Skip to main content
SpringerLink
Log in

Doubly Periodic Contact Problems for a Layer with an Unknown Contact Zone

  • Published:
Mechanics of Solids Aims and scope Submit manuscript

Abstract

Doubly periodic contact problems are considered for an elastic layer with an unknown contact area. One face of the layer is in conditions of sliding or rigid embedding. The problems are reduced to integral equations whose kernels do not contain quadratures. For the case of full contact of the other face of the layer with a two-dimensional sinusoidal rigid surface, the problems have an exact solution, which is used to debug programs that implement the numerical method of non-linear Galanov integral equations, which makes it possible to simultaneously determine the contact area and contact pressures. The mechanical characteristics are calculated for the introduction of a system of elliptical paraboloids and the transition from a discrete to a continuous contact area is studied.

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

Fig. 1.
Fig. 2.

REFERENCES

  1. H. M. Westergaard, “Bearing pressure and cracks,” ASME. J. Appl. Mech. E 6 (1), 43–53 (1939).

    Google Scholar 

  2. D. A. Pozharskii, “Periodic contact and mixed problems of the elasticity theory (review),” Izv. Vyssh. Uchebn. Zaved. Sev.-Kavk. Region. Estestv. Nauki, No. 2, 22–33 (2021).

  3. I. G. Goryacheva, “The periodic contact problem for an elastic half-space,” J. Appl. Math. Mech. 62 (6), 959–966 (1998).

    Article  MathSciNet  Google Scholar 

  4. I. G. Goryacheva, Contact Mechanics in Tribology (Springer, Berlin, 1998).

    Book  Google Scholar 

  5. K. L. Johnson, J. A. Greenwood, and J. G. Higginson, “The contact of elastic regular wavy surfaces,” Int. J. Mech. Sci. 27 (6), 383–396 (1985).

    Article  Google Scholar 

  6. K. L. Johnson, Contact Mechanics (Univ. Press, Cambridge, 1985).

    Book  Google Scholar 

  7. V. A. Yastrebov, G. Anciaux, and J.-F. Molinari, “The contact of elastic regular wavy surfaces revisited,” Tribol. Lett. 56, 171–183 (2014).

    Article  Google Scholar 

  8. V. M. Aleksandrov, “Doubly periodic contact problems for and elastic layer,” J. Appl. Math. Mech. 66 (2), 297–305 (2002).

    Article  MathSciNet  Google Scholar 

  9. I. A. Soldatenkov, “The periodic contact problem of the plane theory of elasticity. Taking friction, wear and adhesion into account,” J. Appl. Math. Mech. 77 (2), 245–255 (2013).

    Article  MathSciNet  Google Scholar 

  10. I. G. Goryacheva and E. V. Torskaya, “Modeling of fatigue wear of a two-layered elastic half-space in contact with periodic system of indenters,” Wear 268 (11-12), 1417–1422 (2010).

    Article  CAS  Google Scholar 

  11. F. Jin, Q. Wan, and X. Guo, “A double-Westergaard model for adhesive contact of a wavy surface,” Int. J. Solids Struct 102–103, 66–76 (2016).

    Article  Google Scholar 

  12. I. G. Goryacheva and Y. Makhovskaya, “Combined effect of surface microgeometry and adhesion in normal and sliding contacts of elastic bodies,” Friction 5 (3), 339–350 (2017).

    Article  Google Scholar 

  13. I. A. Soldatenkov, “The spatial contact problem for an elastic layer and wavy punch when there is friction and wear,” J. Appl. Math. Mech. 78 (1), 99–106 (2014).

    Article  MathSciNet  Google Scholar 

  14. I. Goryacheva and A. Yakovenko, “The periodic contact problem for spherical indenters and viscoelastic half-space,” Tribol. Int. 161, 107078 (2021).

  15. N. B. Zolotov and D. A. Pozharskii, “Periodic contact problems for a half-space with a partially fixed boundary,” Mech. Solids 57 (7), 1758–1765 (2022).

    Article  ADS  Google Scholar 

  16. B. A. Galanov, “The method of boundary equations of the Hammerstein-type for contact problems of the theory of elasticity when the regions of contact are not known,” J. Appl. Math. Mech. 49 (5), 634–640 (1985).

    Article  ADS  MathSciNet  Google Scholar 

  17. A. A. Yakovenko, “Modeling of discrete contact of elastic and viscoelastic bodies,” Candidate’s Dissertation in Mathematics and Physics (Ishlinsky Institute for Problems in Mechanics RAS, Moscow, 2022) [in Russian].

Download references

Funding

This study was supported by the Russian Science Foundation (project code 22-21-00013) and is dedicated to the centenary of applied mathematician Academician E.V. Zolotov (1922–1990).

Author information

Authors and Affiliations

  1. Don State Technical University, Rostov-on-Don, Russia

    N. B. Zolotov & D. A. Pozharskii

Authors
  1. N. B. Zolotov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. A. Pozharskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. A. Pozharskii.

Ethics declarations

The authors declare that they have no conflict of interest.

Additional information

Translated by K. Gumerov

Publisher’s Note.

Allerton Press remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zolotov, N.B., Pozharskii, D.A. Doubly Periodic Contact Problems for a Layer with an Unknown Contact Zone. Mech. Solids 58, 2602–2609 (2023). https://doi.org/10.3103/S0025654423070257

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0025654423070257

Keywords:

Search

Navigation