First Basic Problem of Elasticity Theory for a Layer with Cylindrical Cavities Smoothly Contacting Two Cylindrical Bushings

image_print
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 51-61
Cited by J. of Mech. Eng., 2025, vol. 28, no. 3, pp. 51-61

 

Author

Oleksii O. Ilin, Aerospace University “Kharkiv Aviation Institute” (17, Vadyma Manka str., Kharkiv, 61070, Ukraine), e-mail: parfumer.ua@gmail.com, ORCID: 0009-0005-7852-9873

 

Abstract

A spatial problem of elasticity is solved for a layer with n longitudinal cylindrical cavities, two of which contain thick-walled pipes in smooth contact with the layer. Stresses are given on the surfaces of the layer, the inner surfaces of the pipes, and the cavities. All canonical surfaces do not intersect each other. The material of the layer and cylindrical pipes is homogeneous and isotropic. An analytical and numerical calculation method, which assumes the fulfillment of statics conditions (for the first basic problem of elasticity theory) and is based on the Lamé equation, is proposed. The basic solutions of the Lamé equation are taken in a form that makes it possible to obtain an exact solution for a separate boundary surface in each separate coordinate system. The basic solutions in these coordinate systems (Cartesian for the layer and local cylindrical for the cylindrical inhomogeneities) are interconnected through the mathematical framework of the generalized Fourier method. The fulfillment of boundary conditions on the upper and lower surfaces of the layer, on the inner surfaces of pipes, on cylindrical cavities, as well as the consideration of interface conditions, create an infinite system of integro-algebraic equations, which is reduced to an infinite linear one. In the numerical study, the reduction method is applied to the resulting infinite linear algebraic system of equations. The solution of the system of equations gives the values of the unknown functions. Numerical calculations have shown the rapid convergence of approximate solutions to the exact one. The numerical analysis of the stressed state of the layer and thick-walled pipes showed that the use of polyamide bushings has almost no effect on the stress-strain state of the structure (compared to their absence), the use of steel bushings reduces the stress in the body of the layer in the areas of their location, redistributing the stress to the bushings themselves. The proposed solution method makes it possible to obtain the stress-strain state of structures containing cylindrical cavities and bushings, and the numerical analysis allows to assess the influence of the material on the values of stress distribution in the structures of machines and mechanisms at the design stage.

 

Keywords: fiber composite, generalized Fourier method, Lamé equation, layer with cylindrical inclusions.

 

Full text: Download in PDF

 

References

  1. Savin, G. N. (1968). Raspredeleniye napryazheniy okolo otverstiy [Stress distribution around holes]. Kyiv: Naukova dumka, 1968. 891 p. (in Russian).
  2. Guz, A. N., Kubenko, V. D., & Cherevko, M. A. (1978). Difraktsiya uprugikh voln [Elastic wave diffraction]. Kyiv: Naukova dumka, 307 p. (in Russian).
  3. 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).
  4. 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.
  5. Ansys. (n.d.). Static structural simulation using Ansys Discovery. Ansys Courses. Retrieved April 7, 2025, from https://courses.ansys.com/index.php/courses/structural-simulation.
  6. Zasovenko, A. V. & Fasoliak, A. V. (2023). Matematychne modeliuvannia dynamiky pruzhnoho pivprostoru z tsylindrychnoiu porozhnynoiu, yaka pidkriplena obolonkoiu, pry osesymetrychnykh navantazhenniakh [Mathematical modeling of the dynamics of an elastic half-medium with a cylindrical cavity reinforced by a shell under axisymmetric loads]. Novi materialy i tekhnolohii v metalurhii ta mashynobuduvanni – New Materials and Technologies in Metallurgy and Mechanical Engineering, no. 2, pp. 67–73 (in Ukrainian). https://doi.org/10.15588/1607-6885-2023-2-10.
  7. Morhun, A. S. & Franchuk, O. V. (2016). Metod hranychnykh elementiv v rozrakhunkakh kiltsevykh fundamentiv [Boundary element method in calculations of ring foundations]. Vinnytsia: Vinnytsia National Technical University, 90 p. (in Ukrainian).
  8. Kushnir, R., Pianylo, Ya., & Pianylo, A. (2005). Osoblyvosti zastosuvannia chyslovoho metodu skinchennykh riznyts pry modeliuvanni fizychnykh protsesiv [Features of the application of the numerical finite difference method in the modeling of physical processes]. Fizyko-matematychne modeliuvannia ta informatsiini tekhnolohii – Physico-Mathematical Modelling and Informational Technologies, iss. 2, pp. 58–69 (in Ukrainian).
  9. Aitharaju, V., Aashat, S., Kia, H., Satyanarayana, A., & Bogert, P. (2016). Progressive damage modeling of notched composites. NASA Technical Reports Server. https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20160012242.pdf.
  10. Kondratiev, A. V., Gaidachuk, V. E., & Kharchenko, M. E. (2019). Relationships Between the Ultimate Strengths of Polymer Composites in Static Bending, Compression, and Tension. Mechanics of Composite Materials, vol. 55, pp. 259–266. https://doi.org/10.1007/s11029-019-09808-x.
  11. Ugrimov, S., Smetankina, N., Kravchenko, O., & Yareshchenko, V. (2021). Analysis of Laminated Composites Subjected to Impact. In: Nechyporuk, M., Pavlikov, V., Kritskiy, D. (eds) Integrated Computer Technologies in Mechanical Engineering – 2020. ICTM 2020. Lecture Notes in Networks and Systems, vol. 188, pp. 234–246. https://doi.org/10.1007/978-3-030-66717-7_19.
  12. 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.
  13. 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.
  14. 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.
  15. 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).
  16. 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.
  17. 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.
  18. 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.
  19. 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.
  20. 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.
  21. 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.
  22. 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.
  23. Miroshnikov, V. Yu. (2019). Investigation of the stress strain state of the layer with a longitudinal cylindrical thick-walled tube and the displacements given at the boundaries of the layer. Journal of Mechanical Engineering – Problemy Mashynobuduvannia, vol. 22, no. 2, pp. 44–52. https://doi.org/10.15407/pmach2019.02.044.
  24. Miroshnikov, V. (2023). Rotation of the layer with the cylindrical pipe around the rigid cylinder. In: Altenbach H., et al. Advances in Mechanical and Power Engineering. CAMPE 2021. Lecture Notes in Mechanical Engineering. Cham: Springer, pp. 314–322. https://doi.org/10.1007/978-3-031-18487-1_32.
  25. Miroshnikov, V. Yu., Savin, O. B., Hrebennikov, M. M., & Demenko, V. F. (2023). Analysis of the stress state for a layer with two incut cylindrical supports. Journal of Mechanical Engineering – Problemy Mashynobuduvannia, vol. 26, no. 1, pp. 15–22. https://doi.org/10.15407/pmach2023.01.015.
  26. Miroshnikov, V. Yu., Savin, O. B., Kosenko, M. L., & Ilin, O. O. (2024). Analiz napruzhenoho stanu sharu z dvoma tsylindrychnymy vrizanymy oporamy ta tsylindrychnymy vtulkamy [Analysis of the stress state of a layer with two cylindrical embedded supports and cylindrical bushings]. Vidkryti informatsiini ta kompiuterni intehrovani tekhnolohii – Open Information and Computer Integrated Technologies, no. 101, pp. 112–126 (in Ukrainian). https://doi.org/10.32620/oikit.2024.101.08.
  27. Denshchykov, O. Yu., Pelykh, V. P., Hrebeniuk, Ya. V., & Miroshnikov, V. Yu. (2024). First basic problem of elasticity theory for a composite layer with two thick-walled tubes. Journal of Mechanical Engineering – Problemy Mashynobuduvannia, vol. 27, no. 4, pp. 40–50. https://doi.org/10.15407/pmach2024.04.040.
  28. Miroshnikov, V. Yu., Oleshkevych, S. V., Medvedieva, A. V., & Savin, O. B. (2021). Doslidzhennia pershoi osnovnoi zadachi teorii pruzhnosti dlia sharu z troma pozdovzhnimy tsylindrychnymy porozhnynamy [Investigation of the first basic problem of the theory of elasticity for a layer with three longitudinal cylindrical cavities]. Naukovyi visnyk budivnytstva – Scientific Bulletin of Construction, vol. 103, no. 1, pp. 150–155 (in Ukrainian). https://doi.org/10.29295/2311-7257-2021-103-1-150-155.
  29. Miroshnikov, V. Yu. (2020). Rozviazok osnovnykh ta deiakykh mishanykh zadach teorii pruzhnosti dlia bahatosharovoho seredovyshcha z pozdovzhnimy kruhovymy tsylindrychnymy porozhnynamy ta neodnoridnostiamy [Solution of the basic and some mixed problems of the theory of elasticity for a multilayer medium with longitudinal circular cylindrical cavities and inhomogeneities]: Abstract of the dissertation of the Dr. Sci. (Eng.). National Aerospace University “Kharkiv Aviation Institute”, Kharkiv, 40 p. (in Ukrainian).

 

Received 08 April 2025

Accepted 20 July 2025

Published 30 September 2025