МНОГОУРОВНЕВАЯ СТАТИСТИЧЕСКАЯ КОНСТИТУТИВНАЯ МОДЕЛЬ НА БАЗЕ КОНТИНУАЛЬНОЙ ДИСЛОКАЦИОННОЙ ДИНАМИКИ И ЕЕ ПРИМЕНЕНИЕ ДЛЯ КОМПЛЕКСНОГО ОПИСАНИЯ ТРАНСФОРМАЦИИ СТРУКТУРЫ МЕДИ ПРИ РАВНОКАНАЛЬНОМ УГЛОВОМ ПРЕССОВАНИИ

Романов, Кирилл Андреевич; Швейкин, Алексей Игоревич

doi:10.15593/perm.mech/2025.4.01

МНОГОУРОВНЕВАЯ СТАТИСТИЧЕСКАЯ КОНСТИТУТИВНАЯ МОДЕЛЬ НА БАЗЕ КОНТИНУАЛЬНОЙ ДИСЛОКАЦИОННОЙ ДИНАМИКИ И ЕЕ ПРИМЕНЕНИЕ ДЛЯ КОМПЛЕКСНОГО ОПИСАНИЯ ТРАНСФОРМАЦИИ СТРУКТУРЫ МЕДИ ПРИ РАВНОКАНАЛЬНОМ УГЛОВОМ ПРЕССОВАНИИ

Кирилл Андреевич Романов, Алексей Игоревич Швейкин

Сведения об авторах

Кирилл Андреевич Романов
Пермский национальный исследовательский политехнический университет, Пермь, Российская Федерация
Алексей Игоревич Швейкин
Пермский национальный исследовательский политехнический университет, Пермь, Российская Федерация

https://doi.org/10.15593/perm.mech/2025.4.01

В ходе технологических процессов (ТП) обработки давлением металлов и сплавов, особенно с применением методов интенсивной пластической деформации, происходит изменение их зеренной структуры. При низких гомологических температурах доминирующим процессом перестройки внутренней структуры материала является фрагментация, которая реализуется за счет образования блоков ячеек и постепенной их эволюции в субзерна и новые мелкие зерна за счет разворотов друг относительно друга в ходе деформирования. Развитие вычислительных средств в последние десятилетия позволяет все более эффективно применять для анализа и совершенствования реальных ТП аппарат математического моделирования. При этом для исследования происходящих в ходе ТП изменений внутри материала целесообразно использовать многоуровневые конститутивные модели (КМ), основанные на принципах физической теории пластичности, за счет явного учета в них эволюции внутренней структуры материала и реализующихся механизмов деформирования. В работе предлагается новая многоуровневая статистическая КМ на базе континуальной дислокационной динамики для описания фрагментации зерен металлов с ГЦК-решеткой. Разработанная КМ комплексно описывает процесс фрагментации: учитывает эволюцию плотности статистически накопленных и геометрически необходимых дислокаций, а также её влияние на изменение средних значений углов разориентации, размеров фрагментов и зерен в поликристалле. Верификация результатов моделирования осуществлялась на опытных данных для четырех проходов равноканального углового прессования меди при комнатной температуре по маршруту Вс (с поворотом заготовки). Полученные в КМ результаты предсказания изменения напряженно-деформированного состояния и ключевых параметров структуры (размеров зерен, текстуры, углов разориентации и размеров фрагментов, плотностей дислокаций) находятся в удовлетворительном соответствии с данными экспериментов.

Ключевые слова

многоуровневые статистические конститутивные моделиконтинуальная дислокационная динамикафрагментацияизмельчение зеренРКУПразмер зернаразмер фрагментовстатистически накопленные дислокациигеометрически необходимые дислокацииугол разориентации фрагментов

A statistical crystal plasticity model based on continuum dislocation dynamics and its application to the comprehensive analysis of microstructure evolution in copper during equal-channel angular pressing

During metal forming processes, especially those involving severe plastic deformation techniques, the grain structure of metals and alloys undergoes significant changes. At low homologous temperatures, the predominant mechanism of microstructural reorganization is fragmentation. This process occurs through the formation of dislocation cell blocks and their gradual evolution into subgrains and new fine grains via mutual rotations during deformation. Advances in computational capabilities over recent decades have increasingly enabled the application of mathematical modeling to analyze and optimize technological forming processes. To investigate microstructural transformations during these processes, multilevel constitutive models grounded in the crystal plasticity are particularly suitable. These models explicitly account for the evolution of internal structure and deformation mechanisms. This work proposes a novel multilevel statistical constitutive model based on continuum dislocation dynamics to describe grain fragmentation in face-centered cubic metals. The developed model comprehensively characterizes the fragmentation process by tracking the evolution of statistically stored dislocation and geometrically necessary dislocation densities, quantifying their influence on key microstructural parameters: average misorientation angle, average fragment size and average grain size in polycrystal. Model verification was performed against experimental data from four passes of equal-channel angular pressing of copper at room temperature via route Bc (workpiece rotation between passes). The model demonstrates satisfactory agreement with experimental results in predicting stress-strain states and critical structural parameters (grain size, crystallographic textures, disorientation angle and fragment size, dislocation densities).

Keywords

statistical crystal plasticity modelscontinuum dislocation dynamicsgrain fragmentationgrain refinementECAPgrain sizefragment sizestatistically stored dislocationsgeometrically necessary dislocationsfragment disorientation angle

Литература

1.Начала металлургии: учебник для вузов / В.И. Коротич [и др.] / под ред. В.И. Коротича. – Екатеринбург: УГТУ, 2000. – 392 с. Scholar
2.Металлургические технологии / Ю.Н. Симонов [и др.]. – Пермь: Издательство ПНИПУ, 2012. – 293 с. Scholar
3.Langdon, T.G. Twenty-five years of ultrafine-grained materials: Achieving exceptional properties through grain refinement / T.G. Langdon // Acta Materialia. – 2013. – Vol. 62, no. 19. – P. 7035–7059. DOI: 10.1016/j.actamat.2013.08.018 Scholar
4.Nanomaterials by severe plastic deformation: review of historical developments and recent advances / K. Edalati [et al.] // Materials Research Letters. – 2022. – Vol. 10, no. 4. – P. 163–256. DOI: 10.1080/21663831.2022.2029779 Scholar
5.Ultrafine-Grained Materials / R.Z. Valiev [et al.]. – Cham: Springer International Publishing, 2024. – 177 p. DOI: 10.1007/978-3-031-31729-3 Scholar
6.Plastic working of metals by simple shear / V.M. Segal [et al.] // Russian metallurgy (Metally). – 1981. – Vol. 1. – P. 99–105. Scholar
7.Valiev, R.Z. Principles of equal-channel angular pressing as a processing tool for grain refinement / R.Z. Valiev, T.G. Langdon // Progress in Materials Science. – 2006. – Vol. 51. – P. 881–981. DOI: 10.1016/j.pmatsci.2006.02.003 Scholar
8.Zhilyaev, A.P. Using high-pressure torsion for metal processing: Fundamentals and applications / A.P. Zhilyaev, T.G. Langdon // Progress in Materials Science. – 2008. – Vol. 53. – P. 893–979. DOI: 10.1016/j.pmatsci.2008.03.002 Scholar
9.Edalati, K. A Review on High-Pressure Torsion (HPT) from 1935 to 1988 / K. Edalati, Z. Horita // Materials Science & Engineering A. – 2016. – Vol. 652. – P. 325–352. DOI: 10.1016/j.msea.2015.11.074. Scholar
10.Salishchev, G.A. Formation of submicrocrystalline structure in the titanium alloy VT8 and its influence on mechanical properties / G.A. Salishchev, O.R. Valiakhmetov, R.M. Galeyev // Journal of materials science. – 1993. – Vol. 28. – P. 2898–2902. DOI: 10.1007/BF00354692 Scholar
11.Effect of Pass Strain on Grain Refinement in 7475 Al Alloy during Hot Multidirectional Forging / O. Sitdikov [et al.] // Materials Transactions. – 2004. – Vol. 45, no. 7. – P. 2232–2238. DOI: 10.2320/matertrans.45.2232 Scholar
12.Novel ultra-high straining process for bulk materials – development of the accumulative roll-bonding (ARB) process / Y. Saito [et al.] // Acta Materialia. – 1999. – Vol. 47, no. 2. – P. 579–583. DOI: 10.1016/S1359-6454(98)00365-6 Scholar
13.ARB (Accumulative Roll-Bonding) and other new Techniques to Produce Bulk Ultrafine Grained Materials / N. Tsuji [et al.] // Advanced engineering materials. – 2003. – Vol. 5, no. 5. – P. 338–344. DOI: 10.1002/adem.200310077 Scholar
14.Jamaati, R. Effect of stacking fault energy on nanostructure formation under accumulative roll bonding (ARB) process / R. Jamaati, M.R. Toroghinejad, H. Edris // Materials Science & Engineering A. – 2013. – Vol. 578. – P. 191–196. DOI: 10.1016/j.msea.2013.04.090 Scholar
15.Estrin, Y. Extreme grain refinement by severe plastic deformation: A wealth of challenging science / Y. Estrin, A. Vinogradov // Acta Materialia. – 2013. – Vol. 61. – P. 782–817. DOI: 10.1016/j.actamat.2012.10.038 Scholar
16.Dynamic and post-dynamic recrystallization under hot, cold and severe plastic deformation conditions / T. Sakai [et al.] // Progress in Materials Science. – 2014. – Vol. 60. – P. 130–207. DOI: 10.1016/j.pmatsci.2013.09.002. Scholar
17.Structural evolutions of metallic materials processed by severe plastic deformation / Y. Cao [et al.] // Materials Science & Engineering R. – 2018. – Vol. 133. – P. 1–59. DOI: 10.1016/j.mser.2018.06.001 Scholar
18.Рыбин, В.В. Большие пластические деформации и разрушение металлов / В.В. Рыбин. – М.: Металлургия, 1986. – 224 с. Scholar
19.Останина, Т.В. Измельчение зеренной структуры металлов и сплавов при интенсивном пластическом деформировании: экспериментальные данные и анализ механизмов / Т.В. Останина, А.И. Швейкин, П.В. Трусов // Вестник Пермского национального исследовательского политехнического университета. Механика. – 2020. – № 2. – С. 85–111. DOI: 10.15593/perm.mech/2020.2.08 Scholar
20.Alaneme, K.K. Recrystallization mechanisms and microstructure development in emerging metallic materials: A review / K.K. Alaneme, E.A. Okotete // Journal of Science: Advanced Materials and Devices. – 2019. – Vol. 4, no. 1. – P. 19–33. DOI: 10.1016/j.jsamd.2018.12.007 Scholar
21.Trusov, P. Description of Dynamic Recrystallization by Means of An Advanced Statistical Multilevel Model: Grain Structure Evolution Analysis / P. Trusov, N. Kondratev, A. Podsedertsev // Crystals. – 2022. – Vol. 12. – 653. DOI: 10.3390/ crystals12050653 Scholar
22.Trusov, P. Grain Structure Rearrangement by Means the Advanced Statistical Model Modified for Describing Dynamic Recrystallization / P. Trusov, N. Kondratev, A. Podsedertsev // Metals. – 2023. – Vol. 13. – 113. DOI: 10.3390/met13010113 Scholar
23.Трусов, П.В. Многоуровневая модель для описания пластического и сверхпластического деформирования поликристаллических материалов / П.В. Трусов, Э.Р. Шарифуллина, А.И. Швейкин // Физическая мезомеханика. – 2019. – Т. 22, № 2. – С. 5–23. DOI: 10.24411/1683-805X-2019-12001 Scholar
24.Handbook of Materials Modeling. Methods: Theory and Modeling / ed. by W. Andreoni, S. Yip. – Cham: Springer International Publishing, 2020. – 1971 p. DOI: 10.1007/978-3-319-44677-6 Scholar
25.Hibbitt, H.D. A finite element formulation of large strain and large displacement / H.D. Hibbitt, P.V. Marcal, J.R. Rice // International Journal of Solids and Structures. – 1970. – Vol. 6, no. 8. – P. 1069–1086. DOI: 10.1016/0020-7683(70)90048-X Scholar
26.McMeeking, R.M. Finite-element formulation for problems of large elastic-plastic deformation / R.M. McMeeking, J.R. Rice // International Journal of Solids and Structures. – 1975. – Vol. 11, no. 5. – P. 601–616. DOI: 10.1016/0020-7683(75)90033-5 Scholar
27.Osias, J.R. Finite elasto-plastic deformation–I: Theory and numerical examples / J.R. Osias, J.L. Swedlow // International Journal of Solids and Structures. – 1974. – Vol. 10, no. 3. – P. 321–339. DOI: 10.1016/0020-7683(74)90081-X Scholar
28.McDowell, D.L. A perspective on trends in multiscale plasticity / D.L. McDowell // International Journal of Plasticity. – 2010. – Vol. 26. – P. 1280–1309. DOI: 10.1016/j.ijplas.2010.02.008 Scholar
29.Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications / F. Roters [et al.] // Acta Materialia. – 2010. – Vol. 58. – P. 1152–1211. DOI: 10.1016/j.actamat.2009.10.058. Scholar
30.Diehl, M. Review and outlook: mechanical, thermodynamic, and kinetic continuum modeling of metallic materials at the grain scale / M. Diehl // MRS Communications. – 2017. – Vol. 7. – P. 735–746. doi: 10.1557/mrc.2017.98 Scholar
31.Beyerlein, I. Review of microstructure and micromechanism-based constitutive modeling of polycrystals with a low-symmetry crystal structure / I. Beyerlein, M. Knezevic // Journal of Materials Research. – 2018. – Vol. 33, no. 22. – P. 3711–3738. DOI: 10.1557/jmr.2018.333 Scholar
32.DAMASK – The Düsseldorf Advanced Material Simulation Kit for modeling multi-physics crystal plasticity, thermal, and damage phenomena from the single crystal up to the component scale / F. Roters [et al.] // Computational Materials Science. – 2019. – Vol. 158. – P. 420–478. DOI: 10.1016/j.commatsci.2018.04.030 Scholar
33.Трусов, П.В. Многоуровневые модели моно- и поликристаллических материалов: теория, алгоритмы, примеры применения / П.В. Трусов, А.И. Швейкин. – Новосибирск: Изд-во СО РАН, 2019. – 605 с. DOI: 10.15372/MULTILEVEL2019TPV Scholar
34.Multilevel Models in Physical Mesomechanics of Metals and Alloys: Results and Prospects / P.V. Trusov [et al.] // Physical Mesomechanics. – 2021. – Vol. 24, no. 4. – P. 391–417. DOI: 10.1134/S1029959921040056 Scholar
35.OXFORD-UMAT: An efficient and versatile crystal plasticity framework / E. Demir [et al.] // International Journal of Solids and Structures. – 2025. – Vol. 307. – 113110. DOI: 10.1016/j.ijsolstr.2024.113110 Scholar
36.Horstemeyer, M.F. Historical review of internal state variable theory for inelasticity / M.F. Horstemeyer, D.J. Bammann // International Journal of Plasticity. – 2010. – Vol. 26, no. 9. – P. 1310–1334. DOI: 10.1016/j.ijplas.2010.06.005 Scholar
37.Constitutive relations and their application to the description of microstructure evolution / P.V. Trusov [et al.] // Physical Mesomechanics. – 2010. – Vol. 13, no. 1–2. – P. 38–46. DOI: 10.1016/j.physme.2010.03.005 Scholar
38.Maugin, G.A. The saga of internal variables of state in continuum thermomechanics (1893–2013) / G.A. Maugin // Mechanics Research Communications. – 2015. – Vol. 69. – P. 79–86. DOI: 10.1016/j.mechrescom.2015.06.00 Scholar
39.Трусов, П.В. Многоуровневые физические модели моно- и поликристаллов. Прямые модели / П.В. Трусов, А.И. Швейкин // Физическая мезомеханика. – 2011–2. – Т. 14, № 5. – С. 5–30. Scholar
40.Taylor, G.I. Plastic strain in metals / G.I. Taylor // Journal of the Institute of Metals. – 1938. – Vol. 62 – P. 307–325. Scholar
41.Трусов, П.В. Многоуровневые физические модели моно- и поликристаллов. Статистические модели / П.В. Трусов, А.И. Швейкин // Физическая мезомеханика. – 2011–1. – Т. 14, № 4. – С. 17–28. Scholar
42.Lebensohn, R.A. A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: application to zirconium alloys / R.A. Lebensohn, C.N. Tomé // Acta Metallurgica et Materialia. – 1993. – Vol. 41. – P. 2611–2624. DOI: 10.1016/0956-7151(93)90130-K Scholar
43.Romanov, K. Advanced Statistical Crystal Plasticity Model: Description of Copper Grain Structure Refinement during Equal Channel Angular Pressing / K. Romanov, A. Shveykin, P. Trusov // Metals. – 2023. – Vol. 13. – 953. DOI: 10.3390/ met13050953 Scholar
44.Остапович, К.В. Проектирование рационально текстурированных поликристаллических изделий на основе двухуровневой статистической модели упруговязкопластического деформирования: дис. … канд. физ.-мат. наук / Остапович Кирилл Вадимович, 2023. – 123 с. Scholar
45.A Multilevel Physically Based Model of Recrystallization: Analysis of the Influence of Subgrain Coalescence at Grain Boundaries on the Formation of Recrystallization Nuclei in Metal / P. Trusov [et al.] // Materials. – 2023. – Vol. 16. – 2810. DOI: 10.3390/ma16072810 Scholar
46.Chatterjee, R. Evaluating the influence of deformation variables on dynamic recrystallization behavior using a crystal plasticity model / R. Chatterjee, S.V.S. Narayana Murty, A. Alankar // Modelling and Simulation in Materials Science and Engineering. – 2023. – Vol. 31, no. 4. – 045002. DOI: 10.1088/1361-651X/acc961 Scholar
47.A straightforward 3D polycrystal plasticity finite element method for dynamic/static recrystallization simulation / G. Zhou [et al.] // Journal of Materials Science & Technology. – 2025. – Vol. 220. – P. 180–198. DOI: 10.1016/j.jmst.2024.09.005 Scholar
48.Shveykin, A. Statistical Crystal Plasticity Model Advanced for Grain Boundary Sliding Description / A. Shveykin, P. Trusov, E. Sharifullina // Crystals. – 2020. – Vol. 10. – 822. DOI: 10.3390/cryst10090822 Scholar
49.Crystal plasticity based modeling of grain boundary sliding in magnesium alloy AZ31B sheet / Z. Li [et al.] // Transactions of Nonferrous Metals Society of China. – 2021. – Vol. 31, no. 1. – P. 138–155. DOI: 10.1016/S1003-6326(20)65483-8 Scholar
50.Venkataraman, A. A crystal plasticity model with an atomistically informed description of grain boundary sliding for improved predictions of deformation fields / A. Venkataraman, M.D. Sangid // Computational Materials Science. – 2021. – Vol. 197. – 110589. DOI: 10.1016/j.commatsci.2021.110589 Scholar
51.Li, G. A new set of physically-based unified viscoplastic constitutive equations for superplastic deformation of Al-Zn-Mg-Cu alloy / G. Li, H. Ding, J. Wang // Mechanics Research Communications. – 2022. – Vol. 123. – 103904. DOI: 10.1016/j.mechrescom.2022.103904 Scholar
52.Shveykin, A. Some Issues with Statistical Crystal Plasticity Models: Description of the Effects Triggered in FCC Crystals by Loading with Strain-Path Changes / A. Shveykin, K. Romanov, P. Trusov // Materials. – 2022. – Vol. 15, no. 19. – 6586. DOI: 10.3390/ma15196586 Scholar
53.Trusov, P.V. The three-level elastoviscoplastic model and its application to describing complex cyclic loading of materials with different stacking fault energies / P.V. Trusov, D.S. Gribov // Materials. – 2022. – Vol. 15. – 760 (17 p.). DOI: 10.3390/ma15030760 Scholar
54.Швейкин, А.И. Двухуровневая конститутивная модель металла с комплексным учетом изменяющихся температурно-скоростных условий / А.И. Швейкин, А.А. Вишвкова, П.В. Трусов // Физическая мезомеханика. – 2024. – Т. 27, № 2. – С. 50–68. DOI: 10.55652/1683-805X_2024_27_2_50-68 Scholar
55.Трусов, П.В. Эволюция зеренной структуры металлов и сплавов при интенсивном пластическом деформировании: континуальные модели / П.В. Трусов, Т.В. Останина, А.И. Швейкин // Вестник Пермского национального исследовательского политехнического университета. Механика. – 2022. – № 1. – С. 123–155. DOI: 10.15593/perm.mech/2022.1.11 Scholar
56.Трусов, П.В. Эволюция зеренной структуры металлов и сплавов при интенсивном пластическом деформировании: многоуровневые модели / П.В. Трусов, Т.В. Останина, А.И. Швейкин // Вестник Пермского национального исследовательского политехнического университета. Механика. – 2022. – № 2. – С. 114–146. DOI: 10.15593/perm.mech/2022.2.11 Scholar
57.Beygelzimer, Y. Grain refinement versus voids accumulation during severe plastic deformations of polycrystals: mathematical simulation / Y. Beygelzimer // Mechanics of Materials. – 2005. – Vol. 37. – P. 753–767. DOI: 10.1016/j.mechmat.2004.07.006 Scholar
58.Petryk, H. A quantitative model of grain refinement and strain hardening during severe plastic deformation / H. Petryk, S. Stupkiewicz // Materials Science and Engineering A. – 2007. – Vol. 444. – P. 214–219. DOI: 10.1016/j.msea.2006.08.076 Scholar
59.Adiabatic heating and the saturation of grain refinement during SPD of metals and alloys: experimental assessment and computer modeling / A.P. Zhilyaev [et al.] // Journal of Materials Science. – 2013. – Vol. 48. – P. 4626–4636. DOI: 10.1007/s10853-013-7254-4 Scholar
60.A constitutive model incorporating grain refinement strengthening on metallic alloys / S.C. Cao [et al.] // Journal of Materials Science & Technology. – 2021. – Vol. 88. – P. 233–239. DOI: 10.1016/j.jmst.2021.02.004 Scholar
61.A dislocation-based model for all hardening stages in large strain deformation / Y. Estrin [et al.] // Acta Materialia. – 1998. – Vol. 46, no. 15. – P. 5509–5522. DOI: 10.1016/S1359-6454(98)00196-7 Scholar
62.Borodin, E.N. Non-equilibrium approach to prediction of microstructure evolution for metals undergoing severe plastic deformation / E.N. Borodin, V. Bratov // Materials Characterization. – 2018. – Vol. 141. – P. 267–278. DOI: 10.1016/j.matchar.2018.05.002 Scholar
63.Firouzabadi, S.S. Cell-structure and flow stress investigation of largely strained non-heattreatable Al-alloys using dislocation based model / S.S. Firouzabadi, M. Kazeminezhad // Materials Science & Engineering A. – 2019. – Vol. 739. – P. 167–172. DOI: 10.1016/j.msea.2018.10.007 Scholar
64.Coupled quantitative modeling of microstructural evolution and plastic flow during continuous dynamic recrystallization / F. Chen [et al.] // International Journal of Plasticity. – 2022. – Vol. 156. – 103372. DOI: 10.1016/j.ijplas.2022.103372 Scholar
65.Rezvanian, O. Microstructural modeling of grain subdivision and large strain inhomogeneous deformation modes in f.c.c. crystalline materials / O. Rezvanian, M.A. Zikry, A.M. Rajendran // Mechanics of Materials. – 2006. – Vol. 38. – P. 1159–1169. DOI: 10.1016/j.mechmat.2005.12.006 Scholar
66.Frydrych, K. A three-scale crystal plasticity model accounting for grain refinement in fcc metals subjected to severe plastic deformations / K. Frydrych, K. Kowalczyk-Gajewska // Materials Science & Engineering. – 2016. – Vol. 658. – P. 490–502. DOI: 10.1016/j.msea.2016.01.101 Scholar
67.Frydrych, K. Microstructure evolution in cold-rolled pure titanium: Modeling by the three-scale crystal plasticity approach accounting for twinning / K. Frydrych, K. Kowalczyk-Gajewska // Metallurgical and Materials Transactions A. – 2018. – Vol. 49. – P. 3610–3623. DOI: 10.1007/s11661-018-4676-2 Scholar
68.Continuum dislocation dynamics-based grain fragmentation modeling / A.H. Kobaissy [et al.] // International Journal of Plasticity. – 2019. – Vol. 114. – P. 252–271. DOI: 10.1016/j.ijplas.2018.11.006 Scholar
69.Modeling of the ECAP Induced Strain Hardening Behavior in FCC Metals / A.H. Kobaissy [et al.] // Metallurgical and Materials Transactions A. – 2020. – Vol. 51, no. 10. – P. 5453–5474. DOI: 10.1007/s11661-020-05971-2 Scholar
70.Toth, L.S. Strain hardening at large strains as predicted by dislocation based polycrystal plasticity model / L.S. Toth, A. Molinari, Y. Estrin // Journal of Engineering Materials and Technology. – 2002. – Vol. 124. – P. 71–77. DOI: 10.1115/1.1421350 Scholar
71.Sheikh, H. Modeling the effect of strain reversal on grain refinement and crystallographic texture during simple shear extrusion / H. Sheikh, R. Ebrahimi // International Journal of Solids and Structures. – 2017. – Vol. 126–127. – P. 175–186. DOI: 10.1016/j.ijsolstr.2017.08.004 Scholar
72.Orowan, E. Problems of plastic gliding / E. Orowan // Proceedings of the Physical Society. – 1940. – Vol. 52, no. 1. – 8 p. DOI: 10.1088/0959-5309/52/1/303 Scholar
73.Hall, E.O. The deformation and aging of mild steel. III. Discussion and results / E.O. Hall // Proc. Phys. Soc. of London. – 1951. – Vol. B64. – P. 747–753. DOI: 10.1088/0370-1301/64/9/303 Scholar
74.Petch, N.J. The cleavage strength of polycrystals / N.J. Petch // Journal of the Iron and Steel Institute. – 1953. – Vol. 174. – P. 25–28. Scholar
75.Cordero, Z.C. Six decades of the Hall–Petch effect – a survey of grain-size strengthening studies on pure metals / Z.C. Cordero, B.E. Knight, C.A. Schuh // International Materials Reviews. – 2016. – Vol. 61, no. 8. – P. 495–512. DOI: 10.1080/09506608.2016.1191808 Scholar
76.Bailey, J.E. The dislocation distribution, flow stress, and stored energy in cold-worked polycrystalline silver / J.E. Bailey, P.B. Hirsch // Philosophical Magazine. – 1960. – Vol. 5, no. 23. – P. 485–497. DOI: 10.1080/14786436008238300 Scholar
77.Ohashi, T. A multiscale approach for modeling scale-dependent yield stress in polycrystalline metals / T. Ohashi, M. Kawamukai, H. Zbib // International Journal of Plasticity. – 2007. – Vol. 23. – P. 897–914. DOI: 10.1016/j.ijplas.2006.10.002 Scholar
78.Lyu, H. A dislocation-based model for deformation and size effect in multi-phase steels / H. Lyu, A. Ruimi, H.M. Zbib // International Journal of Plasticity. – 2015. – Vol. 72. – P. 44–59. DOI: 10.1016/j.ijplas.2015.05.005 Scholar
79.Трусов, П.В. О разложении движения, независимых от выбора системы отсчета производных и определяющих соотношениях при больших градиентах перемещений: взгляд с позиций многоуровневого моделирования / П.В. Трусов, А.И. Швейкин, А.Ю. Янц // Физическая мезомеханика. – 2016. – Т. 19, № 2. – С. 49–65. Scholar
80.Shveykin, A.I. Multilevel models of polycrystalline metals: comparison of relations describing the crystallite lattice rotations / A.I. Shveykin, P.V. Trusov // Nanoscience and Technology: An International Journal. – 2019. – Vol. 10, no. 1. – P. 1–20. DOI: 10.1615/NanoSciTechnolIntJ.2018028673 Scholar
81.Trusov, P. Some Issues on Crystal Plasticity Models Formulation: Motion Decomposition and Constitutive Law Variants / P. Trusov, A. Shveykin, N. Kondratev // Crystals. – 2021. – Vol. 11. – 1392. DOI: 10.3390/cryst1111392 Scholar
82.Deformation Texture Prediction: From the Taylor Model to the Advanced Lamel Model / P. Van Houtte [et al.] // International Journal of Plasticity. – 2005. – Vol. 21. – P. 589–624. DOI: 10.1016/j.ijplas.2004.04.011 Scholar
83.Modeling Lattice Strain Evolution at Finite Strains and Experimental Verification for Copper and Stainless Steel Using in Situ Neutron Diffraction / C. Neil [et al.] // International Journal of Plasticity. – 2010. – Vol. 26. – P. 1772–1791. DOI: 10.1016/j.ijplas.2010.03.005 Scholar
84.Zecevic, M. An Implicit Formulation of the Elasto-Plastic Self-Consistent Polycrystal Plasticity Model and its Implementation in Implicit Finite Elements / M. Zecevic, M. Knezevic // Mechanics of Materials. – 2019. – Vol. 136. – 103065. DOI: 10.1016/j.mechmat.2019.103065 Scholar
85.A Comparative Study between Elasto-Plastic Self-Consistent Crystal Plasticity and Anisotropic Yield Function with Distortional Hardening Formulations for Sheet Metal Forming / Z. Feng [et al.] // Mechanics of Materials. – 2020. – Vol. 148. – 103422. DOI: 10.1016/j.mechmat.2020.103422 Scholar
86.Anand, L. Single-Crystal Elasto-Viscoplasticity: Application to Texture Evolution in Polycrystalline Metals at Large Strains / L. Anand // Computer Methods in Applied Mechanics and Engineering. – 2004. – Vol. 193. – P. 5359–5383. DOI: 10.1016/j.cma.2003.12.068 Scholar
87.Latent Hardening and Plastic Anisotropy Evolution in AA6060 Aluminium Alloy / M. Khadyko [et al.] // International Journal of Plasticity. – 2016. – Vol. 76. – P. 51–74. DOI: 10.1016/j.ijplas.2015.07.010 Scholar
88.Lieou, C.K. Thermodynamic Theory of Crystal Plasticity: Formulation and Application to Polycrystal fcc Copper / C.K. Lieou, C.A. Bronkhorst // Journal of the Mechanics and Physics of Solids. – 2020. – Vol. 138. – 103905. DOI: 10.1016/j.jmps.2020.103905 Scholar
89.A Multiplicative Finite Strain Crystal Plasticity Formulation based on Additive Elastic Corrector Rates: Theory and Numerical Implementation / M. Zhang [et al.] // International Journal of Plasticity. – 2021. – Vol. 137. – 102899. DOI: 10.1016/j.ijplas.2020.102899 Scholar
90.Швейкин, А.И. Об одном подходе к численной оценке устойчивости многоуровневых конститутивных моделей материалов / А.И. Швейкин, П.В. Трусов, К.А. Романов // Вычислительная механика сплошных сред. – 2021-1. – Т. 14, № 1. – С. 61–76. DOI: 10.7242/1999-6691/2021.14.1.6 Scholar
91.Швейкин, А.И. Некоторые результаты численной оценки устойчивости двухуровневой конститутивной модели ГЦК-поликристалла / А.И. Швейкин, П.В. Трусов, К.А. Романов // Вычислительная механика сплошных сред. – 2021-2. – Т. 14, № 2. – С. 127–143. DOI: 10.7242/1999-6691/2021.14.2.11 Scholar
92.Shveykin, A. Stability of Crystal Plasticity Constitutive Models: Observations in Numerical Studies and Analytical Justification / A. Shveykin, P. Trusov, K. Romanov // Metals. – 2024. – Vol. 14. – 947. DOI: 10.3390/met14080947 Scholar
93.Romanov, K.A. Two-level statistical constitutive model with integrated ETMB model: description of grain structure refinement of AISI 304 steel in cold bending / K.A. Romanov, A.I. Shveykin // Russian Physics Journal. – 2024. – Vol. 67 – P. 1555–1561 – DOI: 10.1007/s11182-024-03281-3 Scholar
94.Overview no. 96 evolution of f.c.c. deformation structures in polyslip / B. Bay [et al.] // Acta Metallurgica et Materialia. – 1992. – Vol. 40, no. 2. – P. 205–219. DOI: 10.1016/0956-7151(92)90296-Q Scholar
95.Hansen, N. 17 – Plastic Deformation of Metals and Alloys / N. Hansen, C.Y. Barlow // Physical Metallurgy (Fifth Edition). – 2014 – P. 1681–1764. DOI: 10.1016/B978-0-444-53770-6.00017-4 Scholar
96.Rezvanian, O. Statistically stored, geometrically necessary and grain boundary dislocation densities: microstructural representation and modelling / O. Rezvanian, M.A. Zikry, A.M. Rajendran // Proceedings of the royal society A. – 2007. – Vol. 463. – P. 2833–2853. DOI: 10.1098/rspa.2007.0020 Scholar
97.Predicting plastic flow and irradiation hardening of iron single crystal with mechanism-based continuum dislocation dynamics / D. Li [et al.] // International Journal of Plasticity. – 2014. – Vol. 52. – P. 3–17. DOI: 10.1016/j.ijplas.2013.01.015 Scholar
98.A study of the hot and cold deformation of twin-roll cast magnesium alloy AZ31 / H. Askari [et al.] // Philosophical Magazine. – 2014. – Vol. 94, no. 4. – P. 381–403. DOI: 10.1080/14786435.2013.853884 Scholar
99.Devincre, B. Physical analyses of crystal plasticity by DD simulations / B. Devincre, L. Kubin, T. Hoc // Scripta Materialia. – 2006. – Vol. 54, no. 5. – P. 741–746. DOI: 10.1016/j.scriptamat.2005.10.066 Scholar
100.A discrete dislocation dynamics framework for modeling polycrystal plasticity with hardening / T.N. Tak [et al.] // International Journal of Solids and Structures. – 2023. – Vol. 281. – 112442. DOI: 10.1016/j.ijsolstr.2023.112442 Scholar
101.Motaman, S.A.H. Microstructural constitutive model for polycrystal viscoplasticity in cold and warm regimes based on continuum dislocation dynamics / S.A.H. Motaman, U. Prahl // Journal of the Mechanics and Physics of Solids. – 2019. – Vol. 122. – P. 205–243. DOI: 10.1016/j.jmps.2018.09.002 Scholar
102.Models for long-/short-range interactions and cross slip in 3D dislocation simulation of BCC single crystals / M. Rhee [et al.] // Modelling and Simulation in Materials Science and Engineering. – 1998. – Vol. 6, no. 4. – 467. DOI: 10.1088/0965-0393/6/4/012 Scholar
103.A Continuum Dislocation Dynamics Crystal Plasticity Approach to Irradiated Body-Centered Cubic α-Iron / S.A. Pitts [et al.] // Journal of Engineering Materials and Technology. – 2022. – Vol. 144. – 011018. DOI: 10.1115/1.4052256 Scholar
104.A crystallographic dislocation model for describing hardening of polycrystals during strain path changes. Application to low carbon steels / K. Kitayama [et al.] // International Journal of Plasticity. – 2013. – Vol. 46. – P. 54–69. DOI: 10.1016/j.ijplas.2012.09.004 Scholar
105.Nye, J.F. Some geometrical relations in dislocated crystals / J.F. Nye // Acta Metallurgica. – 1953. – Vol. 1, no. 2. – P. 153–162. DOI: 10.1016/0001-6160(53)90054-6 Scholar
106.Shizawa, K. A thermodynamical theory of gradient elastoplasticity with dislocation density tensor. I: Fundamentals / K. Shizawa, H.M. Zbib // International Journal of Plasticity. – 1999. – Vol. 15, no. 9. – P. 899–938. DOI: 10.1016/S0749-6419(99)00018-2 Scholar
107.Burgers, J.M. Geometrical considerations concerning the structural irregularities to be assumed in a crystal / J.M. Burgers // Proceedings of the Physical Society. – 1940. – Vol. 52, no. 1. – P. 23–33. DOI: 10.1088/0959-5309/52/1/304 Scholar
108.Kuhlmann-Wilsdorf, D. Geometrically necessary, incidental and subgrain boundaries / D. Kuhlmann-Wilsdorf, N. Hansen // Scripta Metallurgica et Materialia. – 1991. – Vol. 25, no. 7. – P. 1557–1562. DOI: 10.1016/0956-716X(91)90451-6 Scholar
109.Gourdet, S. A model of continuous dynamic recrystallization / S. Gourdet, F. Montheillet // Acta Materialia. – 2003. – Vol. 51. – P. 2685–2699. DOI: 10.1016/S1359-6454(03)00078-8 Scholar
110.Modeling of continuous dynamic recrystallization of Al-Zn-Cu-Mg alloy during hot deformation based on the internal-state-variable (ISV) method / Z.C. Sun [et al.] // International Journal of Plasticity. – 2018. – Vol. 106. – P. 73–87. DOI: 10.1016/j.ijplas.2018.03.002 Scholar
111.Beyerlein, I.J. Texture evolution in equal-channel angular extrusion / I.J. Beyerlein, L.S. Toth // Progress in Materials Science. – 2009. – Vol. 54, no. 4. – P. 427–510. DOI: 10.1016/j.pmatsci.2009.01.001 Scholar
112.Bronkhorst, C.A. Polycrystalline plasticity and the evolution of crystallographic texture in FCC metals / C.A. Bronkhorst, S.R. Kalidindi, L. Anand // Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences. – 1992. – Vol. 341. – P. 443–477. DOI: 10.1098/rsta.1992.0111 Scholar
113.“Sigma Technology”. Novel Optimization Strategy – IOSO [Electronic resource] URL: https://iosotech.com/ (accessed: 17.08.2025). Scholar
114.Harder, J. FEM-simulation of the hardening behavior of FCC single crystals / J. Harder // Acta Mechanica. – 2001. – Vol. 150. – P. 197–217. DOI: 10.1007/BF01181812 Scholar
115.Thermal stability of ultrafine grains size of pure copper obtained by equal-channel angular pressing / N. Lugo [et al.] // Journal of Materials Science. – 2010. – Vol. 45. – P. 2264–2273. DOI: 10.1007/s10853-009-4139-7 Scholar
116.Influence of stacking fault energy and temperature on microstructures and mechanical properties of fcc pure metals processed by equal-channel angular pressing / F. Liu [et al.] // Materials Science and Engineering: A. – 2016. – Vol. 662. – P. 578–587. DOI: 10.1016/j.msea.2016.03.022 Scholar
117.Prangnell, P.B. Ultra-fine grain structures in aluminium alloys by severe deformation processing / P.B. Prangnell, J.R. Bowen, P.J. Apps // Materials Science and Engineering A. – 2004. – Vol. 375-377. – P. 178–185. DOI: 10.1016/j.msea.2003.10.170 Scholar

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