Journal of Mathematical Sciences

, Volume 240, Issue 2, pp 184–193 | Cite as

Contact Problem for a Rigid Punch and an Elastic Half Space as an Inverse Problem

  • N. I. Obodan
  • T. A. Zaitseva
  • O. D. Fridman

We solve a contact problem of indentation of a punch into an elastic half space with regard for the friction and in the presence of the zones of adhesion, sliding, and separation. The applied approach is based on the statement of the problem in the form of the inverse problem in which the Coulomb law of friction is used as an additional condition in the regions with friction. In the formulation of the inverse problem, we take into account the presence of the zones of adhesion whose sizes are unknown. The correctness of the solution of the inverse problem is analyzed. The proposed approach, in combination with the procedure of discretization, enables us to determine the zones of microsliding alternating with the zones of adhesion and separation.


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  1. 1.
    A. E. Alekseev, “Nonlinear laws of dry friction in contact problems of the linear theory of elasticity,” Prikl. Mekh. Tekh. Fiz., 43, No. 4, 161–169 (2002); English translation: J. Appl. Mech. Tech. Phys., 43, No. 4, 622–629 (2002).Google Scholar
  2. 2.
    A. I. Gasanov, “Computational diagnostics for the determination of the properties of structural materials,” Mat. Model., 1, No. 6, 1–32 (1989).MathSciNetGoogle Scholar
  3. 3.
    I. Hlaváček, J. Haslinger, J. Nečas, and J. Lovíšek, Riešenie variačných nerovností v mechanike. Alfa, Bratislava (1982); MR 0755152.Google Scholar
  4. 4.
    D. A. Kudashkina and R. V. Namm, “Uzawa method for the solution of a contact problem of the elasticity theory with friction,” Uchen. Zamet. TOGU, 5, No. 3, 1–9 (2014).Google Scholar
  5. 5.
    J.-L. Lions and E. Magenes, Problemes aux Limites Non Homogenes et Applications, Dunod, Paris (1968).zbMATHGoogle Scholar
  6. 6.
    V. I. Mossakovskii and A. G. Biskup, “Indentation of a punch in the presence of friction and adhesion,” Dokl. Akad. Nauk SSSR, 206, No. 5, 1068–1070 (1972).Google Scholar
  7. 7.
    V. I. Mossakovskii, V. V. Petrov, and A. V. Sladkovskii, “Study of microsliding under the conditions of compression and shift of an elastic rectangle by rigid plates,” Tren. Iznos, 3, No. 4, 596–602 (1982).Google Scholar
  8. 8.
    R. V. Namm and S. A. Sachkov, “Solving the quasivariational Signorini inequality by the method of successive approximations,” Zh. Vych. Mat. Mat. Fiz., 49, No. 5, 805–814 (2009); English translation: Comput. Math. Math. Phys., 49, No. 5, 776–785 (2009).Google Scholar
  9. 9.
    L. N. Slobodetskii, “Generalized spaces by S. L. Sobolev and their application to boundary-value problems for partial differential equations,” Uchen. Zap. Lenin. Gos. Pedagog. Inst. im. Gertsena, 197, 54–112 (1958).MathSciNetGoogle Scholar
  10. 10.
    L.-L. Ke and Y.-S. Wang, “Two-dimensional contact mechanics of functionally graded materials with arbitrary spatial variations of material properties,” Int. J. Solids Struct., 43, No. 18-19, 5779–5798 (2006).CrossRefzbMATHGoogle Scholar
  11. 11.
    Naveena, J. Ganesh Kumar, and M. D. Mathew, “Finite-element analysis of plastic deformation during impression creep,” J. Mater. Eng. Perform., 24, No. 4, 1741–1753 (2015).CrossRefGoogle Scholar

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

Authors and Affiliations

  • N. I. Obodan
    • 1
  • T. A. Zaitseva
    • 1
  • O. D. Fridman
    • 1
  1. 1.Honchar Dnipro National UniversityDniproUkraine

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