Advertisement

Journal of Mathematical Sciences

, Volume 240, Issue 1, pp 86–97 | Cite as

Solutions of Axisymmetric Problems of Elasticity and Thermoelasticity for an Inhomogeneous Space and a Half Space

  • Yu. V. Tokovyy
Article
  • 14 Downloads

We developed a technique for the construction of solutions of axisymmetric problems of elasticity and thermoelasticity in stresses for a space and a half space whose elastic properties are arbitrary functions of the coordinate z. By using the direct integration method and the Hankel integral transformation, the problems are reduced to governing integral equations accompanied by a local boundary condition and an integral condition in the case of the half space. The solutions of the deduced equations are constructed in the explicit form by using the resolvent-kernel method.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. F. Verlan’ and V. S. Sizikov, Integral Equations: Methods, Algorithms, and Programs: A Handbook [in Russian], Naukova Dumka, Kiev (1986).zbMATHGoogle Scholar
  2. 2.
    B. I. Kogan, “Stresses and strains in coatings with continuously varying modulus of elasticity,” Trudy KhADI, No. 19, 53–66 (1957).Google Scholar
  3. 3.
    B. I. Kogan, “Stresses and strains in multilayer coatings,” Trudy KhADI, No. 14, 33–46 (1953).Google Scholar
  4. 4.
    M. A. Koltunov, Yu. N. Vasil’ev, and V. A. Chernykh, Elasticity and Strength of Cylindrical Bodies [in Russian], Vysshaya Shkola, Moscow (1975).Google Scholar
  5. 5.
    B. G. Korenev, “A die lying on an elastic half space whose modulus of elasticity is a power function of depth,” Dokl. Akad. Nauk SSSR, 112, No. 5, 823–826 (1957).zbMATHGoogle Scholar
  6. 6.
    R. D. Kul’chyts’kyi-Zhyhailo, “Elastic half space with laminated coating of periodic structure under the action of Hertz’s pressure,” Fiz.- Khim. Mekh. Mater., 47, No. 4, 92–98 (2011); English translation: Mater. Sci., 47, No. 4, 527–534 (2012).Google Scholar
  7. 7.
    R. Kul’chyts’kyi-Zhyhailo and G. Rogowski, “Axially symmetric contact problem of pressing of an absolutely rigid ball into an elastic half space with inhomogeneous coating,” Fiz.-Khim. Mekh. Mater., 45, No. 6, 82–92 (2009); English translation: Mater. Sci., 45, No. 6, 845–858 (2009).Google Scholar
  8. 8.
    S. G. Lekhnitskii, “Radial distribution of stresses in a wedge and in a half plane with variable modulus of elasticity,” Prikl. Mat. Mekh., 26, No. 1, 146–151 (1962); English translation: J. Appl. Math. Mech., 26, No. 1, 199–206 (1962).Google Scholar
  9. 9.
    I. A. Molotkov and I. V. Mukhina, “Nonstationary wave propagation in an inhomogeneous half space with the minimum propagation velocity,” Boundary-Value Problems of Mathematical Physics, 4, Trudy MIAN SSSR, 92, 165–181 (1966).Google Scholar
  10. 10.
    V. I. Mossakovskii, “Pressure of a circular die [punch] on an elastic half space whose modulus of elasticity is an exponential function of depth,” Prikl. Mat. Mekh., 22, No. 1, 123–125 (1958); English translation: J. Appl. Math. Mech., 22, No. 1, 168–171 (1958).Google Scholar
  11. 11.
    V. P. Plevako, “The deformation of a nonhomogeneous half space under the action of a surface load,” Prikl. Mekh., 9, No. 6, 16–23 (1973); English translation: Sov. Appl. Mech., 9, No. 6, 593–598 (1973).Google Scholar
  12. 12.
    V. P. Plevako, “A problem concerned with the action of shear forces applied to the surface of an inhomogeneous half space,” Prikl. Mekh., 9, No. 11, 49–55 (1973); English translation: Sov. Appl. Mech., 9, No. 11, 1191–1195 (1973).Google Scholar
  13. 13.
    V. P. Plevako, “Inhomogeneous layer bonded to a half space under the action of internal and external forces,” Prikl. Mat. Mekh., 38, No. 5, 864–875 (1974); English translation: J. Appl. Math. Mech., 38, No. 5, 813–823 (1974).Google Scholar
  14. 14.
    Ya. S. Podstrigach, V. A. Lomakin, and Yu. M. Kolyano, Thermoelasticity of Bodies with Inhomogeneous Structures [in Russian], Nauka, Moscow (1984).zbMATHGoogle Scholar
  15. 15.
    G. Ya. Popov, “On the theory of deflection of plates on an elastic inhomogeneous half space,” Izv. Vyssh. Uchebn. Zaved., Ser. Stroit. Arkhitekt., No. 11–12, 11–19 (1959).Google Scholar
  16. 16.
    V. S. Popovych, H. Yu. Harmatii, and O. M. Vovk, “Thermoelastic state of a thermally sensitive space with spherical cavity under the conditions of convective-radiation heat transfer,” Mat. Metody Fiz.-Mekh. Polya, 49, No. 3, 168–176 (2006).Google Scholar
  17. 17.
    A. K. Privarnikov, Solution of Boundary-Value Problems of the Theory of Elasticity for Multilayer Foundations [in Russian], DGU, Dnepropetrovsk (1976).Google Scholar
  18. 18.
    L. P. Tokova and Ya. V. Yasinskyy, “Approximate solution of a one-dimensional problem of the theory of elasticity for an inhomogeneous solid cylinder,” Mat. Metody Fiz.-Mekh. Polya, 58, No. 4, 107–112 (2015).Google Scholar
  19. 19.
    Ya. S. Uflyand, Survey of Articles on the Application of Integral Transforms in the Theory of Elasticity, North Carolina State Univ., Raleigh (1965).Google Scholar
  20. 20.
    N. A. Tsytovich, Soil Mechanics [in Russian], Gosstroiizdat, Moscow (1963).Google Scholar
  21. 21.
    D. M. Burmister, “The general theory of stresses and displacements in layered systems,” J. Appl. Phys., 16, No. 2, 89–94 (1945).CrossRefGoogle Scholar
  22. 22.
    R. E. Gibson, “Some results concerning displacements and stresses in a nonhomogeneous elastic half space,” Géotechnique, 17, No. 1, 58–67 (1967).CrossRefGoogle Scholar
  23. 23.
    R. E. Gibson and G. C. Sills, “On the loaded elastic half space with a depth varying Poisson’s ratio,” Z. Angew. Math. Phys., 20, No. 5, 691–695 (1969).CrossRefzbMATHGoogle Scholar
  24. 24.
    K. A. Khan and H. H. Hilton, “On inconstant Poisson’s ratios in nonhomogeneous elastic media,” J. Therm. Stresses, 33, No. 1, 29–36 (2010).CrossRefGoogle Scholar
  25. 25.
    L. I. Krenev, S. M. Aizikovich, Yu. V. Tokovyy, and Y.-C. Wang, “Axisymmetric problem on the indentation of a hot circular punch into an arbitrarily nonhomogeneous half space,” Int. J. Solids Struct., 59, 18–28 (2015).CrossRefGoogle Scholar
  26. 26.
    L. I. Krenev, Yu. V. Tokovyy, S. M. Aizikovich, N. M. Seleznev, and S. V. Gorokhov, “A numerical-analytical solution to the mixed boundary-value problem of the heat-conduction theory for arbitrarily inhomogeneous coatings,” Int. J. Therm. Sci., 107, 56–65 (2016).CrossRefGoogle Scholar
  27. 27.
    R. M. Kushnir, V. S. Popovych, and O. M. Vovk, “The thermoelastic state of a thermosensitive sphere and space with a spherical cavity subject to complex heat exchange,” J. Eng. Math., 61, No. 2-4, 357–369 (2008).CrossRefzbMATHGoogle Scholar
  28. 28.
    Yu. V. Tokovyy, “Direct integration method,” in: R. B. Hetnarski (editor), Encyclopedia of Thermal Stresses, Vol. 2, Springer, Dordrecht, etc. (2014), pp. 951–960.Google Scholar
  29. 29.
    Yu. V. Tokovyy, B. M. Kalynyak, and C.-C. Ma, “Nonhomogeneous solids: integral equations approach,” in: R. B. Hetnarski (editor), Encyclopedia of Thermal Stresses, Vol. 7, Springer, Dordrecht, etc. (2014), pp. 3350–3356.Google Scholar
  30. 30.
    Yu. Tokovyy and C.-C. Ma, “An analytical solution to the three-dimensional problem on elastic equilibrium of an exponentially-inhomogeneous layer,” J. Mech., 31, No. 5, 545–555 (2015).CrossRefGoogle Scholar
  31. 31.
    Yu. Tokovyy and C.-C. Ma, “Analytical solutions to the 2D elasticity and thermoelasticity problems for inhomogeneous planes and half planes,” Arch. Appl. Mech., 79, No. 5, 441–456 (2009).CrossRefzbMATHGoogle Scholar
  32. 32.
    Yu. Tokovyy and C.-C. Ma, “Analytical solutions to the axisymmetric elasticity and thermoelasticity problems for an arbitrarily inhomogeneous layer,” Int. J. Eng. Sci., 92, 1–17 (2015).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Yu. V. Tokovyy
    • 1
  1. 1.Pidstryhach Institute for Applied Problems in Mechanics and MathematicsUkrainian National Academy of SciencesLvivUkraine

Personalised recommendations