phone +7 (3412) 91 60 92

Archive of Issues

Russia Smolensk
Section Mathematics
Title Stability of mathematical models of the main problems of the anisotropic theory of elasticity
Author(-s) Yudenkov A.V.a, Volodchenkov A.M.bc
Affiliations Smolensk State Academy of Physical Culture, Sports and Tourisma, Plekhanov Russian University of Economics, Smolensk Branchb, Saratov State Academy of Law, Smolensk Branchc
Abstract The boundary problems of the complex-variable function theory are effectively used while investigating equilibrium of homogeneous elastic mediums. The most complicated systems of the boundary value problems correspond to the case when an elastic body exhibits anisotropic properties. Anisotropy of the medium results in the drift of boundary conditions of the function that in general disrupts analyticity of the functions of interest. The paper studies systems of the boundary value problems with drift for analytic vectors corresponding to the primal elastic problems (first, second and mixed problems). Systems of analytic vectors with drift are reduced to equivalent systems of Hilbert boundary value problems for analytic functions with weak singularity integrators. The obtained general solution of the primal boundary value problems for the anisotropic theory of elasticity allows us to check the above problems for stability with respect to perturbations of boundary value conditions and contour shape. The research is relevant as there is necessity to apply approximate numerical methods to the boundary value problems with drift. The main research result comes to be a proof of stability of the systems of the vector boundary value problems with drift for analytic functions on the Hölder space corresponding to the primal problems of the elastic theory for anisotropic bodies in the case of change in the boundary value conditions and contour shape.
Keywords boundary value problem, analytic function, elasticity theory, Fredholm equation
UDC 517.977
MSC 49N75, 91A23
DOI 10.35634/vm200108
Received 1 December 2019
Language Russian
Citation Yudenkov A.V., Volodchenkov A.M. Stability of mathematical models of the main problems of the anisotropic theory of elasticity, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 1, pp. 112-124.
  1. Balk M.B. Polyanalytic functions and their generalizations, Complex Analysis I, Berlin: Springer, 1997, pp. 195-253.
  2. Vekua I.N. On one solution of the main biharmonic boundary value problem and the Dirichlet problem, Nekotorye problemy matematiki i mekhaniki (Some problems of mathematics and mechanics), Leningrad: Nauka, 1970, pp. 120-127.
  3. Gakhov F.D. Kraevye zadachi (Boundary value problems), Moscow: Nauka, 1977.
  4. Lekhnitskii G.S. Teoriya uprugosti anizotropnogo tela (Elasticity theory for the anisotropic body), Moscow: Nauka, 1977.
  5. Maksimova L.A., Yudenkov A.V. Stochastic potential theory in the two-dimensional elasticity theory, Bulletin of the Yakovlev Chuvash State Pedagogical University. Series: Mechanics of Limit State, 2015, no. 4 (26), pp. 134-142 (in Russian).
  6. Muskhelishvili N.I. Nekotorye osnovnye zadachi matematicheskoi teorii uprugisti (Some primal problems of the mathematical theory of elasticity), Moscow: Nauka, 1966.
  7. Muskhelishvili N.I. Singulyarnye inregral'nye uravnenia (Singular integral equations), Moscow: Nauka, 1968.
  8. Redkozubov S.A., Yudenkov A.V., Volodchenkov A.M. Simulation of the process of linear deformation of the elastic homogeneous body through bianalytic functions, Yakovlev Chuvash State Pedagogical University Bulletin, 2006, no. 1 (49), pp. 128-134 (in Russian).
  9. Savin G.N. Stress distribution around holes, Kiev: Naukova Dumka, 1968.
  10. Yudenkov A.V. Kraevye zadachi so sdvigom dlya polianaliticheskikh funktsii i ikh prilozheniya k voprosam staticheskoi teorii uprugosti (Boundary value problems with drift for polyanalytic functions and their application to the static theory of elasticity), Smolensk: Smyadyn', 2002.
  11. Yudenkov A.V., Volodchenkov A.M. The primal problems of the elasticity theory for bodies with linear anisotropy in the stochastic theory of potential, Uchenye Zapiski. Elektronnyi Nauchnyi Zhurnal Kurskogo Gosudarstvennogo Universiteta, 2013, no. 2 (26), pp. 14-17 (in Russian).
  12. Balk M.B. Polyanalytic functions, Berlin: Akademie Verlag, 1991.
  13. Kuritsyn S.Yu., Rasulov K.M. On a generalized Riemann problem for metaanalytic functions of the second type, Lobachevskii Journal of Mathematics, 2018, vol. 39, no. 1, pp. 97-103.
  14. Rasulov K.M. On the uniqueness of the solution of the Dirichlet boundary value problem for quasiharmonic functions in a non-unit disk, Lobachevskii Journal of Mathematics, 2018, vol. 39, no. 1, pp. 142-145.
Full text
<< Previous article
Next article >>