools, which can accurately capture local strain distribution and local microstructural attributes [33–35]. However, constructing an accurate 3D RVE of the same specimen is a paradox as 3D EBSD requires polishing, slicing, and measurement throughout the specimen. Moreover, different phases frequently cause false detection at the measurement surfaces and the grain boundaries [36].

Therefore, to carry out such a comparison, the 2D RVE results from CP simulation should be reliably transformed to keep the global results the same while transforming the local stress and strain distributions for comparison with the results of the 3D material surface. It is a mathematical challenge that can yield a model for transforming 2D RVE simulation results of local strain distribution for comparison with strain measurements on the surface of a 3D specimen. Recently, Qayyum et al. [37,38] investigated the stress and strain distribution between 2D and 3D RVEs with different total grain numbers. It was shown that the global distribution of the flow curve is similar across different RVEs. However, the local distribution showed obvious variation. The numerical finding of

Qayyum et al. [37,38] underlines the different stress and strain distribution in 2D and 3D via random distribution of grain size and orientation. In the current article, the previous study is carried one step further, and a corresponding local 3D stress and strain distribution on the RVE surface is achieved by transforming 2D stress and strain. The work is based on the alternative error and least square method. The transformation of the ferrite phase stress and strain from the 2D result is iteratively achieved. The iterative constants for different cases of magnification, strain levels, and volume fractions are presented and compared. The proposed new numerical approach is helpful in iteratively transforming the 2D stress and strain if the 3D outcome is known. To analyze and compare the derived 3D distribution with actual results, a statistical analysis of the obtained data is carried out.

The following Section 2 presents the detail of the numerical simulation model, including grain size, boundary condition, and material parameter of crystal plasticity. Then, the proposed iterative method, which is adopted in the current work, is explained in Section 3. Next, the derived 3D local stress and strain distribution calculated using the proposed method are displayed. Meanwhile, similarities between derived and exact 3D distribution are provided by statistical analysis in Section 4. Finally, the discussion and conclusion are presented in Sections 5 and 6.

#### **2. Numerical Simulation Model**

The material adopted in the current investigation is DP steel, which comprises two distinct phases, i.e., soft ferrite matrix and hard martensite island. This large local microstructural heterogeneity yields significant local strain contrast during deformation. The microstructural distribution, grain size, material properties, and grain orientation are adopted from previously published work [39–41]. This article investigates different parameters for varying the overall number of grains, strain levels, and volume fractions. Table 1 demonstrates the grain size distribution of ferrite and martensite, number of total grains, volume fractions, and strain levels for each model. For cases 1 to 3, the varying parameter is the total number of grains; for cases 4 to 6, the varying parameter is strain level; For cases 7 to 9, the varying parameter is volume fraction. Note that the unit of grains size is μm and total grain is a dimensionless value. Based on experimental observation [42], the smaller average grain sizes of ferrite and martensite were assigned as 6.35 and 4.6 in RVE-A, whereas the larger average grain sizes were 14.0 and 12.8 in RVE-C during the crystal plasticity simulation. Hence, the total grains of RVE-A are more than RVE-C.


**Table 1.** Grain size distribution, number of total grain, volume fraction, and strain level for the microstructure [37].

A cubic equiaxed crystal structure and ellipsoid grain shape are assigned to both ferrite and martensite phases. 100 × 100 × 100 voxels RVEs are systematically constructed using DREAM-3D for the sizes as shown in Table 1. Figure 1 shows a schematic diagram and flow chart in this article. The blue arrows indicate the steps involved in carrying out this work. In the beginning, 2D and 3D simulations for different RVEs are carried out. Both ferrite (F) and martensite (M) phases are then individually analyzed. Combined with the

least square and alternative error method, the iterative constants (*<sup>a</sup>*1, *b*1, *a*2, and *b*2, ... ) of the linear equation are calculated while transforming the 2D results into 3D.

**Figure 1.** The schematic diagram represents the flow chart of the current work. The upper half (yellow background) indicates the flow chart of crystal simulation, and the bottom half (blue background) represents numerical analysis. The blue arrow indicates the flow of data. The red arrow represents tensile loading, and the yellow arrow separates the ferrite (F) and martensite (M) phases.

The principal aim of this ongoing study is to develop a robust methodology for deriving the 3D surface stress and strain distribution from the 2D RVE result for ferrite and martensite phases. This work brings us one step closer to that aim. As a representation,

RVE-C in inverse pole figure (IPF) colors is shown in Figure 2a (h = 1), where h indicates the length of z-direction normalized total length of z-direction. Following Qayyum et al. [37], 50- (h = 0.5) and 1-layer (h = 0.01) RVE were sliced from the initial microstructure and regarded as 3D and 2D RVE, respectively. The simulation results for 1- and 50-layers have been selected and defined as 2D and 3D results of the local stress and strain distribution. There are 10,000 Gaussian mesh elements in one layer, meaning 2D and 3D possess 10,000 and 500,000 elements, respectively. All the constructed RVEs were sliced into similar geometries with 1-layer (h = 0.01) RVE and shown in terms of texture style in Figure 2b.

**Figure 2.** (**a**) The resulting RVE-C initial microstructure colored according to the inverse pole figure (IPF). (**b**) All RVE models (RVE-A to RVE-F) were sliced into geometries with 1-layer (h = 0.01) RVE in terms of texture style.

The ferrite and martensite are defined as elastic–viscoplastic deformable phases. The elastic stiffness, shear resistance, hardening behavior, and curve fitting parameter are adapted from previously published work [38] and presented in Table 2.


**Table 2.** Mechanical properties of multiphase (ferrite and martensite) were adopted from [37] for simulation modeling.

The crystallographic orientation, mechanical properties, and phase of ferrite and martensite are included in the RVE geometry definition. A uniaxial load along the x-direction is defined using mixed boundary conditions as follows:

$$
\dot{F}\_{ij} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \* & 0 \\ 0 & 0 & \* \end{bmatrix} \times 10^{-3} \cdot s^{-1} \tag{1}
$$

$$P\_{ij} = \begin{bmatrix} \* & \* & \* \\ \* & 0 & \* \\ \* & \* & 0 \end{bmatrix} Pa \tag{2}$$

where *Fij* is the coefficients of the macroscopic rate of the deformation gradient, *<sup>P</sup>ij* is the first Piola–Kirchhoff stress. . *F*11 = 1/*s* indicates tensile condition, 0 is represented as restricted, and \* is an arbitrary value during the simulation. It should be noted that the strain rate of all simulations is assumed to be 1 × 10−3/*s* in conjunction with periodic boundary conditions in all three directions. The simulations are performed in plain strain mode, where 2D RVE is interpreted as a columnar grain structure.

The spectral method via fast Fourier transform [43] is used to solve the continuum mechanics formulation mentioned above. After completing simulations, the 2D and 3D data are statistically analyzed. The 3D numerical results are used as a reference for iteratively modifying the 2D results by combining the least square and alternative error methods via Matlab programs (2019b, The MathWorks Inc., Natick, MA, USA).
