# The study of the first main problem of the theory of elasticity for a layer with a cylindrical cavity

## Authors

• Vitalii Miroshnikov Kharkiv National University of Construction and Architecture, Ukraine
• Tetiana Denysova Simon Kuznets Kharkiv National University of Economics, Ukraine
• Volodymyr Protsenko National Aerospace University named after N.E. Zhukovsky "Kharkiv Aviation Institute", Ukraine

## Keywords:

cylindrical cavity in a layer, Lame's equation, generalized Fourier method, infinite systems of linear algebraic equations

## Abstract

An analytic - numerical algorithm for solving a substantially spatial problem of the theory of elasticity for a layer with a longitudinal cylindrical cavity and the conditions of the first principal problem, given on the boundary surfaces, is developed.

The solution is based on the generalized Fourier method applied to the Lame equation system. The cavity is considered in the cylindrical coordinate system, the layer - in Cartesian. By satisfying the boundary conditions and using special formulas for the transition between coordinate systems for basic solutions, we create infinite systems of linear algebraic equations, which are solved by the method of cutting. The numerical study of the determinant gives reason to claim that this system of equations has a single solution. As a result, stresses were obtained at different points in the elastic body.

Cutting parameters were chosen so that the accuracy of the boundary conditions reaches 10-2. As the cutoff parameter increases, the accuracy of boundary conditions increases, but the duration of the calculation increases.

The numerical solution of the problem is performed for a layer with a cylindrical cavity and a normal balanced load at the boundary of the layer. The analysis of the stress state gives grounds to state:

1. The solution of the algebraic system of equations can be found with any degree of accuracy by the method of reduction, which is confirmed by the high accuracy of boundary conditions.

2. The presence of a cavity gives rise to a redistribution of stresses, in which maximum values occur on the surface of the cavity, and tensile stresses occur at the top of the layer (which is compressed without a cavity).

This method can be used in the calculation of structures, the calculation scheme of which is a layer with a cylindrical cavity, under given boundary conditions in the form of balanced stresses. Stress analysis enables the selection of geometric characteristics at the initial design stage.

## Author Biographies

### Vitalii Miroshnikov, Kharkiv National University of Construction and Architecture

Candidate of Technical Sciences, Associate Professor, Associate Professor of the Department of Structural Mechanics

### Tetiana Denysova, Simon Kuznets Kharkiv National University of Economics

Candidate of Technical Sciences, Associate Professor of the Department of Higher Mathematics and Economic-Mathematical Methods

### Volodymyr Protsenko, National Aerospace University named after N.E. Zhukovsky "Kharkiv Aviation Institute"

Doctor of Physical and Mathematical Sciences, Professor of the Department of Mathematics and Systems Analysis

## References

Guz’ A.N., Kosmodamianskiy A.S., Shevchenko V.P. and others. Mekhanika kompozitov (Mechanics of composites). Vol 7. Kontsentratsiya napryazheniy (Concentration of stresses). Kiev: Nauk. Dumka. – 1998. – P. 114 – 137. (In Russian).

Vaysfel’d N., Popov G., Reut V. The axisymmetric contact interaction of an infinite elastic plate with an absolutely rigid inclusion / Acta Mech. – 2015. – vol. 226. – P. 797–810. doi: https://doi.org/10.1007/s00707-014-1229-7

Popov G., Vaysfel’d N. Osesimmetrichnaya zadacha teorii uprugosti dlya beskonechnoy plity s tsilindricheskim vklyucheniyem pri uchete yeye udel'nogo vesa (The axisymmetric problem of the theory of elasticity for an infinite plate with a cylindrical inclusion, taking into account its specific gravity) / Prikladnaya mekhanika (Applied mechanics). – 2014. – Vol. 50, № 6. – P. 27–38.

Bobyleva T. Approximate Method of Calculating Stresses in Layered Array / Procedia Engineering. – 2016. – Vol.153. – P.103 – 106. https://doi.org/10.1016/j.proeng.2016.08.087

Grinchenko V.T., Ulitko A.F. Prostranstvennyye zadachi teorii uprugosti i plastichnosti. Ravnovesiye uprugikh tel kanonicheskoy formy (Spatial Problems of the Theory of Elasticity and Plasticity. Equilibrium of elastic bodies of canonical form). – Kiyev: Naukova Dumka. – 1985.– 280 p. (in Russian).

Guz' A.N., Kubenko V.D. , Cherevko M.A. Difraktsiya uprugikh voln (Diffraction of elastic waves). – Kiev: Nauk. Dumka. – 1978. – 307 p. (In Russian).

Grinchenko V.T., Meleshko V.V. Garmonicheskiye kolebaniya i volny v uprugikh telakh (Harmonic vibrations and waves in elastic bodies). – Kiev: Nauk. Dumka. – 1981. – 284 p. (In Russian).

Volchkov V.V., Vukolov D.S., Storogev V.I. 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 non-homogeneities in the form of a cavity and inclusion in an elastic layer with free faces) / Mekhanika tverdogo tela (Solid mechanics). – 2016. – Vol. 46. – P. 119 – 133. (In Russian).

Nikolaev A.G., Protsenko V.S. Obobshchennyy metod Fur'ye v prostranstvennykh zadachakh teorii uprugosti (Generalized Fourier method in spatial problems of the theory of elasticity). –Kharkov: Nats. aerokosm. universitet im. N.Ye. Zhukovskogo «KHAI» (National Aerospace University "KhAI"), 2011. – 344 с. (In Russian).

Protsenko V.S., Nikolaev A.G. Prostranstvennaya zadacha Kirsha (Kirsch spatial problem) / Matematicheskiye metody analiza dinamicheskikh sistem (Mathematical methods for analyzing dynamic systems). – 1982. – Vol. 6. – P. 3 – 11. (In Russian).

Protsenko V.S., Ukrainec N.A. Primeneniye obobshchennogo metoda Fur'ye k resheniyu pervoy osnovnoy zadachi teorii uprugosti v poluprostranstve s tsilindricheskoy polost'yu (Application of the generalized Fourier method to the solution of the first main problem of the theory of elasticity in a half-space with a cylindrical cavity) / Visnyk Zaporizʹkoho natsionalʹnoho universytetu (Bulletin of Zaporizhzhya National University). 2015. Vol. 2. P. 193–202. (In Russian).

Nikolaev A.G., Orlov E.M. Resheniye pervoy osesimmetrichnoy termouprugoy krayevoy zadachi dlya transversal'no-izotropnogo poluprostranstva so sferoidal'noy polost'yu (Solution of the first axisymmetric thermoelastic boundary value problem for a transversely isotropic half-space with a spheroidal cavity) / Problemy obchyslyuvalʹnoyi mekhaniky i mitsnosti konstruktsiy (Problems of Computational Mechanics and Strength of Structures). – 2012. – Vol.20. – P. 253-259. (In Russian).

Miroshnikov V.Yu. First basic elasticity theory problem in a half-space with several parallel round cylindrical cavities / Journal of Mechanical Engineering. – 2018. – Vol. 21, № 2. – P. 12 – 18.

Protsenko V., Miroshnikov V. 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. – 2018. – Vol 4, № 7 (94). – P. 43 – 50.

Nikolaev A.G., Tanchyk E.A. Uprugaya mekhanika mnogokomponentnykh tel (Elastic mechanics of multicomponent bodies). – Kharkov: Nats. aerokosm. universitet im. N.Ye. Zhukovskogo «KHAI» (National Aerospace University "KhAI"). – 2014. – 272 p. (In Russian).

Miroshnikov V. Yu. The study of the second main problem of the theory of elasticity for a layer with a cylindrical cavity / Strength of Materials and Theory of Structures. – 2019. – №102. – P. 77–90. https://doi.org/10.32347/2410-2547.2019.102.77-90

Miroshnikov V.Yu., Medvedeva A.V., Oleshkevich S.V. Determination of the Stress State of the Layer with a Cylindrical Elastic Inclusion / Materials Science Forum. – 2019. – Vol. 968. – pp. 413-420. https://doi.org/10.4028/www.scientific.net/MSF.968.413

Miroshnikov V. 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. – 2019. – Vol. 22, N 2. – P. 44 – 52. https://doi.org/10.15407/pmach2019.02.044