Edge effect in the bending of a thin three-dimensional plate. (English. Russian original) Zbl 0722.73037

J. Appl. Math. Mech. 53, No. 4, 500-507 (1989); translation from Prikl. Mat. Mekh. 53, No. 4, 642-650 (1989).
Summary: The boundary layer near the rigidly clamped edge of a thin three- dimensional plate subjected to bending loads is investigated. It is shown that taking account of the next term in the deflection asymptotic form results in the appearance of inhomogeneities in the boundary conditions on the plate edge. It is proved that far from the edge the difference in the solution of the problem in an invariant formulation and the three- dimensional solution is inversely proportional to the plate thickness (the error for the Kirchhoff solution is inversely proportional to the square of the thickness; near the edge the accuracies of both solutions are identical). A correction term is found in a representation of the eigenfrequencies of the bending vibrations and a comparison is made with the Reissner theory.


74K20 Plates
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
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[1] Gol’denveizer, A. L., Construction of an approximate theory of shells by using asymptotic integration of the elasticity theory equations, PMM, 27, 5 (1962)
[2] Zino, E. I.; Tropp, E. A., Asymptotic Methods in Problems of Heat Conduction and Thermoelasticity Theories (1978), Izd. Leningrad. Gosud. Univ
[3] Berdichevskii, V. L., Variational Principles of the Mechanics of Continuous Media (1983), Nauka: Nauka Moscow · Zbl 0158.46505
[4] Nazarov, S. A., Introduction to the Asymptotic Methods of Elasticity Theory (1983), Izd. Leningrad. Gosud. Univ
[5] Agmon, S.; Nirenberg, L., Properties of solutions of ordinary differential equations in Banach spaces, Comm. Pure and Appl. Math., 16, 2 (1963) · Zbl 0117.10001
[6] Maz’ya, V. G.; Plamenevskii, B. A., On the coefficients in the asymptotic form of solutions of elliptical boundary value problems in domains with conical points, Math. Nachr., 76 (1977) · Zbl 0359.35024
[7] Oleinik, O. A.; Iosif’yan, G. A., On damping conditions and the limits behaviour at infinity of solutions of a system of elasticity theory equations, Dokl. Akad. Nauk SSSR, 258, 3 (1981) · Zbl 0497.73003
[8] Gregory, R. D.; Wan, F. V.M., Decaying states of plane strain in a semi-infinite strip and boundary conditions for plate theory, J. Elast., 14, 1 (1984) · Zbl 0536.73047
[9] Shoikhet, B. A., An energetic identity in physically non-linear elasticity theory and error estimates of the plate equations, PMM, 40, 2 (1976)
[10] Leora, S. N.; Nazarov, S. A.; Proskura, A. V., Derivation of limit equations for elliptical problems in thin domains by using an electronic computer, Zh. vychisl. Matem. Mat. Fiziki, 26, 7 (1986) · Zbl 0626.65129
[11] Agmon, S.; Douglis, A.; Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. 2, Comm. Pure and Appl. Math., 17, 1 (1964) · Zbl 0123.28706
[12] Nazarov, S. A., Structure of solutions of elliptic boundary value problems in thin domains, Vestnik Leningrad. Gosud. Univ., Matem., Mekhan. Astron., 7 (1982) · Zbl 0509.35008
[13] Maz’ya, V. G.; Plamenevskii, B. A., Schauder estimates of solutions of elliptic boundary value problems in domains with ribs on the boundary, Partial Differential Equations, Tr. Seminara S.L. Sobolev, 2 (1978) · Zbl 0423.35021
[14] Kato, T., Perturbation Theory of Linear Operators (1972), Mir: Mir Moscow · Zbl 0247.47009
[15] Ivanov, L. A.; Kotko, L. A.; Krein, S. G., Boundary value problems in invariable domains, (Differential Equations and their Application, 19 (1977), Izd. Akad. Nauk Lit SSR: Izd. Akad. Nauk Lit SSR Vil’nyus) · Zbl 0368.35006
[16] Nazarov, S. A., Asymptotic Expansions of Eigenvalues (1987), Izd. Leningrad. Gosud. Univ · Zbl 0618.35005
[17] Vishik, M. I.; Lyusternik, L. A., Regular degeneration and boundary layer for linear differential equations with a small parameter, Usp. Matem. Nauk, 12, 5 (1957) · Zbl 0087.29602
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