Advertisement

Tribology Letters

, Volume 53, Issue 2, pp 433–448 | Cite as

On the Contact Area and Mean Gap of Rough, Elastic Contacts: Dimensional Analysis, Numerical Corrections, and Reference Data

  • Nikolay Prodanov
  • Wolf B. Dapp
  • Martin H. MüserEmail author
Original Paper

Abstract

The description of elastic, nonadhesive contacts between solids with self-affine surface roughness seems to necessitate knowledge of a large number of parameters. However, few parameters suffice to determine many important interfacial properties as we show by combining dimensional analysis with numerical simulations. This insight is used to deduce the pressure dependence of the relative contact area and the mean interfacial separation \(\Updelta \bar{u}\) and to present the results in a compact form. Given a proper unit choice for pressure p, i.e., effective modulus E * times the root mean square gradient \(\bar{g}\), the relative contact area mainly depends on p but barely on the Hurst exponent H even at large p. When using the root mean square height \(\bar{h}\) as unit of length, \(\Updelta \bar{u}\) additionally depends on the ratio of the height spectrum cutoffs at short and long wavelengths. In the fractal limit, where that ratio is zero, solely the roughness at short wavelengths is relevant for \(\Updelta \bar{u}\). This limit, however, should not be relevant for practical applications. Our work contains a brief summary of the employed numerical method Green’s function molecular dynamics including an illustration of how to systematically overcome numerical shortcomings through appropriate finite-size, fractal, and discretization corrections. Additionally, we outline the derivation of Persson theory in dimensionless units. Persson theory compares well to the numerical reference data.

Keywords

Contact mechanics Surface roughness Relative contact area Mean gap 

Notes

Acknowledgements

We thank the Jülich Supercomputing Centre for computing time on JUGENE, JUQUEEN, and JUROPA. MHM also thanks DFG for support through Grant No. Mu 1694/5-1.

References

  1. 1.
    Boost C++ Libraries. http://boost.org
  2. 2.
    Aifantis, E.C.: The physics of plastic deformation. Int. J. Plast. 3, 211 (1987)CrossRefGoogle Scholar
  3. 3.
    Allen, M.P., Tildesley, D.J.: Computer Simulation of Liquids. Oxford University Press, Oxford (1987)Google Scholar
  4. 4.
    Almqvist, A., Campañá, C., Prodanov, N., Persson, B.N.J.: Interfacial separation between elastic solids with randomly rough surfaces: comparison between theory and numerical techniques. J. Mech. Phys. Solids 59, 2355 (2011)CrossRefGoogle Scholar
  5. 5.
    Barber, J.R.: Incremental stiffness and electrical contact conductance in the contact of rough finite bodies. Phys. Rev. E. 87, 013,203 (2013)CrossRefGoogle Scholar
  6. 6.
    Bush, A.W., Gibson, R.D., Thomas, T.R.: Elastic contact of a rough surface. Wear 35, 87 (1975)CrossRefGoogle Scholar
  7. 7.
    Campañá, C., Müser, M.H.: Practical green’s function approach to the simulation of elastic semi-infinite solids. Phys. Rev. B 74, 075,420 (2006)CrossRefGoogle Scholar
  8. 8.
    Campañá, C., Müser, M.H.: Contact mechanics of real vs. randomly rough surfaces: a Green’s function molecular dynamics study. Europhys. Lett. 77, 38,005 (2007)CrossRefGoogle Scholar
  9. 9.
    Campañá, C., Persson, B.N.J., Müser, M.H.: Transverse and normal interfacial stiffness of solids with randomly rough surfaces. J. Phys.: Condens. Matter 23, 085,001 (2011)Google Scholar
  10. 10.
    Dapp, W.B., Lücke, A., Persson, B.N.J., Müser, M.H.: Self-affine elastic contacts: percolation and leakage. Phys. Rev. Lett. 108, 244,301 (2012)CrossRefGoogle Scholar
  11. 11.
    Frigo, M., Johnson, S.G.: The design and implementation of fftw3. Proceedings of the IEEE 93(2), 216 (2005)CrossRefGoogle Scholar
  12. 12.
    Greenwood, J.A., Williamson, J.B.P.: Contact of nominally flat surfaces. Proc. R. Soc. London A295, 300 (1966)CrossRefGoogle Scholar
  13. 13.
    Hyun, S., Pei, L., Molinari, J.F., Robbins, M.O.: Finite-element analysis of contact between elastic self-affine surfaces. Phys. Rev. E70, 026,117 (2004)Google Scholar
  14. 14.
    Johnson, K.L.: Contact Mechanics. Cambridge University Press, New York (1985)CrossRefGoogle Scholar
  15. 15.
    Kendall, K.: Molecular Adhesion and Its Applications: The Sticky Universe. Kluwer Academic, New York (2001)Google Scholar
  16. 16.
    Kong, L.T., Bartels, G., Campañá, C., Denniston, C., Müser, M.H.: Implementation of greens function molecular dynamics: an extension to lammps. Comput. Phys. Commun. 180, 1004 (2009)CrossRefGoogle Scholar
  17. 17.
    Landau, L.D., Lifshitz, E.M.: Theory of Elasticity, 3rd ed. Pergamon Press, Oxford (1970)Google Scholar
  18. 18.
    Lechenault, F., Pallares, G., George, M., Rountree, C., Bouchaud, E., Ciccotti, M.: Effects of finite probe size on self-affine roughness measurements. Phys. Rev. Lett. 104, 025,502 (2010)CrossRefGoogle Scholar
  19. 19.
    Lorenz, B., Krick, B.A., Mulakaluri, N., Smolyakova, M., Dieluweit, S., Sawyer, W.G., Persson, B.N.J.: Adhesion: role of bulk viscoelasticity and surface roughness. J. Phys.: Condens. Matter 25, 225,004 (2013)Google Scholar
  20. 20.
    Lorenz, B., Persson, B.N.J.: Leak rate of seals: effective-medium theory and comparison with experiment. Eur. Phys. J. 31, 159 (2010)Google Scholar
  21. 21.
    Lorenz, B., Persson, B.N.J.: Time-dependent fluid squeeze-out between solids with rough surfaces. Eur. Phys. J. E32, 281 (2010)Google Scholar
  22. 22.
    Lyashenko, I., Pastewka, L., Persson, B.N.J.: On the validity of the method of reduction of dimensionality: area of contact, average interfacial separation and contact stiffness. Tribol. Lett. 52, 223–229 (2013)CrossRefGoogle Scholar
  23. 23.
    Ma, Z.S., Zhou, Y.C., Long, S., Lu, C.: On the intrinsic hardness of a metallic film/substrate system: Indentation size and substrate effects. Int. J. Plast. 34, 1 (2012)CrossRefGoogle Scholar
  24. 24.
    Pastewka, L., Prodanov, N., Lorenz, B., Müser, M.H., Robbins, M.O., Persson, B.N.J.: Finite-size scaling in the interfacial stiffness of rough elastic contacts. Phys. Rev. E 87, 062,809 (2013)CrossRefGoogle Scholar
  25. 25.
    Persson, B.N.J.: Theory of rubber friction and contact mechanics. J. Chem. Phys. 115, 3840 (2001)CrossRefGoogle Scholar
  26. 26.
    Persson, B.N.J.: Contact mechanics for randomly rough surfaces. Surf. Sci. Rep. 61, 201 (2006)CrossRefGoogle Scholar
  27. 27.
    Persson, B.N.J.: Relation between interfacial separation and load: a general theory of contact mechanics. Phys. Rev. Lett. 99, 125,502 (2007)CrossRefGoogle Scholar
  28. 28.
    Persson, B.N.J., Albohr, O., Tartaglino, U., Volokitin, A.I., Tosatti, E.: On the nature of surface roughness with application to contact mechanics, sealing, rubber friction and adhesion. J. Phys. Condens. Matter 17, R1–R62 (2005)CrossRefGoogle Scholar
  29. 29.
    Persson, B.N.J., Prodanov, N., Krick, B.A., Rodriguez, N., Mulakaluri, N., Sawyer, W.G., Mangiagalli, P.: Elastic contact mechanics: percolation of the contact area and fluid squeeze-out. Eur. Phys. J. E 35, 5 (2012)CrossRefGoogle Scholar
  30. 30.
    Persson, B.N.J., Scaraggi, M.: On the transition from boundary lubrication to hydrodynamic lubrication in soft contacts. J. Phys. Condens. Matter 21, 185,002 (2009)CrossRefGoogle Scholar
  31. 31.
    Persson, B.N.J., Tosatti, E.: The effect of surface roughness on the adhesion of elastic solids. J. Chem. Phys. 115(12), 5597 (2001)CrossRefGoogle Scholar
  32. 32.
    Persson, B.N.J., Yang, C.: Theory of the leak-rate of seals. J. Phys. Condens. Matter 20, 315,011 (2008)CrossRefGoogle Scholar
  33. 33.
    Power, W.L., Tullis, T.E.: Euclidean and fractal models for the description of rock surface roughness. J. Geophys. Res. 96, 415 (1991)CrossRefGoogle Scholar
  34. 34.
    Prodanov, N., Gachot, C., Rosenkranz, A., Mücklich, F., Müser, M.H.: Contact mechanics of laser-textured surfaces: Correlating contact area and friction. Tribol. Lett. 50, 41 (2013)CrossRefGoogle Scholar
  35. 35.
    Putignano, C., Afferrante, L., Carbone, G., Demelio, G.: The influence of the statistical properties of self-affine surfaces in elastic contacts: a numerical investigation. J. Mech. Phys. Solids 60, 973 (2012)CrossRefGoogle Scholar
  36. 36.
    Putignano, C., Afferrante, L., Carbone, G., Demelio, G.: A multiscale analysis of elastic contacts and percolation threshold for numerically generated and real rough surfaces. Tribol. Int. 64, 148 (2013)CrossRefGoogle Scholar
  37. 37.
    Yang, C., Persson, B.N.J.: Contact mechanics: contact area and interfacial separation from small contact to full contact. J. Phys. Condens. Matter 20, 215,214 (2008)CrossRefGoogle Scholar
  38. 38.
    Yastrebov, V.A., Anciaux, G., Molinari, J.F.: Contact between representative rough surfaces. Phys. Rev. E 86, 035,601R (2012)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Nikolay Prodanov
    • 1
    • 2
  • Wolf B. Dapp
    • 1
  • Martin H. Müser
    • 1
    • 2
    Email author
  1. 1.Jülich Supercomputing Centre, Institute for Advanced SimulationFZ JülichJülichGermany
  2. 2.Department of Materials Science and EngineeringUniversität des SaarlandesSaarbrückenGermany

Personalised recommendations