FIRST BASIC ELASTICITY THEORY PROBLEM IN A HALF-SPACE WITH SEVERAL PARALLEL ROUND CYLINDRICAL CAVITIES

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DOI https://doi.org/10.15407/pmach2018.02.012
Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher A. Podgorny Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN 0131-2928 (Print), 2411-0779 (Online)
Issue Vol. 21, no. 2, 2018 (June)
Pages 12-18
Cited by J. of Mech. Eng., 2018, vol. 21, no.2, pp. 12-18

 

Author

 V. Yu. Miroshnikov, Kharkiv National University of Construction and Architecture (40, Sumska str., Kharkiv, 61002, Ukraine), e-mail: mivi30166@gmail.com

 

Abstract

 When designing different kinds of structures and forecasting the strength of mine workings in rock and geotechnical mechanics, there occur problems in which it is necessary to know the stress-strain state of a half-space with cylindrical cavities and take into account the mutual influence of the cavities and the half-space boundaries. The article gives an analytical and numerical solution to the first main spatial problem of the theory of elasticity (stresses are specified on boundaries) for a homogeneous half-space with several circular cylindrical cavities parallel to each other and the boundary of the half-space. The specified stresses are assumed to rapidly decay to zero at great distances from the origin of coordinates, on the boundaries of the cavities at coordinates z and on the boundary of the half-space at coordinates z and x. To solve the problem, the generalized Fourier method is used in relation to a system of the Lame equations in the cylindrical coordinates associated with the cylinders, and the Cartesian coordinates related to the half-space. For transition between the basic solutions of the Lame equation, special formulas for transition between local cylindrical coordinate systems and between the Cartesian and cylindrical coordinate systems were used. Infinite systems of linear algebraic equations for which the problem is reduced is solved by the truncation method. As a result, displacements and stresses were found in an elastic body. As an example, a detailed numerical analysis of the stress-strain state for two parallel cylindrical cavities in a half-space at various values of the geometric parameters of the problem is given. The graphs given show a picture of the distribution of stresses in a body in the most interesting zones and  illustrate the mutual influence of cylindrical cavities as well as the mutual influence of  a half-space and cylindrical cavities, depending on the geometric parameters of the problem.

 

Keywords: cylindrical cavities in a half-space, Lame equations, generalised Fourier method

 

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References

  1. Podilchuk, Yu. N. (1979). Prostranstvennye zadachi teorii uprugosti [Spatial Problems in the Theory of Elasticity]. Kiyev: Nauk. Dumka (in Russian).
  2. Grinchenko, V. T. & Ulitko, A. F. (1985). Prostranstvennye zadachi teorii uprugosti i plastichnosti. Ravnovesie uprugikh tel kanonicheskoy formy [Spatial Problems in the Theory of Elasticity and Plasticity. Equilibrium of Elastic Bodies of Canonical Form]. Kiyev: Nauk. Dumka (in Russian).
  3. Ulitko, A. F. (1979). Metod sobstvennykh vektornykh funktsiy v prostranstvennykh zadachakh teorii uprugosti [Method of Vector Eigen Functions in Spatial Problems in the Theory of Elasticity]. Kiyev: Nauk. Dumka (in Russian).
  4. Ufliand, Ya. S. (1967). Integralnye preobrazovaniya v zadachakh teorii uprugosti [Integral Transforms in the Problems in the Theory of Elasticity]. Leningrad: Nauka (in Russian).
  5. Huz, A. N., Chernyshenko, I. S., & Shnerenko, K. I. (1970). Sfericheskie dnishcha, oslablennye otverstiyami [Spherical Bottoms Weakened by Holes]. Kiyev: Nauk. Dumka (in Russian).
  6. Huz, A. N. & Golovchan, V. T. (1972). Difraktsiya uprugikh voln v mnogosvyaznykh telakh [Diffraction of Elastic Waves in Multiply-Connected Bodies]. Kiyev: Nauk. Dumka (in Russian).
  7. Nikolayev, O. G. (1997). Uzahalnenyi metod Furie v prostorovykh zadachakh teorii pruzhnosti dlia kanonichnykh bahatozviazkovykh til [The Generalized Fourier Method for Spatial Problems in the Theory of Elasticity for Canonical Multiply-Connected Bodies] (Author’s Abstract. Diss. Doc. Phys.-Math. Sci.), Dnipropetrovsk. Ukraine (in Ukrainian).
  8. Nikolayev, A. G. & Protsenko, V. S. (2011). Obobshchennyy metod Fourier v prostranstvennykh zadachakh teorii uprugosti [The Generalised Fourier Method for Spatial Problems in the Theory of Elasticity]. Kharkov: N. Ye. Zhukovskii National Aerospace University ‘KhAI’ (in Russian).
  9. Miroshnikov, V. Yu. (2017). Persha osnovna zadacha teorii pruzhnosti v prostori z N paralelnymy kruhovymy tsylindrychnymy porozhnynamy [The First Basic Problem in the Theory of Elasticity in Space with N Parallel Round Cylindrical Cavities]. Journal of Mechanical Engineering, vol. 20, no. 4. pp. 45–52 (in Ukrainian). https://doi.org/10.15407/pmach2017.04.045
  10. Miroshnikov, V. Yu. (2017). On Computation of the Stress-Strain State of a Space Weakened by a System of Parallel Circular Cylindrical Cavities with Different Edge Conditions. Science and Practice: A New Level of Integration in the Modern World. 4th Intern. Conf. Proc. Scope Academic House. Sheffield, (pp. 77–83), UK.
  11. Shcherbakova, Yu. A. & Shekhvatova, Ye. M. (2015). Sravnitelnyy analiz NDS mnogosvyaznykh transversalno-izotropnykh tel s razlichnymi uprugimi kharakteristikami [Comparative Analysis of the Stress-Strain State of Multiply-Connected Transverse-Isotropic Bodies with Different Elastic Characteristics]. Visnyk Zaporizhskoho Natsional’noho Universytetu – Bull. of Zaporizhia National University, iss. 2, pp. 253–261 (in Russian).
  12. Nikolayev, A. G. & Shcherbakova, Yu. A. (2009). Apparat i prilozheniya obobshchennogo metoda Fure dlya transversalno- izotropnykh tel, ogranichennykh ploskostyu i paraboloidom vrashcheniy [Apparatus and Applications of the Generalised Fourier Method for Transverse-Isotropic Bodies Bounded by a Plane and a Paraboloid of Revolution. Mat. metodi ta fіz.-mekh. polya – Math. Methods and Phys.-Mech. of a Field, vol. 52, no. 3, pp. 160–169 (in Russian).
  13. Nikolayev, A. G. & Shcherbakova, Yu. A. (2010). Obosnovanie metoda Fure v osesimmet-richnykh zadachakh teorii uprugosti dlya transversalno-izotropnykh tel, ogranichennykh poverkhnostyu paraboloida [Substantiation of the Fourier Method in Asymmetrical Problems in the Theory of Elasticity for Transverse-Isotropic Bodies Bounded by a Paraboloid Surface]. Otkrytye informatsionnye i kompyuternye integri-rovannye tekhnologii – Open Informational and Computer-Aided Integrated Technologies: Proc. Kharkov N. Ye. Zhukovskii National Aerospace University ‘KhAI’, iss. 48, pp. 180–190 (in Russian).
  14. Nikolayev, A. G., Shcherbakova, A. Yu, & Yukhno, A. I. (2006). Deystvie sosredotochennoy sily na transversalno-izotropnoe poluprostranstvo s paraboloidalnym vklyucheniem [Action of a Lumped Force on a Transverse-Isotropic Half-Space with a Paraboloid Containment]. Voprosy proektirovaniya i proizvodstva konstruktsiy letatelnykh apparatov – Design and Production of Aircraft Constructions: Proc. N. Ye. Zhukovskii National Aerospace University ‘KhAI’, iss. 2 (45), pp. 47–51 (in Russian).
  15. Nikolayev, A. G. & Orlov, Ye. M. (2012). Reshenie pervoy osesimmetrichnoy termouprugoy kraevoy zadachi dlya transversalno-izotropnogo poluprostranstva so sferoidalnoy polostyu [Solution of the first Axisymmetric Thermal Elasticity Boundary Value Problem for a Transverse-Isotropic Half-Space with a Spheroidal Cavity]. Problemi obchislyuvalnoi mekhaniki i mitsnosti konstruktsiy – Computational Mechanics and Strength of Constructions, iss. 20, pp. 253–259 (in Russian).
  16. Protsenko, V. S. & Ukrainets, N. A. (2015). Primenenie obobshchennogo metoda Fourier k resheniyu pervoy osnovnoy zadachi teorii uprugosti v poluprostranstve s tsilindricheskoy polostyu [Application of the Generalised Fourier Method to Solving the First Basic Problem in the Theory of Elasticity in a Half-Space with a Cylindrical Cavity]. Visnyk Zaporizhskoho Natsional’noho Universytetu – Bull. of Zaporizhia National University, iss. 2, pp. 193–202 (in Russian).

 

Received 17 January 2018

Published 30 June 2018