Analysis of the Stress State of a Layer with Two Cylindrical Elastic Inclusions and Mixed Boundary Conditions

Authors

Vitalii Yu. Miroshnikov, National Aerospace University “Kharkiv Aviation Institute” (17, Chkalov str., Kharkov, 61070, Ukraine), e-mail: v.miroshnikov@khai.edu, ORCID: 0000-0002-9491-0181

Oleksandr B. Savin, National Aerospace University “Kharkiv Aviation Institute” (17, Chkalov str., Kharkov, 61070, Ukraine), e-mail: asavin344@gmail.com, ORCID: 0000-0002-2664-0255

Mykhailo M. Hrebennikov, National Aerospace University “Kharkiv Aviation Institute” (17, Chkalov str., Kharkov, 61070, Ukraine), e-mail: m.grebennikov@khai.edu, ORCID: 0000-0001-7648-3027

Oleksandr A. Pohrebniak, National Aerospace University “Kharkiv Aviation Institute” (17, Chkalov str., Kharkov, 61070, Ukraine), e-mail: pogrebnaksasa@gmail.com, ORCID: 0000-0002-0912-8823

Abstract

The spatial problem of the elasticity theory is studied for a layer with two infinite circular solid cylindrical inclusions that are parallel to each other and to the layer boundaries. The physical characteristics of the layer and the inclusions are different from each other and they are homogeneous, isotropic materials. The spatial function of stresses is given at the upper boundary, and the function of displacements is given at the lower layer boundary. Circular cylindrical elastic inclusions are rigidly connected to the layer. It is necessary to determine the stress-strain state of the composite body. The problem solution is based on the generalized Fourier method, which uses special formulas for the transition between the basic solutions of the Lamé equation in different coordinate systems. Thus, the layer is considered in the Cartesian coordinate system, the inclusions – in the local cylindrical ones. Satisfying the boundary and conjugation conditions, systems of infinite integro-algebraic equations were obtained, which were subsequently reduced to linear algebraic ones. The resulting infinite system is solved by the reduction method. After deter-mining the unknowns, it is possible to find the stress values at any point of the elastic composite body. In numerical studies, a comparative analysis of the stress state in the layer and on the surfaces of inclusions at different distances between them is carried out. The analysis showed that when the inclusions approach each other, the stress state in the layer practically does not change. However, its significant change is observed in the bodies of inclusions, so with dense reinforcement ((R1 + R2) / L > 0.5), it is necessary to take into account the distances between the reinforcing fibers. At stress values from 0 to 1 and the order of the system of equations m=10, the accuracy of meeting the boundary conditions was 10-4. With an increase in the system order, the accuracy of meeting the boundary conditions will increase. The given analytical-numerical solution can be used for high-precision determination of the stress-strain state of the given type of problems, and also as a reference for problems based on numerical methods.

Keywords: composite, cylindrical inclusions in a layer, generalized Fourier method.

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References

1. Aitharaju, V., Aashat, S., Kia, H., Satyanarayana, A., & Bogert, P. (2016). Progressive damage modeling of notched composites. NTRS – NASA Technical Reports Server: Official site. https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20160012242.pdf.
2. Yershova, A. Yu. & Martirosov, M. I. (2014). Eksperimentalnyye issledovaniya polimernykh kompozitov s melkodispersnym napolnitelem (ispytaniya na rastyazheniye-szhatiye) [Experimental studies of polymer composites with a finely dispersed filler (tensile-compression tests)]. Stroitelnaya mekhanika inzhenernykh konstruktsiy i sooruzheniy – Structural Mechanics of Engineering Constructions and Buildings, vol. 5, pp. 61–69 (in Russian).
3. Pelekh, B. L. & Makhnitskii, R. N. (1981). Approximate methods for solving problems on the concentration of stresses around apertures in orthotropic disks made out of composite materials. Mechanics of Composite Materials, vol. 16, iss. 6, pp. 690–693. https://doi.org/10.1007/BF00606258.
4. Pobedrya, B. E. & Gorbachev, V. I. (1984). Stress and strain concentration in composite materials. Mechanics of Composite Materials, vol. 20, iss. 2, pp. 141–148. https://doi.org/10.1007/BF00610353.
5. Annin, B. D. & Maksimenko, V. N. (1989). Evaluation of the failure of plates made of composite materials with holes. Mechanics of Composite Materials, vol. 25, iss. 2, pp. 216–222. https://doi.org/10.1007/BF00616267.
6. Smetankina, N., Kravchenko, I., Merculov, V., Ivchenko, D., & Malykhina, A. (2020). Modelling of bird strike on an aircraft glazing. Integrated Computer Technologies in Mechanical Engineering. Series “Advances in Intelligent Systems and Computing”, vol. 1113, pp. 289–297. https://doi.org/10.1007/978-3-030-37618-5_25.
7. Smetankina, N. V. (2011). Nestatsionarnoye deformirovaniye, termouprugost i optimizatsiya sloistykh plastin i tsilindricheskikh obolochek [Non-stationary deformation, thermal elasticity and optimisation of laminated plates and cylindrical shells]. Kharkiv: Miskdruk Publishers, 376 p. (in Russian).
8. Rodichev, Y. M., Smetankina, N. V., Shupikov, O. M., & Ugrimov, S. V. (2018). Stress-strain assessment for laminated aircraft cockpit windows at static and dynamic load. Strength of Materials, vol. 50, iss. 6, pp. 868–873. https://doi.org/10.1007/s11223-019-00033-4.
9. 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 by internal tunneling cylindrical inhomogeneities in the form of a cavity and an inclusion in an elastic layer with free faces]. Mekhanika Tverdogo Tela – Rigid Body Mechanics, vol. 46, pp. 119–133 (in Russian).
10. Huzʹ, A. N., Kubenko, V. D., & Cherevko, M. A. (1978). Dyfraktsyya upruhykh voln [Diffraction of elastic waves]. Kyiv: Naukova Dumka, 307 p. (in Russian).
11. 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).
12. 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, pp. 31–37. https://doi.org/10.1007/BF00886618.
13. 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 “KhAI”, 344 p. (in Russian).
14. Nykolaev, A. H. & Orlov, E. 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 transversely isotropic half-space with a spheroidal cavity]. Problemy vychislitelnoy mekhaniki i prochnosti konstruktsiy – Problems of computational mechanics and strength of structures, vol. 20, pp. 253–259 (in Russian).
15. Nikolayev, A. G., Shcherbakova, A. Yu., & Yukhno, A. I. (2006). Deystviye sosredotochennoy sily na transversalno-izotropnoye poluprostranstvo s paraboloidalnym vklyucheniyem [Action of concentrated force on a transversely-isotropic half-space with paraboloidal inclusion]. Voprosy proyektirovaniya i proizvodstva konstruktsiy letatelnykh apparatov – Issues of Design and Production of Structures of Aircraft, vol. 2, pp. 47–51 (in Russian).
16. Protsenko, V. & Miroshnikov, V. (2018). Investigating a problem from the theory of elasticity for a half-space with cylindrical cavities for which boundary conditions of contact type are assigned. Eastern-European Journal of Enterprise Technologies. Applied mechanics, vol. 4, no. 7 (94), pp. 43–50. https://doi.org/10.15587/1729-4061.2018.139567.
17. Miroshnikov, V. Yu. (2020). Stress state of an elastic layer with a cylindrical cavity on a rigid foundation. International Applied Mechanics, vol. 56 (3), pp. 372–381. https://doi.org/10.1007/s10778-020-01021-x.
18. Miroshnikov, V. Y., 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.
19. Miroshnikov, V., Denysova, T., & Protsenko, V. (2019). The study of the first main problem of the theory of elasticity for a layer with a cylindrical cavity. Strength of Materials and Theory of Structures, vol. 103, pp. 208–218. https://doi.org/10.32347/2410-2547.2019.103.208-218.
20. 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, vol. 17, pp. 11–21.
21. 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.
22. Nikolaev, A. G. & Tanchik, E. A. (2016). Stresses in an infinite circular cylinder with four cylindrical cavities. Journal of Mathematical Sciences, vol. 217 (3), pp. 299–311. https://doi.org/10.1007/s10958-016-2974-z.
23. 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.
24. 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.
25. Miroshnikov, V. Yu. (2019). Investigation of the stress state of a composite in the form of a layer and a half space with a longitudinal cylindrical cavity at stresses given on boundary surfaces. Journal of Mechanical Engineering – Problemy mashynobuduvannia, vol. 22, no. 4, pp. 24–31. https://doi.org/10.15407/pmach2019.04.024.

Received 04 May 2022

Published 30 June 2022