2.1 EBSD Data Collection and Cleaning by MTex

The three cases of C45EC steel based on different spheroidization degrees considered in this study are shown in Table  2 , with statistical details about ferrite and cementite particle sizes. For calculating spheroidization degree, the micrographs are processed under SEM by ignoring very small cementite particles. During characterization, cementite particles with aspect ratios less than or equal to 1:3 are considered spheroidized and are calculated per unit area to estimate the material's spheroidization degree.

The orientation distribution of the material at the grain level can substantially influence the ductility at a global level.^{[ 22 ]} Therefore, changes in crystallography at the microscale need to be estimated and understood for predicting deformation texture accurately.^{[ 23 ]} 2D EBSD analysis in SEM–being a fast and convenient method to collect real-time microstructural data^{[ 24 ]}–was performed after spheroidization annealing. During the scanning process, local material data were recorded with a step size of 0.5 μm on both the x and y axes. The data obtained from the EBSD scan are recorded in the form of .ang files, which generally contain point-wise details of 2D coordinates, phase, and crystallographic orientation in the form of Euler angles.

The experimentally collected EBSD data have a few discrepancies like with all other experimental systems. It needs careful postscan working to remove the noise, unindexed regions, and reference frame alignment issues. The random error can occur during indexing orientations using a traditional band detection method or pattern matching.^{[ 25 ]} Systematic error may also arise due to imprecise alignment in specimen and pattern center, resulting in nonrealistic data at output. Although small at the time of detection, these errors may become considerable during amplification of the data during further calculations for kernel averaged misorientations (KAM), orientations curvature tensor, or geometrically necessary dislocation (GND) densities.^{[ 26 ]} The process of reducing the random error up to the minimum level assumes that the actual orientations are space-dependent while noise is not.^{[ 27 ]} There exists a denoising threshold of the process surpassing which the small precipitate particles can be engulfed, being considered as noise.^{[ 28 ]}

The open-source toolbox called MTex has recently emerged as a robust tool for data denoising. It has several built-in filters for this purpose. The well-known ones are the mean filter, the Kuwahara filter, the median filter, and variational denoising techniques such as “smoothingSplinefilter.”^{[ 29 ]} The mean filter uses a simple method to average down all the neighboring grains to get the orientations data and assigns this value of direction to the central grain/pixel to be cleaned. This method is assumed to remove the wrongly indexed orientation of a specific pixel located somewhere within the grain of known orientation. If nonindexed pixels further need to be reduced, this can be done by reapplying the filter using more neighboring pixels as the reference orientation information.

Similarly, the missing data can also be filled using the MTex by “filling the missing data in orientation maps” filter, which uses two methods. The first method used the “fill” option, which takes the mean value of orientation from the surrounding pixels and fills out the missing dots, while the other method used “halfQuadraticFilter” for the same purpose. The MATLAB routines used to generate the results by filling data from raw to the cleaned form are attached as Appendix A. MTex also helps with the material characterization in various subroutines available for plotting data concerning crystal orientation, KAM angle, GNDs, and inverse pole figure (IPF) maps. This tool can be easily integrated with MATLAB post-2015 versions and accessed using graphical user interface (GUI) for limited applications. The resulting IPF map for three spheroidization states of the steel samples obtained from EBSD analysis obtained is shown in Figure  2 .

2.2 Crystal Plasticity-Based Modeling and Simulations

The data after MTex postprocessing is converted into geometry input files for full phase simulations. For this purpose, a very interactive tool called Dream.3D is used for converting clean EBSD files into geometry descriptions and associated material configuration files. Then, these EBSD-based files and load definition files are fed as input into the crystal plasticity and continuum mechanics-based phase-field numerical simulation model called DAMASK. Being a crystal-level simulator, it solves coupled equations already defined for various crystalline phases, i.e., ferrite and cementite in this case.^{[ 30 ]}

All 2D RVEs considered in this study are of 85 × 85 × 1 μm³ size. During EBSD analysis, it was assured that the magnification was enough to capture multiple ferrite grains for constructing a polycrystalline RVE. Therefore, the step size was chosen to be 0.5 μm for recording EBSD with appropriate resolution. The final size of each RVE is 476 × 476 × 1 pixels. This specific resolution is selected to assure that the number of elements for calculations is large enough to assign more than one element to individual cementite particles, which at some positions are slightly bigger than the selected step size.

Here, $h_0$, $S_{\mathrm{s}}$ are reference hardening parameter and saturation stress, respectively. Phase parameters with the corresponding fitting parameters used in the material model are shown in Table 3. These phase parameters have been previously reported after calibration for C45EC steel.^{[ 15, 31 ]}

The complementary conditions represented with “*” are applied on various coefficients of the second rank tensors in (1) and (2). Generally, for the cubic crystal structure of the ferrite phase, there are 24 slip systems dominantly involved during plastic deformation by tensile loading, 12 slip systems of the type {110} and 12 of type {112} with<111> direction family makeup 24 slip planes which are the most prone to the deformation.^{[ 32, 33 ]}

The first step in the solution is executed when the material model homogenizes the individual point stress in the overall mean stress at the crystal level. The second step is solving the elastic–plastic material problem at the crystallite or grain level by using the constitutive laws as detailed in Equation (1-2)–(3). The ferrite phase is defined with elastic and plastic properties, whereas the cementite phase is assigned elastic properties. The details about the elastic and plastic phase parameters calculated using the nanoindentation method are adopted from the literature^{[ 13 ]} and for reference are provided in Table 3. Several fitting parameters such as n and w need calibration to address the variability in the thermomechanical processing route of the material. This is done by using an already published technique^{[ 34 ]} of material model calibration.

A robust, fast Fourier transform (FFT)-based spectral solver is implemented in the DAMASK framework to calculate the individual material point results using coupled crystal plasticity constitutive equations. The output is recorded in the form of local stress and strain maps and global stress–strain curves. In addition, the output data are used to withdraw valuable information for analyzing local deformation behavior due to varied microstructural attributes of second phase particles. The results and discussion sections contain details about the valuable insights obtained from this study.

3.1 Hot-Rolled (HR) State

This material state is industrially produced C45EC by hot rolling and analyzed as-received (HR). Due to 0.45% carbon in chemical composition and thermomechanical processing of the material, a microstructure with an overall irregular periphery of ferrite grains is obtained, shown in Figure  3a as an IPF map. It is observed that, generally, the material is based on a ferrite matrix in which cementite particles are embedded. Ferrite grains are of 8.85 μm average equivalent diameter, and cementite particles are of 1.17 μm average equivalent diameter. Due to a specific processing route, the distribution of cementite particles in nonhomogeneous spaces with a clustered appearance in some areas is highlighted with boxes in Figure 3a. In Figure 3a _1, note that the cementite particles are present within the ferrite grains with a slight concentration on the grain boundaries.

The orientation information of the microstructure helps in understanding how it affects the local deformation behavior at a particular material point. As reported in the literature, the triple points are essential to note where the damage would initiate due to localized stress and strain.^{[ 15 ]} It is shown in Figure 3b that the orientation distribution of the RVE is relatively homogeneous, with a slight concentration on the $[\overline{1}01]$ plane. Initial isotropy does not imply that the material remains isotropic during deformation. The anisotropy condition in RVE simulations is necessary because it ensures the material behavior to be a true representative of the overall material.

Another key attribute to consider is the grain size and aspect ratio data for each phase. For HR state, this data is shown in Figure  4 . It is observed that the aspect ratio of ferrite grains majorly lies between 1 and 3. The ferrite grains (90%) are smaller than 20 μm sphere diameter, with few grains large than 25 μm equivalent diameter. Due to the very small cementite particle size and relatively large step size, the aspect ratio of the recorded cementite grains is around 1.5. Most cementite particles have an equivalent diameter between 0.5 and 1 μm which is relatively small compared with the ferrite grain diameter.

Most of the cementite particles recorded in the EBSD data are close to the step size of the electron beam and, therefore, are square, comprising a few pixels. It is also important to note that the data shown in Figure 4 and Figure  5 are from the middle of the sample, and it might slightly change in different areas. The microstructural attributes presented here help understand how the material behaves when loaded experimentally or in a numerical simulation model.

The grain orientation and the presence of the second phase particles have a critical significance in deciding the micromechanical behavior of polycrystalline materials. The IPF map shown in Figure 5a has an over-laid crystal orientation represented by 3D cubes representing the relative rotation of each grain in the 3D space. It can be observed clearly that the grains’ random orientations suggest an interesting outcome upon deformation due to tensile loading. The distribution of the cementite particles in the RVE is quite heterogeneous and not evenly distributed in the whole RVE. Several ferrite grains are free from the cementite particles, as shown in Figure 5b; however, they can be seen to have accumulated higher local strains even at 5% of the global strain. Figure 5 shows local stress and strain evolution in the RVE as the material is subjected to tensile load in the mentioned direction. The local stress and strain maps show an overall heterogeneous behavior during evolution.

The stress maps are shown in Figure 5(right) for only ferrite grains because the magnitude of the stress shown by the hard cementite particles is ignored to very high values. The average local stress in ferrite grain is around 600 MPa even at 25% of global strain, while the stress in cementite particles surpasses 5 GPa. A general contrast in the stress map can be observed as the global strain evolves from 5% to 25%. It is observed that the stresses start to grow in the small ferrite grains and continue to grow as compared with the large grains as the global strain evolve. It can be observed that the stress concentration is more in the grains, which are at approximately 40–50° disoriented as compared with the neighboring grains, and the sharp difference is distinguished at the interface. Stresses in the grains majorly evolve in a similar trend starting from 5% to 25% of the global strain. Interestingly, grains show stress concentration at the ferrite–ferrite interface, and the rest of the grains are at average stress. It has been observed that the stress concentration is primarily controlled by the crystallographic orientation, which is distinguished by the grain boundary at various positions in the RVE.

As received (HR), local strain distribution maps have been shown in Figure 5(left) in the evolution of global strain for 5%, 15%, and 25%. Although the local strains are distributed heterogeneously, they are majorly controlled by the difference in crystallographic orientations distinguished by grain boundaries. The strains are observed in bands at an angle of 45° to the line of application of tensile load. Strains notably originate from the interface and evolve in narrow paths along with the ferrite–ferrite interface. Large ferrite grains with no cementite grains embedded in them are seen to undergo a large strain. The strain concentration gets intense at the areas where cementite particles are distributed with the grain in close proximity, and the grain is ≈45° disoriented from the surrounding grains.

The close-up of locally evolved stress and strain is shown in Figure  6 . The marked and magnified zone 1 shows the stress and strain distribution in the close vicinity of cementite particles clustered together inside ferrite grains. It is observed that the finely distributed cementite particles hinder the large deformations and cause localized high-stress zones.

These areas with high stress and high strain zones are critical in defining the load-bearing capacity of a particular material. In contrast, zone 2 has been magnified and shown with slight clustering of cementite particles on a grain boundary. Due to these particles’ presence on the grain boundaries, they reduce the local strain evolution without causing any high-stress localization.

Stress and strain along the line 3 and 4 have been plotted to quantitatively compare the effect of grain orientation and cementite particle presence. While comparing the two line graphs of stress and strain distributions, the trends change abruptly at the grain boundaries. This is because a high shear occurs due to a cementite cluster as the model is sensitive to the defined orientation of a solution point. As a result, a sharp change in the stress and strain behavior across the grain boundaries is observed.

3.2 Hot-Rolled, Cold-Rolled, Spheroidized (HR-CR-S) State

For this state, the bulk material was hot-rolled, followed by cold rolling. Finally, the samples were spheroidization annealed to manufacture material with slightly higher strength with elongated grain in one direction. This suggests the overall dominance of the anisotropic behavior of the properties.

The material recrystallizes after heat treatment, but the grains are elongated in the rolling direction due to cold rolling passes, as shown in Figure  7 . The IPF map is highlighted to show high cementite particle concentration zones. The phase map of the material in Figure 7b shows the clustering of cementite particles on the grain boundaries with some areas with uniform cementite particle distribution and other areas with concentrated cementite particle distribution on the grain boundaries. The ferrite grains orientation distribution map shown in Figure 7c highlights the point that the grains are biased toward $[001]$ and $[\overline{1}11]$ orientations directions.

Analyzing each phase's statistical grain size distribution in Figure  8 , it is observed that the aspect ratio of the majority of the ferrite grains lies between one and two. In contrast, the ferrite matrix consists of multipolar grain size distribution, with most of the grains lying between 1 and 3 μm. A few grains are between 10 and 15 μm, and a few are even bigger than 20 μm. This uneven grain size distribution is probably due to the material processing route adopted to manufacture this material. It is also observed that there is a considerable variation in the grain size distribution of the ferrite phase.

The cementite particles have a very small size between 0.5 and 1 μm, with most aspect ratios close to 1 and 1.5. These small cementite particles are seen clustered together on the grain boundaries, which pin the grain boundary slip during plastic deformation but are highly susceptible sites for void nucleation and consequent damage propagation in high deformation regimes.

The evolution of local stress and strain after the application of uniaxial tensile load on the geometry with elongated grains biased towards two contrasting $[001]$ and $[\overline{1}11]$ directions is shown in Figure  9 . Discontinuous strain evolution is observed as the high degree of cementite particle clustering (in the form of bands) on the grain boundaries restricts deformation flow throughout the matrix.

A high-stress contrast is observed even at 5% of global strain due to grains not aligned in the most favorable direction for plastic deformation. In this case, the high-stress zones are mostly the least deformed areas saturated with cementite clustering. For a detailed insight, the thorough analysis of the local stress and strain behavior at 25% of global strain is shown in Figure  10 .

It is not only the cementite particle clustering or banding that dictates local stress and strain evolution. Instead, it is the combination of grain orientation and neighboring media which plays the critical role. For example, in Figure 10 comparing zone 1 and zone 2, comparable strain distribution in the ferrite matrix, very high-stress contrast is observed in zone 1 specifically around cementite clusters within the same grain.

On line 3, stress and strain have been plotted across a cementite particle band positioned on a grain boundary. The strain is lower and stress higher in the ferrite matrix close to the cementite particle band regardless of the grain orientation. The trend on line 4 shows a smooth transition of stress across the grain boundaries with a sharp variation of strain distribution in the absence of the cementite particle clusters.

3.3 Hot-Rolled, Spheroidized, Cold-Rolled, Spheroidized (HR-S-CR-S) State

The material was spheroidized after hot rolling. Then, cold-rolled to the desired caliber. It was spheroidized again to manufacture a material with a relatively homogeneous microstructure and even distribution of globular cementite particles in the ferrite matrix. The IPF map of material microstructure is shown in Figure  11a. It is observed that the ferrite grains are of relatively even grain size, and the cementite particles are evenly distributed within the matrix. A close-up of a highlighted area in Figure 11a ₁ shows that the spherical shape cementite particles are present on the grain boundaries and within the ferrite grains. The orientation distribution map in Figure 11c shows that the grain orientation is relatively homogeneous with even distribution in all planes. The readers might observe the 45° rotation of the RVE regarding the loading direction. This was done intentionally to run simulations with comparable output as per other RVEs.

The statistical distribution of grain size and aspect ratio of both phases is shown in Figure  12 . It is observed that due to the spheroidization annealing process after hot rolling and after cold rolling, the ferrite grains have a relatively similar aspect ratio, majorly lying between 1 and 2 with the equivalent radius of grain below 10 μm. There are very few grains bigger than 10 μm. The aspect ratio of cemented particles majorly lies between 1 and 2, with particle equivalent radius averaging at 1 μm.

It is expected that the homogeneous distribution of relatively larger cementite particles slightly elongated in the rolling direction within the relatively finer ferrite grains would result in relatively homogeneous stress and strain distributions within the matrix during deformation.

The grain orientation and second phase particles critically affect the micromechanical behavior of polycrystalline, multiphase materials. In Figure  13a,b, it is observed that the distribution of the cementite particles in the nearly equal-sized ferrite grains is quite homogeneous. Therefore, the overlaid crystal shapes over each grain according to the IPF representation in Figure 13a will be pretty helpful in understanding the local stress and strain evolution. For this RVE specifically, the local stress and strain maps show an overall homogeneous behavior during evolution compared with previously presented cases.

The stress maps shown in Figure 13(right) are only representing stress in ferrite grains. A general contrast in the stress map can be observed as the global strain evolves from 5% to 25%. It is observed that the stresses start to grow in the small ferrite grains, usually on the triple points, and continue to grow as the global strain evolves. It can be observed that the stress concentration is more in the grains, which are at approximately 40–50° disoriented as compared with the neighboring grains, and the sharp difference is distinguished at the interface. The stress concentration is controlled mainly by the crystallographic orientation distinguished by the grain boundary at various positions in the RVE.

The evolution of strain distribution is shown in Figure 13(left). Relatively homogeneous distribution of local strain is observed throughout the matrix with slight localization at high global strains of 25%. This homogeneous distribution of the strain is due to the evenly orientation distribution function (ODF), as shown in Figure 11c. The strain concentration gets intense around the areas where cementite particles are present, and the grain is at ≈45° disoriented from the surrounding grains.

The quantitative analysis of the local stress and strain at 25% global strain is shown in Figure  14 . While comparing the local distributions in zone 1 and 2, it is observed that regardless of the similar cementite particle presence in these areas, the strain distribution is relatively homogenous, whereas significant stress variations are observed. To quantify this observation, line maps of stress and strain on 3 and 4 are shown in Figure 14. These plots verified that the stress and strain partitioning within a specific grain is dependent mainly on the crystal orientation.