539.3 Modeling of finite deformations of composite materials based on universal An models and the asymptotic averaging method
Dimitrienko Y. I. (Bauman Moscow State Technical University), Karimov S. B. (Bauman Moscow State Technical University), Dimitrienko A. Y. (Lomonosov Moscow State University)
COMPOSITES, FINITE DEFORMATIONS, ASYMPTOTIC AVERAGING METHOD, UNIVERSAL AN MODELS, PERIODICITY CELL, FINITE ELEMENT METHOD
doi: 10.18698/2309-3684-2024-2-1734
The paper is devoted to modeling the deformation of composite materials with finite deformations. The so-called universal models of constitutive relations for composite components are considered, defining several classes of nonlinear relationship between the Piola-Kirchhoff stress tensor and the strain gradient within different energy pairs of stress-strain tensors. The method of asymptotic averaging is applied and local problems are formulated to solve the problem of determining the averaged properties of composites with finite deformations. A variational formulation of the original deformation problem, the so-called local problems on a periodicity cell and the averaged problem for a composite is considered, which makes it possible to use FEM for the numerical solution of these classes of problems. A software module has been developed as part of the SMCM software package, which implements the proposed numerical algorithm. An example of the numerical solution of problems on a periodicity cell for a 3D orthogonally reinforced composite is given, taking into account large deformations of the matrix and fibers, and composite deformation diagrams are calculated for various variants of universal models of constitutive relations.
[1] Kwietniewskia M., Miedzińska D. Review of elastomeric materials for application to composites reinforced by auxetics fabrics. Procedia Structural Integrity, 2019, vol. 17, pp. 154-161.
[2] Peel L.D., Mejia J., Narvaez B., Thompson K., Lingala M. Development of a simple morphing wing using elastomeric composites as skins and actuators. Journal of Mechanical Design, 2009, vol. 131, no. 9, pp. 839-847.
[3] Christensen R.M. Mechanics of composite materials. New York, John Wiley & Sons, 1979, 324 p.
[4] Malmeister A.K., Tamuzh V.P., Teters G.A. Soprotivlenie polimernyh i kompozitnyh materialov [Resistance of polymer and composite materials]. Riga, Zinatne Publ., 1980, 572 p.
[5] Jones R.M. Mechanics of Composite Materials. USA, Taylor&Francis Publ., 1999, 520 p.
[6] Vasiliev V.V., Tarnopolsky Yu.M. Kompozicionnye materialy: spravochnik [Composite materials: handbook]. Moscow, Mashinostroenie Publ., 1989, 510 p.
[7] Vanin G.A. Mikromekhanika kompozicionnyh materialov [Micromechanics of composite materials]. Kiev, Naukova dumka Publ., 1985, 300 p.
[8] Bakhvalov N.S., Panasenko G.P. Osrednenie protsessov v periodicheskikh sredakh. Matematicheskie zadachi mekhaniki kompozitsionnykh materialov [Averaging processes in periodic media. Mathematical problems of the composite material mechanics]. Moscow, Nauka Publ., 1984, 356 p.
[9] Bensoussan A., Lions J.L., Papanicolaou G. Asymptotic analysis for periodic structures. North-Holland, 1978, 721 p.
[10] Pobedrya B.E. Mekhanika kompozitsionnykh materialov [Mechanics of composite materials]. Moscow, Lomonosov Moscow State University Publ., 1984, 336 p.
[11] Sanchez-Palencia E. Non-Homogeneous Media and Vibration Theory. New York, Springer Publ., 1980.
[12] Dimitrienko Yu.I., Kashkarov A.I. Finite element method for calculating the effective characteristics of spatially reinforced composites. Vestnik MGTU im. N.E. Baumana. Ser. Estestvennye nauki — Herald of Bauman Moscow State Technical University. Series Natural Sciences, 2002, no. 2, pp. 95–108.
[13] Dimitrienko Yu.I. Modelling of nonlinear-elastic properties of composites with finite deformations by asymptotic homogenization method. Izvestiya vyshikh uchebnykh zavedeny. Mashinostroenie — Proceedings of Higher Educational Institutions. Machine Building, 2015, no. 11, pp. 68–77. DOI 10.18698/0536-1044-2015-11-68-77
[14] Dimitrienko Yu.I, Gubareva E.A., Kolzhanova D.Yu., Karimov S.B. Modelirovanie neszhimaemyh sloistyh kompozitov s konechnymi deformaciyami na osnove metoda asimptoticheskogo osredneniya [Modeling of incompressible layered composites with finite deformations based on the asymptotic averaging method]. Matematicheskoe modelirovanie i chislennye metody [Mathematical Modeling and Computational Methods], 2017, no. 1, pp. 32–54. DOI: 10.18698/2309-3684-2017-1-3254
[15] Dimitrienko Y.I., Gubareva E.A., Karimov S.B., Kolzhanova D.Y. Modeling the effective characteristics of transversely isotropic incompressible composites with finite strains. Маthematical Modeling and Coтputational Methods, 2017, no. 4, pp. 72–92.
[16] Dimitrienko Y.I., Karimov S.B., Kolzhanova D.Y. Modeling of the effective universal constitutive relations for elastic laminated composites with finite strains. IOP Conference Series: Material Science and Engineering, 2019, vol. 683, art no. 012006. DOI: 10.1088/1757-899X/683/1/012006
[17] Dimitrienko Y.I., Gubareva E.A., Karimov S.B., Kolzhanova D.Y. Universal models of the constitutive relations for transversely isotropic compressible composites with finite strains. IOP Conference Series: Material Science and Engineering, 2020, vol. 934, art no. 012012. DOI: 10.1088/1757-899X/934/1/012012
[18] Abodi J., Arnold S.M. Micromechanical modeling of the finite deformation of thermoelastic multiphase composites. Mathematics and Mechanics of Solids, 2000, vol. 5, no. 1, pp. 75-99.
[19] Costa S., Fagerström M., Olsson R. Development and validation of a finite deformation fibre kinking model for crushing of composites. Composites Science and Technology, 2020, vol. 197, art. 108236.
[20] Qingsheng Yang, Fang Xu. Numerical modeling of nonlinear deformation of polymer composites based on hyperelastic constitutive law. Frontiers of Mechanical Engineering in China, 2009, vol. 4, pp. 284–288.
[21] Aboudi J. Finite strain micromechanical modeling of multiphase composites. Int. J. Multiscale Comput, 2008, vol. 6 (2008), pp. 411–434.
[22] Bin Zhang, Xiaoming Yu, Boqin Gu. Micromechanical modeling of large deformation in sepiolite reinforced rubber sealing composites under transverse tension. Polym Compos, 2015. DOI: 10.1002/pc.23596
[23] Qi Ge, Xiaofan Luo, Christian B. Iversen, Hossein Birjandi Nejad, Patrick T. Mather, Martin L. Dunn, H. Jerry Qi. A finite deformation thermomechanical constitutive model for triple shape polymeric composites based on dual thermal transitions. International Journal of Solids and Structures, 2014, vol. 51, pp. 2777–2790.
[24] Dimitrienko Yu.I. Mekhanika sploshnoy sredy. V 4 tomakh. T. 2. Universalnye zakony mekhaniki i elektrodinamiki sploshnykh sred [Continuum Mechanics. In 4 vols. Vol. 2. Universal lows of continuum mechanics and electrodynamics]. Moscow. BMSTU Publ., 2011, 560 p.
[25] Dimitrienko Yu.I. Mekhanika sploshnoy sredy. V 4 tomakh. Tom 4. Osnovy mekaniki tverdykh sred [Continuum mechanics. In 4 vols. Vol. 4. Fundamentals of mechanics of solid mechanics]. Moscow, BMSTU Publ., 2013, 624 p.
[26] Dimitrienko Yu.I. Mekhanika sploshnoj sredy. T. 1. Tenzornyj analiz [Continuum mechanics. In 4 vols. Vol. 1. Tensor analysis]. Moscow, BMSTU Publ., 2011, 367 p.
[27] Oden J. Konechnye elementy v nelinejnoj mekhanike sploshnyh sred [Finite elements in nonlinear continuum mechanics]. Moscow, Mir Publ., 1976, 464 p.
[28] Certificate no. 2018614767 Programma MultiScale_SMCM dlya mnogomasshtabnogo modelirovaniya napryazhenno-deformirovannogo sostoyaniya konstrukcij iz kompozicionnyh materialov, na osnove metoda mnogourovnevoj asimptoticheskoj gomogenizacii i konechno-elementnogo resheniya trekhmernyh zadach teorii uprugosti [MultiScale_SMCM program for multiscale modeling of the stress-strain state of structures made of composite materials, based on the method of multilevel asymptotic homogenization and finite element solution of three-dimensional problems of elasticity theory]: certificate of ofic. registration of computer programs/ Yu.I. Dimitrienko, S.V. Sborshchikov, Yu.V. Yurin; applicant and copyright holder: BMSTU — no. 2018677684; application 21.02.2018; registered in the register of computer programs 17.04.2018 — [1].
Димитриенко Ю.И., Каримов С.Б., Димитриенко А.Ю. Моделирование конечных деформаций композиционных материалов на основе универсальных моделей Аn и метода асимптотического осреднения. Математическое моделирование и численные методы, 2024, № 2, с. 17–34.
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