539.3 Modeling of stresses in a composite nonlinear elastic panel under cylindrical bending

Dimitrienko Y. I. (Bauman Moscow State Technical University), Gubareva E. A. (Bauman Moscow State Technical University), Karimov S. B. (Bauman Moscow State Technical University), Kolzhanova D. Y. (Bauman Moscow State Technical University)

FINITE DEFORMATIONS, COMPOSITES, CYLINDRICAL BENDING, COMPOSITE PANEL, INCOMPRESSIBLE MEDIA, TRANSVERSELY ISOTROPIC MEDIUM, ASYMPTOTIC AVERAGING, UNIVERSAL MODELS OF ELASTIC MEDIA, CAUCHY STRESS TENSOR


doi: 10.18698/2309-3684-2021-1-330


The problem of calculating the stress–strain state of a composite laminated panel during cylindrical bending under conditions of finite deformations is considered. To solve the problem, the method of asymptotic averaging of periodic nonlinear elastic structures with finite deformations was applied, which was developed in detail earlier in the previous works of the authors. A feature of this problem is the use of universal models of constitutive relations for isotropic components of the composite, as well as for the composite as a whole, which is a transversely isotropic nonlinear elastic medium. Universal models make it possible to obtain solution of problems within the framework of a single solution algorithm simultaneously for several classes of models of nonlinear elastic media corresponding to different conjugate pairs of stress tensors–deformation. An analytical solution is obtained for the problem of cylindrical bending of a composite panel. A numerical analysis of the solution is carried out using the example of a composite, the periodicity cell of which consists of two layers: polyurethane and rubber. It is shown that for thin panels the stresses, both averaged and true, practically do not depend on the class of the model of the constitutive relations. At the same time, for thicker panels, the stresses differ significantly for different classes of models of composite layers.


Christensen R.M. Mechanics of composite materials. New York, John Wiley&Sons Publ., 1979, 324 p.
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.
Jones R.M. Mechanics of Composite Materials. USA, Taylor&Francis Publ., 1999, 520 p.
Vasiliev V.V., Tarnopolsky Yu.M. Kompozicionnye materialy: spravochnik [Composite materials: handbook]. Moscow, Mashinostroenie Publ., 1989, 510 p.
Vanin G.A. Mikromekhanika kompozicionnyh materialov [Micromechanics of composite materials]. Kiev, Naukova dumka Publ., 1985, 300 p.
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, 352 p.
Bensoussan A., Lions J.L., Papanicolaou G. Asymptotic analysis for periodic structures. North-Holland, 1978, 721 p.
Pobedrya B.E. Mekhanika kompozitsionnykh materialov [Mechanics of composite materials]. Moscow, Lomonosov Moscow State University Publ., 1984, 324 p.
Sanches-Palensiya E. Neodnorodnye sredy i teoriya kolebaniy [Nonhomogeneous media and vibration theory]. Moscow, Mir Publ., 1984, 472 p.
Dimitrienko Yu.I., Kashkarov A.I. Finite element method for calculating the effective characteristics of spatially reinforced composites. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2002, no.2, pp.95–108.
Dimitrienko Yu.I. Modelling of nonlinear-elastic properties of composites with finite deformations by asymptotic homogenization method. Higher Educational Institutions. Маchine Building, 2015, no.11, pp.68–77.
Dimitrienko Yu.I., Gubareva E.A., Kolzhanova D.Yu. Modeling of laminated composites with finite deformations by asymptotic homogenization method. Engineering Journal: Science and Innovation: Electronic Science and Engineering Publication, 2015, no.5 (41), pp.1–5.
Dimitrienko Y.I., Gubareva E.A., Kolzhanova D.Y., Karimov S.B. Incompressible layered composites with finite deformations on the basis of the asymptotic averaging method. Маthematical Modeling and Coтputational Methods, 2017, no.1, pp.32–54.
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.
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
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
Yang Q., Xu F. Numerical modeling of nonlinear deformation of polymer composites based on hyperelastic constitutive law. Frontiers of Mechanical Engineering in China, 2009, no.4, pp.284–288.
Aboudi J. Finite strain micromechanical modeling of multiphase composites. International Journal for Multiscale Computational Engineering, 2008, no.6, pp.411–434.
Zhang B., Yu X., Gu B. Micromechanical modeling of large deformation in sepiolite reinforced rubber sealing composites under transverse tension. Polymer Composites, 2017, vol.38, iss.2, pp.381–388.
Ge Q., Luo X., Iversen C.B., Nejad H.B., Mather P.T., Dunn M.L., Jerry Qi H. 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, iss.15–16, pp.2777–2790.
Dimitrienko Yu.I. Mekhanika sploshnoy sredy. Tom 4. Osnovy mekhaniki tverdogo tela [Continuum Mechanics. Vol.4. Fundamentals of solid mechanics]. Moscow, BMSTU Publ., 2013, 624 p.
Dimitrienko Yu.I. Mekhanika sploshnoy sredy. Tom 2. Universal'nye zakony mekhaniki i elektrodinamiki sploshnoj sredy [Continuum Mechanics. Vol.2. Universal laws of mechanics and electrodynamics of a continuous medium]. Moscow, BMSTU Publ., 2011, 560 p.
Chernykh K.F. Nelinejnaya teoriya uprugosti v mashinostroitel'nyh raschetah [Nonlinear theory of elasticity in mechanical engineering calculations]. Leningrad, Mashinostroenie Publ., 1986, 336 p.
Lurie A.I. Nelinejnaya teoriya uprugosti [Nonlinear theory of elasticity]. Moscow, Nauka Publ., 1980, 512 p.


Димитриенко Ю.И., Губарева Е.А., Каримов С.Б., Кольжанова Д.Ю. Моделирование напряжений в композитной нелинейно упругой панели при цилиндрическом изгибе. Математическое моделирование и численные методы, 2021, № 1, с. 3–30.



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