SpringerLink
Log in

On the Problem of Solvability of Nonlinear Boundary Value Problems for Shallow Isotropic Shells of Timoshenko Type in Isometric Coordinates

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

The solvability of a boundary value problem for a system of five nonlinear second-order partial differential equations under given nonlinear boundary conditions, which describes the equilibrium state of elastic flat inhomogeneous isotropic shells with loose edges in the framework of the Timoshenko shear model, referred to isometric coordinates, is studied. The boundary value problem is reduced to a nonlinear operator equation with respect to generalized displacements in a Sobolev space, with the solvability of this equation being established using the contraction map** principle.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

REFERENCES

  1. I. I. Vorovich, Nonlinear Theory of Shallow Shells (Nauka, Moscow, 1989; Springer, New York, 1999). https://doi.org/10.1007/b98970

  2. N. F. Morozov, Selected Two-Dimensional Problems of Elasticity Theory (Leningrad. Gos. Univ., Leningrad, 1978).

    Google Scholar 

  3. M. M. Karchevsky, “Investigation of solvability of the nonlinear equilibrium problem of a shallow unfixed shell,” Uchenye Zap. Kazanskogo Univ. Ser. Fiziko-Mat. Nauki 155 (3), 105–110 (2013).

    Google Scholar 

  4. V. N. Paimushin, S. A. Kholmogorov, and I. B. Badriev, “Consistent equations of nonlinear multilayer shells theory in the quadratic approximation,” Lobachevskii J. Math. 40, 349–363 (2019). https://doi.org/10.1134/s1995080219030156

    Article  MathSciNet  Google Scholar 

  5. S. N. Timergaliev, Existence Theorems in Nonlinear Theory of Thin Elastic Shells (Izd-vo Kazansk. Univ., Kazan, 2011).

    Google Scholar 

  6. S. N. Timergaliev, “On the existence of solutions of a nonlinear boundary value problem for the system of partial differential equations of the theory of Timoshenko type shallow shells with free edges,” Differ. Equations 51, 376–390 (2015). https://doi.org/10.1134/S0012266115030088

    Article  MathSciNet  Google Scholar 

  7. S. N. Timergaliev, “A method of integral equations in nonlinear boundary-value problems for flat shells of the Timoshenko type with free edges,” Russ. Math. 61 (4), 49–64 (2017). https://doi.org/10.3103/s1066369x17040089

    Article  MathSciNet  Google Scholar 

  8. S. N. Timergaliev and A. N. Uglov, “Application of Riemann–Hilbert problem solutions to a study of nonlinear boundary value problems for Timoshenko type inhomogeneous shells with free edges,” Lobachevskii J. Math. 39, 855–865 (2018). https://doi.org/10.1134/s1995080218060203

    Article  MathSciNet  Google Scholar 

  9. S. N. Timergaliev, “Method of integral equations for studying the solvability of boundary value problems for the system of nonlinear differential equations of the theory of Timoshenko type shallow inhomogeneous shells,” Differ. Equations 55, 243–259 (2019). https://doi.org/10.1134/s0012266119020095

    Article  MathSciNet  Google Scholar 

  10. S. N. Timergaliev, “On existence of solutions of nonlinear equilibrium problems on shallow inhomogeneous anisotropic shells of the Timoshenko type,” Russ. Math. 63 (8), 38–53 (2019). https://doi.org/10.3103/S1066369X1908005X

    Article  MathSciNet  Google Scholar 

  11. S. N. Timergaliev, “On the problem of solvability of nonlinear boundary value problems for arbitrary isotropic shallow shells of the Timoshenko type with free edges,” Russ. Math. 65 (4), 81–97 (2021). https://doi.org/10.3103/S1066369X21040071

    Article  MathSciNet  Google Scholar 

  12. S. N. Timergaliev, “On the problem of solvability of nonlinear boundary value problems for arbitrary isotropic shallow shells of the Timoshenko type with free edges,” Russ. Math. 65 (4), 81–97 (2021). https://doi.org/10.3103/S1066369X21040071

    Article  MathSciNet  Google Scholar 

  13. K. Z. Galimov, Fundamentals of Nonlinear Theory of Thin Shells (Izd-vo Kazansk. Univ., Kazan, 1975).

    Google Scholar 

  14. I. N. Vekua, Generalized Analytic Functions (Nauka, Moscow, 1988; Pergamon, 1962). https://doi.org/10.1016/c2013-0-05289-9

  15. N. I. Muskhelishvili, Singular Integral Equations (Nauka, Moscow, 1962; Springer, Dordrecht, 1958). https://doi.org/10.1007/978-94-009-9994-7

  16. S. Prössdorf, Einige Klassen singulärer Gleichungen, Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften, Vol. 46 (Birkhäuser Basel, Basel, 1974). https://doi.org/10.1007/978-3-0348-5827-4

  17. F. D. Gakhov, Boundary Value Problems, 2nd ed. (Fizmatgiz, 1963; Pergamon, 1966). https://doi.org/10.1016/C2013-0-01739-2

  18. M. A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations (Gostekhizdat, Moscow, 1956).

    Google Scholar 

Download references

Funding

This work is supported by the Russian Scientific Foundation (grant no. 23-21-00212).

Author information

Authors and Affiliations

  1. Kazan State University of Architecture and Engineering, 420043, Kazan, Russia

    S. N. Timergaliev

Authors
  1. S. N. Timergaliev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. N. Timergaliev.

Ethics declarations

The author of this work declares that he has no conflicts of interest.

Additional information

Translated by M. Talacheva

Publisher’s Note.

Allerton Press remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Timergaliev, S.N. On the Problem of Solvability of Nonlinear Boundary Value Problems for Shallow Isotropic Shells of Timoshenko Type in Isometric Coordinates. Russ Math. 68, 43–60 (2024). https://doi.org/10.3103/S1066369X2470004X

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X2470004X

Keywords:

Search

Navigation