An approach to numerical estimating the stability of multilevel constitutive models
DOI:
https://doi.org/10.7242/1999-6691/2021.14.1.6Keywords:
multilevel constitutive model, mathematical model stability, perturbation sensitivityAbstract
Multilevel constitutive models of materials give the possibility to explicitly describe the mechanisms of inelastic deformation, evolution of a material structure and changes in the physical and mechanical properties of materials determined by their chemical composition and their internal structure. Therefore, these models seem to be very effective for improving metal processing and forming techniques. A study of the solutions (response change histories) obtained using constitutive models under perturbation of input data (influence history and initial conditions) and model operator is actual due to the fact that the material mechanical characteristics (including the lower scale level properties), physical processes occurring during deformation (for example, acts of interaction of defect structures at the microscale level) and the resulting influences (produced by stochastic boundary conditions) are stochastic in nature. Finding the solution to this problem is particularly important when researchers need to justify the use of new constitutive models for describing modern technological processes of thermomechanical treatment, in particular, those focused on creation of functional materials. The disadvantages of traditional analytical approaches (Lyapunov methods) taken to analyze the stability of multilevel material models have been briefly discussed. The definition of the solution stability is introduced; in contrast to the traditional definition, it takes into account the parametric perturbation of operators and the perturbation of the history of influences, which determine the right-hand side of the system of equations. A procedure for the model stability numerical assessment includes consideration of solutions stability for various values of the parameters that determine the operator and input data. The description of the program of computational experiments for the implementation of the proposed approach is presented. This program can be used to study various perturbations of initial conditions, the influence history, the operator, as well as to analyze the norms of their deviations and the integral norm of deviation of perturbed solutions from the base ones obtained in calculations with unperturbed parameters.
Downloads
References
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.
Agoshkov V.I., Parmuzin E.I., Shutyaev V.P. Observational data assimilation in the problem of Black Sea circulation and sensitivity analysis of its solution. Izv. Atmos. Ocean. Phys., 2013, vol. 49, pp. 592-602. https://doi.org/10.1134/S0001433813060029">https://doi.org/10.1134/S0001433813060029
Nurislamova L.F., Gubaydullin I.M. Reduction of detailed schemes for chemical transformations of formaldehyde and hydrogen oxidation reactions based on a sensitivity analysis of a mathematical model. Vychislitel’nyye metody i programmirovaniye – Numerical methods and programming, 2014, vol. 15, no. 4, pp. 685-696.
Bashkirtseva I.A., Ryashko L.B., Tsvetkov I.N. Stochastic sensitivity of equilibrium and cycles of 1D discrete maps. Izv. vuzov. PND – Izvestiya VUZ. Applied Nonlinear Dynamics, 2009, vol. 17, no. 6, pp. 74-85. https://doi.org/10.18500/0869-6632-2009-17-6-74-85">https://doi.org/10.18500/0869-6632-2009-17-6-74-85
Nossent J., Elsen P., Bauwens W. Sobol’ sensitivity analysis of a complex environmental model. Environ. Model. Software, 2011, vol. 26, pp. 1515-1525. https://doi.org/10.1016/j.envsoft.2011.08.010">https://doi.org/10.1016/j.envsoft.2011.08.010
Gan Y., Duan Q., Gong W., Tong C., Sun Y., Chu W., Ye A., Miao C., Di Z. A comprehensive evaluation of various sensitivity analysis methods: A case study with a hydrological model. Environ. Model. Software, 2014, vol. 51, pp. 269-285. https://doi.org/10.1016/j.envsoft.2013.09.031">https://doi.org/10.1016/j.envsoft.2013.09.031
Haug E.J., Choi K.K., Komkov V. Design sensitivity analysis of structural systems. Academic Press, 1986. 370 p.
Kleiber M., Hien T.D., Postek E. Incremental finite element sensitivity analysis for non-linear mechanics applications. Int. J. Numer. Meth. Eng., 1994, vol. 37, pp. 3291-3308. https://doi.org/10.1002/nme.1620371906">https://doi.org/10.1002/nme.1620371906
Gutiérrez M.A., de Borst R. Simulation of size-effect behaviour through sensitivity analyses. Eng. Fract. Mech., 2003, vol. 70, pp. 2269-2279. https://doi.org/10.1016/S0013-7944(02)00221-7">https://doi.org/10.1016/S0013-7944(02)00221-7
Khaledi K., Mahmoudi E., Datcheva M., König D., Schanz T. Sensitivity analysis and parameter identification of a time dependent constitutive model for rock salt. J. Comput. Appl. Math., 2016, vol. 293, pp. 128-138. https://doi.org/10.1016/j.cam.2015.03.049">https://doi.org/10.1016/j.cam.2015.03.049
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
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
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
Trusov P.V., Ashikhmin V.N., Volegov P.S., Shveykin A.I. Constitutive relations and their application to the description of microstructure evolution. Phys. Mesomech., 2010, vol. 13, pp. 38-46. https://doi.org/10.1016/j.physme.2010.03.005">https://doi.org/10.1016/j.physme.2010.03.005
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
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
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
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
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">https://doi.org/10.1002/adem.201700956
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
Vasin R.A. Svoystva funktsionalov plastichnosti u metallov, opredelyayemyye v eksperimentakh na dvuzvennykh trayektoriyakh deformatsii [Properties of plasticity functionals for metals determined in experiments on two-link deformation trajectories] // Uprugost’ i neuprugost’ [Elasticity and inelasticity]. Moscow, MGU, 1987. Pp. 115-127.
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.
Zubchaninov V.G. Mekhanika sploshnykh deformiruyemykh sred [Mechanics of continuous deformable media]. Tver, Izd-vo TGTU, ChuDo, 2000. 703 p.
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.
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., Ashikhmin V.N., Shveykin A.I. Physical elastoplastic analysis of deformation of FCC metals. Phys. Mesomech., 2011, vol. 14, pp. 40-48. https://doi.org/10.1016/j.physme.2011.04.006">https://doi.org/10.1016/j.physme.2011.04.006
Peregudov F.I., Tarasenko F.P. Vvedeniye v sistemnyy analiz [Introduction to systems analysis]. Moscow, Vysshaya shkola, 1989. 367 p.
Lurie A.I. Nonlinear theory of elasticity. Elsevier, 1990. 617 p.
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.
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.
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
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
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
Trusov P.V., 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/S1029959913010037">https://doi.org/10.1134/S1029959913010037
Trusov P.V., 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/S1029959913020021">https://doi.org/10.1134/S1029959913020021
Trenogin V.A. Funktsional’nyy analiz [Functional analysis]. Moscow, Nauka, 1980. 495 p.
Tikhonov A.N., Arsenin V.Ya. Metody resheniya nekorrektnykh zadach [Methods for solving ill-posed problems]. Moscow, Nauka, 1986. 287 p.
Il’yushin A.A. Mekhanika sploshnoy sredy [Continuum mechanics]. Moscow, Izd-vo MGU, 1990. 310 p.
Shveykin A.I. Mnogourovnevyye modeli polikristallicheskikh metallov: sopostavleniye opredelyayushchikh sootnosheniy dlya kristallitov [Multilevel models of polycrystalline metals: comparison of constitutive relations for crystallites]. Probl. prochnosti i plastichnosti – Probl. of strength and plasticity, 2017, vol. 79, no. 4, pp. 385-397.
Shveikin A.I., Trusov P.V. Correlation between geometrically nonlinear elastoviscoplastic constitutive relations formulated in terms of the actual and unloaded configurations for crystallites. Phys. Mesomech., 2018, vol. 21, pp. 193-202. https://doi.org/10.1134/S1029959918030025">https://doi.org/10.1134/S1029959918030025
Shveykin A.I., Trusov P.V. Multilevel models of polycrystalline metals: Comparison of relations describing the rotations of crystallite lattice. Nanoscience and Technology: An International Journal, 2019, vol. 10, pp. 1-20. https://doi.org/10.1615/NanoSciTechnolIntJ.2018028673">https://doi.org/10.1615/NanoSciTechnolIntJ.2018028673
Lyapunov A.M. Obshchaya zadacha ob ustoychivosti dvizheniya [General problem of motion stability]. Moscow, Leningrad, Gosudarstvennoye izd-vo tekhniko-teoreticheskoy literatury, 1950. 470 p.
Cesari L. Asymptotic behavior and stability problems in ordinary differential equations. Springer-Verlag, 1959. 278 p. https://doi.org/10.1007/978-3-662-40368-6">https://doi.org/10.1007/978-3-662-40368-6
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
Azbelev N.V., Simonov P.M. Ustoychivost’ resheniy uravneniy s obyknovennymi proizvodnymi [Stability of solutions of equations with ordinary derivatives]. Perm, Izd-vo Perm. un-ta, 2001. 229 p.
Lin C.C. The theory hydrodynamic stability. Cambridge University Press, 1955. 176 p.
Gershuni G.Z., Zhukhovitskiy E.M., Nepomnyashchiy A.A. Ustoychivost’ konvektivnykhtecheniy [Stabilityofсonvectiveаlows]. Moscow, Nauka, 1989. 318 p.
Balandin A.S., Sabatulina T.L. The local stability of a population dynamics model in conditions of deleterious effects. Sibirskiye elektronnyye matematicheskiye izvestiya – Siberian electronic mathematical reports, 2015, vol. 12, pp. 610-624. https://doi.org/10.17377/semi.2015.12.049">https://doi.org/10.17377/semi.2015.12.049
Demidova A.V., Druzhinina O.V., Masina O.N. Stability research of population dynamics model on the basis of construction of the stochastic self-consistent models and the principle of the reduction. Vestnik RUDN. Ser. Matematika. Informatika. Fizika – Discrete and Continuous Models and Applied Computational Science, 2015, no. 3, pp. 18-29.
Timoshenko S.P., Gere Dzh. Mekhanika materialov [Mechanics of materials]. Moscow, Lan’, 2002. 672 p.
Krasovskiy N.N. Nekotoryye zadachi teorii ustoychivosti dvizheniya [Some problems of the theory of stability of motion.]. Moscow, Fizmatgiz, 1959. 211 p.
Chetayev N.G. Ustoychivost’ dvizheniya [Stability of movement]. Moscow, Nauka, 1955. 176 p.
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
Georgievskij D.V., Kvachyov K.V. Metod Lyapunova – Movchana v zadachah ustojchivosti techenij i processov deformirovaniya [Lyapunov - Movchan method in problems of stability of flows and deformation processes]. Prikladnaya matematika i mekhanika – Applied Mathematics and Mechanics, 2014, no. 6, pp. 862-885.
Kolmogorov A.N., Fomin S.V. Elementy teorii funktsiy i funktsional’nogo analiza [Elements of the theory of functions and functional analysis]. Moscow, Fizmatlit, 2009. 572 p.
Wilkinson J.H. The algebraic eigenvalue problem. Clarendon Press, 1965. 662 p.
Gitman M.B. Vvedeniye v stokhasticheskuyu optimizatsiyu [Introduction to stochastic optimization]. Perm, Izd-vo Perm. nats. issled. politekhn. un-ta, 2014. 104 p.
Romanov K.A., Shveykin A.I. Investigation of HCP metal mesolevel model sensitivity to lattice orientation perturbations. AIP Conference Proceedings, 2020, vol. 2216, 020010. https://doi.org/10.1063/5.0003386">https://doi.org/10.1063/5.0003386
Downloads
Published
Issue
Section
License
Copyright (c) 2021 Computational Continuum Mechanics
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.