Skip to main content
Log in

Contact with Intermolecular Interaction Forces for a Viscoelastic Layer (Self-Consistent Approach): Calculation of the Stress-Strain State and Energy Dissipation

  • Published:
Mechanics of Solids Aims and scope Submit manuscript

Abstract

The contact between an infinitely extended plane indentor and a viscoelastic layer is considered within the Derjaguin self-consistent approach with the surface (traditional formulation) and bulk (refined formulation) application of intermolecular interaction forces. Some analytical expressions are derived for the stress-strain state. Using the first law of thermodynamics, the energy dissipation in a viscoelastic layer is calculated. It is shown that the traditional and refined problem formulations may lead to essentially different results of calculation for the characteristics of contact between an indentor and a viscoelastic layer.

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

Access this article

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

Instant access to the full article PDF.

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.

Similar content being viewed by others

REFERENCES

  1. B. Derjaguin, “Untersuchungen über die Reibung und Adhäsion, IV. Theorie des Anhaftens kleiner Teilchen,” Kolloid-Z. 69 (2), 155–164 (1934).

    Article  Google Scholar 

  2. K. L. Johnson, K. Kendall, and A. D. Roberts, “Surface energy and the contact of elastic solids,” Proc. R. Soc. London, Ser. A 324 (1558), 301–313 (1971).

    Article  ADS  Google Scholar 

  3. B. V. Derjaguin, V. M. Muller, and Yu. P. Toporov, “Effect of contact deformations on the adhesion of particles,” J. Colloid Interface Sci. 53 (2), 314–326 (1975).

    Article  ADS  Google Scholar 

  4. F. M. Borodich, B. A. Galanov, N. V. Perepelkin, and D. A. Prikazchikov, “Adhesive contact problems for a thin elastic layer: asymptotic analysis and the JKR theory,” Math. Mech. Solids 24 (5), 1405–1424 (2018).

    Article  MathSciNet  Google Scholar 

  5. J. A. Greenwood and K. L. Johnson, “The mechanics of adhesion of viscoelastic solids,” Philos. Mag. A 43 (3), 697–711 (1981).

    Article  ADS  Google Scholar 

  6. I. G. Goryacheva, M. M. Gubenko, and Yu. Yu. Makhovskaya, “Sliding of a spherical indenter on a viscoelastic foundation with the forces of molecular attraction taken into account,” J. Appl. Mech. Techn. Phys. 55 (1), 81–89 (2014).

    Article  ADS  Google Scholar 

  7. Y. Y. Lin and C. Y. Hui, “Mechanics of contact and adhesion between viscoelastic spheres: an analysis of hysteresis during loading and unloading,” J. Polym. Sci. Part B: Polym. Phys. 40, 772–793 (2002).

    Article  ADS  Google Scholar 

  8. G. Haiat, M. C. Phan Huy, and E. Barthel, “The adhesive contact of viscoelastic spheres,” J. Mech. Phys. Solids 51 (1), 69–99 (2003).

    Article  ADS  MathSciNet  Google Scholar 

  9. V. M. Muller, V. S. Yushchenko, and B. V. Derjaguin, “On the influence of molecular forces on the deformation of an elastic sphere and its sticking to a rigid plane,” J. Colloid Interface Sci. 77 (1), 91–101 (1980).

    Article  ADS  Google Scholar 

  10. P. Attard and J. L. Parker, “Deformation and adhesion of elastic bodies in contact,” Phys. Rev. A 46 (12), 7959–7971 (1992).

    Article  ADS  Google Scholar 

  11. J. A. Greenwood, “Adhesion of small spheres,” Philos. Mag. 89 (11), 945–965 (2009).

    Article  ADS  Google Scholar 

  12. I. A. Soldatenkov, “The use of the method of successive approximations to calculate an elastic contact in the presence of molecular adhesion,” J. Appl. Math. Mech. 76 (5), 597–603 (2012).

    Article  MathSciNet  Google Scholar 

  13. R. A. Sauer and S. Li, “A contact mechanics model for quasi-continua,” Int. J. Numer. Meth. Eng. 71 (8), 931–962 (2007).

    Article  MathSciNet  Google Scholar 

  14. L. H. He, “Stress and deformation in soft elastic bodies due to intermolecular forces,” J. Mech. Phys. Solids 61 (6), 1377–1390 (2013).

    Article  ADS  MathSciNet  Google Scholar 

  15. I. A. Soldatenkov, “The contact problem with the bulk application of intermolecular interaction forces (a refined formulation),” J. Appl. Math. Mech. 77 (6), 629–641 (2013).

    Article  MathSciNet  Google Scholar 

  16. G. A. Tomlinson, “Molecular cohesion,” Philos. Mag. 6 (37), 695–712 (1928).

    Article  Google Scholar 

  17. R. S. Bradley, “The cohesive force between solid surfaces and the surface energy of solids,” Philos. Mag. 13 (86), 853–862 (1932).

    Article  Google Scholar 

  18. H. C. Hamaker, “The London–van der Waals attraction between spherical particles,” Physica 4 (10), 1058–1072 (1937).

    Article  ADS  Google Scholar 

  19. I. G. Kaplan, Intermolecular Interactions: Physical Picture, Computational Methods and Model Potentials (Wiley, Chichester, 2006).

    Book  Google Scholar 

  20. J. N. Israelachvili, Intermolecular and Surface Forces, 3rd ed. (Academic, London, 2011).

    Google Scholar 

  21. R. M. Christensen, Theory of Viscoelasticity. An Introduction (Acad. Press, New York, 1971).

    Google Scholar 

  22. P. M. Ogibalov, V. A. Lomakin, and B. P. Kishkin, Mechanics of Polymers (MSU, Moscow, 1975) [in Russian].

    Google Scholar 

  23. A. A. Adamov, V. P. Matveenko, N. A. Trufanov, and I. N. Shardakov, Methods of Applied Viscoelasticity (Ural Branch RAS, Yekaterinburg, 2003) [in Russian].

    Google Scholar 

  24. V. M. Aleksandrov, “Asymptotic solution of the contact problem for a thin elastic layer,” J. Appl. Math. Mech. 33 (1), 49–63 (1969).

    Article  MathSciNet  Google Scholar 

  25. Yu. V. Sidorov, M. V. Fedoryuk, and M. I. Shabunin, Lectures on the Theory of Functions of a Complex Variable (Mir, Moscow, 1985).

  26. J. A. Collins, Failure of Materials in Mechanical Design. Analysis, Prediction, Prevention (Ohio State University, John Wiley & Sons, New York, 1981).

    Google Scholar 

  27. H. G. Hahn, Elastizitätstheorie. Grundlagen der linearen Theorie und Anwendungen auf eindimensionale, ebene und räumliche Probleme (Teubner, Stuttgart, 1985).

    MATH  Google Scholar 

  28. G. M. Fikhtengol’ts, Course of Differential and Integral Calculus. In 3 Vols. (Fizmatlit, Moscow, 2001), Vol. 3 [in Russian].

    Google Scholar 

Download references

Funding

This work was supported by the Russian Foundation for Basic Research (projects nos. 18-58-00014, 18-08-00558).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to I. A. Soldatenkov.

Additional information

Translated by E. Glushachenkova

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Soldatenkov, I.A. Contact with Intermolecular Interaction Forces for a Viscoelastic Layer (Self-Consistent Approach): Calculation of the Stress-Strain State and Energy Dissipation. Mech. Solids 55, 1077–1092 (2020). https://doi.org/10.3103/S0025654420070195

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0025654420070195

Keywords:

Navigation