Skip to main content
SpringerLink
Log in

Exact Solutions to Inhomogeneous Boundary Value Problems of the Theory of Elasticity in a Rectangle

  • MECHANICS
  • Published:
Doklady Physics Aims and scope Submit manuscript

Abstract

A method is proposed for building exact solutions to boundary value problems of the theory of elasticity in a rectangle with stiffeners located inside a region (the inhomogeneous problem). The solutions are presented as series in Papkovich–Fadle eigenfunctions with explicitly determined coefficients. The method is based on the Papkovich orthogonality relation and the developed theory of expansions in the Papkovich–Fadle eigenfunctions in homogeneous boundary value problems of the theory of elasticity in a rectangle (the biharmonic problem). The solution sequence is demonstrated by the example of an even symmetric problem for a rectangle in which the sides are free and an external load acts along a stiffener located on the axis of symmetry of the rectangle.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.

REFERENCES

  1. P. F. Papkovich, Dokl. Akad. Nauk SSSR 27, 335 (1940).

    Google Scholar 

  2. G. A. Grinberg, Prikl. Mat. Mekh. 17, 211 (1953).

    Google Scholar 

  3. V. K. Prokopov, Prikl. Mat. Mekh. 28, 351 (1964).

    Google Scholar 

  4. R. W. Little and S. B. Childs, Q. Appl. Math. 25, 261 (1967).

    Article  Google Scholar 

  5. W. Flügge and V. S. Kelkar, Int. J. Solids Struct. 4, 397 (1968).

    Article  Google Scholar 

  6. R. W. Little, J. Appl. Mech. 36, 320 (1969).

    Article  ADS  Google Scholar 

  7. A. V. Kostarev and V. K. Prokopov, Prikl. Mat. Mekh. 34, 945 (1970).

    Google Scholar 

  8. M. D. Kovalenko and T. D. Shulyakovskaya, Mech. Solids 46, 721 (2011).

    Article  ADS  Google Scholar 

  9. M. D. Kovalenko, I. V. Men’shova, and T. D. Shulyakovskaya, Mech. Solids 48, 584 (2013).

    Article  ADS  Google Scholar 

  10. M. D. Kovalenko, I. V. Men’shova, A. P. Kerzhaev, and T. D. Shulyakovskaya, Mech. Solids 56, 1232 (2021).

    Article  ADS  Google Scholar 

  11. B. Ya. Levin, Distribution of Roots of Entire Functions (GITTL, Moscow, 1956) [in Russian].

    Google Scholar 

  12. V. Ketch and P. Teodorescu, Introduction to the Theory of Generalized Functions with Applications in Engineering (Wiley, New York, 1978).

    Google Scholar 

  13. S. P. Timoshenko and J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 2010).

    Google Scholar 

  14. M. A. Naimark, Linear Differential Operators (Fizmatlit, Moscow, 2010; George G. Harrap, London, 1968).

  15. N. Wiener and R. C. Paley, Fourier Transforms in the Complex Domain (AMS, Providence, 1934).

    Book  Google Scholar 

  16. M. F. Baburchenkov and N. M. Borodachev, Mech. Solids 51, 436 (2016).

    Article  ADS  Google Scholar 

  17. V. Yu. Salamatova, Mech. Solids 44, 57 (2009).

    Article  ADS  Google Scholar 

  18. N. Vaysfel’d, G. Popov, and V. Reut, Acta Mech. 226, 797 (2015).

    Article  MathSciNet  Google Scholar 

Download references

Funding

The study by M.D. Kovalenko and Yu.N. Karnet was carried out within a State Assignment of the Ministry of Science and Higher Education of the Russian Federation for the Institute of Applied Mechanics, Russian Academy of Sciences. The study by A.P. Kerzhaev and I.V. Menshova was carried out within a State Assignment of the Ministry of Science and Higher Education of the Russian Federation for the Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences.

Author information

Authors and Affiliations

  1. Institute of Applied Mechanics, Russian Academy of Sciences, 125040, Moscow, Russia

    M. D. Kovalenko & Yu. N. Karnet

  2. Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences, 117997, Moscow, Russia

    A. P. Kerzhaev & I. V. Menshova

  3. Bauman Moscow State Technical University, 105005, Moscow, Russia

    I. V. Menshova

Authors
  1. M. D. Kovalenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. P. Kerzhaev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. I. V. Menshova
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yu. N. Karnet
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. D. Kovalenko.

Ethics declarations

The authors of this work declare that they have no conflicts of interest.

Additional information

Translated by E. Bondareva

Publisher’s Note.

Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kovalenko, M.D., Kerzhaev, A.P., Menshova, I.V. et al. Exact Solutions to Inhomogeneous Boundary Value Problems of the Theory of Elasticity in a Rectangle. Dokl. Phys. 68, 382–386 (2023). https://doi.org/10.1134/S102833582311006X

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S102833582311006X

Keywords:

Search

Navigation