Asymptotic Integration of One Narrow Plate Problem

  • Valentina O. Finiukova
  • Alexander M. Stolyar
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 15)


The derivation of a narrow plate model is carried out in the paper. It has been based on the method of asymptotic integration in connection with the boundary layer method and applied to one non-linear dynamic problem. It allows to reduce the solution of the given problem to a sequence of linear or nonlinear one-dimensional initial boundary-value problems (as a result of the first iteration process) and a sequence of two-dimensional linear problems (second order process) and get the solution of a given problem with high accuracy. The small parameter is equal to the relation of the lengths of adjacent sides of the plate. It has been obtained that the main term of the asymptotic expansion satisfies the known equation of a beam theory. Some examples are calculated in order to find the limits of asymptotics obtained.


Asymptotic integration Narrow plate 


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  1. 1.
    Aghalovyan, L.A.: On the boundary layer of the orthotropic plates. (in Russian). Izv. AN Arm. SSR. Mechanics. 26(2), 27–43 (1973)Google Scholar
  2. 2.
    Ciarlet, P.G., Rabier, P.: Les equations de von Kármán. Springer-Verlag, Berlin-Heidelberg- New York (1980)Google Scholar
  3. 3.
    Gol’denweiser, A.L.: Theory of Thin Shells (in Russian). Nauka, Moscow (1976)Google Scholar
  4. 4.
    Grinberg, G.A.: On the method, suggested by P. F. Papkovich for solution of the plane problem of the theory of elasticity in rectangular domain and the problem of bending of rectangular thin plate with two fastened edges and its some generalizations (in Russian).. Prikl. Mat. Mech. 17(2), 211–228 (1953)Google Scholar
  5. 5.
    Kucherenko, V.V., Popov, V.A.: Asymptotics of solution of elasticity theory problems in thin domains (in Russian). Dokl. AN SSSR. 274(1), 58–61 (1984)Google Scholar
  6. 6.
    Russell, D.L., White, L.W.: Formulation and validation of dynamical models for narrow plate motion. Appl. Math. Comp. 58, 103–141 (1993)CrossRefGoogle Scholar
  7. 7.
    Russell, D.L., White, L.W.: The lowed narrow plate model. Electronic J. Diff. Equations. 27, 1–19 (2000)Google Scholar
  8. 8.
    Srubshchik, L.S., Stolyar, A.M., Tsibulin, V.G.: Asymptotic integration of nonlinear equations of cylindrical panel vibrations. J. Appl. Math. Mech. 52(4), 511–518 (1988)CrossRefGoogle Scholar
  9. 9.
    Stolyar, A.M.: Asymptotic integration of the equation of vibrations of long rectangular plate (in Russian). Izv. Sev.-Kavk. Nauchn. Tsentra Vyssh. Shk., Estestv. Nauki, 4(56), 46–50 (1986)Google Scholar
  10. 10.
    Stolyar, A.M.: Asymptotic analysis of the problems of statics and dynamics of narrow rectangular plates. (in Russian). Izv. Vyssh. Ucheb. Zaved., Sev.-Kavk. Region, Estestv. Nauki, Specvypusk Pseudo-differential Equations and Some Problems of Mathematical Physics, pp. 107–111 (2005)Google Scholar
  11. 11.
    Stolyar, A.M.: Representation of two real functions as series of P. F. Papkovich functions in one problem of limiting transition (in Russian). Izv. Vyssh. Ucheb. Zaved., Sev.-Kavk. Region, Estestv. Nauki, Specvypusk Actual Problems of Mathematical Hydrodynamics, pp. 203–206 (2009)Google Scholar
  12. 12.
    Sugimoto, N.: Nonlinear theory for flexural motions of thin elastic plate. Trans. ASME. J. Appl. Mech. 48(2), 377–390 (1981)CrossRefGoogle Scholar
  13. 13.
    Ustinov, Yu.A., Yudovich, V.I.: On the completeness of a system of elementary solutions of the biharmonic equation in a semi-strip. J. Appl. Math. Mech. 37(4), 665–674 (1973)CrossRefGoogle Scholar
  14. 14.
    Ustinov, Yu.A.: Boundary problems and the problem of limiting transition from three dimensional problems of the theory of elasticity to two-dimensional problems for heterogeneous plates. Dr. habil. Thesis (in Russian). Moscow (1973)Google Scholar
  15. 15.
    Vasil’ev, V.V., Lur’e, S.A.: Plane problem of elasticity theory for orthotropic clamped strip (in Russian). Izv. AN SSSR. MTT. 5, 125–135 (1984)Google Scholar
  16. 16.
    Vorovich, I.I.: Some mathematical problems of the theory of plates and shells (in Russian). In Trans. of II All-Union Congress on Theoretical and Applied Mechanics, pp. 116–136. Moscow (1966)Google Scholar
  17. 17.
    Vorovich, I.I.: Some results and problems of the asymptotic theory of plates and shells. (in Russian). In Trans. of I All-Union school on theory and numerical methods of calculation of plates and shells, pp. 51–149. Tbilisi (1975)Google Scholar

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Authors and Affiliations

  • Valentina O. Finiukova
    • 1
  • Alexander M. Stolyar
    • 1
  1. 1.Southern Federal University,Rostov-on-DonRussia

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