539.3 Asymptotic theory of thin two-layer elastic plates with layer slippage

Dimitrienko Y. I. (Bauman Moscow State Technical University), Gubareva E. A. (Bauman Moscow State Technical University)


doi: 10.18698/2309-3684-2019-1-326

The problem of deformation of thin two-layer plates, for which a slip condition is speci-fied at the interface between the layers, instead of the classical case of ideal contact, is considered. The method of asymptotic analysis of the general equations of the 3-dimensional theory of elasticity is used to solve this problem under the influence of transverse pressure, longitudinal and shear forces on the end surfaces. Asymptotic analysis is performed using a small geometric parameter representing the ratio of thickness to the characteristic length of the plate. Recurrent formulations of local quasi-one-dimensional problems of elasticity theory with slippage are obtained. For these problems, explicit analytical solutions are obtained. The averaged equations of elastic equilibrium of a two-layer plate with slippage of layers are derived. It is shown that, due to slippage, the order of the averaged equations of the theory of plates increases to 5 orders of magnitude, in contrast to the classical 4th order, which takes place in the theory of Kirchhoff – Love plates. Additional boundary conditions to this 5th order system are formulated and its analytical solution is obtained for the case of a rectangular plate under the influence of uniform pressure. A numerical analysis of the solution of the averaged problem is carried out. It is shown that the presence of layer slippage significantly increases the deflection of the plate in comparison with the conditions of ideal contact of the layers.

[1] Love A.E.H. A treatise on the mathematical theory of elasticity. Cambridge. University Press, 1927, 674 p. [In Russ.: Love A.E.H. Matematicheskaya teoriya uprugosti. Moscow, ONTI Publ., 1935, 674 p.].
[2] Timoshenko S.P., Voinovsky-Krieger S. Plastinki i obolochki [Plates and shells]. Moscow, Nauka Publ., 1966, 635 p.
[3] Vasiliev V.V. Mekhanika konstruktsii iz kompozitsionnykh materialov [Mechan-ics of composite materials structures]. Moscow, Mashinostroenie Publ., 1988, 272 p.
[4] Grigolyuk E.I., Kulikov G.M. Mekhanika kompozitnykh materialov –Mechanics of Composite Materials, 1989, vol. 24, no. 4, pp. 698–704.
[5] Sheshenin S.V. Izvestiya RAN. Mekhanika tverdogo tela – Mechanics of Solids, 2006, no. 6, pp. 71–79.
[6] Scott W Hansen A model for a two-Layered Plate with interfacial slip. Interna-tional Series of Numerical Mathematics. V.118. 1994. pp. 143-170.
[7] Kohn R.V., Vogelius M. A new model of thin plates with rapidly varying thick-ness. Int. J. Solids and Struct, 1984, pp. 333–350.
[8] F. Gruttmann, W. Wagner. Shear correction factors in Timoshenko’s beam theo-ry for arbitrary shaped cross–sections. Computational mechanics, 2001, vol. 27, pр. 199–207.
[9] Y.M. Ghugal, R.P. Shmipi. A review of refined shear deformation theories for isotropic and anisotropic laminated beams. Journal of Reinforced Plastics and Composites, 2001, vol. 20, no. 3, pp. 255–272.
[10] Francesco Tornabene. Free vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ method. Comput. Methods Appl. Mech. Engrg, 2011, no. 200, pp. 931–952.
[11] Zveryaev E.M., Makarov G.I. Prikladnaya matematika i mekhanika – Journal of applied mathematics and mechanics, 2008, vol. 72, no. 2, pp. 308–321.
[12] Nazarov S.A, Sweers G.H, Slutskiy A.S. Matematicheskiy sbornik – Sbornik: Mathematics, 2011, vol. 202, no. 8, pp. 41–80.
[13] Dimitrienko Yu.I. Vestnik MGTU im. N.E. Baumana. Ser. Estestvennye nauki — Herald of the Bauman Moscow State Technical University. Series Natural
Sciences, 2012, no. 3, pp. 86–99.
[14] Dimitrienko Yu.I., Yakovlev D.O. Mekhanika kompositsionnykh materialov i konstruktsiy — Mechanics of composite materials and structures, 2014, vol. 20,
no. 2, pp. 260–282.
[15] Dimitrienko Yu.I., Yurin Yu.V., Gubareva E.A. Matematicheskoe modelirovanie
i chislennye metody – Mathematical Modeling and Computational Methods, 2014, no. 4, pp. 36–57.
[16] Dimitrienko Yu.I., Gubareva E.A., Sborshchikov S.V. Matematicheskoe modeli-
rovanie i chislennye metody — Mathematical Modeling and Computational Methods, 2014, no.1 (1), pp. 36–56.
[17] Dimitrienko Yu.I., Gubareva E.A., Shalygin I.S. Inzhenernyy zhurnal: nauka
i innovatsii – Engineering Journal: Science and Innovation, 2015, no. 5 (41),
pp. 1–19.
[18] Dimitrienko Yu.I., Gubareva E.A., Yurin Yu.V. Inzhenernyy zhurnal: nauka
i innovatsii – Engineering Journal: Science and Innovation, 2016, no. 12 (60), pp. 1-25. DOI: 10.18698/2308-6033-2016-12-1557
[19] Dimitrienko Yu.I., Gubareva E.A., Yakovlev D.O. Nauka i Obrazovanie.
Elektronny zhurnal – Science and Education: scientific edition of Bau-man MSTU, 2014, no. 10. DOI: 10.7463/1014.0730105. pp. 359-382.
[20] Yu.I. Dimitrienko, I. D. Dimitrienko, S.V. Sborschikov. Multiscale Hierarchical Modeling of Fiber Reinforced Composites by Asymptotic Homogenization Method. Applied Mathematical Sciences, 2015, vol. 9, no. 145, рр. 7211–7220.
[21] Yu.I. Dimitrienko, I.D. Dimitrienko. Modeling of the thin composite laminates with general anisotropy under harmonic vibrations by the asymptotic homogenization method. Journal for Multiscale Computational Engineering, 2017,
vol. 15 (3), pp. 219–237.
[22] Dimitrienko Yu.I. Mekhanika sploshnoy sredy [Continuum mechanics]. In
4 vols. Vol. 4. Osnovy mekhaniki tverdykh sred [Foundations of mechanics of solid media]. BMSTU Publ., 2013, 624 p.
[23] Dimitrienko Yu.I. Tensor analysis and Nonlinear Tensor Functions. Kluwer Academic Publishers, Dordrecht/Boston/London, 2002, 680 p. [In Russ.: Dimitrienko Yu.I. Tenzornoe ischislenie. Moscow, Vysshaya shkola Publ., 2001, 576 p.].

Димитриенко Ю.И., Губарева Е.А. Асимптотическая теория тонких двухслой-ных упругих пластин с проскальзыванием слоев. Математическое моделирование и численные методы. 2019. № 1. с. 3–26.

Download article

Количество скачиваний: 133