Representation of solutions to the plane elasticity problems for a rectangular domain via Vihak’s functions

Authors

  • Yu. V. Tokovyy Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NAS of Ukraine, 79060, Lviv, 3-B Naukova Str. https://orcid.org/0000-0003-1610-0113
  • M. Yo. Yuzvyak Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NAS of Ukraine, 79060, Lviv, 3-B Naukova Str.
  • A. V. Yasinskyy Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NAS of Ukraine, 79060, Lviv, 3-B Naukova Str.

DOI:

https://doi.org/10.17721/1812-5409.2021/3.24

Keywords:

elastic equilibrium, plane problem, rectangular domain, direct integration method, Vihak’s functions

Abstract

The paper presents the generalization of the direct integration method for the governing equations of the basic elasticity problems for the bounded domains with corner points. An important stage in the realization of the method is the representation of the unknown stress-tensor components via the key functions. The selection of these functions is motivated by some specific features of the problems and thus was regarded as a weakest part of the solution algorithm. Herein, we suggest an universal approach for the selection of the key functions, which we started to call the Vihak functions (to honor Prof. Vasyl M. Vihak, the founder and developer of the direct integration method) by using the integral relationships derived from the equilibrium equations. The approach is illustrated by the solution of a plane elasticity problem for an elastic rectangle. The relationship between Vihak’s function for the considered problem and the classical biharmonic Airy stress function is shown.

Pages of the article in the issue: 123 - 126

Language of the article: Ukrainian

References

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VIHAK, V., TOKOVYI, Yu., & RYCHAHIVSKYY, A. (2002) Exact solution of the plane problem of elasticity in a rectangular region. Journal of Computational and Applied Mechanics. 3(2). p. 193–206.

KUSHNIR, R.M., TOKOVYY, YU.V., YUZVYAK, M.YO., & YASINSKYY, A.V. (2021) Zvedennya dvovymirnykh zadach termopruzhnosti dlya til z kutovymy tochkamy do klyuchovykh integro-differentialnykh rivnyan’. Ukrainian Mathematical Journal, 73(10).

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No 3 (2021)

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Published

2021-12-07

How to Cite

Tokovyy, Y. V., Yuzvyak, M. Y., & Yasinskyy, A. V. (2021). Representation of solutions to the plane elasticity problems for a rectangular domain via Vihak’s functions. Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences, (3), 123–126. https://doi.org/10.17721/1812-5409.2021/3.24

Issue

No. 3 (2021)

Section

Differential equations, mathematical physics and mechanics

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