|Journal||Journal of Mechanical Engineering – Problemy Mashynobuduvannia|
|Publisher||Anatolii Pidhornyi Institute for Mechanical Engineering Problems
of National Academy of Science of Ukraine
|ISSN||2709-2984 (Print), 2709-2992 (Online)|
|Issue||Vol. 26, no. 1, 2023 (March)|
|Cited by||J. of Mech. Eng., 2023, vol. 26, no. 1, pp. 15-22|
The stress state of a homogeneous isotropic layer under the action of a spatial static external load is studied. Two circular cylindrical supports are cut into the body of the layer parallel to its borders. The supports and body of the layer are rigidly coupled. The spatial problem theory of elasticity is solved using the analytical-numerical generalized Fourier method. The layer is considered in the Cartesian coordinate system, the supports are considered in the local cylindrical coordinates. Stresses are set on the upper and lower surfaces of the layer. The supports are considered as cylindrical cavities in a layer with zero displacements set on their surfaces. Satisfying the boundary conditions on the upper and lower surfaces of the layer, as well as on the cylindrical surfaces of the cavities, a system of infinite integro-algebraic equations, which are further reduced to linear algebraic ones, is obtained. An infinite system is solved by the reduction method. In the numerical studies, the parameters of integration oscillatory functions are analyzed, problems at different distances between supports are solved. A unit load in the form of a rapidly decreasing function is applied to the upper boundary between the supports. For these cases, an analysis of the stress state was performed on the surfaces of the layer between the supports and on the cylindrical surfaces in contact with the supports. The numerical analysis showed that when the distance between the supports increases, the stresses σx on the lower and upper surfaces of the layer and the stresses τρφ on the surfaces of the cavities increase. The use of the analytical-numerical method made it possible to obtain a result with an accuracy of 10-4 for stress values from 0 to 1 at the order of the system of equations m=6. As the order of the system increases, the accuracy of fulfilling the boundary conditions will increase. The presented analytical-numerical solution can be used for high-precision determination of the stress-strain state of the presented problems type, as well a reference for problems based on numerical methods.
Keywords: cylindrical cavities in a layer, generalized Fourier method, Lamé equation
Full text: Download in PDF
- Azarov, A. D., Zhuravlev, G. A., & Piskunov, A. S. (2015). Sravnitelnyy analiz analiticheskogo i chislennogo metodov resheniya ploskoy zadachi o kontakte uprugikh tsilindrov [Comparative analysis of analytical and numerical methods for solving the plane problem of elastic cylinder contact]. Innovatsionnaya nauka – Innovative Science, no. 1–5, pp. 5–13 (in Russian).
- Guz, A. N., Kosmodamianskiy, A. S., Shevchenko, V. P., Nemish, Yu. N., & Avdyushina, Ye. V. (1998). Mekhanika kompozitov [Mechanics of composites]: in 12 vols. Vol. 7. Kontsentratsiya napryazheniy [Stress concentration]. Kyiv: Naukova dumka, 387 p. (in Russian).
- Vaysfel’d, N., Popov, G., & Reut, V. (2015). The axisymmetric contact interaction of an infinite elastic plate with an absolutely rigid inclusion. Acta Mechanica, vol. 226, iss. 3, pp. 797–810. https://doi.org/10.1007/s00707-014-1229-7.
- Popov, G. Y. & Vaisfel’d, N. D. (2014). Solving an axisymmetric problem of elasticity for an infinite plate with a cylindrical inclusion with allowance for its specific weight. International Applied Mechanics, vol. 50, iss. 6, pp. 627–636. https://doi.org/10.1007/s10778-014-0661-7.
- Bobyleva, T. (2016). Approximate method of calculating stresses in layered array. Procedia Engineering, vol. 153, pp. 103–106. https://doi.org/10.1016/j.proeng.2016.08.087.
- Guz, A. N., Kubenko, V. D., & Cherevko, M. A. (1978). Difraktsiya uprugikh voln [Diffraction of elastic waves]. Kyiv: Naukova dumka, 307 p. (in Russian).
- Grinchenko, V. T. & Meleshko, V. V. (1981). Garmonicheskiye kolebaniya i volny v uprugikh telakh [Harmonic oscillations and waves in elastic bodies]. Kyiv: Naukova dumka, 284 p. (in Russian).
- Volchkov, V. V., Vukolov, D. S., & Storozhev, V. I. (2016). Difraktsiya voln sdviga na vnutrennikh tunnel’nykh tsilindricheskikh neodnorodnostyakh v vide polosti i vklyucheniya v uprugom sloye so svobodnymi granyami [Diffraction of shear waves on internal tunnel cylindrical inhomogeneities in the form of a cavity and inclusion in the elastic layer with free face]. Mekhanika tverdogo tela – Mechanics of Rigid Bodies, iss. 46, pp. 119–133 (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.
- Nikolayev, A. G. & Protsenko, V. S. (2011). Obobshchennyy metod Furye v prostranstvennykh zadachakh teorii uprugosti [The 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, iss. 2, pp. 148–158. https://doi.org/10.1134/S1995423915020068.
- Nikolaev, A. G. & Tanchik, E. A. (2016). Stresses in an infinite circular cylinder with four cylindrical cavities. Journal of Mathematical Sciences, vol. 217, iss. 3, pp. 299–311. https://doi.org/10.1007/s10958-016-2974-z.
- 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, iss. 2, pp. 177–188. https://doi.org/10.1007/s11029-016-9571-6.
- 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, iss. 6, pp. 1141–1149. https://doi.org/10.1134/S0021894416060237.
- Nikolayev, A. G. & Orlov, Ye. M. (2012). Resheniye pervoy osesimmetrichnoy termouprugoy krayevoy zadachi dlya transversalno-izotropnogo poluprostranstva so sferoidalnoy polostyu [Solution of the first axisymmetric thermoelastic boundary value problem for a transversally isotropic half-space with a spheroidal cavity]. Problemy vychislitelnoy mekhaniki i prochnosti konstruktsiy – Problems of computational mechanics and strength of structures, iss. 20, pp. 253–259 (in Russian).
- 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., 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. & Protsenko, V. S. (2019). Determining the stress state of a layer on a rigid base weakened by several longitudinal cylindrical cavities. Journal of Advanced Research in Technical Science, iss. 17, pp. 11–21. https://doi.org/10.26160/2474-5901-2019-17-11-21.
- Miroshnikov, V. Yu., Medvedeva, A. V., & Oleshkevich, S. V. (2019). Determination of the stress state of the layer with a cylindrical elastic inclusion. Materials Science Forum, vol. 968, pp. 413–420. https://doi.org/10.4028/www.scientific.net/MSF.968.413.
- Miroshnikov, V. Yu., Savin, O. B., Hrebennikov, M. M., & Pohrebniak, O. A. (2022). Analysis of the stress state of a layer with two cylindrical elastic inclusions and mixed boundary conditions. Journal of Mechanical Engineering – Problemy mashynobuduvannia, vol. 25, no. 2, pp. 22–29. https://doi.org/10.15407/pmach2022.02.022.
- Hrebennikov, M. M. & Myronov, K. V. (2021). Analiz napruzhenoho stanu sharu z pozdovzhnoiu porozhnynoiu ta zadanymy nevlasno mishanymy hranychnymy umovamy [Analysis of the stress state of a layer with a longitudinal cavity and given improperly mixed boundary conditions]. Science, theory and practice: Abstracts of XXIX International Scientific and Practical Conference, Japan, Tokyo, pp. 536–540 (in Ukrainian).
- Miroshnikov, V. (2023). Rotation of the layer with the cylindrical pipe around the rigid cylinder. In: 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.
Received 23 February 2023
Published 30 March 2023