Some results of a numerical estimating of the stability of the FCC metal two-level constitutive model
DOI:
https://doi.org/10.7242/1999-6691/2021.14.2.11Keywords:
multilevel constitutive model, mathematical model stability, sensitivity to perturbationAbstract
An important property of constitutive models is the stability of the response change histories obtained under various perturbations of an input data (history of influences and initial conditions) and a model operator. It is associated with the stochastic nature of material properties at different structural-scale levels and thermomechanical influences. Stability analysis is especially significant to justify the applicability of new constitutive models for describing modern technological processes, for instance, those focused on the design of novel functional materials. Multilevel physically-oriented constitutive models of materials hold the most promise for solving such problems. They are able to provide an explicit description of the inelastic deformation mechanisms, the material structure rebuilding and the changes in the physical and mechanical properties of the material determined by its state. Use of the approach developed by the authors and described in detail in the paper in the previous issue of the journal made it possible to evaluate the stability of multilevel constitutive material models under various perturbations of the initial conditions, the history of influences, and parametric operator perturbations. It includes an analysis of the norms of their deviations and the integral norm of deviation of perturbed solutions from the base ones obtained in calculations with unperturbed parameters. In this paper, the application of the proposed approach has been illustrated by studying a two-level constitutive model of the FCC polycrystal. The obtained results demonstrate the stability of this model to the calculated perturbations.
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References
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">https://doi.org/10.1016/j.ijplas.2010.02.008
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 Materialia, 2010, vol. 58, pp. 1152-1211. https://doi.org/10.1016/j.actamat.2009.10.058">https://doi.org/10.1016/j.actamat.2009.10.058
Diehl M. Review and outlook: mechanical, thermodynamic, and kinetic continuum modeling of metallic materials at the grain scale. MRS Communications, 2017, vol. 7, pp. 735-746. https://doi.org/10.1557/mrc.2017.98">https://doi.org/10.1557/mrc.2017.98
Beyerlein I., 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">https://doi.org/10.1557/jmr.2018.333
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. https://doi.org/10.15372/MULTILEVEL2019TPV">https://doi.org/10.15372/MULTILEVEL2019TPV
Trusov P.V., Shveikin A.I., Kondratyev N.S., Yants A.Yu. Multilevel models in physical mesomechanics of metals and alloys: results and prospects. Phys. Mesomech., 2020, vol. 23, no. 6, pp. 33-62. https://doi.org/10.24411/1683-805X-2020-16003">https://doi.org/10.24411/1683-805X-2020-16003
Trusov P.V. Classical and multi-level constitutive models for describing the behavior of metals and alloys: Problems and Prospects (as a matter for discussion). Mech. Solids, 2021, vol. 56, pp. 55-64. https://doi.org/10.3103/S002565442101012X">https://doi.org/10.3103/S002565442101012X
Truesdell C. A first course in rational continuum mechanics. USA, Maryland, Baltimore, The Johns Hopkins University, 1972. 304 p.
Astarita G., Marrucci G. Principles of non-Newtonian fluid mechanics. McGraw-Hill, 1974. 296 p.
Guo Y.B., Wen Q., 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, pp. 1423-1441. https://doi.org/10.1016/j.ijmecsci.2005.04.015">https://doi.org/10.1016/j.ijmecsci.2005.04.015
Maugin G.A. The saga of internal variables of state in continuum thermos-mechanics (1893-2013). Mech. Res. Comm., 2015, vol. 69, pp. 79-86. https://doi.org/10.1016/j.mechrescom.2015.06.009">https://doi.org/10.1016/j.mechrescom.2015.06.009
Rice J.R. Inelastic constitutive relations for solids: An internal-variable theory and its application to metal plasticity. J. Mech. Phys. Solid., 1971, vol. 19, pp. 433-455. https://doi.org/10.1016/0022-5096(71)90010-X">https://doi.org/10.1016/0022-5096(71)90010-X
Mandel J. Equations constitutives et directeurs dans les milieux plastiques et viscoplastiquest. Int. J. Solid. Struct., 1973, vol. 9, pp. 725-740. https://doi.org/10.1016/0020-7683(73)90120-0">https://doi.org/10.1016/0020-7683(73)90120-0
Aravas N. Finite elastoplastic transformations of transversely isotropic metals. Int. J. Solids Struct., 1992, vol. 29, pp. 2137-2157. https://doi.org/10.1016/0020-7683(92)90062-X">https://doi.org/10.1016/0020-7683(92)90062-X
Aravas N. Finite-strain anisotropic plasticity and the plastic spin. Modelling Simul. Mater. Sci. Eng., 1994, vol. 2, pp. 483-504. https://doi.org/10.1088/0965-0393/2/3A/005">https://doi.org/10.1088/0965-0393/2/3A/005
Dafalias Y.F. On multiple spins and texture development. Case study: kinematic and orthotropic hardening. Acta Mechanica, 1993, vol. 100, pp. 171-194. https://doi.org/10.1007/BF01174788">https://doi.org/10.1007/BF01174788
Trusov P.V. (ed.) Vvedeniye v matematicheskoye modelirovaniye [Introduction to mathematical modeling]. Moscow, Logos, 2007. 440 p.
Sobol’ I.M. Ob otsenke chuvstvitel’nosti nelineynykh matematicheskikh modeley [On the estimation of the sensitivity of nonlinear mathematical models]. Matem. modelirovaniye – Mathematical Models and Computer Simulations, 1990, vol. 2, no. 1, pp. 112-118.
Archer G.E.B., Saltelli A., Sobol I.M. Sensitivity measures, ANOVA-like techniques and the use of bootstrap. J. Stat. Comput. Simulat., 1997, vol. 58, pp. 99-120. https://doi.org/10.1080/00949659708811825">https://doi.org/10.1080/00949659708811825
Saltelli A., Tarantola S., Chan K.P.-S. A quantitative model-independent method for global sensitivity analysis of model output. Technometrics, 1999, vol. 41, pp. 39-56. https://doi.org/10.1080/00401706.1999.10485594">https://doi.org/10.1080/00401706.1999.10485594
Sobol I.M. Global sensitivity indices for the investigation of nonlinear mathematical models. Matem. modelirovaniye – Mathematical Models and Computer Simulations, 2005, vol. 17, no. 9, pp. 43-52.
Saltelli A., Ratto M., Andres T., Campolongo F., Cariboni J., Gatelli D., Saisana M., Tarantola S. Global sensitivity analysis. The Primer. John Wiley & Sons Ltd., 2008. 292 p.
Yang Z., Elgamal A. Application of unconstrained optimization and sensitivity analysis to calibration of a soil constitutive model. Int. J. Numer. Anal. Meth. Geomech., 2003, vol. 27, pp. 1277-1297. https://doi.org/10.1002/nag.320">https://doi.org/10.1002/nag.320
Qu J., Xu B., Jin Q. Parameter identification method of large macro-micro coupled constitutive models based on identifiability analysis. CMC, 2010, vol. 20, pp. 119-157. https://doi.org/10.3970/cmc.2010.020.119">https://doi.org/10.3970/cmc.2010.020.119
Shutov A.V., Kaygorodtseva A.A. Parameter identification in elasto-plasticity: distance between parameters and impact of measurement errors. ZAMM, 2019, vol. 99, e201800340. https://doi.org/10.1002/zamm.201800340">https://doi.org/10.1002/zamm.201800340
Kotha S., Ozturk D., Ghosh S. Parametrically homogenized constitutive models (PHCMs) from micromechanical crystal plasticity FE simulations, part I: Sensitivity analysis and parameter identification for titanium alloys. Int. J. Plast., 2019, vol. 120, pp. 296-319. https://doi.org/10.1016/j.ijplas.2019.05.008">https://doi.org/10.1016/j.ijplas.2019.05.008
Shveykin A.I., Sharifullina E.R., Trusov P.V., Pushkov D.A. About estimation of sensitivity of statistical multilevel polycrystalline metal models to parameter variations. Vychisl. mekh. splosh. sred – Computational Continuum Mechanics, 2018, vol. 11, no. 2, pp. 214-231. https://doi.org/10.7242/1999-6691/2018.11.2.17">https://doi.org/10.7242/1999-6691/2018.11.2.17
Shveykin A.I., Trusov P.V., Romanov K.A. An approach to numerical estimating the stability of multilevel constitutive models. Vychisl. mekh. splosh. sred – Computational Continuum Mechanics, 2021, vol. 14, no. 1, pp. 61-76. https://doi.org/10.7242/1999-6691/2021.14.1.6">https://doi.org/10.7242/1999-6691/2021.14.1.6
Lyapunov A.M. Obshchaya zadacha ob ustoychivosti dvizheniya [General problem of motion stability]. Moscow, Leningrad, Gosudarstvennoye izd-vo tekhniko-teoreticheskoy literatury, 1950. 470 p.
Barbashin E.A. Vvedeniye v teoriyu ustoychivosti [Introduction to the theory of stability]. Moscow, Nauka, 1967. 223 p.
Demidovich B.P. Lektsii po matematicheskoy teorii ustoychivosti [Lectures on the mathematical theory of stability]. Moscow, Nauka, 1967. 472 p.
Berge P., Pomeau Y., Vidal C. L'ordre dans le chaos. Vers une approche déterministe de la turbulence [Order in chaos. On a deterministic approach to turbulence]. Hermann, Editeurs des sciences et des arts, 1988. 353 p.
Lyapunov A.M. The general problem of the stability of motion. Int. J. Contr., 1992, vol. 55, pp. 531-534. https://doi.org/10.1080/00207179208934253">https://doi.org/10.1080/00207179208934253
Precup R.-E., Tomescu M.-L., Preitl St. Fuzzy logic control system stability analysis based on Lyapunov’s direct method. International Journal of Computers, Communications & Control, 2009, vol. 4, pp. 415-426. https://doi.org/10.15837/ijccc.2009.4.2457">https://doi.org/10.15837/ijccc.2009.4.2457
Li Y., Chen Y.Q., Podlubny I. Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability. Comput. Math. Appl., 2010, vol. 59, pp. 1810-1821. https://doi.org/10.1016/j.camwa.2009.08.019">https://doi.org/10.1016/j.camwa.2009.08.019
Aguila-Camacho N., Duarte-Mermoud M.A., Gallegos J.A. Lyapunov functions for fractional order systems. Comm. Nonlinear Sci. Numer. Simulat., 2014, vol. 19, pp. 2951-2957. https://doi.org/10.1016/j.cnsns.2014.01.022">https://doi.org/10.1016/j.cnsns.2014.01.022
Georgievskii D.V., Kvachev K.V. The Lyapunov–Movchan method in problems of the stability of flows and deformation processes. J. Appl. Math. Mech., 2014, vol. 78, pp. 621-633. https://doi.org/10.1016/j.jappmathmech.2015.04.010">https://doi.org/10.1016/j.jappmathmech.2015.04.010
Habraken A.M. Modelling the plastic anisotropy of metals. Arch. Computat. Methods Eng., 2004, vol. 11, pp. 3-96. https://doi.org/10.1007/BF02736210">https://doi.org/10.1007/BF02736210
Van Houtte P. Crystal plasticity based modelling of deformation textures. Microstructure and Texture in Steels, ed. A. Haldar, S. Suwas, D. Bhattacharjee. Springer, 2009. Р. 209-224. https://doi.org/10.1007/978-1-84882-454-6_12">https://doi.org/10.1007/978-1-84882-454-6_12
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, vol. 66, pp. 3-30. https://doi.org/10.1016/j.ijplas.2014.02.003">https://doi.org/10.1016/j.ijplas.2014.02.003
Lebensohn R.A., Ponte Castañeda P., Brenner R., Castelnau O. Full-field vs. homogenization methods to predict microstructure–property relations for polycrystalline materials. Computational Methods for Microstructure-Property Relationships, ed. S. Ghosh, D. Dimiduk. Springer, 2011. Р. 393-441. https://doi.org/10.1007/978-1-4419-0643-4_11">https://doi.org/10.1007/978-1-4419-0643-4_11
Trusov P.V., Shveykin A.I., Nechaeva E.S., Volegov P.S. Multilevel models of inelastic deformation of materials and their application for description of internal structure evolution. Phys. Mesomech., 2012, vol. 15, pp. 155-175. https://doi.org/10.1134/S1029959912020038">https://doi.org/10.1134/S1029959912020038
Pozdeyev A.A., Trusov P.V., Nyashin Yu.I. Bol’shiye uprugoplasticheskiye deformatsii: teoriya, algoritmy, prilozheniya [Large elastoplastic deformations: theory, algorithms, applications]. Moscow, Nauka, 1986. 232 p.
Levitas V.I. Bol’shiye uprugoplasticheskiye deformatsii materialov pri vysokom davlenii [Large elastoplastic deformations of materials at high pressure]. Kiev, Naukova dumka, 1987. 232 p.
Kondaurov V.I., Nikitin L.V. Teoreticheskiye osnovy reologii geomaterialov [Theoretical foundations of geomaterial rheology]. Moscow, Nauka, 1990. 207 p.
Korobeynikov S.N. Nelineynoye deformirovaniye tverdykh tel [Nonlinear deformation of solids]. Novosibirsk, Izd-vo SO RAN, 2000. 262 p.
Rogovoy A.A. Formalizovannyy podkhod k postroyeniyu modeley mekhaniki deformiruyemogo tverdogo tela. Ch. 1. Osnovnyye sootnosheniya mekhaniki sploshnykh sred [A formalized approach to the construction of models of solid mechanics. Part 1. Basic relations of continuum mechanics]. Moscow, Izd-vo IKI, 2021. 288 p.
Markin A.A., Sokolova M.Yu. Termomekhanika uprugoplasticheskogo deformirovaniya [Thermomechanics of elastoplastic deformation]. Moscow, Fizmatlit, 2013. 319 p.
Brovko G.L. Opredelyayushchiye sootnosheniya mekhaniki sploshnoy sredy: razvitiye matematicheskogo apparata i osnov obshchey teorii [Constitutive relations of continuum mechanics: Development of the mathematical apparatus and foundations of the general theory]. Moscow, Nauka, 2017. 432 p.
Trusov P.V., Shveykin A.I., Yanz A.Yu. Motion decomposition, frame-indifferent derivatives, and constitutive relations at large displacement gradients from the viewpoint of multilevel modeling. Phys. Mesomech., 2017, vol. 20, pp. 357-376. https://doi.org/10.1134/S1029959917040014">https://doi.org/10.1134/S1029959917040014
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">https://doi.org/10.1134/S1029959917040026
Trusov P.V., Shveykin A.I., Kondratev N.S. Multilevel metal models: formulation for large displacements gradients. Nanoscience and Technology: An International Journal, 2017, vol. 8, pp. 133-166. https://doi.org/10.1615/NanoSciTechnolIntJ.v8.i2.40">https://doi.org/10.1615/NanoSciTechnolIntJ.v8.i2.40
Anand L. Single-crystal elasto-viscoplasticity: application to texture evolution in polycrystalline metals at large strains. Comput. Meth. Appl. Mech. Eng., 2004, vol. 93, pp. 5359-5383. https://doi.org/10.1016/j.cma.2003.12.068">https://doi.org/10.1016/j.cma.2003.12.068
Trusov P.V., Sharifullina E.R., Shveykin A.I. Multilevel model for the description of plastic and superplastic deformation of polycristalline materials. Phys. Mesomech., 2019, vol. 22, pp. 402-419. https://doi.org/10.1134/S1029959919050072">https://doi.org/10.1134/S1029959919050072
Shveykin A., Trusov P., Sharifullina E. Statistical crystal plasticity model advanced for grain boundary sliding description. Crystals, 2020, vol. 10(9), 822. https://doi.org/10.3390/cryst10090822">https://doi.org/10.3390/cryst10090822
Estrin Y., Tóth L.S., Molinari A., Bréchet Y. A dislocation-based model for all hardening stages in large strain deformation. Acta Mater., 1998, vol. 46, pp. 5509-5522. https://doi.org/10.1016/S1359-6454(98)00196-7">https://doi.org/10.1016/S1359-6454(98)00196-7
Staroselsky A., Anand L. Inelastic deformation of polycrystalline face centered cubic materials by slip and twinning. J. Mech. Phys. Solid., 1998, vol. 46, pp. 671-696. https://doi.org/10.1016/S0022-5096(97)00071-9">https://doi.org/10.1016/S0022-5096(97)00071-9
Kalidindi S.R. Modeling anisotropic strain hardening and deformation textures in low stacking fault energy fcc metals. Int. J. Plast., 2001, vol. 17, pp. 837-860. https://doi.org/10.1016/S0749-6419(00)00071-1">https://doi.org/10.1016/S0749-6419(00)00071-1
Beyerlein I.J., 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.2007.07.017">https://doi.org/10.1016/j.ijplas.2007.07.017
Bronkhorst C.A., Kalidindi S.R., Anand L. Polycrystalline plasticity and the evolution of crystallographic texture in FCC metals. Phil. Trans. Math. Phys. Eng. Sci., 1992, vol. 341, pp. 443-477. https://doi.org/10.1098/rsta.1992.0111">https://doi.org/10.1098/rsta.1992.0111
Harder J. FEM-simulation of the hardening behavior of FCC single crystals. Acta Mechanica, 2001, vol. 150, pp. 197-217. https://doi.org/10.1007/BF01181812">https://doi.org/10.1007/BF01181812
Shveikin A.I., Sharifullina E.R. Analysis of constitutive relations for intragranular dislocation sliding description within two-level elasto-viscoplastic model of FCC-polycrystals. Vestnik Tambovskogo universiteta. Seriya Estestvennyye i tekhnicheskiye nauki – Tambov University Reports. Series: Natural and Technical Sciences, 2013, vol. 18, no. 4-2, pp. 1665-1666.
Trenogin V.A. Funktsional’nyy analiz [Functional analysis]. Moscow, Nauka, 1980. 495 p.
Rocks U.F., Canova G.R., Jonas J.J. Yield vectors in f.c.c. crystals. Acta metall., 1983, vol. 31, pp. 1243-1252. https://doi.org/10.1016/0001-6160(83)90186-4">https://doi.org/10.1016/0001-6160(83)90186-4
Kuhlman-Wilsdorf D., Kulkarni S.S., Moore J.T., Starke E.A. (Jr.) Deformation bands, the LEDS theory, and their importance in texture development: Part I. Previous evidence and new observations. Metall. Mater. Trans. A, 1999, vol. 30, pp. 2491-2501. https://doi.org/10.1007/s11661-999-0258-7">https://doi.org/10.1007/s11661-999-0258-7
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