Two-Level Constitutive Model of Metal with a Comprehensive Account of Temperature and Strain Rate Changes
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Abstract
Abstract An important goal of industrial development is to improve the forming and thermomechanical processing technologies, both in order to get the best characteristics of finished products and to reduce energy costs and material consumption. The key step in solving such problems is the correct formulation of a material constitutive model. The temperature and strain rate attained in particular metal forming processes can vary significantly and have a strong influence on the structural evolution of the material and, consequently, on the resulting physical and mechanical properties. However, there are almost no processes in which the temperature and strain rate are constant and equal at all points of the processed product. In this regard, it is relevant to build constitutive models that correctly account for the influence of changing temperature and strain rate on the material response. Based on our previous review, we propose here a modified two-level statistical model that correctly accounts for the temperature and strain rate effects on intragranular dislocation slip and the associated material response. The model parameters are determined for an fcc polycrystal of Al 2024-T351 alloy using literature data on the compression behavior of this alloy at different temperatures and strain rates. A detailed description is given for an algorithm developed to identify the model parameters using data from constant temperature and constant strain rate experiments. The proposed model showed adequate results for loading conditions with changing temperature and strain rate.
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OpenAlex topics: Microstructure and mechanical properties · High-Velocity Impact and Material Behavior · Metallurgy and Material Forming
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References (100+)
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1. Rittel, D., Ravichandran, G., and Lee, S., Large Strain Constitutive Behavior of OFHC Copper over a Wide Range of Strain Rates Using the Shear Compression Specimen, Mech. Mater., 2002, vol. 34, no. 10, pp. 627–642. https://doi.org/10.1016/S0167-6636(02)00164-310.1016/S0167-6636(02)00164-3 ↗
2. Kocks, U.F. and Mecking, H., Physics and Phenomenology of Strain Hardening: The FCC Case, Progr. Mater. Sci., 2003, vol. 48, pp. 171–273. https://doi.org/10.1016/S0079-6425(02)00003-810.1016/S0079-6425(02)00003-8 ↗
3. Xu, L.X., Wu, H.B., and Wang, X.T., Influence of Microstructural Evolution on the Hot Deformation Behavior of an Fe–Mn–Al Duplex Lightweight Steel, Acta Metall. Sin. Engl. Lett., 2018, vol. 31, pp. 389–400. https://doi.org/10.1007/s40195-017-0655-710.1007/s40195-017-0655-7 ↗
4. Xia, L., Yan, L., Zhang, H., Li, Y., Jiang, Z., and Li, G., Effect of Temperature, Strain Rate and Chromium Content on the Flow Behavior of High-Manganese Steels, Mater. Res. Express., 2022, vol. 9, p. 106517. https://doi.org/10.1088/2053-1591/ac96d210.1088/2053-1591/ac96d2 ↗
5. May, J., Höppel, H.W., and Göken, M., Strain Rate Sensitivity of Ultrafine-Grained FCC- and BCC-Type Metals, Mater. Sci. Forum, 2006, vol. 503–504, pp. 781–786. http://dx.doi.org/10.4028/www.scientific.net/MSF.503-504.78110.4028/www.scientific.net/MSF.503-504.781 ↗
6. Abedrabbo, N., Pourboghrat, F., and Carsley, J., Forming of AA5182-O and AA5754-O at Elevated Temperatures Using Coupled Thermo-Mechanical Finite Element Models, Int. J. Plast., 2007, vol. 23, pp. 841–875. https://doi.org/10.1016/j.ijplas.2006.10.00510.1016/j.ijplas.2006.10.005 ↗
7. Kochubey, A.Ya., Medvedev, P.N., Klochkov, G.G., and Avtaev, V.V., Mechanisms of Texture Formation in Open Die Upsetting of Al-Cu-Li Alloy, Tekhnol. Legkikh Splavov, 2016, no. 1, pp. 74–79.
8. Senseny, P.E., Duffy, J., and Hawley, R.H., Experiments on Strain Rate History and Temperature Effects during the Plastic Deformation of Close-Packed Metals, J. Appl. Mech., 1978, vol. 45, no. 1, pp. 60–66. https://doi.org/10.1115/1.342427410.1115/1.3424274 ↗
9. Shabani, M., Indeck, J., Hazeli, K., Jablonski, P., and Pataky, G., Effect of Strain Rate on the Tensile Behavior of CoCrFeNi and CoCrFeMnNi High-Entropy Alloys, J. Mater. Eng. Perform., 2019, vol. 28, pp. 4348–4356. https://doi.org/10.1007/s11665-019-04176-y10.1007/s11665-019-04176-y ↗
10. Shchetinina, N.D., Rudchenko, A.S., and Selivanov, A.A., Application of Mathematical Modeling Methods in the Development of Deformation Modes of Aluminum-Lithium Alloys (Review), Tr. VIAM, 2020, no. 8(90), pp. 20–34.10.18577/2307-6046-2020-0-8-20-34 ↗
11. Kudryavtsev, E.A., Zherebtsov, S.V., Kostyuchenko, S.A., and Salishchev, G.A., Microstructure Evolution and Mechanical Behavior of VT6 Alloy during Multiaxial Isothermal Deformation at 550 and 800°C, Nauch. Vedomosti Belgorod. Gos. Univ. Mat. Fiz., 2011, no. 23(118), pp. 208–214.
12. Rodríguez-Galán, D., Sabirov, I., and Segurado, J., Temperature and Strain Rate Effect on the Deformation of Nanostructured Pure Titanium, Int. J. Plast., 2015, vol. 70, pp. 191–205. https://doi.org/10.1016/j.ijplas.2015.04.00210.1016/j.ijplas.2015.04.002 ↗
13. Chen, S.R. and Gray, G.T., Influence of Twinning on the Constitutive Reponses of Zr: Experiments and Modeling, J. Phys. IV. Proc., 1997. https://doi.org/10.1051/jp4:1997312610.1051/jp4:19973126 ↗
14. Lennon, A.M. and Ramesh, K.T., The Thermoviscoplastic Response of Polycrystalline Tungsten in Compression, Mater. Sci. Eng. A, 2000, vol. 276, pp. 9–21. https://doi.org/10.1016/S0921-5093(99)00517-110.1016/S0921-5093(99)00517-1 ↗
15. Pryakhin, V.V., Low-Cycle Deformation, Fracture and the Unsteady Loading Effect on the Damage Behavior of Die Materials at Operating Temperatures, Mezhd. Zh. Prikl. Fund. Issl., 2014, no. 11-2, pp. 169–176.
16. Wildemann, V.E. and Tretyakov, M.P., Experimental Study of Postcritical Deformation and Failure of Steels at High Temperature, PNRPU Mech. Bull., 2019, no. 1, pp. 27–37. https://doi.org/10.15593/perm.mech/2019.1.0310.15593/perm.mech/2019.1.03 ↗
17. Zhang, L., Yuan, S., Wang, J., Chen, L., and Jin, P., Hot Deformation Behavior Processing Map, Microstructure Evolution and Dynamic Recrystallization Mechanism of Mg-5Al-0.6Sc Alloy, J. Alloys Compd, 2022, vol. 922, p. 166244. https://doi.org/10.2139/ssrn.410113810.2139/ssrn.4101138 ↗
18. Shveykin, A.I., Vshivkova, A.A., and Trusov, P.V., Methods of Accounting for Temperature and Strain Rate Variation in Multilevel Constitutive Models of Metal Deformation (Analytical Review), Phys. Mesomech., 2024, vol. 27, no. 2, pp. 133–151. https://doi.org/10.1134/S102995992402003610.1134/S1029959924020036 ↗
19. Truesdell, C., A First Course in Rational Continuum Mechanics, New York: Academic Press, 1977.
20. Huang, K. and Logé, R.E., A Review of Dynamic Recrystallization Phenomena in Metallic Materials, Mater. Des., 2016, vol. 111, pp. 548–574. https://doi.org/10.1016/j.matdes.2016.09.01210.1016/j.matdes.2016.09.012 ↗
21. Rollett, A., Rohrer, G.S., and Humphreys, J., Recrystallization and Related Annealing Phenomena, Oxford: Elsevier, 2017.
22. Petit-Grostabussiat, S., Taleb, L., and Jullien, J.-F., Experimental Results on Classical Plasticity of Steels Subjected to Structural Transformations, Int. J. Plast., 2004, vol. 20, pp. 1371–1386. https://doi.org/10.1016/j.ijplas.2003.07.00310.1016/j.ijplas.2003.07.003 ↗
23. Barbe, F. and Quey, R., A Numerical Modelling of 3D Polycrystal-to-Polycrystal Diffusive Phase Transformations Involving Crystal Plasticity, Int. J. Plast., 2011, vol. 27, pp. 823–840. https://doi.org/10.1016/j.ijplas.2010.09.00810.1016/j.ijplas.2010.09.008 ↗
24. Vasin, R.A. and Enikeev, F.U., Introduction to the Mechanics of Superplasticity, vol. 1, Ufa: Gilem, 1998.
25. Mulyukov, R.R., Imaev, R.M., Nazarov, A.A., Imaev, V.M., Imaev, M.F., Valitov, V.A., Galeev, R.M., Dmitriev, S.V., Korznikov, A.V., Kruglov, A.A., Lutfullin, R.Ya., Markushev, M.V., Safiullin, R.V., Sitdikov, O.Sh., Trifonov, V.G., and Utyashev, F.Z., Superplasticity of Ultrafine-Grained Alloys: Experiment, Theory, Technology, Moscow: Nauka, 2014.
26. Polukhin, P.I., Gorelik, S.S., and Vorontsov, V.K., Physical Fundamentals of Plastic Deformation, Moscow: Metallurgiya, 1982.
27. Xing, H., Recent Development in the Mechanics of Superplasticity and Its Applications, J. Mater. Process. Technol., 2004, vol. 151, nos. 1–3, pp. 196–202. https://doi.org/10.1016/j.jmatprotec.2004.04.03910.1016/j.jmatprotec.2004.04.039 ↗
28. Shorshorov, M.Kh., Bazyk, A.S., and Kazakov, M.V., Superplasticity of Steels and Alloys and Alternative Metal Forming Technologies, Tula: Izd-vo TGU, 2018.
29. Trusov, P.V. and Shveykin, A.I., Multilevel Crystal Plasticity Models of Single- and Polycrystals. Statistical Models, Phys. Mesomech., 2013, vol. 16, no. 1, pp. 23–33. https://doi.org/10.1134/S102995991301003710.1134/S1029959913010037 ↗
30. Trusov, P.V. and Shveykin, A.I., Multilevel Crystal Plasticity Models of Single- and Polycrystals. Direct Models, Phys. Mesomech., 2013, vol. 16, no. 2, pp. 99–124. https://doi.org/10.1134/S102995991302002110.1134/S1029959913020021 ↗
31. Panin, V.E. and Egorushkin, V.E., Basic Physical Mesomechanics of Plastic Deformation and Fracture of Solids as Hierarchically Organized Nonlinear Systems, Phys. Mesomech., 2015, vol. 18, no. 4, pp. 377–390. https://doi.org/10.1134/S102995991504010410.1134/S1029959915040104 ↗
32. Khadyko, M., Dumoulin, S., Cailletaud, G., and Hopperstad, O.S., Latent Hardening and Plastic Anisotropy Evolution in AA6060 Aluminium Alloy, Int. J. Plast., 2016, vol. 76, pp. 51–74. https://doi.org/10.1016/j.ijplas.2015.07.01010.1016/j.ijplas.2015.07.010 ↗
33. Beyerlein, I. and Knezevic, M., Review of Microstructure and Micro-Mechanism-Based Constitutive Modeling of Polycrystals with a Low-Symmetry Crystal Structure, J. Mater. Res., 2018, vol. 33, no. 22, pp. 3711–3738. https://doi.org/10.1557/jmr.2018.33310.1557/jmr.2018.333 ↗
34. Romanova, V.A., Balokhonov, R.R., Panin, A.V., Batukhtina, E.E., Kazachenok, M.S., and Shakhijanov, V.S., Micromechanical Model of Deformation-Induced Surface Roughening in Polycrystalline Materials, Phys. Mesomech., 2017, vol. 20, no. 3, pp. 324–333. https://doi.org/10.1134/S102995991703008010.1134/S1029959917030080 ↗
35. Trusov, P.V. and Shveykin, A.I., Multilevel Models of Single- and Polycrystalline Materials: Theory, Algorithms, Application Examples, Novosibirsk: Izd-vo SO RAN, 2019.
36. Roters, F., Diehl, M., Shanthraja, P., Eisenlohr, P., Reuber, C., Wong, S.L., Maiti, T., Ebrahimi, A., Hochrainer, T., Fabritius, H.-O., Nikolov, S., Friáka, M., Fujita, N., Grilli, N., Janssens, K.G.F., Jia, N., Kok, P.J.J., Mam, D., Meier, F., Werner, E., Stricker, M., Weygand, D., and Raabe, D., 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, Comput. Mater. Sci., 2019, vol. 158, pp. 420–478. https://doi.org/10.1016/j.commatsci.2018.04.03010.1016/j.commatsci.2018.04.030 ↗
37. Panin, V.E., Foundations of Physical Mesomechanics, Phys. Mesomech., 1998, vol. 1, no. 1, pp. 5–20.
38. Rice, J.R., Inelastic Constitutive Relations for Solids: An Internal-Variable Theory and Its Application to Metal Plasticity, J. Mech. Phys. Solids, 1971, vol. 19, pp. 433–455. https://doi.org/10.1016/0022-5096(71)90010-X10.1016/0022-5096(71)90010-X ↗
39. McDowell, D.L., Internal State Variable Theory, in Handbook of Materials Modeling, Yip, S., Ed., Netherlands: Springer, 2005, pp. 1151–1169. https://doi.org/10.1007/978-1-4020-3286-8_5810.1007/978-1-4020-3286-8_58 ↗
40. Guo, Y.B., Wen, Q., and Horstemeyer, M.F., An Internal State Variable Plasticity-Based Approach to Determine Dynamic Loading History Effects on Material Property in Manufacturing Processes, Int. J. Mech. Sci., 2005, vol. 47, no. 9, pp. 1423–1441. https://doi.org/10.1016/j.ijmecsci.2005.04.01510.1016/j.ijmecsci.2005.04.015 ↗
41. Trusov, P.V., Ashikhmin, V.N., Volegov, P.S. and Shveykin, A.I., Constitutive Relations and Their Application to the Description of Microstructure Evolution, Phys. Mesomech., 2010, vol. 13, no. 1–2, pp. 38–46. https://doi.org/10.1016/j.physme.2010.03.00510.1016/j.physme.2010.03.005 ↗
42. Horstemeyer, M.F., Potirniche, G.P., and Marin, E.B., Crystal Plasticity, in Handbook of Materials Modeling, Yip, S., Ed., Netherlands: Springer, 2005, pp. 1133–1149.10.1007/978-1-4020-3286-8_57 ↗
43. Maugin, G.A., The Saga of Internal Variables of State in Continuum Thermo-Mechanics (1893–2013), Mech. Res. Communicat., 2015, vol. 69, pp. 79–86. https://doi.org/10.1016/j.mechrescom.2015.06.00910.1016/j.mechrescom.2015.06.009 ↗
44. Romanova, V.A., Balokhonov, R.R., and Karpenko, N.I., Simulation of the Mechanical Behavior of Materials with Regard to the 3D Internal Structure, Fiz. Mezomekh., 2004, vol. 7, no. 2, pp. 71–79.
45. Trusov, P., Shveykin, A., and Kondratev, N., Some Issues on Crystal Plasticity Models Formulation: Motion Decomposition and Constitutive Law Variants, Crystals, 2021, vol. 11, p. 1392. https://doi.org/10.3390/cryst1111139210.3390/cryst11111392 ↗
46. Essmann, U. and Mughrabi, H., Annihilation of Dislocations during Tensile and Cyclic Deformation and Limits of Dislocation Densities, Philos. Mag. A, 1979, vol. 40(6), pp. 731–756.10.1080/01418617908234871 ↗
47. Cheong, K.S. and Busso, E.P., Discrete Dislocation Density Modelling of Single Phase FCC Polycrystal Aggregates, Acta Mater., 2004, vol. 52, pp. 5665–5675. http://dx.doi.org/10.1016/j.actamat.2004.08.04410.1016/j.actamat.2004.08.044 ↗
48. Austin, R. and McDowell, D., A Dislocation-Based Constitutive Model for Viscoplastic Deformation of FCC Metals at Very High Strain Rates, Int. J. Plast., 2011, vol. 27, no. 1, pp. 1–24. https://doi.org/10.1016/j.ijplas.2010.03.00210.1016/j.ijplas.2010.03.002 ↗
49. Grilli, N., Janssens, K.G.F., Nellessen, J., Sandlobes, S., and Raabe, D., Multiple Slip Dislocation Patterning in a Dislocation-Based Crystal Plasticity Finite Element Method, Int. J. Plast., 2018, vol. 100, pp. 104121. https://doi.org/10.1016/j.ijplas.2017.09.01510.1016/j.ijplas.2017.09.015 ↗
50. Gribov, D.S. and Trusov, P.V., Three-Level Dislocation-Based Model for Describing the Deformation of Polycrystals: Structure, Implementation Algorithm, Examples for Studying Nonproportional Cyclic Loading, Phys. Mesomech., 2022, vol. 25, no. 6, pp. 557–567. https://doi.org/10.1134/S102995992206008X10.1134/S102995992206008X ↗
51. Trusov, P.V., Shveykin, A.I., Kondratyev, N.S., and Yants, A.Yu., Multilevel Models in Physical Mesomechanics of Metals and Alloys: Results and Prospects, Phys. Mesomech., 2021, vol. 24, no. 4, pp. 391–417. https://doi.org/10.1134/S102995992104005610.1134/S1029959921040056 ↗
52. Van Houtte, P., Li, S., Seefeldt, M., and Delannay, L., Deformation Texture Prediction: From the Taylor Model to the Advanced Lamel Model, Int. J. Plast., 2005, vol. 21, pp. 589–624. https://doi.org/10.1016/j.ijplas.2004.04.01110.1016/j.ijplas.2004.04.011 ↗
53. Zhao, D., Zhu, Y., Ying, L., Hu, P., and Zhang, W., Modeling and Experimental Verification of Thermo-Mechanical Coupled Behavior of Face-Centered-Cubic Polycrystals, Mater. Des., 2013, vol. 52, pp. 289–294. https://doi.org/10.1016/j.matdes.2013.05.09210.1016/j.matdes.2013.05.092 ↗
54. Zhang, K., Holmedal, B., Hopperstad, O.S., Dumoulin, S., Gawad, J., Van Bael, A., and 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, vol. 66, pp. 3–30. https://doi.org/10.1016/j.ijplas.2014.02.00310.1016/j.ijplas.2014.02.003 ↗
55. Alleman, C., Luscher, D.J., Bronkhorst, C., and Ghosh, S., Distribution-Enhanced Homogenization Framework and Model for Heterogeneous Elasto-Plastic Problems, J. Mech. Phys. Solids, 2015, vol. 85, pp. 176–202. https://doi.org/10.1016/j.jmps.2015.09.01210.1016/j.jmps.2015.09.012 ↗
56. Habraken, A.M., Modelling the Plastic Anisotropy of Metals, Arch. Comput. Meth. Engng, 2004, vol. 11, no. 1, pp. 3–96. https://doi.org/10.1007/BF0273621010.1007/BF02736210 ↗
57. Perdahcioğlu, E.S., Constitutive Modeling of Metastable Austenitic Stainless Steel: PhD Thesis, Enschede, The Netherlands, 2008. https://doi.org/10.3990/1.978903652769910.3990/1.9789036527699 ↗
58. Lebensohn, R.A., Ponte Castañeda, P., Brenner, R., and Castelnau, O., Full-Field Versus Homogenization Methods to Predict Microstructure–Property Relations for Polycrystalline Materials, in Computational Methods for Microstructure–Property Relationships, Ghosh, S. and Dimiduk, D., Eds., Springer Science + Business Media, LLC, 2011, pp. 393–441. https://doi.org/10.1007/978-1-4419-0643-41110.1007/978-1-4419-0643-411 ↗
59. McDowell, D.L., Viscoplasticity of Heterogeneous Metallic Materials, Mater. Sci. Eng. R, 2008, vol. 62, pp. 67–123. https://doi.org/10.1016/j.mser.2008.04.00310.1016/j.mser.2008.04.003 ↗
60. Watanabe, I. and Setoyama, D., Multiscale Characterization of a Polycrystalline Aggregate Subjected to Severe Plastic Deformation with the Finite Element Method, Mater. Trans. Spec. Iss. Advanced Materials Science in Bulk Nanostructured Metals III, 2016, pp. 1–7. https://doi.org/10.2320/matertrans.MH20151410.2320/matertrans.MH201514 ↗
61. Ha, S., Jang, J.-H., and Kim, K.T., Finite Element Implementation of Dislocation-Density-Based Crystal Plasticity Model and Its Application to Pure Aluminum Crystalline Materials, Int. J. Mech. Sci., 2017, vol. 120, pp. 249–262. https://doi.org/10.1016/j.ijmecsci.2016.11.01110.1016/j.ijmecsci.2016.11.011 ↗
62. Shveykin, A.I., Trusov, P.V., and Sharifullina, E.R., Statistical Crystal Plasticity Model Advanced for Grain Boundary Sliding Description, Crystals, 2020, vol. 10, no. 9, p. 822. https://doi.org/10.3390/cryst1009082210.3390/cryst10090822 ↗
63. Trusov, P., Kondratev, N., and Podsedertsev, A., Description of Dynamic Recrystallization by Means of an Advanced Statistical Multilevel Model: Grain Structure Evolution Analysis, Crystals, 2022, vol. 12, no. 5, p. 653. https://doi.org/10.3390/cryst1205065310.3390/cryst12050653 ↗
64. Kondratev, N., Trusov, P., Podsedertsev, A., and Baldin, M., Subgrain Coalescence Simulation by Means of an Advanced Statistical Model of Inelastic Deformation, Materials, 2022, vol. 15, no. 15, p. 5406. https://doi.org/10.3390/ma1515540610.3390/ma15155406 ↗
65. Terentyev, D., Xiao, X., Dubinko, A., Bakaeva, A., and Duan, H., Dislocation-Mediated Strain Hardening in Tungsten: Thermo-Mechanical Plasticity Theory and Experimental Validation, J. Mech. Phys. Solids, 2015, vol. 85, pp. 1–15. http://dx.doi.org/10.1016/j.jmps.2015.08.01510.1016/j.jmps.2015.08.015 ↗
66. Trusov, P.V., Shveykin, A.I., and Kondratev, N.S., Multilevel Metal Models: Formulation for Large Displacements Gradients, Nanosci. Technol. Int. J., 2017, vol. 8, no. 2, pp. 133–166. https://doi.org/10.1615/NanoSciTechnolIntJ.v8.i2.4010.1615/NanoSciTechnolIntJ.v8.i2.40 ↗
67. Beyerlein, I.J. and Tome, C.N., A Dislocation-Based Constitutive Law for Pure Zr Including Temperature Effects, Int. J. Plast., 2008, vol. 24, pp. 867–895. https://doi.org/10.1016/j.ijplas.2007.07.01710.1016/j.ijplas.2007.07.017 ↗
68. Trusov, P.V. and Shveykin, A.I., On Motion Decomposition and Constitutive Relations in Geometrically Nonlinear Elastoviscoplasticity of Crystallites, Phys. Mesomech., 2017, vol. 20, no. 4, pp. 377–391. https://doi.org/10.1134/S102995991704002610.1134/S1029959917040026 ↗
69. Mishra, A., Kad, B.K., Gregori, F., and Meyers, M.A., Microstructural Evolution in Copper Subjected to Severe Plastic Deformation: Experiments and Analysis, Acta Mater., 2007, vol. 55, pp. 13–28. https://doi.org/10.1016/j.actamat.2006.07.00810.1016/j.actamat.2006.07.008 ↗
70. Shveykin, A.I. and Trusov, P.V., Multilevel Models of Polycrystalline Metals: Comparison of Relations Describing the Rotations of Crystallite Lattice, Nanosci. Technol. Int. J., 2019, vol. 10, no. 1, pp. 120. https://doi.org/10.1615/NanoSciTechnolIntJ.201802867310.1615/NanoSciTechnolIntJ.2018028673 ↗
71. Anand, L., Single-Crystal Elasto-Viscoplasticity: Application to Texture Evolution in Polycrystalline Metals at Large Strains, Comput. Methods Appl. Mech. Eng., 2004, vol. 193, pp. 5359–5383. https://doi.org/10.1016/j.cma.2003.12.06810.1016/j.cma.2003.12.068 ↗
72. Maresca, F., Kouznetsova, V.G., and Geers, M.G.D., Reduced Crystal Plasticity for Materials with Constrained Slip Activity, Mech. Mater., 2016, vol. 92, pp. 198–210. https://doi.org/10.1016/j.mechmat.2015.09.01110.1016/j.mechmat.2015.09.011 ↗
73. Shveykin, A.I., Multilevel Models of Polycrystalline Metals: Comparison of Constitutive Relations for Crystallites, Probl. Prochnost. Plastich., 2017, vol. 79, no. 4, pp. 385–397.
74. Shveykin, A.I. and Sharifullina, E.R., Development of Multilevel Models Based on Crystal Plasticity: Description of Grain Boundary Sliding and Evolution of Grain Structure, Nanomechanics Sci. Technol. Int. J., 2015, vol. 6, no. 4, pp. 281–298. https://doi.org/10.1615/NanomechanicsSciTechnolIntJ.v6.i4.3010.1615/NanomechanicsSciTechnolIntJ.v6.i4.30 ↗
75. Toth, L.S., Skrotzki, W., Zhao, Y., Pukenas, A., Braun, C., and Birringer, R., Revealing Grain Boundary Sliding from Textures of a Deformed Nanocrystalline Pd–Au Alloy, Materials, 2018, vol. 11, no. 2, p. 190. https://doi.org/10.3390/ma1102019010.3390/ma11020190 ↗
76. Zhao, Y., Toth, L., Massion, R., and Skrotzki, W., Role of Grain Boundary Sliding in Texture Evolution for Nanoplasticity, Adv. Eng. Mater., 2018, vol. 20, p. 1700212. https://doi.org/10.1002/adem.20170021210.1002/adem.201700212 ↗
77. Trusov, P.V., Sharifullina, E.R., and Shveykin, A.I., Multilevel Model for the Description of Plastic and Superplastic Deformation of Polycrystalline Materials, Phys. Mesomech., 2019, vol. 22, no. 5, pp. 402–419. https://doi.org/10.1134/S102995991905007210.1134/S1029959919050072 ↗
78. Tang, T., Zhou, G., Li, Z., Li, D., Peng, L., Peng, Y., Wu, P., Wang, H., and Lee, M.-G., A Polycrystal Plasticity Based Thermo-Mechanical-Dynamic Recrystallization Coupled Modeling Method and Its Application to Light Weight Alloys, Int. J. Plast., 2019, vol. 116, pp. 159–191. https://doi.org/10.1016/j.ijplas.2019.01.00110.1016/j.ijplas.2019.01.001 ↗
79. Zecevic, M., Knezevic, M., McWilliams, B., and Lebensohn, R.A., Modeling of the Thermo-Mechanical Response and Texture Evolution of WE43 Mg Alloy in the Dynamic Recrystallization Regime Using a Viscoplastic Self-Consistent Formulation, Int. J. Plast., 2020, vol. 130, p. 102705. https://doi.org/10.1016/j.ijplas.2020.10270510.1016/j.ijplas.2020.102705 ↗
80. Trusov, P.V., Kondratev, N.S., and Yanz, A.Yu., A Model for Static Recrystallization through Strain-Induced Boundary Migration, Phys. Mesomech., 2020, vol. 23, no. 2, pp. 97–108. https://doi.org/10.1134/S102995992002001010.1134/S1029959920020010 ↗
81. Cyr, E.D., Mohammadi, M., Mishra, R.K., and Inal, K., A Three Dimensional (3D) Thermo-Elasto-Viscoplastic Constitutive Model for FCC Polycrystals, Int. J. Plast., 2015, vol. 70, pp. 166–190. https://doi.org/10.1016/j.ijplas.2015.04.00110.1016/j.ijplas.2015.04.001 ↗
82. Forest, S. and Rubin, M.B., A Rate-Independent Crystal Plasticity Model with a Smooth Elastic-Plastic Transition and No Slip Indeterminacy, Eur. J. Mech. A Solids, 2016, vol. 55, pp. 278–288. https://doi.org/10.1016/j.euromechsol.2015.08.01210.1016/j.euromechsol.2015.08.012 ↗
83. Li, J., Romero, I., and Segurado, J., Development of a Thermo-Mechanically Coupled Crystal Plasticity Modeling Framework: Application to Polycrystalline Homogenization, Int. J. Plast., 2019, vol. 119, pp. 313–330. https://doi.org/10.1016/j.ijplas.2019.04.00810.1016/j.ijplas.2019.04.008 ↗
84. Xie, M. and Chen, G., A Developed Crystal Plasticity Model for Viscoplastic Mechanical Behavior of SAC305 Solder under Thermomechanical Coupled Cyclic Loading, Int. J. Plast., 2022, vol. 159, p. 103465. https://doi.org/10.1016/j.ijplas.2022.10346510.1016/j.ijplas.2022.103465 ↗
85. Romanov, K., Shveykin, A., and Trusov, P., Advanced Statistical Crystal Plasticity Model: Description of Copper Grain Structure Refinement during Equal Channel Angular Pressing, Metals, 2023, vol. 13, no. 5, p. 953. https://doi.org/10.3390/met1305095310.3390/met13050953 ↗
86. Yu, H., Xin, Y., Wang, M., and Liu, Q., Hall–Petch Relationship in Mg Alloys: A Review, J. Mater. Sci. Technol., 2018, vol. 34, no. 2, pp. 248–256. https://doi.org/10.1016/j.jmst.2017.07.02210.1016/j.jmst.2017.07.022 ↗
87. Liu, T., Li, Y., and Ren, Y., Effect of Pr Inoculation and Crystal Size on the Hall–Petch Relationship for Al-30 wt % Mg2Si Composites, Mater. Lett., 2018, vol. 214, pp. 6–9. https://doi.org/10.1016/j.matlet.2017.11.10510.1016/j.matlet.2017.11.105 ↗
88. Chong, Y., Deng, G., Gao, S., Yi, J., Shibata, A., and Tsuji, N., Yielding Nature and Hall–Petch Relationships in Ti-6Al-4V Alloy with Fully Equiaxed and Bimodal Microstructures, Scripta Mater., 2019, vol. 172, pp. 77–82. https://doi.org/10.1016/j.scriptamat.2019.07.01510.1016/j.scriptamat.2019.07.015 ↗
89. Chen, S., Aitken, Z.H., Wu, Z., Yu, Z., Banerjee, R., and Zhang, Y.-W., Hall–Petch and Inverse Hall–Petch Relations in High-Entropy CoNiFeAlxCu1–x Alloys, Mater. Sci. Eng. A, 2020, vol. 773, p. 138873. https://doi.org/10.1016/j.msea.2019.13887310.1016/j.msea.2019.138873 ↗
90. Rubio, R.A., Haouala, S., and LLorca, J., Grain Boundary Strengthening of FCC Polycrystals, J. Mater. Res., 2019, vol. 34, pp. 2263–2274. https://doi.org/10.1557/jmr.2019.5810.1557/jmr.2019.58 ↗
91. Khan, A.S. and Liu, H., Variable Strain Rate Sensitivity in an Aluminum Alloy: Response and Constitutive Modeling, Int. J. Plast., 2012, vol. 36, pp. 1–14. https://doi.org/10.1016/j.ijplas.2012.02.00110.1016/j.ijplas.2012.02.001 ↗
92. Dursun, T. and Soutis, C., Recent Developments in Advanced Aircraft Aluminium Alloys, Mater. Design, 2014, vol. 56, pp. 862–871. https://doi.org/10.1016/j.matdes.2013.12.00210.1016/j.matdes.2013.12.002 ↗
93. Khan, M.K., Hainsworth, S.V., Fitzpatrick, M.E., and Edwards, L., A Combined Experimental and Finite Element Approach for Determining Mechanical Properties of Aluminium Alloys by Nanoindentation, Comput. Mater. Sci., 2010, vol. 49, pp. 751–760. https://doi.org/10.1016/j.commatsci.2010.06.01810.1016/j.commatsci.2010.06.018 ↗
94. Himmelblau, D.M., Applied Nonlinear Programming, New York: McGraw-Hill, 1972.
95. Minoux, M., Mathematical Programming: Theory and Algorithms, Chichester–New York: Wiley, 1986.
96. Gao, F. and Han, L., Implementing the Nelder–Mead Simplex Algorithm with Adaptive Parameters, Comput. Optim. Appl., 2012, vol. 51, no. 1, pp. 259–277. https://doi.org/10.1007/s10589-010-9329-310.1007/s10589-010-9329-3 ↗
97. Diniz-Ehrhardt, M., Martínez, J.M., and Pedroso, L., Derivative-Free Methods for Nonlinear Programming with General Lower-Level Constraints, Comput. Appl. Math., 2016, vol. 30, pp. 19–52. https://doi.org/10.1590/S1807-0302201100010000310.1590/S1807-03022011000100003 ↗
98. Voglis, C., Parsopoulos, K.E., Papageorgiou, D.G., Lagaris, I.E., and Vrahatis, M.N., MEMPSODE: A Global Optimization Software Based on Hybridization of Population-Based Algorithms and Local Searches, Comput. Phys. Commun., 2012, vol. 183, no. 2, pp. 1139–1154. https://doi.org/10.1016/j.cpc.2012.01.01010.1016/j.cpc.2012.01.010 ↗
99. Rios-Coelho, A.C., Sacco, W.F., and Henderson, N., A Metropolis Algorithm Combined with Hooke–Jeeves Local Search Method Applied to Global Optimization, Appl. Math. Comput., 2010, vol. 217, no. 2, pp. 843–853. https://doi.org/10.1016/j.amc.2010.06.02710.1016/j.amc.2010.06.027 ↗
100. Sulimov, V.D., Shkapov, P.M., and Nosachev, S.K., Local Search by the Hooke–Jeeves Method in a Hybrid Global Optimization Algorithm, Nauk. Obrazov. Nauch. Izd. MGTU Baumana, 2014, no. 6, pp. 107–123.