Skip to main content
SpringerLink
Log in

On Contact Problems with a Deformable Punch and Variable Rheology

  • ON THE ANNIVERSARY OF ALEXANDER KONSTANTINOVICH BELYAEV
  • Published:
Vestnik St. Petersburg University, Mathematics Aims and scope Submit manuscript

Abstract

The paper presents for the first time one of the methods for studying and solving contact problems with a deformed stamp for those cases when there is a need to change the rheology of the stamp material. It is based on a new universal modeling method previously published by the authors, which is used in boundary-value problems for systems of partial differential equations. With its help, solutions of complex-vector boundary-value problems for systems of differential equations can be decomposed into solutions of scalar boundary-value problems for individual differential equations. Among them, the Helmholtz equations are the simplest. The solutions to the scalar boundary-value problems are represented as fractals, self-similar mathematical objects, first introduced by the American mathematician B. Mandelbrot. The role of fractals is performed by packed block elements. The transition from systems of differential equations in partial derivatives to individual equations is carried out using the transformation of Academician B.G. Galerkin or representation by potentials. It is known that the solutions of dynamic contact problems with a deformable stamp of complex rheology are cumbersome and their study is always difficult. The problem is complicated by the presence of discrete resonant frequencies in such problems, which were once discovered by Academician I.I. Vorovich. A contact problem with a deformable punch admits the construction of a solution if it is possible to solve the contact problem for an absolutely rigid punch and construct a solution to the boundary problem for a deformable punch. In earlier works of the authors, the deformable stamp was described by a separate Helmholtz equation. In this paper, we consider a contact problem on the action of a semiinfinite stamp on a multilayer base, described by the system of Lame equations. One of the methods of transition to other rheologies is shown when describing the properties of a deformable stamp in contact problems.

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

REFERENCES

  1. A. Papangelo, M. Ciavarella, and J. R. Barber, “Fracture mechanics implications for apparent static friction coefficient in contact problems involving slip-weakening laws,” Proc. R. Soc. A 471, 20150271 (2015). https://doi.org/10.1098/rspa.2015.0271

    Article  MathSciNet  Google Scholar 

  2. S. Zhou and X. L. Gao, “Solutions of half-space and half-plane contact problems based on surface elasticity,” Z. Angew. Math. Phys. 64, 145–166 (2013). https://doi.org/10.1007/s00033-012-0205-0

    Article  MathSciNet  Google Scholar 

  3. A. Almqvist, “An LCP solution of the linear elastic contact mechanics problem” (2013). https://www.mathworks.com/matlabcentral/fileexchange/43216-an-lcp-solution-of-the-linearelastic-contact-mechanics-problem. https://doi.org/10.13140/RG.2.1.3960.7200

  4. M. Cocou, “A class of dynamic contact problems with Coulomb friction in viscoelasticity,” Nonlinear Anal.: Real World Appl. 22, 508–519 (2015). https://doi.org/10.1016/j.nonrwa.2014.08.012

    Article  MathSciNet  Google Scholar 

  5. M. Ciavarella, “The generalized Cattaneo partial slip plane contact problem. I. Theory,” Int. J. Solids Struct. 35, 2349–2362 (1998). https://doi.org/10.1016/S0020-7683(97)00154-6

    Article  MathSciNet  Google Scholar 

  6. M. Ciavarella, “The generalized Cattaneo partial slip plane contact problem. II. Examples,” Int. J. Solids Struct. 35, 2363–2378 (1998). https://doi.org/10.1016/S0020-7683(97)00155-8

    Article  MathSciNet  Google Scholar 

  7. M. A. Guler and F. Erdogan, “The frictional sliding contact problems of rigid parabolic and cylindrical stamps on graded coatings,” Int. J. Mech. Sci. 49, 161–182 (2007). https://doi.org/10.1016/j.ijmecsci.2006.08.006

    Article  Google Scholar 

  8. L.-L. Ke and Y.-S. Wang, “Two-dimensional sliding frictional contact of functionally graded materials,” Eur. J. Mech. A/Solids 26, 171–188 (2007). https://doi.org/10.1016/j.euromechsol.2006.05.007

    Article  Google Scholar 

  9. A. Almqvist, F. Sahlin, R. Larsson, and S. Glavatskih, “On the dry elasto-plastic contact of nominally flat surfaces,” Tribol. Int. 40, 574–579 (2007). doi 10.1016 /j.triboint.2005.11.008

  10. L. E. Andersson, “Existence results for quasistatic contact problems with Coulomb friction,” Appl. Math. Optim. 42, 169–202 (2000). https://doi.org/10.1007/s002450010009

    Article  MathSciNet  Google Scholar 

  11. M. Cocou and R. Rocca, “Existence results for unilateral quasistatic contact problems with friction and adhesion,” Math. Modell. Numer. Anal. 34, 981–1001 (2000).

    Article  MathSciNet  Google Scholar 

  12. N. Kikuchi and J. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods (SIAM, Philadelphia, Penn., 1988).

    Book  Google Scholar 

  13. M. Raous, L. Cangémi, and M. Cocou, “A consistent model coupling adhesion, friction, and unilateral contact,” Comput. Meth. Appl. Mech. Eng. 177, 383–399 (1999). https://doi.org/10.1016/S0045-7825(98)00389-2

    Article  MathSciNet  Google Scholar 

  14. M. Shillor, M. Sofonea, and J. J. Telega, Models and Analysis of Quasistatic Contact: Variational Methods (Springer-Verlag, Berlin, 2004), in Ser.: Lecture Notes in Physics, Vol. 655. https://doi.org/10.1007/b99799

  15. V. A. Babeshko, O. V. Evdokimova, and O. M. Babeshko, “On contact problems with a deformable stamp,” Probl. Prochn. Plast. 84, 25–34 (2022). https://doi.org/10.32326/1814-9146-2022-84-1-25-34

    Article  Google Scholar 

  16. I. I. Vorovich and V. A. Babeshko, Dynamic Mixed Problems of Elasticity Theory for Non-Classical Domains (Nauka, Moscow, 1979) [in Russian].

    Google Scholar 

  17. V. A. Babeshko, O. V. Evdokimova, and O. M. Babeshko, “Fractal properties of block elements and a new universal modeling method,” Dokl. Phys. 499, 218–222 (2021). https://doi.org/10.1134/S1028335821080012

    Article  Google Scholar 

  18. V. A. Babeshko, O. V. Evdokimova, and O. M. Babeshko, “On a method for solving boundary value problems of the dynamic theory of elasticity in a quarter plane,” Prikl. Mat. Mekh. 85, 275–282 (2021). https://doi.org/10.31857/S0032823521030024

    Article  Google Scholar 

  19. V. Novatskii, Theory of Elasticity (Mir, Moscow, 1975) [in Russian].

  20. V. A. Babeshko, O. V. Evdokimova, and O. M. Babeshko, “The Hilbert–Wiener factorization problem and the block-element method,” Dokl. Phys. 59, 591–595 (2014).

    Article  Google Scholar 

  21. V. A. Babeshko, O. V. Evdokimova, and O. M. Babeshko, “On the problem of evaluating the behavior of multicomponent materials in mixed boundary conditions in contact problems,” Mater. Phys. Mech. 48, 379–385 (2022). https://doi.org/10.18720/MPM.48(3)2022_8

    Article  Google Scholar 

  22. I. I. Vorovich, “Spectral properties of a boundary value problem of elasticity theory for a nonuniform band,” Dokl. Akad. Nauk SSSR 245, 817–820 (1979).

    MathSciNet  Google Scholar 

  23. I. I. Vorovich, “Resonance properties of an elastic inhomogeneous strip,” Dokl. Akad. Nauk SSSR 245, 1076–1079 (1979).

    MathSciNet  Google Scholar 

Download references

Funding

The work was supported by the Russian Science Foundation, project no. 22-21-00129.

Author information

Authors and Affiliations

  1. Southern Scientific Center of the Russian Academy of Sciences, 344006, Rostov-on-Don, Russia

    V. A. Babeshko

  2. Kuban State University, 350040, Krasnodar, Russia

    V. A. Babeshko, O. V. Evdokimova, O. M. Babeshko, M. V. Zaretskaya & V. S. Evdokimov

Authors
  1. V. A. Babeshko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. V. Evdokimova
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. O. M. Babeshko
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. V. Zaretskaya
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. V. S. Evdokimov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to V. A. Babeshko, O. V. Evdokimova, O. M. Babeshko, M. V. Zaretskaya or V. S. Evdokimov.

Ethics declarations

The authors declare that they have no conflict of interest.

Additional information

Translated by K. Gumerov

Publisher’s Note.

Pleiades Publishing 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

Babeshko, V.A., Evdokimova, O.V., Babeshko, O.M. et al. On Contact Problems with a Deformable Punch and Variable Rheology. Vestnik St.Petersb. Univ.Math. 56, 416–423 (2023). https://doi.org/10.1134/S1063454123040027

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1063454123040027

Keywords:

Search

Navigation