DOI | |
Journal | Journal of Mechanical Engineering – Problemy Mashynobuduvannia |
Publisher | Anatolii Pidhornyi Institute of Power Machines and Systems of National Academy of Science of Ukraine |
ISSN | 2709-2984 (Print), 2709-2992 (Online) |
Issue | Vol. 28, no. 3, 2025 (September) |
Pages | 42-50 |
Cited by | J. of Mech. Eng., 2025, vol. 28, no. 3, pp. 42-50 |
Authors
Tetiana M. Aloshechkina, O. M. Beketov National University of Urban Economy in Kharkiv (17, Chornohlazivska str., Kharkiv, 61002, Ukraine), e-mail: atn4042@gmail.com, ORCID: 0000-0001-7234-1558
Natalia A. Ukrainets, National Aerospace University “Kharkiv Aviation Institute” (17, Vadyma Manka str., Kharkiv, 61070, Ukraine), e-mail: n.ukrayinets@khai.edu, ORCID: 0000-0001-7406-5809
Vitalii Yu. Miroshnikov, National Aerospace University “Kharkiv Aviation Institute” (17, Vadyma Manka str., Kharkiv, 61070, Ukraine), e-mail: v.miroshnikov@khai.edu, ORCID: 0000-0002-9491-0181
Abstract
In aerospace and mechanical engineering, elements that are loaded by periodic loads (periodic function) are used. In problems for a layer with cylindrical inhomogeneities, it is difficult to take such loads into account. Therefore, there is a need to develop a methodology for calculating the stress state for a layer with a cylindrical cavity and taking into account the boundary conditions in the form of a periodic function. In this paper, we propose a solution to the problem of elasticity theory for a layer with a cylindrical cavity within the framework of the generalized Fourier method. Stresses are given at the upper boundary of the layer and on the surface of the cylindrical cavity, and displacements are given at the lower boundary of the layer. The layer and cylindrical cavity are considered in different coordinate systems (Cartesian and cylindrical). The redistribution functions of the generalized Fourier method are applied to the Lamé equations. The problem is reduced to the sum of two solutions – an auxiliary problem and the main problem. Both problems are reduced to infinite systems of linear algebraic equations, which allow the application of the reduction method to them. After finding the unknowns in the auxiliary problem, the stresses at the geometric location of the cavity are found. The main problem is solved for the layer with the cavity, on which stresses obtained from the auxiliary problem are set with the reverse sign. The complete solution consists of the auxiliary and main problems. Having calculated all the unknowns, it is possible to obtain the stress-strain state at any point of the body with a given accuracy. Numerical analysis of the stress state showed high accuracy of the boundary conditions and dependence on periodic loading. Thus, the stresses sx and sz at the upper boundary of the layer have extremes in the places of maximum values sy and their negative values increase at the location of the cavity. At the same time, the stresses sx exceed the specified sy.
Keywords: periodic loading, layer with a cylindrical cavity, Lamé equation, generalized Fourier method.
Full text: Download in PDF
References
- Tekkaya, A. E. & Soyarslan, C. (2014). Finite element method. In: Laperrière, L., Reinhart, G. (eds) CIRP Encyclopedia of Production Engineering. Berlin, Heidelberg: Springer, pp. 508–514. https://doi.org/10.1007/978-3-642-20617-7_16699.
- Ansys. (n.d.). Static structural simulation using Ansys Discovery. Ansys Courses. Retrieved February 27, 2025, from https://courses.ansys.com/index.php/courses/structural-simulation.
- Guz, A. N., Kubenko, V. D., & Cherevko, M. A. (1978). Difraktsiya uprugikh voln [Elastic wave diffraction]. Kyiv: Naukova dumka, 307 p. (in Russian).
- Grinchenko, V. T. & Meleshko, V. V. (1981). Garmonicheskiye kolebaniya i volny v uprugikh telakh [Harmonic vibrations and waves in elastic bodies]. Kyiv: Naukova dumka, 284 p. (in Russian).
- Grinchenko, V. T. & Ulitko, A. F. (1968). An exact solution of the problem of stress distribution close to a circular hole in an elastic layer. Soviet Applied Mechanics, vol. 4, iss. 10, pp. 31–37. https://doi.org/10.1007/BF00886618.
- Fesenko, A. & Vaysfel’d, N. (2019). The wave field of a layer with a cylindrical cavity. In: Gdoutos, E. (eds) Proceedings of the Second International Conference on Theoretical, Applied and Experimental Mechanics. ICTAEM 2019. Structural Integrity, vol. 8. Cham: Springer, pp. 277–282. https://doi.org/10.1007/978-3-030-21894-2_51.
- Fesenko, A. & Vaysfel’d, N. (2021). The dynamical problem for the infinite elastic layer with a cylindrical cavity. Procedia Structural Integrity, vol. 33, pp. 509–527. https://doi.org/10.1016/j.prostr.2021.10.058.
- Jafari, M., Chaleshtari, M. H. B., Khoramishad, H., & Altenbach H. (2022). Minimization of thermal stress in perforated composite plate using metaheuristic algorithms WOA, SCA and GA. Composite Structures, vol. 304, part 2, article 116403. https://doi.org/10.1016/j.compstruct.2022.116403.
- Malits, P. (2021). Torsion of an elastic half-space with a cylindrical cavity by a punch. European Journal of Mechanics – A/Solids, vol. 89, article 104308. https://doi.org/10.1016/j.euromechsol.2021.104308.
- Khechai, A., Belarbi, M.-O., Bouaziz, A., & Rekbi, F. M. L. (2023). A general analytical solution of stresses around circular holes in functionally graded plates under various in-plane loading conditions. Acta Mechanica, vol. 234, pp. 671–691. https://doi.org/10.1007/s00707-022-03413-1.
- Snitser, A. R. (1996). The reissner-sagoci problem for a multilayer base with a cylindrical cavity. Journal of Mathematical Sciences, vol. 82, iss. 3, pp. 3439–3443. https://doi.org/10.1007/bf02362661.
- Nikolayev, A. G. & Protsenko, V. S. (2011). Obobshchennyy metod Furye v prostranstvennykh zadachakh teorii uprugosti [Generalized Fourier method in spatial problems of the theory of elasticity]. Kharkiv: National Aerospace University “Kharkiv Aviation Institute”, 344 p. (in Russian).
- Nikolaev, A. G. & Tanchik, E. A. (2015). The first boundary-value problem of the elasticity theory for a cylinder with N cylindrical cavities. Numerical Analysis and Applications, vol. 8, pp. 148–158. https://doi.org/10.1134/S1995423915020068.
- Nikolaev, A. G. & Tanchik, E. A. (2016). Stresses in an elastic cylinder with cylindrical cavities forming a hexagonal structure. Journal of Applied Mechanics and Technical Physics, vol. 57, pp. 1141–1149. https://doi.org/10.1134/S0021894416060237.
- Nikolaev, A. G. & Tanchik, E. A. (2016). Model of the stress state of a unidirectional composite with cylindrical fibers forming a tetragonal structure. Mechanics of Composite Materials, vol. 52, pp. 177–188. https://doi.org/10.1007/s11029-016-9571-6.
- Ukrayinets, N., Murahovska, O., & Prokhorova, O. (2021). Solving a one mixed problem in elasticity theory for half-space with a cylindrical cavity by the generalized Fourier method. Eastern-European Journal of Enterprise Technologies, vol. 2, no. 7 (110), pp. 48–57. https://doi.org/10.15587/1729-4061.2021.229428.
- Miroshnikov, V. Yu. (2019). Doslidzhennia druhoi osnovnoi zadachi teorii pruzhnosti dlia sharu z tsylindrychnoiu porozhnynoiu [Investigation of the second fundamental problem of the theory of elasticity for a layer with a cylindrical cavity]. Opir materialiv i teoriia sporud – Strength of Materials and Theory of Structures, no. 102, pp. 77–90 (in Ukrainian). https://doi.org/10.32347/2410-2547.2019.102.77-90.
- Miroshnikov, V., Denysova, T., & Protsenko, V. (2019). Doslidzhennia pershoi osnovnoi zadachi teorii pruzhnosti dlia sharu z tsylindrychnoiu porozhnynoiu [The study of the first main problem of the theory of elasticity for a layer with a cylindrical cavity]. Opir materialiv i teoriia sporud – Strength of Materials and Theory of Structures, no. 103, pp. 208–218 (in Ukrainian). https://doi.org/10.32347/2410-2547.2019.103.208-218.
- Miroshnikov, V. Yu. (2020). Stress state of an elastic layer with a cylindrical cavity on a rigid foundation. International Applied Mechanics, vol. 56, iss. 3, pp. 372–381. https://doi.org/10.1007/s10778-020-01021-x.
- Miroshnikov, V., Younis, B., Savin, O., & Sobol, V. (2022). A linear elasticity theory to analyze the stress state of an infinite layer with a cylindrical cavity under periodic load. Computation, vol. 10, iss. 9, article 160. https://doi.org/10.3390/computation10090160.
Received 12 March 2025
Accepted 20 May 2025
Published 30 September 2025