Tribology Letters

, 64:14 | Cite as

On the Contact Area of Nominally Flat Hertzian Contacts

Original Paper


In a recent paper, Pastewka and Robbins (Appl Phys Lett 108:221–601, 2016) state an analytical expression for the real contact area of a Hertzian tip with small-scale roughness. We confirm that their formula predicts real contact areas quite well—with less than 10 % error. Nonetheless, the complementary contact area does not show the proper scaling to the continuum results at large loads. This shortcoming is fixed in the present work by abandoning a mean-field approximation made in the original work. Analytical results can even be made essentially perfect with a relation giving the accurate dependence of contract area on pressure for contacts between solids with nominally flat surfaces.


Contact mechanics Surface roughness Analysis and models Molecular dynamics 


  1. 1.
    Akarapu, S., Sharp, T., Robbins, M.O.: Stiffness of contacts between rough surfaces. Phys. Rev. Lett. 106, 204,301 (2011)CrossRefGoogle Scholar
  2. 2.
    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, 2355–2369 (2011)CrossRefGoogle Scholar
  3. 3.
    Bowden, F.P., Tabor, D.: Friction and Lubrication. Wiley, New York (1956)Google Scholar
  4. 4.
    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
  5. 5.
    Campañá, C., Müser, M.H.: Contact mechanics of real vs. randomly rough surfaces: a Green’s function molecular dynamics study. EPL 77, 38,005 (2007)CrossRefGoogle Scholar
  6. 6.
    Campañá, C., Robbins, M.O., Müser, M.H.: Elastic contact between self-affine surfaces: comparison of numerical stress and contact correlation functions with analytic predictions. J. Phys. Condens. Matter 20, 354,013 (2008)CrossRefGoogle Scholar
  7. 7.
    Cheng, S., Robbins, M.O.: Defining contact at the atomic scale. Tribol. Lett. 39, 329–348 (2010)CrossRefGoogle Scholar
  8. 8.
    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
  9. 9.
    Dapp, W.B., Prodanov, N., Müser, M.H.: Systematic analysis of Persson’s contact mechanics theory of randomly rough elastic surfaces. J. Phys. Condens. Matter 226, 355,002 (2014)CrossRefGoogle Scholar
  10. 10.
    Hertz, G.: Ueber die Berührung fester elastischer Körper. J. Reine Angew. Math. 92, 156–171 (1881)Google Scholar
  11. 11.
    Hu, X., Martini, A.: Atomistic simulation of the effect of roughness on nanoscale wear. Comput. Mater. Sci. 102, 208–212 (2015)CrossRefGoogle Scholar
  12. 12.
    Hyun, S., Pei, L., Molinari, J.F., Robbins, M.O.: Finite-element analysis of contact between elastic self-affine surfaces. Phys. Rev. E 70, 026,117 (2004)CrossRefGoogle Scholar
  13. 13.
    Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1985)CrossRefGoogle Scholar
  14. 14.
    Karpov, E.G., Wagner, G.J., Liu, W.K.: A green’s function approach to deriving non-reflecting boundary conditions in molecular dynamics simulations. Int. J. Numer. Method Eng. 62(9), 1250–1262 (2005)CrossRefGoogle Scholar
  15. 15.
    Luan, B.Q., Robbins, M.O.: The breakdown of continuum models for mechanical contacts. Nature 435, 929–932 (2005). doi:10.1038/nature03700 CrossRefGoogle Scholar
  16. 16.
    Manners, W., Greenwood, J.A.: Some observations on persson’s diffusion theory of elastic contact. Wear 261, 600–610 (2006). doi:10.1016/j.wear.2006.01.007 CrossRefGoogle Scholar
  17. 17.
    Mo, Y., Turner, K.T., Szlufarska, I.: Friction laws at the nanoscale. Nature 457, 1116–1119 (2009). doi:10.1038/nature07748 CrossRefGoogle Scholar
  18. 18.
    Mulakaluri, N., Persson, B.N.J.: Adhesion between elastic solids with randomly rough surfaces: comparison of analytical theory with molecular-dynamics simulations. EPL 96, 66,003 (2011)CrossRefGoogle Scholar
  19. 19.
    Pastewka, L., Prodanov, N., Lorenz, B., Müser, M.H., Robbins, M.O., Persson, B.N.J.: Finite-size effects in contacts between self-affine surfaces. Phys. Rev. E 87, 062,809 (2013)CrossRefGoogle Scholar
  20. 20.
    Pastewka, L., Robbins, M.O.: Contact area of rough spheres: large scale simulations and simple scaling laws. Appl. Phys. Lett. 108, 221,601 (2016). doi:10.1063/1.4950802 CrossRefGoogle Scholar
  21. 21.
    Pastewka, L., Sharp, T.A., Robbins, M.O.: Seamless elastic boundaries for atomistic calculations. Phys. Rev. B 86, 075,459 (2012)CrossRefGoogle Scholar
  22. 22.
    Persson, B.N.J.: Theory of rubber friction and contact mechanics. J. Chem. Phys. 115, 3840–3861 (2001)CrossRefGoogle Scholar
  23. 23.
    Persson, B.N.J.: Contact mechanics for randomly rough surfaces. Surf. Sci. Rep. 61, 201–227 (2006)CrossRefGoogle Scholar
  24. 24.
    Persson, B.N.J.: On the elastic energy and stress correlation in the contact between elastic solids with randomly rough surfaces. J. Phys. Condens. Matter 20, 312,001 (2008)CrossRefGoogle Scholar
  25. 25.
    Persson, B.N.J.: On the fractal dimension of rough surfaces. Tribol. Lett. 54, 99–106 (2014)CrossRefGoogle Scholar
  26. 26.
    Persson, B.N.J., Yang, C.: Theory of the leak-rate of seals. J. Phys. Condens. Matter 20, 315,011 (2008)CrossRefGoogle Scholar
  27. 27.
    Prodanov, N., Dapp, W.B., Müser, M.H.: On the contact area and mean gap of rough, elastic contacts: dimensional analysis, numerical corrections and reference data. Tribol. Lett. 53, 433–448 (2014)CrossRefGoogle Scholar
  28. 28.
    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. Sol. 60, 973–982 (2012)CrossRefGoogle Scholar
  29. 29.
    Wenning, L., Müser, M.H.: Friction laws for elastic nanoscale contacts. Europhys. Lett. 54, 693–699 (2001). doi:10.1209/epl/i2001-00371-6 CrossRefGoogle Scholar
  30. 30.
    Yastrebov, V.A., Anciaux, G., Molinari, J.F.: From infinitesimal to full contact between rough surfaces: evolution of the contact area. Int. J. Solids Struct. 52, 83 (2015)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Materials Science and EngineeringSaarland UniversitySaarbrückenGermany

Personalised recommendations