Numerical Analysis and Applications

, Volume 8, Issue 2, pp 148–158 | Cite as

The first boundary-value problem of the elasticity theory for a cylinder with N cylindrical cavities

Article

Abstract

An effective method for the analytical-numerical solution to a nonaxisymmetric boundary-value problem of the elasticity theory for a multiconnected body in the form of a cylinder with N cylindrical cavities is proposed. The solution is constructed as a superposition of exact basis solutions of the Lame equation for a cylinder in coordinate systems fitted to the centers of the boundary surfaces of the body. The boundary conditions are exactly satisfied with the help of the apparatus of the generalized Fourier method. As a result, the original problem reduces to an infinite system of linear algebraic equations, which has a Fredholm operator in the Hilbert space l 2. The resolving system is numerically solved by the reduction method. The rate of convergence of the reduction method is investigated. The numerical analysis of stresses in the areas of their greatest concentration is carried out. The reliability of the results obtained is confirmed by comparing them for two cases: a cylinder with sixteen cylindrical cavities and a cylinder with four cylindrical cavities.

Keywords

boundary-value problem multiconnected body generalized Fourier method resolving system cylindrical boundary addition theorems 

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References

  1. 1.
    Abramyan, B.L., On the Problem of Axisymmetric Deformation of a Circular Cylinder, Dokl. Akad. Nauk Arm. SSR, 1954, Vol. 19, No. 1, pp. 3–12.MATHMathSciNetGoogle Scholar
  2. 2.
    Arutyunyan, N.Kh., Movchan, A.B., and Nazarov, S.A., Behavior of Solutions of Elasticity Theory Problems in Unbounded Regions with Paraboloidal and Cylindrical Inclusions or Cavities, Usp. Mekh., 1987, Vol. 10, No. 4, pp. 3–91.MathSciNetGoogle Scholar
  3. 3.
    Valov, G.M., On Axisymmetric Deformation of a Finite-Length Solid Circular Cylinder, Prikl. Mat. Mekh., 1962, Vol. 26, No. 4, pp. 650–667.MathSciNetGoogle Scholar
  4. 4.
    Vanin, G.A., Mikromekhanika kompozitnykh materialov (Micromechanics of Composite Materials), Kiev: Naukova Dumka, 1985.Google Scholar
  5. 5.
    Gomilko, A.M., Grinchenko, V.T., and Meleshko, V.V., Homogeneous Solutions in the Problem of Equilibrium of a Finite-Length Elastic Cylinder, Teor. Prikl. Mekh., 1989, no. 20, pp. 3–9.Google Scholar
  6. 6.
    Grinchenko, V.T., Axisymmetric Problem of the Elasticity Theory for a Semi-Infinite Circular Cylinder, Prikl. Mekh., 1965, Vol. 1, No. 1, pp. 109–119.Google Scholar
  7. 7.
    Grinchenko, V.T., Axisymmetric Problem of the Elasticity Theory for a Finite-Length Thick-Walled Cylinder, Prikl. Mekh., 1967, Vol. 3, No. 8, pp. 93–103.Google Scholar
  8. 8.
    Grinchenko, V.T., Ravnovesie i ustanovivshiesya kolebaniya uprugikh tel konechnykh razmerov (Equilibrium and Steady-State Oscillations of Finite-Size Elastic Bodies), Kiev: Naukova Dumka, 1978.Google Scholar
  9. 9.
    Kantorovich, L.V. and Akilov, G.L., Funktsionalnyi analiz (Functional Analysis), Moscow: Nauka, 1977.Google Scholar
  10. 10.
    Christensen, R.M., Mechanics of Composite Materials, New York: Wiley, 1979.Google Scholar
  11. 11.
    Lebedev, N.N., Spetsialnye funktsii i ikh prilozheniya (Special Functions and Their Applications), Moscow: Fizmatlit, 1963.Google Scholar
  12. 12.
    Lur’e, A.I., Prostranstvennye zadachi teorii uprugosti (Spatial Problems of the Elasticity Theory), Moscow: Gostekhizdat, 1955.Google Scholar
  13. 13.
    Nikolaev, A.G. and Tanchik, E.A., Stress-Strain State in a Cylindrical Sample with Two Parallel Cylindrical Fibers, Aviats.-Kosm. Tekh. Tekhnol., 2013, no. 6(103), pp. 32–38.Google Scholar
  14. 14.
    Nikolaev, A.G. and Protsenko, V.S., Obobshchennyi metod Fur’e v prostranstvennykh zadachakh teorii uprugosti (Generalized Fourier Method in Spatial Problems of the Elasticity Theory), Kharkiv: Zhukovskii National Aerospace University “Kharkiv Aviation Institute,” 2011.Google Scholar
  15. 15.
    Nikolaev, A.G., Justification of the Fourier Method in the Basic Boundary-Value Problems of the Elasticity Theory for Some Spatial Canonical Domains, Dokl. Nats. Akad. Nauk Ukrainy, 1998, no. 2, pp. 78–83.MathSciNetGoogle Scholar
  16. 16.
    Nikolaev, A.G. and Tanchik, E.A., Stress Distribution in a Cylindrical Sample of a Material with Two Cylindrical Cavities, in Issues of Design and Production of Flying Vehicle Structures, no. 4(76), Kharkiv: Zhukovskii National Aerospace University “Kharkiv Aviation Institute,” 2013, pp. 26–35.Google Scholar
  17. 17.
    Nikolaev, A.G., Addition Theorems of Solutions of the Lame Equation, Kharkov Aviation Institute, Kharkov, State Scientific Technical Library of Ukraine, dep. 21.06.93, no. 117–Uk 93, 1993.Google Scholar
  18. 18.
    Prokopov, V.K., Axisymmetric Problem of the Elasticity Theory for an Isotropic Cylinder, Tr. Leningrad. Politekhn. Inst., 1950, no. 2, pp. 286–304.Google Scholar
  19. 19.
    Tokovyy, Yu.V., Axisymmetric Loading of a Circular Cylinder with Normal Tension Uniformly Distributed over the Side Surface, Prikl. Probl. Mekh. Mat., 2010, no. 8, pp. 144–151.Google Scholar
  20. 20.
    Meleshko, V.V. and Tokovyy, Yu.V., Equilibrium of an Elastic Finite Cylinder under Axisymmetric Discontinuous Normal Loadings, J. Eng.Math., 2013, Vol. 78, pp. 143–166.MathSciNetCrossRefGoogle Scholar
  21. 21.
    Vihak, V.M., Yasinskyy, A.V., Tokovyy, Yu.V., and Rychahivskyy, A.V., Exact Solution of the Axisymmetric Thermoelasticity Problem for a Long Cylinder Subjected to Varying with-Respect-to-Length Loads, J. Mech. Behav. Mater., 2007, no. 18, pp. 141–148.CrossRefGoogle Scholar
  22. 22.
    Williams, D.K. and Ranson, W.F., Pipe-Anchor Discontinuity Analysis Utilizing Power Series Solutions, Bessel Functions, and Fourier Series, Nucl. Eng. Des., 2003, no. 220, pp. 1–10.CrossRefGoogle Scholar
  23. 23.
    Zhong, Z. and Sun, Q.P., Analysis of a Transversely Isotropic Rod Containing a Single Cylindrical Inclusion with Axisymmetric Eigenstrains, Int. J. Solids Struct., 2002, Vol. 39, No. 23, pp. 5753–5765.MATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Zhukovskii National Aerospace UniversityKharkivUkraine

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