Three-level model based on physical theories of plasticity: Formulation, implementation algorithms, results of application to the study of cyclic loading
DOI:
https://doi.org/10.7242/1999-6691/2022.15.3.21Keywords:
three-level dislocation-based model, simple and complex cyclic loading, additional cyclic hardening effectAbstract
The physical and mechanical properties of metals and alloys, as well as, the operational characteristics of products made of them, are known to be significantly determined by the meso- and microstructure of materials. Therefore, physically oriented models that can be used to analyze the material structure evolution have been intensively developed and widely applied in recent decades for the study of thermomechanical processing of metals and alloys. These models are based on the introduction of internal variables, theories of crystal plasticity (elastoviscous plasticity), and a multilevel approach. In this paper, we consider the structure, mathematical formulation and algorithm for implementing a three-level (macro-, meso-1, and meso-2 levels) dislocation-oriented model designed to study the behavior of a representative macrovolume (macrosample) of mono- and polycrystalline alloys under cyclic deformation paths. The loading is given, and the Voigt (Taylor) hypothesis is used to connect the macro- and mesolevel-1. The mesolevel-2 submodel operates with the densities and velocities of full and split edge dislocations on slip systems. Interactions of dislocations of various systems, such as annihilation, hardening due to forest dislocations, formation of barriers of a dislocation nature (Lomer–Cottrell, Hirt) are taken into account. At mesolevel-1, the description is carried out in terms of shear stresses and shear rates along slip systems, determined by the Orowan equation using the mesolevel-2 data. A rigid moving coordinate system associated with a crystal lattice is introduced to evaluate the rotation of crystallites. The response of the material at the macro level is determined by averaging the stresses in crystallites. The results of applying the model to the study of deformation of alloy macrosamples with different values of stacking fault energy along simple and complex deformation trajectories are presented. It is shown that the materials with low stacking fault energy demonstrate the effect of additional cyclic hardening when loaded along complex deformation trajectories.
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References
Likhachev V.A., Malinin V.G. Strukturno-analiticheskaya teoriya prochnosti [Structural-analytical theory of strength]. Saint Petersburg, Nauka, 1993. 471 p.
Panin V.E. (ed.) Fizicheskaya mezomekhanika i komp’yuternoye konstruirovaniye materialov [Physical mesomechanics and the computer design of materials]. Novosibirsk, Nauka, 1995. Vol. 1, 298 p.
Panin V.E. (ed.) Fizicheskaya mezomekhanika i komp’yuternoye konstruirovaniye materialov [Physical mesomechanics and the computer design of materials]. Novosibirsk, Nauka, 1995. Vol. 2, 320 p.
Trusov P.V., Shveykin A.I. Teoriya plastichnosti [Plasticity theory]. Perm, Izd-vo PNIPU, 2011. 419 p.
Il’yushin A.A. Plastichnost’. Osnovy obshchey matematicheskoy teorii [Plasticity. Fundamentals of mathematical theory]. Moscow, AN SSSR, 1963. 272 p.
Kachanov L.M. Osnovy teorii plastichnosti [Fundamentals of the theory of plasticity]. Moscow, Nauka, 1969. 420 p.
Sokolovskiy V.V. Teoriya plastichnosti [Plasticity theory]. Moscow, Vysshaya shkola, 1969. 608 p.
Vasin R.A. Opredelyayushchiye sootnosheniya teorii plastichnosti [Constitutive relations of the theory of plasticity]. Itogi nauki i tekhniki. Ser. Mekhanika deformiruemogo tverdogo tela, 1990, vol. 21, pp. 3-75.
Bondar' V.S. Neuprugost’. Varianty teorii [Inelasticity. Theorys]. Moscow, Fizmatlit, 2004. 144 p.
Vasin R.A. Svoystva funktsionalov plastichnosti u metallov, opredelyayemyye v eksperimentakh na dvuzvennykh trayektoriyakh deformatsii [Properties of metals plasticity functionals, determined in experiments on two-link strain trajectories]. Uprugost’ i neuprugost’ [Elasticity and inelasticity]. Moscow, MGU, 1987. Pp. 115-127.
Zubchaninov V.G. Osnovy teorii uprugosti i plastichnosti [Fundamentals of the theory of elasticity and plasticity]. Moscow, Vysshaya shkola, 1990. 368 p.
Annin B.D., Zhigalkin V.M. Povedeniye materialov v usloviyakh slozhnogo nagruzheniya [Behavior of materials under complex loading conditions]. Novosibirsk, Izd-vo SO RAN, 1999. 342 p.
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. Continuum mechanics of electromagnetic solids. North Holland, Amsterdam, 1988. 598 p.
McDowell D.L. Internal state variable theory. Handbook of materials modeling, ed. S. Yip. Springer, 2005. Pp. 1151-1169. https://doi.org/10.1007/978-1-4020-3286-8_58
Ashihmin V.N., Volegov P.S., Trusov P.V. Konstitutivnyye sootnosheniya s vnutrennimi peremennymi: obshchaya struktura i prilozheniye k teksturoobrazovaniyu v polikristallakh [Constitutive relations with internal variables: General structure and application to texture formation in polycrystals]. Vestnik PGTU. Matematicheskoye modelirovaniye sistem i protsessov, 2006, no. 14, pp. 11-26.
Horstemeyer M.F., Bammann D.J. Historical review of internal state variable theory for inelasticity. Int. J. Plast., 2010, vol. 26, pp. 1310-1334. https://doi.org/10.1016/j.ijplas.2010.06.005
Maugin G.A. The saga of internal variables of state in continuum thermo-mechanics (1893-2013). Mech. Res. Comm., 2015, vol. 69, pp. 79-86. https://doi.org/10.1016/j.mechrescom.2015.06.009
Taylor G.I. Plastic strain in metals. J. Inst. Metals, 1938, vol. 62, pp. 307-324.
Lin P., El-Azab A. Implementation of annihilation and junction reactions in vector density-based continuum dislocation dynamics. Modelling Simul. Mater. Sci. Eng., 2020, vol. 28, 045003. https://doi.org/10.1088/1361-651X/ab7d90
Asaro R.J., Needleman A. Texture development and strain hardening in rate dependent polycrystals. Acta Metall., 1985, vol. 33, pp. 923-953. https://doi.org/10.1016/0001-6160(85)90188-9
Horstemeyer M.F. Multiscale modeling: A review. Practical aspects of computational chemistry, ed. J. Leszczynski, M.K. Shukla. Springer, 2009. Рp. 87-135. https://doi.org/10.1007/978-90-481-2687-3_4
McDowell D.L. A perspective on trends in multiscale plasticity. Int. J. Plast., 2010, vol. 26, pp. 1280-1309. https://doi.org/10.1016/j.ijplas.2010.02.008
Roters F. Advanced material models for the crystal plasticity finite element method: Development of a general CPFEM framework. Aachen: RWTH Aachen, 2011. 226 р.
Trusov P.V., Shveykin A.I. Mnogourovnevyye modeli mono- i polikristallicheskikh materialov: teoriya, algoritmy, primery primeneniya [Multilevel models of mono- and polycrystalline materials: theory, algorithms, examples of application]. Novosibirsk, Izd-vo SO RAN, 2019. 605 p.
Dupuy L., Fivel M.C. A study of dislocation junctions in FCC metals by an orientation dependent line tension model. Acta Mater., 2002, vol. 50, pp. 4873-4885. https://doi.org/10.1016/s1359-6454(02)00356-7
Krempl E., Lu H. Comparison of the stress responses of an aluminum alloy tube to proportional and alternate axial and shear strain paths at room temperature. Mech. Mater., 1983, vol. 2, pp. 183-192. https://doi.org/10.1016/0167-6636(83)90013-3
Tanaka E., Murakami S., Ooka M. Effects of plastic strain amplitudes on non-proportional cyclic plasticity. Acta Mech., 1985, vol. 57, pp. 167-182. https://doi.org/10.1007/BF01176916
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. https://doi.org/10.1016/0956-7151(93)90325-M
Zubchaninov V.G., Okhlopkov N.L. Hardening of structural materials in the process of complex deformation along closed plane trajectories. Strength of Mater., 1997, vol. 29, pp. 220-228. https://doi.org/10.1007/BF02767438
Itoh T., Nakata T., Sakane M., Ohnami M. Nonproportional low cycle fatigue of 6061 aluminum alloy under 14 strain paths. European Structural Integrity Society, 1999, vol. 25, pp. 41-54. https://doi.org/10.1016/S1566-1369(99)80006-5
Anes V., Reis L., Li B., de Freitas M. New approach to evaluate non-proportionality in multiaxial loading conditions. Fatig. Fract. Eng. Mater. Struct., 2014, vol. 37, pp. 1338-1354. https://doi.org/10.1111/ffe.12192
Anes V., Reis L., Li B., Fonte M., de Freitas M. New approach for analysis of complex multiaxial loading paths. Int. J. Fatig., 2014, vol. 62, pp. 21-33. https://doi.org/10.1016/j.ijfatigue.2013.05.004
Rohatgi A., Vecchio K.S., Gray G.T. III The influence of stacking fault energy on the mechanical behavior of Cu and Cu-Al alloys: Deformation twinning, work hardening, and dynamic recovery. Metall. Mater. Trans. A, 2001, vol. 32, pp. 135-145. https://doi.org/10.1007/s11661-001-0109-7.
Friedel J. Dislocations. Pergamon Press, 1964. 512 p
Honeycombe R.W.K. The plastic deformation of metals. Edward Arnold Ltd., 1968. 483 p.
Novikov I.I. Defekty kristallicheskogo stroyeniya metallov [Defects in the crystal structure of metals]. Moscow, Metallurgiya, 1975. 208 p.
Kanazawa K., Miller K.J., Brown M.W. Cyclic deformation of 1% Cr-Mo-V steel under out-of-phase loads. Fatig. Fract. Eng. Mater. Struct., 1979, vol. 2, pp. 217-228. https://doi.org/10.1111/j.1460-2695.1979.tb01357.x
Ohashi Y., Tanaka E., Ooka M. Plastic deformation behavior of type 316 stainless steel subject to out-of-phase strain cycles. J. Eng. Mater. Technol., 1985, vol. 107, pp. 286-292. https://doi.org/10.1115/1.3225821
Krempl E., Lu H. The hardening and rate-dependent behavior of fully annealed AISI type 304 stainless steel under biaxial in-phase and out-of-phase strain cycling at room temperature. J. Eng. Mater. Technol., 1984, vol. 106, pp. 376-382. https://doi.org/10.1115/1.3225733
Doong S.-H., Socie D.F., Robertson I.M. Dislocation substructures and nonproportional hardening. J. Eng. Mater. Technol., 1990, vol. 112, pp. 456-464. https://doi.org/10.1115/1.2903357
Doong S.-H., Socie D.F. Constitutive modeling of metals under nonproportional cyclic loading. J. Eng. Mater. Technol., 1991, vol. 113, pp. 23-30. https://doi.org/10.1115/1.2903379
Borodii M.V., Shukaev S.M. Additional cyclic strain hardening and its relation to material structure, mechanical characteristics, and lifetime. Int. J. Fatig., 2007, vol. 29, pp. 1184-1191. https://doi.org/10.1016/j.ijfatigue.2006.06.014
Bees M.R., Pattison S.J., Fox N., Whittaker M.T. The non-proportional behaviour of a nickel-based superalloy at room temperature, and characterisation of the additional hardening response by a modified cyclic hardening curve. Int. J. Fatig., 2014, vol. 67, pp. 134-141. https://doi.org/10.1016/j.ijfatigue.2014.02.023
Shang D.-G., Wang D.-J., Yao W.-X. A simple approach to the description of multiaxial cyclic stress-strain relationship. Int. J. Fatig., 2000, vol. 22, pp. 251-256. https://doi.org/10.1016/S0142-1123(99)00117-6
Gates N.R., Fatemi A. A simplified cyclic plasticity model for calculating stress-strain response under multiaxial non-proportional loadings. Eur. J. Mech. Solids, 2016, vol. 59, pp. 344-355. http://dx.doi.org/10.1016/j.euromechsol.2016.05.001
Yuan F., Chen P., Feng Y., Jiang P., Wu X. Strain hardening behaviors and strain rate sensitivity of gradient-grained Fe under compression over a wide range of strain rates. Mech. Mater., 2016, vol. 95, pp. 71-82. https://doi.org/10.1016/j.mechmat.2016.01.002
Lei C., Deng X., Li X., Wang Z., Wang G., Misra R.D.K. Mechanical properties and strain hardening behavior of phase reversion-induced nano/ultrafine Fe-17Cr-6Ni austenitic structure steel. J. Alloy. Comp., 2016, vol. 689, pp. 718-725. http://dx.doi.org/10.1016/j.jallcom.2016.08.020
Shao C.W., Zhang P., Zhu Y.K., Zhang Z.J., Tian Y.Z., Zhang Z.F. Simultaneous improvement of strength and plasticity: Additional work-hardening from gradient microstructure. Acta Mater., 2018, vol. 145, pp. 413-428. https://doi.org/10.1016/j.actamat.2017.12.028
Xia S., El-Azab A. Computational modelling of mesoscale dislocation patterning and plastic deformation of single crystals. Modelling Simul. Mater. Sci. Eng., 2015, vol. 23, 055009. https://doi.org/10.1088/0965-0393/23/5/055009
Knezevic M., Beyerlein I. Multiscale modeling of microstructure-property relationships of polycrystalline metals during thermo-mechanical deformation. Adv. Eng. Mater., 2018, vol. 20, 1700956. https://doi.org/10.1002/adem.201700956
Bergström Y. A dislocation model for the stress-strain behaviour of polycrystalline α-Fe with special emphasis on the variation of the densities of mobile and immobile dislocations. Materials Science and Engineering, 1970, vol. 5, pp. 193-200. https://doi.org/10.1016/0025-5416(70)90081-9
Sudmanns M., Bach J., Weygand D., Schulz K. Data-driven exploration and continuum modeling of dislocation networks. Modelling Simul. Mater. Sci. Eng., 2020, vol. 28, 065001. https://doi.org/10.1088/1361-651X/ab97ef
Trusov P.V., Shveykin A.I. On motion decomposition and constitutive relations in geometrically nonlinear elastoviscoplasticity of crystallites. Phys. Mesomech., 2017, vol. 20, pp. 377-391. https://doi.org/10.1134/S1029959917040026
Khan A.S., Liu J. A deformation mechanism based crystal plasticity model of ultrafine grained/nanocrystalline FCC polycrystals. Int. J. Plast., 2016, vol. 86, pp. 56-69. https://doi.org/10.1016/j.ijplas.2016.08.001
Orowan E. Problems of plastic gliding. Proc. Phys. Soc., 1940, vol. 52, pp. 8-22. https://doi.org/10.1088/0959-5309/52/1/303
Kocks U.F. Constitutive behavior based on crystal plasticity. Unified constitutive equations for creep and plasticity, ed. A.K. Miller. Springer, 1987. Pp. 1-88. https://doi.org/10.1007/978-94-009-3439-9_1
Trusov P.V., 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, 760. https://doi.org/10.3390/ma15030760
Orlov A.N. Vvedeniye v teoriyu defektov v kristallakh [Introduction to the theory of defects in crystals]. Moscow, Vysshaya shkola, 1983. 144 p.
Cho J., Molinari J.-F., Anciaux G. Mobility law of dislocations with several character angles and temperatures in FCC aluminum. Int. J. Plast., 2017, vol. 90, pp. 66-75. https://doi.org/10.1016/j.ijplas.2016.12.004
Shtremel' M.A. Prochnost’ splavov. Ch. I. Defekty reshetki [Alloy strength. Part I. Lattice defects]. Moscow, MISIS, 1999. 384 p.
Yi H.Y., Yan F.K., Tao N.R., Lu K. Work hardening behavior of nanotwinned austenitic grains in a metastable austenitic stainless steel. Scripta Mater., 2016, vol. 114, pp.133-136. https://doi.org/10.1016/j.scriptamat.2015.12.0211
Madec R., Devincre B., Kubin L.P. Simulation of dislocation patterns in multislip. Scripta Mater., 2002, vol. 47, pp. 689-695. https://doi.org/10.1016/S1359-6462(02)00185-9
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