Abstract
A three-level constitutive model is proposed for describing the deformation of polycrystalline materials, which is based on crystal elasto-viscoplasticity and the introduction of internal variables. The structure, mathematical formulation, and implementation algorithm of the model are discussed. The element of the upper structural-scale level is the representative macrovolume. The elements of mesolevels 1 and 2, which are identical in scale, are crystallites (grains, subgrains, fragments, depending on the required element size). The description at mesolevel 1 is performed in terms of thermomechanical variables (stresses, strains, strain rates). The behavior of meso-2 elements is described in terms of dislocation densities and velocities. Particular attention is paid to the formation of barriers on split dislocations. As an example, the model is applied to study proportional and nonproportional cyclic loading of samples with substantially different stacking fault energies. It is shown that barriers are more readily formed in materials with low stacking fault energy, leading to their additional cyclic hardening under nonproportional loading.
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REFERENCES
Ilyushin, A.A., Plasticity: Fundamentals of General Mathematical Theory, Moscow: AN SSSR, 1963.
Sokolovskii, V.V., Theory of Plasticity, Moscow: Vyssh. Shkola, 1969.
Malinin, N.N., Applied Theory of Plasticity and Creep, Moscow: Mashinostroenie, 1968.
Kachanov, L.M., Foundations of the Theory of Plasticity, North-Holland Pub. Co., Amsterdarm, 1971.
Pozdeev, A.A., Trusov, P.V., and Nyashin, Yu.I., Large Elastoplastic Deformations: Theory, Algorithms, Applications, Moscow: Nauka, 1986.
Laird, C., Charsley, P., and Mughrabi, H., Low Energy Dislocation Structures Produced by Cyclic Deformation, Mater. Sci. Eng., 1986, vol. 81, pp. 433–450.
Vasin, R.A., Constitutive Relations in the Theory of Plasticity, Itogi Nauk. Tekhn. Mekh. Deform. Tv. Tela. VINITI, 1990, vol. 21, pp. 3–75.
Doquet, V., Twinning and Multiaxial Cyclic Plasticity of a Low Stacking-Fault-Energy F.C.C. Alloy, Acta Metall. Mater., 1993, vol. 41, pp. 2451–2459.
Rogovoy, A.A., Formalized Approach to the Construction of Solid Mechanics Models. Part 1. Basic Equations of Continuum Mechanics, Moscow: Institute of Computer Science, 2021.
Taylor, G.I., Plastic Strain in Metals, J. Inst. Met., 1938, vol. 62, pp. 307–324.
Bishop, J.F. and Hill, R., A Theory of the Plastic Distortion of a Polycrystalline Aggregate under Combined Stresses, Philos. Mag. Ser. 7, 1951, vol. 42, no. 327, pp. 414–427. https://doi.org/10.1080/14786445108561065
Bishop, J.F.W. and Hill, R., A Theoretical Derivation of the Plastic Properties of a Polycrystalline Face-Centered Metal, Philos. Mag. Ser. 7, 1951, vol. 42, no. 334, pp. 1298–1307. https://doi.org/10.1080/14786444108561385
Physical Mesomechanics of Heterogeneous Media and Computer-Aided Design of Materials, Panin, V.E., Ed., Cambridge: Cambridge Interscience Publishing, 1998.
Krivtsov, A.M., Deformation and Fracture of Solids with Microstructure, Moscow: FIZMATLIT, 2007.
Horstemeyer, M.F., Multiscale Modeling: A Review, in Practical Aspects of Computational Chemistry, Leszczynski, J. and Shukla, M.K., Eds., Heidelberg: Springer, 2009, pp. 87–135. https://doi.org/10.1007/978-90-481-2687-3_4
Roters, F., Advanced Material Models for the Crystal Plasticity Finite Element Method: Development of a General CPFEM Framework, Aachen: RWTH Aachen, 2011.
Li, P., Li, S.X., Wang, Z.G., and Zhang, Z.F., Fundamental Factors on Formation Mechanism of Dislocation Arrangements in Cyclically Deformed FCC Single Crystals, Progr. Mater. Sci., 2011, vol. 56, pp. 328–377. https://doi.org/10.1016/J.PMATSCI.2010.12.001
Cho, J., Molinari, J.-F., and Anciaux, G., Mobility Law of Dislocations with Several Character Angles and Temperatures in FCC Aluminum, Int. J. Plasticity, 2017, vol. 90, pp. 66–75. https://doi.org/10.1016/j.ijplas.2016.12.004
Romanova, V.A., Balokhonov, R.R., Batukhtina, E.E., Emelyanova, E.S., and Sergeev, M.V., On the Solution of Quasi-Static Micro- and Mesomechanical Problems in a Dynamic Formulation, Phys. Mesomech., 2019, vol. 22, no. 4, pp. 296–306. https://doi.org/10.1134/S1029959919040052
Bisht, A., Kumar, L., Subburaj, J., Jagadeesh, G., and Suwas, S., Effect of Stacking Fault Energy on the Evolution of Microstructure and Texture during Blast Assisted Deformation of FCC Materials, J. Mater. Process. Technol., 2019, vol. 271, pp. 568–583. https://doi.org/10.1016/j.jmatprotec.2019.04.029
Liang, Q., Weng, S., Fu, T., Hu, S., and Peng, X., Dislocation Reaction-Based Formation Mechanism of Stacking Fault Tetrahedra in FCC High-Entropy Alloy, Mater. Chem. Phys., 2022, vol. 282, p. 125997. https://doi.org/10.1016/j.matchemphys.2022.125997
McDowell, D.L., A Perspective on Trends in Multiscale Plasticity, Int. J. Plasticity, 2010, vol. 26, pp. 1280–1309. https://doi.org/10.1016/j.ijplas.2010.02.008
Beyerlein, I. and Knezevic, M., Review of Microstructure and Micromechanism-Based Constitutive Modeling of Polycrystals with a Low-Symmetry Crystal Structure, J. Mater. Res., 2018, vol. 33, pp. 3711–3738. https://doi.org/10.1557/jmr.2018.333
Trusov, P.V. and Shveikin, A.I., Multilevel Models of Single- and Polycrystalline Materials: Theory, Algorithms, Application Examples, Novosibirsk: Izd-vo SO RAN, 2019.
Trusov, P.V. and Gribov, D.S., The Three-Level Elastoviscoplastic Model and Its Application to Describing Complex Cyclic Loading of Materials with Different Stacking Fault Energies, Materials, 2022, vol. 15(3). https://doi.org/10.3390/ma15030760
Coleman, B.D. and Gurtin, M.E., Thermodynamics with Internal State Variables, J. Chem. Phys., 1967, vol. 47, pp. 597–613. https://doi.org/10.1063/1.1711937
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-X
Maugin, G.A., Mechanics of Electromagnetic Solids, Norwell: Kluwer Academic Publishers, 2003.
Ashikhmin, V.N., Volegov, P.S., and Trusov, P.V., Constitutive Equations with Internal Variables: General Structure and Application to Texture Formation in Polycrystals, PNRPU Bull. Mat. Modelir. Sistem Protsess., 2006, no. 14, pp. 11–26.
Trusov, P.V. and Shveikin, A.I., Theory of Plasticity, Perm: Izd-vo PNRPU, 2011.
Maugin, G.A., The Saga of Internal Variables of State in Continuum Thermo-Mechanics (1893–2013), Mech. Res. Commun., 2015, vol. 69, pp. 79–86. https://doi.org/10.1016/j.mechrescom.2015.06.00
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/S1029959917040026
Orowan, E., Problems of Plastic Gliding, Proc. Phys. Soc., 1940, vol. 52, pp. 1926–1948. https://doi.org/10.1088/0959-5309/52/1/303
Kocks, U.F., Constitutive Behavior Based on Crystal Plasticity, in Unified Constitutive Equations for Creep and Plasticity, Miller, A.K., Ed., Dordrecht: Springer, 1987, pp. 1–88. https://doi.org/10.1007/978-94-009-3439-9_1
Orlov, A.N., Introduction to the Theory of Defects in Crystals, Moscow: Vyssh. Shkola, 1983.
Arsenlis, A. and Parks, D.M., Modeling the Evolution of Crystallographic Dislocation Density in Crystal Plasticity, J. Mech. Phys. Solids, 2002, vol. 50, pp. 1979–2009. https://doi.org/10.1016/S0022-5096(01)00134-X
Shtremel, M.A., Strength of Alloys. Part I. Lattice Defects, Moscow: MISIS, 1999.
Cottrell, A.H., Dislocations and Plastic Flow in Crystals, Oxford University Press, New York, 1953.
Franciosi, P., The Concepts of Latent Hardening and Strain Hardening in Metallic Single Crystals, Acta Metall., 1985, vol. 33, pp. 1601–1612. https://doi.org/10.1016/0001-6160(85)90154-3
Benallal, A. and Marquis, D., Effects of Non-Proportional Loadings in Cyclic Elasto-Viscoplasticity: Experimental, Theoretical and Numerical Aspects, Eng. Comput., 1988, vol. 5, pp. 241–247. https://doi.org/10.1108/eb023742
Benallal, A., Le Gallo, P., and Marquis, D., An Experimental Investigation of Cyclic Hardening of 316 Stainless Steel and of 2024 Aluminium Alloy under Multiaxial Loadings, Nucl. Eng. Design, 1989, vol. 114, pp. 345–353. https://doi.org/10.1016/0029-5493(89)90112-x
Xia, Z. and Ellyin, F., Nonproportional Multiaxial Cyclic Loading: Experiments and Constitute Modeling, J. Appl. Mech., 1991, vol. 58, pp. 317–325. https://doi.org/10.1115/1.2897188
Aubin, V., Quaegebeur, P., and Degallaix, S., Cyclic Behaviour of a Duplex Stainless Steel under Multiaxial Loading: Experiments and Modelling, Eur. Struct. Integr. Soc., 2003, vol. 31, pp. 401–422. https://doi.org/10.1016/S1566-1369(03)80022-5
Zhang, J. and Jiang, Y., An Experimental Investigation on Cyclic Plastic Deformation and Substructures of Polycrystalline Copper, IJOP, 2005, vol. 21, pp. 2191–2211. https://doi.org/10.1016/j.ijplas.2005.02.004
Funding
The work was carried out with financial support from the Ministry of Science and Higher Education of the Russian Federation (as basic part of the government statement of work for PNRPU, Project No. FSNM-020-0027) and the Russian Foundation for Basic Research (Project No. 20-41-596002 r (Perm Scientific and Educational Center)).
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Gribov, D.S., 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 25, 557–567 (2022). https://doi.org/10.1134/S102995992206008X
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DOI: https://doi.org/10.1134/S102995992206008X