# Modeling Bainite Dual-Phase Steels: A High-Resolution Crystal Plasticity Simulation Study

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## Abstract

**:**

## 1. Introduction

## 2. Modeling Framework

## 3. Materials and Methods

## 4. Simulation Setup

#### 4.1. Fitting of CP Parameters

#### 4.2. Representative Volume Element—RVE

- “3D-RVE”: described in Section 4.1.
- “3D-RVE no-substructure”: similar to “3D-RVE” but the implementation of the experimental grain size distribution is substituted by a Voronoi tesellation of 80 random seeds. The main difference with “3D-RVE” is the lower accuracy in the statistical representation of the texture, i.e., 80 grains compared to 10,000 grains.
- “3D-RVE gb-substructure”: based on “3D-RVE no-substructure”. PF grains are kept unchanged. GB grains of “3D-RVE no-substructure” are used as prior austenite grains which is the input required in the microstructure generation tool of Gallardo-Basile et al. [31]. This tool is used to generate a lath-martensite substructure for the GB (justification for this at the end of this section). The following values for the input parameters of the tool are used: ${V}_{\mathrm{lath}}$ = 647 A.U. (arbitrary units of volume), ${t}_{\mathrm{subblock}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}10$ A.U. (arbitrary units of length), ${l}_{\mathrm{lath}}\ge {w}_{\mathrm{lath}}\ge {t}_{\mathrm{lath}}$ = 9:3:1, and ${\theta}_{\mathrm{max}}$ = 3°.
- “2D-RVE measured”: a direct 2D takeover of the measured crystallographic orientation of each pixel of the EBSD scan in Figure 4d. The phase is assigned according to the classification tool. Cleaning is performed with OIM software [32] for assigning a phase and an orientation to non-indexed pixels and the ones indexed as austenite (most of them are measuring errors).

_{$\alpha $}, which is parallel to the corresponding close-packed direction of the parent austenite [$\stackrel{-}{1}$ $0$ $1$]

_{$\gamma $}[50], analogous to lath martensite. The surface $l\times t$ is in the plane (1 0 1)

_{$\gamma $}, as it is in martensite. The habit plane ($l\times w$) of bainitic lath has been measured to be close to (2 3 2)

_{$\gamma $}≈ ($\stackrel{-}{1}$ $5$ $4$)

_{$\alpha $}[51], which deviates from the one in lath martensite (1 1 1)

_{$\gamma $}.

#### 4.3. Mechanical Behavior of Materials S_{1} and S_{2}

#### 4.4. Microstructural Study of Ferritic Bainite

- Type-I voids at the PF–GB boundary.
- Type-II voids at the PF–PF boundary or in the PF grain.
- Type-III voids at the GB–GB boundary.

## 5. Results and Discussion

#### 5.1. Fitting of CP Parameters

#### 5.2. Representative Volume Element—RVE

- Using a 2D- or a 3D-RVE. The 2D vs. 3D transition shows the biggest differences among all RVEs, but still the deviation is not significant. A small difference in the macroscopic results achieved from a 2D-RVE vs. a 3D-RVE has already been reported by Gallardo-Basile et al. [31]. There, a comparison was made between a 2D-RVE based on a direct measurement and a 3D-RVE with lath-like microstructure (similar to “2D-RVE measured” and “3D-RVE gb-substructure” in this manuscript). Additionally, a 2D-RVE with lath-like microstructure is created by cutting a slice of the 3D-RVE. In all the cases, the stress–strain curves showed small deviations from each other.
- Using the experimental grain size distribution or the one created from Voronoi tesellation of random seeds. Similar results were obtained from the “3D-RVE”, where the experimental grain size distribution was used and the “3D-RVE no-substructure”, where Voronoi tesellation is used. This is expected since the constitutive law used is not size dependent.
- Representing the experimental texture (low texture index) with 80 or 10 k grains. Similar results were obtained from “3D-RVE” and “3D-RVE no-substructure”.
- Including the substructure modeling of bainitic ferrite or not. Similar results were obtained from “3D-RVE no-substructure” and “3D-RVE gb-substructure”.

#### 5.3. Mechanical Behavior S_{1} and S_{2}

_{2}is higher than that of S

_{1}, i.e., S

_{2}is closer to the iso-strain behavior while S

_{1}is closer to the iso-stress behavior. The faster decay of β for S

_{2}leads to the same value of β ≈ 100 GPa at a strain of approximately 0.004. Beyond an applied strain of approximately 0.005 up to the final deformation, S

_{2}has a slightly lower β.

#### 5.4. Microstructural Study of Ferritic Bainite

## 6. Conclusions

- The CP parameters of PF and GB are determined for both materials. For GB, they are determined following an inverse modeling procedure based on measuring a post mortem nanoindentation imprint by atomic force microscopy and comparing it to a simulated one. For PF, they are determined by fitting the macroscopic stress–strain curves of the tensile tests.
- The heterogeneity of the microscopic results among the different RVEs is discussed. The von Mises stress–strain curves show no significant differences among the RVEs. This suggested that for the utilized CP model, the main inputs required to predict the macroscopic behavior are the phase fraction and the CP parameters of each phase. In contrast to the macroscopic response, the microscopic responses (stress distributions are shown) are clearly different for the different RVEs.
- The average stress and the plastic strain are analyzed for each phase of both materials. It is shown that the onset of the plastic deformation of PF is almost the same for both materials, but the GB plasticity is delayed for S${}_{2}$, which contains a higher PF fraction.
- The contributions per slip plane family to the total cumulative plastic shear are calculated. The main difference in both materials is exhibited at the early stage of the deformation. For material S${}_{2}$, the partioning of strain is made between {1 2 3} and {1 1 2}, while for material S${}_{1}$, the {1 1 0} contributes 0.18% from the start.
- The strain partioning is calculated to be similar for both materials, except at the early stage of deformation, where material S
_{2}is closer to iso-strain behavior while S_{1}is closer to iso-stress behavior. Both materials followed the iso-work assumption with only small deviations compared to the iso-strain and iso-stress which exhibited significant deviations. - Ductile damage initiation is indicated by the fraction of points where the ductile damage parameter, ${\u03f5}_{p}$, surpasses a certain threshold value at the final deformation stage. The contribution of both phases is shown to be significant. For GB, no direct correlation can be seen between damage initiation and sub-block size. For PF, the damage decreases with the sub-block size. It is shown that ductile damage initiation can be linked to a coarse bainitic substructure because of the accumulation of plastic strain at PF–PF boundaries (type-II voids).
- Material S${}_{2}$ has a higher PF phase fraction but a lower ultimate elongation compared to S${}_{1}$. It is hypothesized that this can be explained by the preference of S${}_{2}$ to form type-I voids (PF–GB), which may interfere with the full utilization of the plastic capacity of PF.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Ishikawa, N.; Shinmiya, T.; Shikanai, N.; Muraoka, R.; Kakihara, S. Recent advance in high strength linepipes for heavy sour service. In Proceedings of the Pipeline Technology Conference, Hannover, Germany, 22–23 April 2009; EITEP: Hannover, Germany, 2009; pp. 12–14. [Google Scholar]
- Kobayashi, K.; Omura, T.; Takahashi, N.; Minato, I.; Yamamoto, A. Advanced technologies for manufacturing high strength sour grade UOE line pipe. In Proceedings of the 2010 8th International Pipeline Conference, Calgary, AB, Canada, 27 September–1 October 2010; Volume 44212, pp. 289–297. [Google Scholar]
- Rosado, D.B.; Waele, W.D.; Vanderschueren, D.; Hertelé, S. Latest developments in mechanical properties and metallurgical features of high strength line pipe steels. Int. J. Sustain. Constr. Des.
**2013**, 4. [Google Scholar] [CrossRef] [Green Version] - Javaheri, V.; Pohjonen, A.; Asperheim, J.I.; Ivanov, D.; Porter, D. Physically based modeling, characterization and design of an induction hardening process for a new slurry pipeline steel. Mater. Des.
**2019**, 182, 108047. [Google Scholar] [CrossRef] - Bhadeshia, H.K.D.H. Bainite in Steels; Institute of Materials, IOM Communications Ltd.: London, UK, 2001. [Google Scholar]
- Entezari, E.; González-Velxaxzquez, J.L.; López, D.R.; Zúñiga, M.A.B.; Szpunar, J.A. Review of Current Developments on High Strength Pipeline Steels for HIC Inducing Service. Frat. Integrità Strutt.
**2022**, 16, 20–45. [Google Scholar] [CrossRef] - Xu, X.N.; Tian, Y.; Ye, Q.B.; Misra, R.D.K.; Wang, Z.D. The Significant Impact of the Characteristics of Granular Structure and Granular Bainite on the Mechanisms Contributing to Strength–Ductility Combination. J. Mater. Eng. Perform.
**2022**, 30, 7479–7487. [Google Scholar] [CrossRef] - Fang, H.-s.; Feng, C.; Zheng, Y.-k.; Yang, Z.-g.; Bai, B.-z. Creation of Air-Cooled Mn Series Bainitie Steels. J. Iron Steel Res. Int.
**2008**, 15, 1–9. [Google Scholar] [CrossRef] - Akbarpour, M.R.; Ekrami, A. Effect of ferrite volume fraction on work hardening behavior of high bainite dual phase (DP) steels. Mater. Sci. Eng. A
**2008**, 477, 306–310. [Google Scholar] [CrossRef] - Ishikawa, N.; Yasuda, K.; Sueyoshi, H.; Endo, S.; Ikeda, H.; Morikawa, T.; Higashida, K. Microscopic deformation and strain hardening analysis of ferrite–bainite dual-phase steels using micro-grid method. Acta Mater.
**2015**, 97, 257–268. [Google Scholar] [CrossRef] - Tang, C.-j.; Liu, S.-l.; Shang, C.-j. Micromechanical behavior and failure mechanism of F/B multi-phase high performance steel. J. Iron Steel Res. Int.
**2016**, 23, 489–494. [Google Scholar] [CrossRef] - Tu, X.; Ren, Y.; Shi, X.; Yan, W.; Shi, Q.; Shan, Y.; Li, C. Effect of distribution characters of ferrite/bainite on deformation compatibility in dual-phase pipeline steel: Experimental and numerical study. Mater. Today Commun.
**2022**, 33, 104923. [Google Scholar] [CrossRef] - Tu, X.; Shi, X.; Shan, Y.; Yan, W.; Shi, Q.; Li, Y.; Li, C.; Yang, K. Tensile deformation damage behavior of a high deformability pipeline steel with a ferrite and bainite microstructure. Mater. Sci. Eng. A
**2020**, 793, 139889. [Google Scholar] [CrossRef] - Tu, X.; Shi, X.; Yan, W.; Li, C.; Shi, Q.; Shan, Y.; Yang, K. Tensile deformation behavior of ferrite-bainite dual-phase pipeline steel. Mater. Sci. Eng. A
**2022**, 831, 142230. [Google Scholar] [CrossRef] - Roters, F.; Eisenlohr, P.; Hantcherli, L.; Tjahjanto, D.D.; Bieler, T.R.; Raabe, D. Overview of constitutive laws, kinematics, homogenization, and multiscale methods in crystal plasticity finite element modeling: Theory, experiments, applications. Acta Mater.
**2010**, 58, 1152–1211. [Google Scholar] [CrossRef] - Roters, F. Advanced Material Models for the Crystal Plasticity Finite Element Method—Development of a General CPFEM Framework. Ph.D. Dissertation, RWTH Aachen, Aachen, Germany, 2011. [Google Scholar]
- Raabe, D.; Sachtleber, M.; Zhao, Z.; Roters, F.; Zaefferer, S. Micromechanical and macromechanical effects in grain scale polycrystal plasticity experimentation and simulation. Acta Mater.
**2001**, 49, 3433–3441. [Google Scholar] [CrossRef] - Zaafarani, N.; Raabe, D.; Singh, R.N.; Roters, F.; Zaefferer, S. Three-dimensional investigation of the texture and microstructure below a nanoindent in a Cu single crystal using 3D EBSD and crystal plasticity finite element simulations. Acta Mater.
**2006**, 54, 1863–1876. [Google Scholar] [CrossRef] - Raabe, D.; Roters, F. Using texture components in crystal plasticity finite element simulations. Int. J. Plast.
**2004**, 20, 339–361. [Google Scholar] [CrossRef] - Tasan, C.C.; Hoefnagels, J.P.M.; Diehl, M.; Yan, D.; Roters, F.; Raabe, D. Strain localization and damage in dual phase steels investigated by coupled in situ deformation experiments-crystal plasticity simulations. Int. J. Plast.
**2014**, 63, 198–210. [Google Scholar] [CrossRef] [Green Version] - Tasan, C.C.; Diehl, M.; Yan, D.; Zambaldi, C.; Shanthraj, P.; Roters, F.; Raabe, D. Integrated experimental-numerical analysis of stress and strain partitioning in multi-phase alloys. Acta Mater.
**2014**, 81, 386–400. [Google Scholar] [CrossRef] - Roters, F.; Diehl, M.; Shanthraj, P.; Eisenlohr, P.; Reuber, C.; Wong, S.L.; Maiti, T.; Ebrahimi, A.; Hochrainer, T.; Fabritius, H.-O.; et al. DAMASK—The Düsseldorf Advanced Material Simulation Kit for Modelling Multi-Physics Crystal Plasticity, Damage, and Thermal Phenomena from the Single Crystal up to the Component Scale. Comput. Mater. Sci.
**2019**, 158, 420–478. [Google Scholar] [CrossRef] - Hutchinson, J.W. Bounds and self-consistent estimates for creep of polycrystalline materials. Proc. R. Soc. Lond. Ser. A
**1976**, 348, 101–127. [Google Scholar] [CrossRef] - Jentner, R.M.; Srivastava, K.; Scholl, S.; Best, J.P.; Kirchlechner, C.; Dehm, G. Local strength of bainitic and ferritic HSLA steel constituents understood using correlative electron microscopy and microcompression testing. SSRN Electron. J.
**2023**. Available online: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4408322 (accessed on 5 April 2023). - Jentner, R.M.; Tsai, S.-P.; Welle, A.; Srivastava, K.; Scholl, S.; Best, J.P.; Kirchlechner, C.; Dehm, G. Automated Classification of Granular Bainite and Polygonal Ferrite by Electron Backscatter Diffraction Verified through Local Structural and Mechanical Analyses. SSRN Electron. J.
**2023**. [Google Scholar] [CrossRef] - Jentner, R.M.; Srivastava, K.; Scholl, S.; Gallardo-Basile, F.-J.; Best, J.P.; Kirchlechner, C.; Dehm, G. Unsupervised clustering of nanoindentation data for microstructural reconstruction: Challenges in phase discrimination. Materialia
**2023**, 28, 101750. [Google Scholar] [CrossRef] - Bachmann, F.; Hielscher, R.; Schaeben, H. Texture Analysis with MTEX—Free and Open Source Software Toolbox. Solid State Phenom.
**2010**, 160, 63–68. [Google Scholar] [CrossRef] [Green Version] - Gallardo-Basile, F.-J.; Roters, F.; Jentner, R.M.; Best, J.P.; Kirchlechner, C.; Srivastava, K.; Scholl, S.; Diehl, M. Application of a nanoindentation-based approach for parameter identification to a crystal plasticity model for bcc metals. MSEA, 2022; submitted. [Google Scholar]
- Groeber, M.A.; Jackson, M.A. DREAM.3D: A Digital Representation Environment for the Analysis of Microstructure in 3D. Integr. Mater. Manuf. Innov.
**2014**, 3, 56–72. [Google Scholar] [CrossRef] [Green Version] - Ostoja-Starzewski, M. Material spatial randomness: From statistical to representative volume element. Probabilistic Eng. Mech.
**2006**, 21, 112–132. [Google Scholar] [CrossRef] - Gallardo-Basile, F.-J.; Naunheim, Y.; Roters, F.; Diehl, M. Lath Martensite Microstructure Modeling: A High-Resolution Crystal Plasticity Simulation Study. Materials
**2021**, 14, 691. [Google Scholar] [CrossRef] - EDAX. TSL OIM Analysis 7, Version 7.3.1; EDAX: Pleasanton, CA, USA, 2017.
- Bhadeshia, H.; Honeycombe, R. Formation of Martensite. In Steels: Microstructure and Properties; Butterworth-Heinemann: Oxford, UK; Elsevier: London, UK, 2017; pp. 135–177. [Google Scholar] [CrossRef]
- Bhadeshia, H.K.D.H.; Christian, J.W. Bainite in steels. Metall. Trans. A
**1990**, 21, 767–797. [Google Scholar] [CrossRef] - Krauss, G. Martensite in steel: Strength and structure. Mater. Sci. Eng. A
**1999**, 273–275, 40–57. [Google Scholar] [CrossRef] - Wayman, C.; Bhadeshia, H. Phase transformations, nondiffusive. In Physical Metallurgy; Butterworth-Heinemann: Oxford, UK; Elsevier: London, UK, 1996; pp. 1507–1554. [Google Scholar] [CrossRef]
- Murata, Y. Formation Mechanism of Lath Martensite in Steels. Mater. Trans.
**2018**, 59, 151–164. [Google Scholar] [CrossRef] [Green Version] - Morito, S.; Tanaka, H.; Konishi, R.; Furuhara, T.; Maki, T. The morphology and crystallography of lath martensite in Fe-C alloys. Acta Mater.
**2003**, 51, 1789–1799. [Google Scholar] [CrossRef] - Morito, S.; Huang, X.; Furuhara, T.; Maki, T.; Hansen, N. The morphology and crystallography of lath martensite in alloy steels. Acta Mater.
**2006**, 54, 5323–5331. [Google Scholar] [CrossRef] - Maki, T.; Tsuzaki, K.; Tamura, I. The Morphology of Microstructure Composed of Lath Martensites in Steels. Trans. ISIJ
**1980**, 20, 207–214. [Google Scholar] [CrossRef] [Green Version] - Nishiyama, Z. X-ray investigation of the mechanism of the transformation from face centered cubic lattice to body centered cubic. Sci. Rep. Tohoku Univ.
**1934**, 23, 637–664. [Google Scholar] - Wassermann, G. Ueber den Mechanismus der α→γ-Umwandlung des Eisens. Mitt. K.-Wilh.-Inst. Eisenforsch.
**1935**, 17, 149–155. [Google Scholar] - Kurdjumow, G.; Sachs, G. Über den mechanismus der stahlhärtung. Z. Phys.
**1930**, 64, 325–343. [Google Scholar] [CrossRef] - Greninger, A.B.; Troiano, A.R. The mechanism of Martensite formation. JOM
**1949**, 1, 590–598. [Google Scholar] [CrossRef] - Josefsson, B. Microscopy and Microanalysis of Bainitic Weld Metal. Ph.D. Dissertation, Chalmers University, Gothenburg, Sweeden, 1989. [Google Scholar]
- De-Castro, D.; Eres-Castellanos, A.; Vivas, J.; Caballero, F.G.; San-Martín, D.; Capdevila, C. Morphological and crystallographic features of granular and lath-like bainite in a low carbon microalloyed steel. Mater. Charact.
**2022**, 184, 111703. [Google Scholar] [CrossRef] - Aaronson, H.I.; Wells, C. Sympathetic Nucleation of Ferrite. JOM
**1956**, 8, 1216–1223. [Google Scholar] [CrossRef] - Chilton, J.; Barton, C.; Speich, G. Martensite transformation in low-carbon steels. J. Iron Steel Inst.
**1970**, 208, 184–193. [Google Scholar] - Zhang, M.-X.; Kelly, P.M. Crystallography of carbide-free bainite in a hard bainitic steel. Mater. Sci. Eng. A
**2006**, 438–440, 272–275. [Google Scholar] [CrossRef] - Davenport, E.S.; Bain, E.C. Transformation of austenite at constant subcritical temperatures. Metall. Mater. Trans. B
**1970**, 1, 3503–3530. [Google Scholar] [CrossRef] - Davenport, A. The crystallography of upper bainite. Repub. Steel Res. Rep. Proj.
**1974**, 12051, 1–35. [Google Scholar] - Landheer, H.; Offerman, S.E.; Petrov, R.H.; Kestens, L.A.I. The role of crystal misorientations during solid-state nucleation of ferrite in austenite. Acta Mater.
**2009**, 57, 1486–1496. [Google Scholar] [CrossRef] - Abbasi, M.; Kim, D.-I.; Nelson, T.W.; Abbasi, M. EBSD and reconstruction of pre-transformation microstructures, examples and complexities in steels. Mater. Charact.
**2014**, 95, 219–231. [Google Scholar] [CrossRef] - Tamura, M.O.I.; Tomota, Y. In Proceedings of the 3rd International Conference on the Strength of Metals and Alloys, Cambridge, UK, 20–25 August 1973.
- Voigt, W. Über die Beziehung zwischen den beiden Elastizitätskonstanten isotroper Körper. Ann. Phys.
**1889**, 38, 573–587. [Google Scholar] [CrossRef] [Green Version] - Reuss, A. Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. Z. Angew. Math. Und Mech.
**1929**, 9, 49–58. [Google Scholar] [CrossRef] - Bouaziz, O.; Buessler, P. Mechanical behaviour of multiphase materials: An intermediate mixture law without fitting parameter. Rev. Métall.
**2002**, 99, 71–77. [Google Scholar] [CrossRef] - Furuhara, T.; Kawata, H.; Morito, S.; Maki, T. Crystallography of upper bainite in Fe–Ni–C alloys. Mater. Sci. Eng. A
**2006**, 431, 228–236. [Google Scholar] [CrossRef] - Lou, Y.; Huh, H. Prediction of ductile fracture for advanced high strength steel with a new criterion: Experiments and simulation. J. Mater. Process. Technol.
**2013**, 213, 1284–1302. [Google Scholar] [CrossRef]

**Figure 1.**Experimental macroscopic tensile stress–strain curves in the rolling direction (RD) for the materials obtained from applying different cooling rates to a high-strength, low-alloy steel.

**Figure 2.**An EBSD scan and a phase map are shown in (

**a**,

**b**) for material S${}_{1}$ and in (

**c**,

**d**) for material S${}_{2}$. EBSD scans are colored by their IPF. Phase maps, showing polygonal ferrite and granular bainite phases, were calculated from the EBSD scans with the classification tool described in work by Jentner et al. [25]. The grain boundaries are colored in black.

**Figure 3.**Pole figures for each phase in materials ${\mathrm{S}}_{1}$ (

**a**,

**b**) and ${\mathrm{S}}_{2}$ (

**c**,

**d**). Contour lines are plotted in blue for polygonal ferrite (

**a**,

**c**) and in orange for granular bainite (

**b**,

**d**). The white-to-black scale bar indicates the “multiples of random density”.

**Figure 4.**3D-RVE (

**a**), 3D-RVE no-substructure (

**b**), 3D-RVE gb-substructure (

**c**), and 2D-RVE measured (

**d**) for material ${\mathrm{S}}_{2}$. IPFs along the RD (horizontal direction in the 2D-RVE) are displayed in the left. Phase maps are shown in the right, where blue corresponds to polygonal ferrite and orange corresponds to granular bainite.

**Figure 5.**Bainitic packets of type-A (

**a**), type-B (

**b**), and type-C (

**c**) in Fe-9Ni-C alloys. Taken from Furuhara et al. [58]. In (

**c**), the originally labeled “block” is renamed to “sub-block”.

**Figure 6.**IPF representations of type-B sb $\to \infty $ (

**a**), type-B sb = 5 (

**b**), type-C sb = 5 (

**c**), and type-C sb = 15 (

**d**) RVEs of material S${}_{1}$. The RVEs are clipped with a plane on the left to show the same single grain extracted which is magnified on the right.

**Figure 7.**True (left) and engineering (right) stress–strain curves for materials ${\mathrm{S}}_{1}$ (

**a**) and ${\mathrm{S}}_{2}$ (

**b**). The simulation results were obtained using a 3D-RVE with the fitted CP parameters of each individual phase.

**Figure 8.**Macroscopic stress–strain curves for the different RVEs used in this work. Polygonal ferrite is shown in blue and granular bainite is shown in in orange.

**Figure 9.**Stress distributions at the final applied strain for 2D-RVE measured (

**a**), 3D-RVE (

**b**), 3D-RVE no-substructure(

**c**), and 3D-RVE gb-substructure (

**d**) RVEs. Polygonal ferrite is colored in blue tones and granular bainite in orange tones.

**Figure 10.**Average stress per phase (

**a**) and plastic strain per phase (

**b**) in materials ${\mathrm{S}}_{1}$ and S${}_{2}$.

**Figure 11.**Contributions per slip plane family to the total cumulative plastic shear in materials ${\mathrm{S}}_{1}$ (

**a**) and ${\mathrm{S}}_{2}$ (

**b**). Polygonal ferrite is shown in blue and granular bainite is shown in orange.

**Figure 12.**Contributions per slip plane family to the total cumulative plastic shear at the initial (

**a**) and final (

**b**) deformation. Polygonal ferrite is shown in blue and granular bainite is shown in orange.

**Figure 14.**Stress–strain curves for the simulation output (left column), for the iso-work assumption (middle), and for the iso-stress and iso-strain assumptions (right column) in materials ${\mathrm{S}}_{1}$ (

**a**) and ${\mathrm{S}}_{2}$ (

**b**).

**Figure 15.**The ductile damage parameter based on the equivalent plastic for RVEs with microstructural variations in materials ${\mathrm{S}}_{1}$ (

**a**) and ${\mathrm{S}}_{2}$ (

**b**).

**Figure 16.**IPF representation (left) and plastic strain map (right) for type-B sb $\to \infty $ (

**a**) and Type-B sb = 5 (

**b**) RVEs of material S${}_{2}$. Two PF–PF boundaries are indicated with light green arrows. Two PF–GB boundaries are indicated with pink arrows.

**Table 1.**Summary of experimental results for materials ${\mathrm{S}}_{1}$ and ${\mathrm{S}}_{2}$ from [24].

Property | ${\mathit{\sigma}}_{\mathbf{y}}$ (MPa) | UTS (MPa) | UE (%) | Grain Size ($\mathsf{\mu}$m) | ${\mathbf{T}}_{\mathbf{ODF}}$ (mrd) | CRSS (MPa) | ${\mathit{\rho}}_{\mathbf{d}}$ (10${}^{13}$/m${}^{2}$) | Cooling Rate | |
---|---|---|---|---|---|---|---|---|---|

Material—Phase | |||||||||

${\mathrm{S}}_{1}$—Polygonal ferrite (25%) | 495 | 593 | 26 | 4.7 ± 0.6 | 1.34 | 153 ± 4 | 0.9 ± 0.1 | low | |

${\mathrm{S}}_{1}$—Granular bainite (75%) | 1.66 | 183 ± 7 | 2.1 ± 0.9 | ||||||

${\mathrm{S}}_{2}$—Polygonal ferrite (41%) | 518 | 605 | 23 | 4.2 ± 0.6 | 1.43 | 194 ± 6 | 2.8 ± 0.3 | high | |

${\mathrm{S}}_{2}$—Granular bainite (59%) | 1.91 | 221 ± 7 | 7.1 ± 0.7 |

**Table 2.**Micropillar compression test results grouped by slip plane families for materials ${\mathrm{S}}_{1}$ and S${}_{2}$.

Material Phase | ${\mathbf{S}}_{1}$—Polygonal Ferrite | ${\mathbf{S}}_{1}$—Granular Bainite | ${\mathbf{S}}_{2}$—Polygonal Ferrite | ${\mathbf{S}}_{2}$—Granular Bainite | |
---|---|---|---|---|---|

Property | |||||

CRSS for {1 1 0} / MPa | 151 ± 3 | 180 ± 10 | 188 ± 6 | - | |

CRSS for {1 1 2} / MPa | 163 ± 9 | - | 197 ± 10 | 222 ± 11 | |

CRSS for {1 2 3} / MPa | 148 ± 6 | 187 ± 9 | 195 ± 10 | 221 ± 9 |

Property | ${\mathit{C}}_{11}$ (GPa) | ${\mathit{C}}_{12}$ (GPa) | ${\mathit{C}}_{44}$ (GPa) | ${\mathit{\xi}}_{\left\{110\right\}}^{0}={\mathit{\xi}}_{\left\{112\right\}}^{0}={\mathit{\xi}}_{\left\{123\right\}}^{0}$ (MPa) | ${\mathit{\xi}}_{\left\{110\right\}}^{\mathit{\infty}}={\mathit{\xi}}_{\left\{112\right\}}^{\mathit{\infty}}={\mathit{\xi}}_{\left\{123\right\}}^{\mathit{\infty}}$ (MPa) | ${\mathit{h}}_{0}$ (GPa) | ${\dot{\mathit{\gamma}}}_{0}$ (−) | n (−) | a (−) | |
---|---|---|---|---|---|---|---|---|---|---|

Material—Phase | ||||||||||

${\mathrm{S}}_{1}$—Polygonal ferrite (25%) | 233.3 | 135.5 | 118.0 | 101.9 | 341.7 | 446.3 | 0.04 | 20 | 6.05 | |

${\mathrm{S}}_{1}$—Granular bainite (75%) | 143.8 | 662.7 | 506.9 | 10.3 | ||||||

${\mathrm{S}}_{2}$—Polygonal ferrite (41%) | 119.2 | 408.1 | 237.2 | 6.5 | ||||||

${\mathrm{S}}_{2}$—Granular bainite (59%) | 209.8 | 497.7 | 417.1 | 5.7 |

Property | Experimental ${\mathit{\sigma}}_{y}$ (MPa) | Simulation ${\mathit{\sigma}}_{y}$ (MPa) | Experimental ${\mathit{\sigma}}_{\mathrm{eng}}({\mathit{\u03f5}}_{\mathbf{eng}}=0.12)$ (MPa) | Simulation ${\mathit{\sigma}}_{\mathrm{eng}}({\mathit{\u03f5}}_{\mathbf{eng}}=0.12)$ (MPa) | |||
---|---|---|---|---|---|---|---|

Material—Phase | |||||||

S${}_{1}$ | Polygonal ferrite (25%) | 495 | 413 | 373 | 593 | 598 | 514 |

Granular bainite (75%) | 428 | 629 | |||||

S${}_{2}$ | Polygonal ferrite (41%) | 518 | 455 | 409 | 604 | 606 | 549 |

Granular bainite (59%) | 487 | 649 |

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## Share and Cite

**MDPI and ACS Style**

Gallardo-Basile, F.-J.; Roters, F.; Jentner, R.M.; Srivastava, K.; Scholl, S.; Diehl, M.
Modeling Bainite Dual-Phase Steels: A High-Resolution Crystal Plasticity Simulation Study. *Crystals* **2023**, *13*, 673.
https://doi.org/10.3390/cryst13040673

**AMA Style**

Gallardo-Basile F-J, Roters F, Jentner RM, Srivastava K, Scholl S, Diehl M.
Modeling Bainite Dual-Phase Steels: A High-Resolution Crystal Plasticity Simulation Study. *Crystals*. 2023; 13(4):673.
https://doi.org/10.3390/cryst13040673

**Chicago/Turabian Style**

Gallardo-Basile, Francisco-José, Franz Roters, Robin M. Jentner, Kinshuk Srivastava, Sebastian Scholl, and Martin Diehl.
2023. "Modeling Bainite Dual-Phase Steels: A High-Resolution Crystal Plasticity Simulation Study" *Crystals* 13, no. 4: 673.
https://doi.org/10.3390/cryst13040673