Abstract
In this paper we study the procedure of reducing the three-dimensional problem of elasticity theory for a thin inhomogeneous anisotropic plate to a two-dimensional problem in the median plane. The plate is in equilibrium under the action of volume and surface forces of general form. À notion of internal force factors is introduced. The equations for force factors (the equilibrium equations in the median plane) are obtained from the thickness-averaged three-dimensional equations of elasticity theory. In order to establish the relation between the internal force factors and the characteristics of the deformed middle surface, we use some prior assumptions on the distribution of displacements along the thickness of the plate. To arrange these assumptions in order, the displacements of plate points are expanded into Taylor series in the transverse coordinate with consideration of the physical hypotheses on the deformation of a material fiber being originally perpendicular to the median plane. The well-known Kirchhoff—Love hypothesis is considered in detail. À closed system of equations for the theory of inhomogeneous anisotropic plates is obtained on the basis of the Kirchhoff—Love hypothesis. The boundary conditions are formulated from the Lagrange variational principle.
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References
I. N. Vekua, General Methods of Constructing Different Versions of Shell Theory (Nauka, Moscow, 1982) [in Russian].
N. A. Kil'chevskii, Fundamentals of the Analytical Mechanics of Shells (Izd. Akad. Nauk UkrSSR, Kiev, 1963) [in Russian].
V. Z. Vlasov, Selected Works Vol. 1 (Izd. Akad. Nauk SSSR, Moscow, 1962) [in Russian].
V. V. Vlasov, Method of Initial Functions in Problems of the Theory of Elasticity and Structural Mechanics (Stroiizdat, Moscow, 1975) [in Russian].
L. A. Agolovyari, Asymptotic Theory of Anisotropic Plates and Shells (Nauka, Moscow, 1997) [in Russian].
I. I. Vorovich, "Some Results and Problems of Asymptotic Theory of Plates and Shells," in Proc. 1st All- Union School on Theory and Numerical Methods for Shells and Plates, Gegechkori, Georgia, October 1 10, 1974 (Tbilisi Univ., Tbilisi, 1975), pp. 51–149.
A. L. Goldenveiser, Theory of Elastic Thin Shells (Nauka, Moscow, 1976; Pergamori, Oxford, 1961).
S. A. Ambartsumyari, General Theory of Anisotropic Shells (Nauka, Moscow, 1974) [in Russian].
A. A. Dudcheriko, S. A. Lurie, and L F. Obraztsov, "Anisotropic Multilayered Plates arid Shells," (VINITI, Moscow, 1983), Itogi Nauki Tekli.,Ser.: Mekh. Def. Tverd. Tela, Vol. 15, pp. 3–68.
A. L. Lurie, Statics of Thin-Walled Elastic Shells (Gostekhizdat, Moscow—Leningrad, 1947) [in Russian].
Ya. M. Grigoreriko and A. T. Vasileriko, Static Problems for Anisotropic Inhomogeneous Shells (Nauka, Moscow, 1992) [in Russian].
V. I. Gorbachev and V. A. Simakov, "An Operator Method for Solving the Equilibrium Problem for an Elastic Inhomogeneous Anisotropic Slab," Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 55–64 (2004) [Mech. Solids 39 (2),43–49 (2004)].
V. I. Gorbachev and O. Yu. Tolstykh, “On an Approach to the Construction of Engineering Theory of Inhomogeneous Anisotropic Beams,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 6, 137–121 (2005) [Mech. Solids 40 (6), 101–106 (2005)].
V. I. Gorbachev and L. L. Firsov, "New Statement of the Elasticity Problem for a Layer," Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 1, 114–121 (2011) [Mech. Solids 46 (1), 89–95 (2011)].
V. V. Vasil'ev, Mechanics of Composite Structures (Mashiriostroeriie, Moscow, 1988; Taylor arid Francis, London, 1993).
B. E. Pobedrya and D. V. Georgievskii, Lectures on Elasticity Theory (Editorial, Moscow, 1999) [in Russian].
V. G. Zubchariiriov, Foundations of the Theory of Elasticity and Plasticity (Vysshaya Shkola, Moscow, 1990) [in Russian].
P. M. Ogibalov aND M. A. Kolturiov, Shells and Plates (Mosk. Gos. Univ., Moscow, 1969) [in Russian].
S. P. Timoshenko, History of Strength of Materials (McGraw, New York, 1983; Librokom, Moscow, 2009).
K. Rektorys, Variational Methods in Mathematics, Science and Engineering (Reidel, Dordrecht, 1980; Mir, Moscow, 1985).
B. E. Pobedrya, Numerical Methods in the Theory of Elasticity and Plasticity 2nd ed. (Mosk. Gos. Univ., Moscow, 1995) [in Russian].
V. L. Berdichevskii, Variational Principles of Continuum Mechanics (Nauka, Moscow, 1983) [in Russian].
S. G. Mikhliri, Variational Methods in Mathematical Physics (Gos. Izd. Tekh. Teor. Lit., Moscow, 1957; Macmillanm, New York, 1964).
A. N. Aridreev and Yu. V. Nemirovskii, Multilayer Anisotropic Shells and Plates: Bending, Stability, and Vibrations (Nauka, Novosibirsk, 2001) [in Russian].
V. I. Gorbachev, “The Engineering Theory of the Deforming of the Nonuniform Plates from Composite Materials,” Mekh. Komposits. Mater. Konstruct. 22 (4), 585–601 (2016).
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Original Russian Text © V.I. Gorbachev. L.A. Kabanova. 2018. published, in Vestnik Moskovskogo Universiteta. Matematika. Mekhanika, 2018, Vol 73, No. 3, pp. 43–50.
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Gorbachev, V.I., Kabanova, L.A. Formulation of Problems in the General Kirchhoff—Love Theory of Inhomogeneous Anisotropic Plates. Moscow Univ. Mech. Bull. 73, 60–66 (2018). https://doi.org/10.3103/S0027133018020020
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DOI: https://doi.org/10.3103/S0027133018020020