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Reducing three-dimensional elasticity problems to two-dimensional problems by approximating stresses and displacements by legendre polynomials

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Abstract

Shell equations are constructed in orthogonal curvilinear coordinates using approximations of stresses and displacements by Legendre polynomials. The order of the constructed system of differential equations is independent of whether stresses and displacements or their combination are specified on the shell surfaces, which provides the correct formulation of the surface conditions in terms of both displacements and stresses. This allows the system of differential equations of laminated shells to be constructed using matching conditions for displacements and stresses on the contact surfaces.

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References

  1. S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, New York (1959).

    Google Scholar 

  2. A. I. Soler, “Higher-order theories for structural analysis using Legendre polynomial expansions,” Trans. ASME, Ser. E: J. Appl. Mech., 36, No. 4, 757–762 (1969).

    MATH  Google Scholar 

  3. G. V. Ivanov, “Solution of the plane mixed problem of the theory of elasticity in the form of a series in Legendre polynomials,” J. Appl. Mech. Tech. Phys., No. 6, 856–866 (1976).

  4. G. V. Ivanov, Theory of Plates and Shells [in Russian], Izd. Novosib. Univ., Novosibirsk (1980).

    Google Scholar 

  5. B. L. Pelekh, Laminated Anisotropic Plates and Shells with Stress Concentrators [in Russian], Naukova Dumka, Kiev (1982).

    Google Scholar 

  6. L. A. Dergileva, “Method of solving a two-dimensional contact problem for an elastic layer,” in: Dynamics of Continuous Media (collected scientific papers) [in Russian], No. 25, Inst. of Hydrodynamics, Sib. Div., Acad. of Sci. of the USSR, Novosibirsk (1976), pp. 24–32.

    Google Scholar 

  7. Yu. M. Volchkov and L. A. Dergileva, “Solution of elastic-layer problems using approximate equations and a comparison with elastic-theory solutions,” in: Dynamics of Continuous Media (collected scientific papers) [in Russian], No. 28, Inst. of Hydrodynamics, Sib. Div., Acad. of Sci. of the USSR, Novosibirsk (1977), pp. 43–54.

    Google Scholar 

  8. D. V. Vajeva and Yu. M. Volchkov, “The equations for determination of stress-deformed state of multilayered shells,” in: Proc. of the 9th Russian-Korean Symp. on Sci. and Technol. (Novosibirsk, June 26—July 2, 2005), Novosib. State Univ., Novosibirsk (2005), pp. 547–550.

    Chapter  Google Scholar 

  9. Yu. M. Volchkov, “Finite elements with matching conditions at their faces,” in: Dynamics of Continuous Media (collected scientific papers) [in Russian], No. 116, Inst. of Hydrodynamics, Sib. Div., Russian Acad. of Sci., Novosibirsk (2000), pp. 175–180.

    Google Scholar 

  10. Yu. M. Volchkov, L. A. Dergileva, and G. V. Ivanov, “Numerical modeling of stress states in two-dimensional problems of elasticity by the layers method,” J. Appl. Mech. Tech. Phys., 35, No. 6, 936–941 (1994).

    Article  Google Scholar 

  11. Yu. M. Volchkov and L. A. Dergileva, “Edge effects in the stress state of a thin elastic interlayer,” J. Appl. Mech. Tech. Phys., 40, No. 2, 189–195 (1999).

    Article  MATH  Google Scholar 

  12. A. E. Alekseev, “Bending of a three-layer orthotropic beam,” J. Appl. Mech. Tech. Phys., 36, No. 3, 458–465 (1995).

    Article  Google Scholar 

  13. A. E. Alekseev and A. G. Demeshkin, “Detachment of a beam glued to a rigid plate,” J. Appl. Mech. Tech. Phys., 44, No. 4, 577–583 (2003).

    Article  Google Scholar 

  14. A. E. Alekseev, “Derivation of equations for a layer of variable thickness based on expansions in terms of Legendre’s polynomials,” J. Appl. Mech. Tech. Phys., 35, No. 4, 612–622 (1994).

    Article  MathSciNet  Google Scholar 

  15. A. E. Alekseev, V. V. Alekhin, and B. D. Annin, “A plane elastic problem for an inhomogeneous layered body,” J. Appl. Mech. Tech. Phys., 42, No. 6, 1038–1042 (2001).

    Article  Google Scholar 

  16. A. E. Alekseev and B. D. Annin, “Equations of deformation of an elastic inhomogeneous laminated body of revolution,” J. Appl. Mech. Tech. Phys., 44, No. 3, 432–437 (2003).

    Article  Google Scholar 

  17. Yu. M. Volchkov and L. A. Dergileva, “Equations of elastic anisotropic layer,” J. Appl. Mech. Tech. Phys., 45, No. 2, 301–309 (2004).

    Article  Google Scholar 

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Authors and Affiliations

  1. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk, 630090

    Yu. M. Volchkov & L. A. Dergileva

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  1. Yu. M. Volchkov
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  2. L. A. Dergileva
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Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 179–190, May–June, 2007.

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Volchkov, Y.M., Dergileva, L.A. Reducing three-dimensional elasticity problems to two-dimensional problems by approximating stresses and displacements by legendre polynomials. J Appl Mech Tech Phys 48, 450–459 (2007). https://doi.org/10.1007/s10808-007-0056-1

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  • DOI: https://doi.org/10.1007/s10808-007-0056-1

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