and Computational Methods

doi: 10.18698/2309-3684-2022-2-2862

The problem of development a theory for calculating the stress-strain state of thin multilayer elastic plates, for which linearized slip conditions are specified at the interface between the layers, is considered. The solution of this problem is constructed using an asymptotic analysis of the general equations of the 3-dimensional theory of elasticity with the conditions of non-ideal contact of the layers. The asymptotic analysis is carried out with respect to a small geometric parameter representing the ratio of the plate thickness to its characteristic length. Recurrent formulations of local quasi-one-dimensional problems of the theory of elasticity with slip are obtained. Explicit analytical solutions are obtained for these problems. The derivation of the averaged equations of elastic equilibrium of multilayer plates is presented, taking into account the slippage of the layers. It is shown that due to the effect of slippage of layers, the system of averaged equations of the theory of multilayer plates has an increased - 5th order of derivatives, in contrast to the classical 4th order, which takes place in the theory of Kirchhoff-Love plates. It is shown that the asymptotic theory makes it possible to obtain an explicit analytical expression for all 6 components of the stress tensor in the layers of the plate. As a special case, the problem of calculating the stress-strain state of a 4-layer plate under uniform pressure bending with one slip coefficient is considered. A complete analytical solution of this problem is obtained, including explicit expressions for all non-zero components of the stress tensor. A numerical analysis of the solution of the averaged problem for a composite plate is carried out, in which the layers are unidirectional reinforced fibrous materials oriented at different angles. A comparative analysis of the influence of the fiber reinforcement angles and the slip coefficient of the layers on the displacement of the plate and the distribution of stresses in the layers was carried out. It is shown that the problem of bending a plate with slip admits the existence of a spectrum of critical values of the slip coefficient, when passing through which the displacements and stresses in the layers of the plate change significantly, and these critical values depend on the angle of reinforcement of the composite layers.

Sai kumar M., Bhaskara Reddy C. Modelling and structural analysis of leaf spring using finite element method. International Research Journal of Engineering and Technology, 2017, vol. 4, iss. 1, pp. 1155–1161.

Scott W.H. A model for a two-layered plate with interfacial slip. International Series of Numerical Mathematics, 1994, vol. 118, pp.143–170.

Sedighi M., Shirazib K.H., Naderan-Tahanb K. Stick-slip analysis in vibrating two-layer beams with frictional interface. Latin American Journal of Solid and Structures, 2013, no. 10, pp. 1025–1042.

Damisa O., Olunloyo V.O.S., Osheku C.A., Oyediran A.A. Dynamic analysis of slip damping in clamped layered beams with non-uniform pressure distribution at the interface. Journal of Sound and Vibration, 2008, vol. 309, pp. 349–374.

Awrejcewicz J., Krysko A.V. An iterative algorithm for solution of contact problems of beams, plates and shells. Mathematical Problems in Engineering, 2006, vol. 2006, art. no. 71548. DOI 10.1155/MPE/2006/71548.

Awrejcewicz J., Krysko A.V., Ovsiannikova O. Novel procedure to compute a contact zone magnitude of vibrations of two-layered uncoupled plates. Mathematical Problems in Engineering, 2005, vol. 2005, iss. 4, pp. 425–435. DOI: 10.1155/MPE.2005.425

Lyav A. Matematicheskaya teoriya uprugosti [Mathematical theory of elasticity]. Moscow, ONTI Publ., 1935, 674 p.

Timoshenko S., Woinowsky-Krieger S. Theory of Plates and Shells. USA, McGraw-Hill College, 1976, 580 p.

Vasiliev V.V. Mekhanika konstrukcij iz kompozicionnyh materialov [Mechanics of structures made of composite materials]. Moscow, Mashinostroenie Publ., 1988, 272 p.

Grigolyuk E.I., Kulikov G.M. Generalized model of the mechanics of thinwalled structures made of composite materials. Mechanics of Composite Materials, 1989, vol. 24, iss. 4, pp. 537–543.

Dimitrienko Y.I., Gubareva E.A. Asymptotic theory of thin two-layer elastic plates with layer slippage. Маthematical Modeling and Coтputational Methods, 2019, no. 1, pp. 3–26.

Kohn R.V., Vogelius M. A new model for thin plates with rapidly varying thickness. International Journal of Solids and Structures, 1984, vol. 20, iss. 4, pp. 333–350.

Sheshenin S.V. Asymptotic analysis of plates with periodic cross-sections. Mechanics of Solids, 2006, vol. 41, no. 6, pp. 57–63.

Nazarov S.A., Sweers G.H., Slutskij A.S. Homogenization of a thin plate reinforced with periodic families of rigid rods. Sbornik: Mathematics, 2011, vol. 202, iss. 8, pp. 1127–1168.

Dimitrienko Yu.I. Asymptotic theory of multilayer thin plates. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2012, no. 3, pp. 86–99.

Dimitrienko Yu.I., Yakovlev D.O. Asymptotic theory of thermoelasticity of multilayer composite plates. Mekhanika kompozitsionnykh materialov i konstruktsii, 2014, vol. 20, no. 2, pp. 260–282.

Dimitrienko Y.I., Gubareva E.A., Yurin Y.V. Asymptotic theory of thermocreepfor multilayer thin plates. Маthematical Modeling and Coтputational Methods, 2014, no. 4, pp. 18–36.

Dimitrienko Y.I., Gubareva E.A., Sborschikov S.V. Asymptotic theory of constructive-orthotropic plates with two-periodic structures. Маthematical Modeling and Coтputational Methods, 2014, no. 1, pp. 36–56.

Dimitrienko Y.I., Dimitrienko I.D., Sborschikov S.V. Multiscale hierarchical modeling of fiber reinforced composites by asymptotic homogenization method. Applied Mathematical Sciences, 2015, vol. 9. iss. 145–148, pp. 7211–7220

Dimitrienko Y.I., Dimitrienko I.D. 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, iss. 3, pp. 219–237.

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. Tenzornoe ischislenie [Tensor calculus]. Moscow, Higher School Publ., 2001, 576 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.

Димитриенко Ю.И., Губарева Е.А. Асимптотическая теория многослойных тонких упругих пластин с проскальзыванием слоев. Математическое моделирование и численные методы, 2022, № 2, с. 30–64

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