Skip to main content
SpringerLink
Log in

Contact Problem for Inhomogeneous Cylinders with Variable Poisson’s Ratio

  • Published:
Mechanics of Solids Aims and scope Submit manuscript

Abstract

In cylindrical coordinates, the system of two elastic-equilibrium differential equations is studied under the assumption of axial symmetry and the assumption that the Poisson’s ratio is an arbitrary, sufficiently smooth, function of the radial coordinate and the modulus of rigidity is constant. It turns out that the elastic coefficient is variable with respect to the radial coordinate in this case. We propose a general representation of the solution of this system, leading to the vector Laplace equation and scalar Poisson equation such that its right-hand side depends on the Poisson’s ratio. Being projected, the vector Laplace equation is reduced to two differential equations such that one of them is the scalar Laplace equation. Using the Fourier integral transformation, we construct exact general solutions of the Laplace and Poisson equations in quadratures. We obtain the integral equation of the axially symmetric contact problem on the interaction of a rigid band with an inhomogeneous cylinder and find its regular and singular asymptotic solutions by means of the Aleksandrov method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Borodachev, A.N. and Dudinskii, V.I., Rigid punch on an elastic semispace with a depth-varying Poisson’s ratio, Prikl. Mekh., 1985, vol. 21, no. 8, pp. 34–39.

    Google Scholar 

  2. Borodachev, A.N., Elastic equilibrium in a layer inhomogeneous with depth, Prikl. Mekh., 1988, vol. 24, no. 8, pp. 30–35.

    MathSciNet  MATH  Google Scholar 

  3. Pozharskii, D.A., The elastic equilibrium of an inhomogeneous wedge with varying Poisson’s ratio, J. Appl. Math. Mech., 2016, vol. 80, no. 5, pp. 433–438.

    Article  MathSciNet  Google Scholar 

  4. Kuznetsov, E.A., Round cylinder pressure onto semispace with a depth-varying Poisson’s ratio, Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, 1985, no. 1, pp. 73–86.

  5. Lur’e, A.I., Prostranstvennye zadachi teorii uprugosti (Spatial Problems on Theory of Elasticity), Moscow: Gos. Izd. Tekhniko-Teoreticheskoi Literatury, 1955.

    Google Scholar 

  6. Aleksandrov, V.M. and Belokon’, A.V., Asymptotic solution of a class of integral equations and its application to contact problems for cylindrical elastic bodies, J. Appl. Math. Mech., 1967, vol. 31, no. 4, pp. 718–724.

    Article  Google Scholar 

  7. Aleksandrov, V.M. and Romalis, B.L., Kontaktnye zadachi v mashinostroenii (Contact Problems on Machinery Manufacture), Moscow: Mashinostroenie, 1986.

    Google Scholar 

  8. Aleksandrov, V.M. and Pozharskii, D.A., An asymptotic method in contact problems, J. Appl. Math. Mech., 1999, vol. 63, no. 2, pp. 283–290.

    Article  MathSciNet  Google Scholar 

  9. Alexandrov, V.M. and Pozharskii, D.A., Three-Dimensional Contact Problems, Dordrecht: Kluwer Academic Publ., 2001.

    Book  Google Scholar 

  10. Lomakin, V.A., Teoriya uprugosti neodnorodnykh tel (Theory of Elasticity of Heterogeneous Bodies), Moscow: Lenand, 2014.

    Google Scholar 

  11. Kalinchuk, V.V. and Belyankova, T.I., Dinamika poverkhnosti neodnorodnykh sred (Dynamics of Heterogeneous Media’ Surface), Moscow: Fizmatlit, 2009.

    Google Scholar 

  12. Belyankova, T.I. and Kalinchuk, V.V., The dynamic contact problem for a prestressed cylindrical tube filled with a fluid, J. Appl. Math. Mech., 2009, vol. 73, no. 2, pp. 209–219.

    Article  Google Scholar 

  13. Belyankova, T.I., Kalinchuk, V.V., and Lyzhov, V.A., Peculiarities of dynamics of three-layered hollow cylinder, Ekol. Vestn. Nauchn. Tsentr. Chernomorsk. Ekon. Sotr., 2015, no. 4, pp. 19–32.

  14. Gurtin, M.E., The Linear Theory of Elasticity. Handbuch der Physik, Berlin: Springer, 1972, vol. 6a/2, pp. 1–295.

    Google Scholar 

  15. Prudnikov, A.P., Brychkov, Yu.A., and Marichev, O.I., Integraly i ryady. Spetsial’nye funktsii (Integrals and Series. Special Functions), Moscow: Nauka, 1983.

    MATH  Google Scholar 

  16. Handbook of Mathematical Functions, Abramowitz, M. and Stigan, I., Eds., New York: Dover, 1964.

    MATH  Google Scholar 

Download references

Funding

This work was supported by the Russian Foundation for Basic Research, project no. 18-01-00017.

Author information

Authors and Affiliations

  1. Don State Technical University, Azov, 344002, Russia

    D. A. Pozharskii

Authors
  1. D. A. Pozharskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. A. Pozharskii.

Additional information

Russian Text © The Author(s), 2019, published in Prikladnaya Matematika i Mekhanika, 2019, Vol. 83, No. 2, pp. 323–330.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pozharskii, D.A. Contact Problem for Inhomogeneous Cylinders with Variable Poisson’s Ratio. Mech. Solids 54, 709–716 (2019). https://doi.org/10.3103/S0025654419050133

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0025654419050133

Keywords

Search

Navigation