# Description of Dynamic Recrystallization by Means of An Advanced Statistical Multilevel Model: Grain Structure Evolution Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Advanced Statistical Multilevel Model for Describing Inelastic Deformation during Discontinuous Dynamic Recrystallization

## 3. Rearrangement of the Grain Structure Formed during High-Temperature Deformation with Recrystallization

## 4. Simulation Results and Their Analysis

^{–3}s

^{–1}for a polycrystal of commercial pure copper, which was pre-treated by annealing at 973 K for two hours [78]. The values of the anisotropic elastic moduli of the grain ${\u043f}_{ijkl}$ [76] and the shear modulus of the polycrystal $G$ [77] correspond to the absolute temperature 775 K adopted for the numerical experiment. In order to perform specific calculations by applying an advanced statistical model, the grain structure of a polycrystal should be given in the reference configuration (the procedure for grain structure formation is described in [79]). Based on the available experimental data, statistic laws were derived for the distribution of normalized grain size ${d}_{eq}$ and sphericity ${\mathsf{\psi}}_{g}$ of the copper polycrystal grains. The average polycrystalline copper grain size ${d}_{0}$ was 78 µm [78]. To approximate the grain size distribution, the lognormal law was used [80,81]. According to the data collected from the microsectional analysis [78], there is a scattering in grain sizes, which is characteristic of annealed materials. The following parameters are needed to specify a log-normal distribution: μ = 3.86, σ = 1 such that the mathematical expectation exp(μ + σ

^{2}/2) be equal to the average grain size d

_{0}= 78 μm. If the representative experimental data on grain size distribution exist, then there is no difficulty in solving the optimization problem of determining the statistical distribution law and identifying its parameters, as was done in [82] for subgrain size distributions. We suppose that in the initial state, the copper sample after annealing has predominantly equiaxed grains with an average sphericity $\langle {\mathsf{\psi}}_{g}\rangle $ = 0.90; the data on sphericity are given in [83]. Figure 6 presents the grain size distribution d

_{eq}and grain sphericity ${\mathsf{\psi}}_{g}$ histograms obtained based on existing experimental data [78,83].

^{–3}s

^{–1}and temperature $\mathsf{\theta}$ = 775 K. The loading diagram (with and without the recrystallization process) is given in Figure 7a, and the corresponding deformation textures formed after the end of plastic deformation are shown in Figure 7b. We emphasize a significant contribution of recrystallization to the deformation texture “blurring”, which is attributed to the appearance of new DRX grains with an orientation consistent with the parent grain—numerous points for DRX grains are seen on the pole figure near the point reflecting the orientation of the parent grain.

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Hsu, E.; Carsley, J.E.; Verma, R. Development of Forming Limit Diagrams of Aluminum and Magnesium Sheet Alloys at Elevated Temperatures. J. Mater. Eng. Perform.
**2008**, 17, 288–296. [Google Scholar] [CrossRef] - Zhang, R.; Shao, Z.; Lin, J. A Review on Modelling Techniques for Formability Prediction of Sheet Metal Forming. Int. J. Lightweight Mater. Manuf.
**2018**, 1, 115–125. [Google Scholar] [CrossRef] - Zheng, G.; Pang, T.; Sun, G.; Wu, S.; Li, Q. Theoretical, Numerical, and Experimental Study on Laterally Variable Thickness (LVT) Multi-Cell Tubes for Crashworthiness. Int. J. Mech. Sci.
**2016**, 118, 283–297. [Google Scholar] [CrossRef] - Sun, G.; Zhang, H.; Wang, R.; Lv, X.; Li, Q. Multiobjective Reliability-Based Optimization for Crashworthy Structures Coupled with Metal Forming Process. Struct. Multidisc. Optim.
**2017**, 56, 1571–1587. [Google Scholar] [CrossRef] - Humphreys, F.J.; Hatherly, M. Recrystallization and Related Annealing Phenomena; Elsevier: Amsterdam, The Netherlands, 2012; ISBN 978-0-08-098388-2. [Google Scholar]
- Trusov, P.V.; Shveykin, A.I. Multilevel Models of Mono- and Polycrystalline Materials: Theory, Algorithms, Application Examples; SB RAS: Novosibirsk, Russia, 2019. (In Russian) [Google Scholar]
- Montheillet, F. Moving Grain Boundaries During Hot Deformation of Metals: Dynamic Recrystallization. In Moving Interfaces in Crystalline Solids; Fischer, F.D., Ed.; CISM International Centre for Mechanical Sciences; Springer: Vienna, Austria, 2005; pp. 203–256. ISBN 978-3-211-27404-0. [Google Scholar]
- Zhou, G.; Li, Z.; Li, D.; Peng, Y.; Zurob, H.S.; Wu, P. A Polycrystal Plasticity Based Discontinuous Dynamic Recrystallization Simulation Method and Its Application to Copper. Int. J. Plast.
**2017**, 91, 48–76. [Google Scholar] [CrossRef] - Chen, S.F.; Li, D.Y.; Zhang, S.H.; Han, H.N.; Lee, H.W.; Lee, M.G. Modelling Continuous Dynamic Recrystallization of Aluminum Alloys Based on the Polycrystal Plasticity Approach. Int. J. Plast.
**2020**, 131, 102710. [Google Scholar] [CrossRef] - Liu, F.; Fa, T.; Chen, P.H.; Wang, J.T. Steady-State Characteristics of Fcc Pure Metals Processed by Severe Plastic Deformation: Experiments and Modelling. Philos. Mag.
**2020**, 100, 62–83. [Google Scholar] [CrossRef] - Hansen, N. Cold Deformation Microstructures. Mater. Sci. Technol.
**1990**, 6, 1039–1047. [Google Scholar] [CrossRef] - Asaro, R.J. Micromechanics of Crystals and Polycrystals. Adv. Appl. Mech.
**1983**, 23, 1–115. [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] - McDowell, D.L. A Perspective on Trends in Multiscale Plasticity. Int. J. Plast.
**2010**, 9, 1280–1309. [Google Scholar] [CrossRef] - Acar, P.; Ramazani, A.; Sundararaghavan, V. Crystal Plasticity Modeling and Experimental Validation with an Orientation Distribution Function for Ti-7Al Alloy. Metals
**2017**, 7, 459. [Google Scholar] [CrossRef] [Green Version] - Galán-López, J.; Hidalgo, J. Use of the Correlation between Grain Size and Crystallographic Orientation in Crystal Plasticity Simulations: Application to AISI 420 Stainless Steel. Crystals
**2020**, 10, 819. [Google Scholar] [CrossRef] - Taylor, G.I. Plastic Strain in Metals. J. Inst. Met.
**1938**, 62, 307–324. [Google Scholar] - Kocks, F.; Mecking, H. Physics and Phenomenology of Strain Hardening: The FCC Case. Prog. Mater. Sci.
**2003**, 48, 171–273. [Google Scholar] [CrossRef] - Zhang, K.; Holmedal, B.; Hopperstad, O.S.; Dumoulin, S.; Gawad, J.; Van Bael, A.; Van Houtte, P. Multi-Level Modelling of Mechanical Anisotropy of Commercial Pure Aluminium Plate: Crystal Plasticity Models, Advanced Yield Functions and Parameter Identification. Int. J. Plast.
**2015**, 66, 3–30. [Google Scholar] [CrossRef] - Lebensohn, R.A.; Liu, Y.; Ponte Castañeda, P. On the Accuracy of the Self-Consistent Approximation for Polycrystals: Comparison with Full-Field Numerical Simulations. Acta Mater.
**2004**, 52, 5347–5361. [Google Scholar] [CrossRef] - Lebensohn, R.A.; Tomé, C.N.; CastaÑeda, P.P. Self-Consistent Modelling of the Mechanical Behaviour of Viscoplastic Polycrystals Incorporating Intragranular Field Fluctuations. Philos. Mag.
**2007**, 87, 4287–4322. [Google Scholar] [CrossRef] - Beyerlein, I.J.; Knezevic, M. Review of Microstructure and Micromechanism-Based Constitutive Modeling of Polycrystals with a Low-Symmetry Crystal Structure. J. Mater. Res.
**2018**, 33, 3711–3738. [Google Scholar] [CrossRef] - Yaghoobi, M.; Ganesan, S.; Sundar, S.; Lakshmanan, A.; Rudraraju, S.; Allison, J.E.; Sundararaghavan, V. PRISMS-Plasticity: An Open-Source Crystal Plasticity Finite Element Software. Comput. Mater. Sci.
**2019**, 169, 109078. [Google Scholar] [CrossRef] - Feather, W.G.; Lim, H.; Knezevic, M. A Numerical Study into Element Type and Mesh Resolution for Crystal Plasticity Finite Element Modeling of Explicit Grain Structures. Comput. Mech.
**2021**, 67, 33–55. [Google Scholar] [CrossRef] - Trusov, P.V.; Kondratev, N.S.; Yanz, A.Y. A Model for Static Recrystallization through Strain-InducedBoundary Migration. Phys. Mesomech.
**2020**, 23, 97–108. [Google Scholar] [CrossRef] - Shveykin, A.; Trusov, P.; Sharifullina, E. Statistical Crystal Plasticity Model Advanced for Grain Boundary Sliding Description. Crystals
**2020**, 10, 822. [Google Scholar] [CrossRef] - Trusov, P.V.; Shveykin, A.I. Multilevel Crystal Plasticity Models of Single- and Polycrystals. Statistical Models. Phys. Mesomech.
**2013**, 16, 23–33. [Google Scholar] [CrossRef] - Trusov, P.V.; Shveykin, A.I. Multilevel Crystal Plasticity Models of Single- and Polycrystals. Direct Models. Phys. Mesomech.
**2013**, 16, 99–124. [Google Scholar] [CrossRef] - Rollett, A.D. Overview of Modeling and Simulation of Recrystallization. Prog. Mater. Sci.
**1997**, 42, 79–99. [Google Scholar] [CrossRef] - Huang, K.; Logé, R.E. A Review of Dynamic Recrystallization Phenomena in Metallic Materials. Mater. Des.
**2016**, 111, 548–574. [Google Scholar] [CrossRef] - Tan, K.; Li, J.; Guan, Z.; Yang, J.; Shu, J. The Identification of Dynamic Recrystallization and Constitutive Modeling during Hot Deformation of Ti55511 Titanium Alloy. Mater. Des.
**2015**, 84, 204–211. [Google Scholar] [CrossRef] - Irani, M.; Joun, M. Determination of JMAK Dynamic Recrystallization Parameters through FEM Optimization Techniques. Comput. Mater. Sci.
**2018**, 142, 178–184. [Google Scholar] [CrossRef] - Puchi-Cabrera, E.S.; Guérin, J.D.; La Barbera-Sosa, J.G.; Dubar, M.; Dubar, L. Plausible Extension of Anand’s Model to Metals Exhibiting Dynamic Recrystallization and Its Experimental Validation. Int. J. Plast.
**2018**, 108, 70–87. [Google Scholar] [CrossRef] - Vandermeer, R.A.; Rath, B.B. Microstructural Modeling of Recrystallization in Deformed Iron Single Crystals. Metall. Mater. Trans. A
**1989**, 20, 1933–1942. [Google Scholar] [CrossRef] - Momeni, A.; Dehghani, K. Prediction of Dynamic Recrystallization Kinetics and Grain Size for 410 Martensitic Stainless Steel during Hot Deformation. Met. Mater. Int.
**2010**, 16, 843–849. [Google Scholar] [CrossRef] - Lin, F.; Zhang, Y.; Tao, N.; Pantleon, W.; Juul Jensen, D. Effects of Heterogeneity on Recrystallization Kinetics of Nanocrystalline Copper Prepared by Dynamic Plastic Deformation. Acta Mater.
**2014**, 72, 252–261. [Google Scholar] [CrossRef] [Green Version] - Matsumoto, H.; Velay, V. Mesoscale Modeling of Dynamic Recrystallization Behavior, Grain Size Evolution, Dislocation Density, Processing Map Characteristic, and Room Temperature Strength of Ti-6Al-4V Alloy Forged in the (A+β) Region. J. Alloy. Compd.
**2017**, 708, 404–413. [Google Scholar] [CrossRef] [Green Version] - Arun Babu, K.; Prithiv, T.S.; Gupta, A.; Mandal, S. Modeling and Simulation of Dynamic Recrystallization in Super Austenitic Stainless Steel Employing Combined Cellular Automaton, Artificial Neural Network and Finite Element Method. Comput. Mater. Sci.
**2021**, 195, 110482. [Google Scholar] [CrossRef] - Beltran, O.; Huang, K.; Logé, R.E. A Mean Field Model of Dynamic and Post-Dynamic Recrystallization Predicting Kinetics, Grain Size and Flow Stress. Comput. Mater. Sci.
**2015**, 102, 293–303. [Google Scholar] [CrossRef] - Zecevic, M.; Lebensohn, R.A.; McCabe, R.J.; Knezevic, M. Modelling Recrystallization Textures Driven by Intragranular Fluctuations Implemented in the Viscoplastic Self-Consistent Formulation. Acta Mater.
**2019**, 164, 530–546. [Google Scholar] [CrossRef] - Chuan, W.; He, Y.; Wei, L.H. Modeling of Discontinuous Dynamic Recrystallization of a Near-α Titanium Alloy IMI834 during Isothermal Hot Compression by Combining a Cellular Automaton Model with a Crystal Plasticity Finite Element Method. Comput. Mater. Sci.
**2013**, 79, 944–959. [Google Scholar] [CrossRef] - Zhao, P.; Wang, Y.; Niezgoda, S.R. Microstructural and Micromechanical Evolution during Dynamic Recrystallization. Int. J. Plast.
**2018**, 100, 52–68. [Google Scholar] [CrossRef] - Ruiz Sarrazola, D.A.; Pino Muñoz, D.; Bernacki, M. A New Numerical Framework for the Full Field Modeling of Dynamic Recrystallization in a CPFEM Context. Comput. Mater. Sci.
**2020**, 179, 109645. [Google Scholar] [CrossRef] - Yu, P.; Wu, C.; Shi, L. Analysis and Characterization of Dynamic Recrystallization and Grain Structure Evolution in Friction Stir Welding of Aluminum Plates. Acta Mater.
**2021**, 207, 116692. [Google Scholar] [CrossRef] - Raabe, D. Introduction of a Scalable Three-Dimensional Cellular Automaton with a Probabilistic Switching Rule for the Discrete Mesoscale Simulation of Recrystallization Phenomena. Philos. Mag. A
**1999**, 79, 2339–2358. [Google Scholar] [CrossRef] - Zhu, H.; Chen, F.; Zhang, H.; Cui, Z. Review on Modeling and Simulation of Microstructure Evolution during Dynamic Recrystallization Using Cellular Automaton Method. Sci. China Technol. Sci.
**2020**, 63, 357–396. [Google Scholar] [CrossRef] - Chang, K.; Chen, L.-Q.; Krill, C.E.; Moelans, N. Effect of Strong Nonuniformity in Grain Boundary Energy on 3-D Grain Growth Behavior: A Phase-Field Simulation Study. Comput. Mater. Sci.
**2017**, 127, 67–77. [Google Scholar] [CrossRef] [Green Version] - Maire, L.; Scholtes, B.; Moussa, C.; Bozzolo, N.; Muñoz, D.P.; Settefrati, A.; Bernacki, M. Modeling of Dynamic and Post-Dynamic Recrystallization by Coupling a Full Field Approach to Phenomenological Laws. Mater. Des.
**2017**, 133, 498–519. [Google Scholar] [CrossRef] - Mora, L.A.B.; Gottstein, G.; Shvindlerman, L.S. Three-Dimensional Grain Growth: Analytical Approaches and Computer Simulations. Acta Mater.
**2008**, 56, 5915–5926. [Google Scholar] [CrossRef] - Mellbin, Y.; Hallberg, H.; Ristinmaa, M. A Combined Crystal Plasticity and Graph-Based Vertex Model of Dynamic Recrystallization at Large Deformations. Model. Simul. Mater. Sci. Eng.
**2015**, 23, 045011. [Google Scholar] [CrossRef] - Glezer, A.M.; Kozlov, E.V.; Koneva, N.A.; Popova, N.A.; Kurzina, I.A. Plastic Deformation of Nanostructured Materials; CRC Press: Boca Raton, FL, USA, 2017; ISBN 978-1-315-11196-4. [Google Scholar]
- Dyakonov, G.S.; Mironov, S.; Semenova, I.P.; Valiev, R.Z.; Semiatin, S.L. EBSD Analysis of Grain-Refinement Mechanisms Operating during Equal-Channel Angular Pressing of Commercial-Purity Titanium. Acta Mater.
**2019**, 173, 174–183. [Google Scholar] [CrossRef] - Hall, E.O. The Deformation and Ageing of Mild Steel: III Discussion of Results. Proc. Phys. Soc. B
**1951**, 64, 747–753. [Google Scholar] [CrossRef] - Petch, N.J. The Cleavage Strength of Polycrystals. J. Iron Steel Inst.
**1953**, 174, 25–28. [Google Scholar] - Armstrong, R.W. The Influence of Polycrystal Grain Size on Several Mechanical Properties of Materials. Metall. Trans.
**1970**, 1, 1169–1176. [Google Scholar] [CrossRef] - Knezevic, M.; Beyerlein, I.J. Multiscale Modeling of Microstructure-Property Relationships of Polycrystalline Metals during Thermo-Mechanical Deformation. Adv. Eng. Mater.
**2018**, 20, 1700956. [Google Scholar] [CrossRef] - Zhou, X.; Feng, Z.; Zhu, L.; Xu, J.; Miyagi, L.; Dong, H.; Sheng, H.; Wang, Y.; Li, Q.; Ma, Y.; et al. High-Pressure Strengthening in Ultrafine-Grained Metals. Nature
**2020**, 579, 67–72. [Google Scholar] [CrossRef] [PubMed] - Tan, J.C.; Tan, M.J. Superplasticity and Grain Boundary Sliding Characteristics in Two Stage Deformation of Mg–3Al–1Zn Alloy Sheet. Mater. Sci. Eng. A
**2003**, 339, 81–89. [Google Scholar] [CrossRef] - Quey, R.; Renversade, L. Optimal Polyhedral Description of 3D Polycrystals: Method and Application to Statistical and Synchrotron X-Ray Diffraction Data. Comput. Methods Appl. Mech. Eng.
**2018**, 330, 308–333. [Google Scholar] [CrossRef] [Green Version] - Roberts, W.; Boden, H.; Ahlblom, B. Dynamic Recrystallization Kinetics. Metal. Sci.
**1979**, 13, 195–205. [Google Scholar] [CrossRef] - Ponge, D.; Gottstein, G. Necklace Formation during Dynamic Recrystallization: Mechanisms and Impact on Flow Behavior. Acta Mater.
**1998**, 46, 69–80. [Google Scholar] [CrossRef] - Zurob, H.S.; Bréchet, Y.; Dunlop, J. Quantitative Criterion for Recrystallization Nucleation in Single-Phase Alloys: Prediction of Critical Strains and Incubation Times. Acta Mater.
**2006**, 54, 3983–3990. [Google Scholar] [CrossRef] - Cram, D.G.; Zurob, H.S.; Brechet, Y.J.M.; Hutchinson, C.R. Modelling Discontinuous Dynamic Recrystallization Using a Physically Based Model for Nucleation. Acta Mater.
**2009**, 57, 5218–5228. [Google Scholar] [CrossRef] - Beck, P.A.; Sperry, P.R. Strain Induced Grain Boundary Migration in High Purity Aluminum. J. Appl. Phys.
**1950**, 21, 150–152. [Google Scholar] [CrossRef] - Hallberg, H.; Wallin, M.; Ristinmaa, M. Simulation of Discontinuous Dynamic Recrystallization in Pure Cu Using a Probabilistic Cellular Automaton. Comput. Mater. Sci.
**2010**, 49, 25–34. [Google Scholar] [CrossRef] [Green Version] - Kondratev, N.S.; Trusov, P.V.; Podsedertsev, A.N. Multilevel model of polycrystals: Application to assessing the effect of texture and grains misorientation on the critical deformation of the dynamic recrystallization initiation. PNRPU Mech. Bull.
**2021**, 4, 83–97. (In Russian) [Google Scholar] [CrossRef] - Kondratev, N.S.; Trusov, P.V.; Podsedertsev, A.N. Multilevel physical-oriented model: Applicationto the description of the initial stageof dynamic recrystallization of polycrystals. Probl. Strength Plast.
**2021**, 83, 451–461. [Google Scholar] [CrossRef] - Lebensohn, R.A.; Brenner, R.; Castelnau, O.; Rollett, A.D. Orientation Image-Based Micromechanical Modelling of Subgrain Texture Evolution in Polycrystalline Copper. Acta Mater.
**2008**, 56, 3914–3926. [Google Scholar] [CrossRef] - Kondratev, N.S.; Trusov, P.V. Description of Hardening Slip Systems Due to the Boundaries of the Crystallines in a Polycrystalline Aggregate. PNRPU Mech. Bull.
**2012**, 3, 78–97. (In Russian) [Google Scholar] - Trusov, P.V.; Shveykin, A.I. On Motion Decomposition and Constitutive Relations in Geometrically Nonlinear Elastoviscoplasticity of Crystallites. Phys. Mesomech.
**2017**, 20, 377–391. [Google Scholar] [CrossRef] - Anand, L. Single-Crystal Elasto-Viscoplasticity: Application to Texture Evolution in Polycrystalline Metals at Large Strains. Comput. Methods Appl. Mech. Eng.
**2004**, 193, 5359–5383. [Google Scholar] [CrossRef] - Bronkhorst, C.A.; Kalidindi, S.R.; Anand, L. Polycrystalline Plasticity and the Evolution of Crystallographic Texture in FCC Metals. Phil. Trans. R. Soc. Lond. A
**1992**, 341, 443–477. [Google Scholar] [CrossRef] - Bailey, J.E.; Hirsch, P.B. The Recrystallization Process in Some Polycrystalline Metals. In Proceedings of the Royal Society of London. Series, A. Mathematical and Physical Sciences; The Royal Society: London, UK, 1962; Volume 267, pp. 11–30. [Google Scholar] [CrossRef]
- Mitchell, M. An Introduction to Genetic Algorithms; MIT Press: Massachusetts, USA, 1998; ISBN 978-0-262-63185-3. [Google Scholar]
- Pozdeev, A.A.; Trusov, P.V.; Nyashin, Y.I. Large Elastoplastic Deformations: Theory, Algorithms, Applications; Nauka: Moscow, Russia, 1986. (In Russian) [Google Scholar]
- Chang, Y.A.; Himmel, L. Temperature Dependence of the Elastic Constants of Cu, Ag, and Au above Room Temperature. J. Appl. Phys.
**1966**, 37, 3567–3572. [Google Scholar] [CrossRef] [Green Version] - Ledbetter, H.M.; Naimon, E.R. Elastic Properties of Metals and Alloys. II. Copper. J. Phys. Chem. Ref. Data
**1974**, 3, 897–935. [Google Scholar] [CrossRef] - Blaz, L.; Sakai, T.; Jonas, J.J. Effect of Initial Grain Size on Dynamic Recrystallization of Copper. Metal. Sci.
**1983**, 17, 609–616. [Google Scholar] [CrossRef] - Kondratev, N.S.; Trusov, P.V.; Podsedertsev, A.N. The polycrystals grain structure formation for modified two-level crystal plasticity statistical models. Procedia Struct. Integr.
**2022**, in press. [Google Scholar] - Vaz, M.F.; Fortes, M.A. Grain Size Distribution: The Lognormal and the Gamma Distribution Functions. Scr. Metall.
**1988**, 22, 35–40. [Google Scholar] [CrossRef] - Raeisinia, B.; Sinclair, C.W. A Representative Grain Size for the Mechanical Response of Polycrystals. Mater. Sci. Eng. A
**2009**, 525, 78–82. [Google Scholar] [CrossRef] - Kondratev, N.S.; Trusov, P.V. To Determination a Distribution Law of Subgrain Sizes Formed in the Cold Plastic Deformation Process. AIP Conf. Proc.
**2020**, 2216, 040010. [Google Scholar] [CrossRef] - Suresh, K.S.; Rollett, A.D.; Suwas, S. Evolution of Microstructure and Texture During Deformation and Recrystallization of Heavily Rolled Cu-Cu Multilayer. Metall. Mater. Trans. A
**2013**, 44, 3866–3881. [Google Scholar] [CrossRef]

**Figure 1.**Scheme for the formation of a fine-grained structure during DDRX at various degrees of deformation $\mathsf{\epsilon}$: $\mathsf{\epsilon}<{\mathsf{\epsilon}}_{c}$ (

**a**), $\mathsf{\epsilon}\approx {\mathsf{\epsilon}}_{c}$ (

**b**), ${3/4\mathsf{\epsilon}}_{x}<\mathsf{\epsilon}<{\mathsf{\epsilon}}_{s}$ (

**c**), $\mathsf{\epsilon}\approx {\mathsf{\epsilon}}_{s}$ (

**d**) (it was developed based on [61]).

**Figure 2.**Scale levels and the connection between the structural elements of a multilevel model (thin lines correspond to subgrain boundaries, bold lines show grains boundaries).

**Figure 3.**Polyhedral representation of polycrystal grains (

**a**) and scheme of the grain structure in the statistical model (

**b**).

**Figure 4.**Scheme used for modeling a cluster of grains (a “consumed” defective grain surrounded by DRX grains).

**Figure 6.**Normalized grain size distribution d

_{eq}(non-dimensional) (

**a**) and grain sphericity ${\mathsf{\psi}}_{g}$ (non-dimensional) (

**b**) histograms.

**Figure 7.**Loading diagram for the copper polycrystal at deformation velocity $\dot{\gamma}$ = 10

^{–3}s

^{–1}and temperature $\mathsf{\theta}$ = 775 K (dashed curve—experiment [78], solid curve—calculations) (

**a**). Calculated pole figures {100}, {110}, and {111} at the end of deformation (

**b**).

**Figure 8.**Evolution of average grain size (

**a**) and average grain facet area (

**b**) and the ratio of the average grain volume to the facet area (

**c**) during plastic deformation.

**Figure 9.**Polyhedral grain structure obtained using Neper at the instants of deformation ${\mathsf{\epsilon}}_{m}$ (

**a**) and ${\mathsf{\epsilon}}_{f}$ (

**b**).

**Figure 10.**Grain size distribution ${r}_{g}$ (

**a**,

**b**) and grain sphericity ${\mathsf{\psi}}_{g}$ (

**c**,

**d**) histograms at different instants of deformation ${\mathsf{\epsilon}}_{m}$ and ${\mathsf{\epsilon}}_{f}$, respectively.

Parameter | Value | Literature Source |
---|---|---|

${\u043f}_{1111}$ | 140 GPa | [76] |

${\u043f}_{1122}$ | 104 GPa | [76] |

${\u043f}_{1313}$ | 63 GPa | [76] |

$G$ | 39 GPa | [77] |

${\mathsf{\tau}}_{c0}$ | 10 MPa | Identification procedure |

${\mathsf{\tau}}_{sat}$ | 27 MPa | Identification procedure |

${h}_{0}$ | 200 MPa | Identification procedure |

$a$ | 1.4 | Identification procedure |

${q}_{lat}$ | 1.4 | [72] |

${\dot{\gamma}}_{0}$ | 0.001 s^{–1} | [72] |

$m$ | 83 | [72] |

$\mathsf{\eta}$ | 1.0 | Identification procedure |

$\mathsf{\alpha}$ | 0.0012 | Identification procedure |

${e}_{gb}$ | 0.625 J/m^{2} | [5] |

${r}_{0}$ | 0.25 µm | [62] |

$Q$ | 121 kJ/mol | [5] |

${m}_{0}$ | 1.5 × 10^{–6} s·m^{2}/kg | [5] |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 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/).

## Share and Cite

**MDPI and ACS Style**

Trusov, P.; Kondratev, N.; Podsedertsev, A.
Description of Dynamic Recrystallization by Means of An Advanced Statistical Multilevel Model: Grain Structure Evolution Analysis. *Crystals* **2022**, *12*, 653.
https://doi.org/10.3390/cryst12050653

**AMA Style**

Trusov P, Kondratev N, Podsedertsev A.
Description of Dynamic Recrystallization by Means of An Advanced Statistical Multilevel Model: Grain Structure Evolution Analysis. *Crystals*. 2022; 12(5):653.
https://doi.org/10.3390/cryst12050653

**Chicago/Turabian Style**

Trusov, Peter, Nikita Kondratev, and Andrej Podsedertsev.
2022. "Description of Dynamic Recrystallization by Means of An Advanced Statistical Multilevel Model: Grain Structure Evolution Analysis" *Crystals* 12, no. 5: 653.
https://doi.org/10.3390/cryst12050653