Tribology Letters

, Volume 50, Issue 3, pp 331–347 | Cite as

A Numerical Contact Model Based on Real Surface Topography

  • Can K. Bora
  • Michael E. Plesha
  • Robert W. Carpick
Original Paper


A numerical finite element contact model is developed to make use of the high precision surface topography data obtained at the nanoscale by atomic force microscopy or other imaging techniques while minimizing computational complexity. The model uses degrees of freedom that are normal to the surface, and uses the Boussinesq solution to relate the normal load to the long-range surface displacement response. The model for contact between two rough surfaces is developed in a step-by-step manner, taking into account the far-field effects of the loads developed at asperities that have come to contact in previous steps. Method accuracy is verified by comparison to simple test cases with well-defined analytical solutions. Agreement was found to be within 1 % for a wide range of practical loads for the high precision models. Applicability of extrapolation from lower precision models is presented. The real contact area estimates for micrometer-size tribology test machine surfaces are calculated and convergence behavior with mesh refinement is investigated.


Contact mechanics Finite elements Boussinesq solution 



We acknowledge Graham Wabiszewksi (University of Pennsylvania) for the MEMS surface images, the Microelectronics Development Laboratory at Sandia National Laboratories for the samples, Matthew A. Hamilton (Exactech, Inc), and W. Gregory Sawyer (University of Florida) for useful discussions. This work was partially supported by the National Science Foundation, grant CMMI 1200019, and by the US Department of Energy, BES-Materials Sciences, under Contract DE-FG02-02ER46016.


  1. 1.
    Greenwood, J.A., Williamson, J.B.P.: Contact of nominally flat surfaces. Proc. Roy. Soc. Lond. A 295, 300–319 (1966)CrossRefGoogle Scholar
  2. 2.
    McCool, J.I.: Comparison of models for the contact of rough surfaces. Wear 107, 37–60 (1986)CrossRefGoogle Scholar
  3. 3.
    Yan, W., Komvopoulos, K.: Contact analysis of elastic–plastic fractal surfaces. J. Appl. Phys. 84, 3617–3624 (1998)CrossRefGoogle Scholar
  4. 4.
    Majumdar, A., Bhushan, B.: Fractal model of elastic–plastic contact between rough surfaces. ASME J. Tribol. 113, 1–11 (1991)CrossRefGoogle Scholar
  5. 5.
    Hyun, S., Pei, L., Molinari, J.F., Robbins, M.O.: Finite-element analysis of contact between elastic self-affine surfaces. Phys. Rev. E 70, 26117 (2004)CrossRefGoogle Scholar
  6. 6.
    Persson, B.N.J., Bucher, F., Chiaia, B.: Elastic contact between randomly rough surfaces: comparison of theory with numerical results. Phys. Rev. B 65, 184106 (2002)CrossRefGoogle Scholar
  7. 7.
    Polonsky, I.A., Keer, L.M.: A numerical method for solving rough contact problems based on the multi-level multi-summation and conjugate gradient techniques. Wear 231, 206–219 (1999)CrossRefGoogle Scholar
  8. 8.
    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, 1–62 (2005)CrossRefGoogle Scholar
  9. 9.
    Boussinesq, J.: Application des potentiels à l’étude de l’équilibre et du mouvement des solides élastiques (Application of potentials to the study of equilibrium and motion of elastic solids.). Gauthier Villars, Paris (1885)Google Scholar
  10. 10.
    Love, A.E.H.: The stress produced in a semi-infinite solid by pressure on part of the boundary. Proc. Roy. Soc. Lond. A 228, 377–420 (1929)Google Scholar
  11. 11.
    Webster, M.N., Sayles, R.S.: A numerical model for the elastic frictionless contact of real rough surfaces. ASME J. Tribol. 108, 314–320 (1986)CrossRefGoogle Scholar
  12. 12.
    Poon, C.Y., Sayles, R.S.: Numerical contact model of a smooth ball on an anisotropic rough surface. ASME J. Tribol. 116, 194–201 (1994)CrossRefGoogle Scholar
  13. 13.
    Ren, N., Lee, S.C.: Contact simulation of three-dimensional rough surfaces using moving grid method. ASME J. Tribol. 115, 597–601 (1993)CrossRefGoogle Scholar
  14. 14.
    Liu, G., Wang, Q., Liu, S.: A three-dimensional thermal-mechanical asperity contact model for two nominally flat surfaces in contact. ASME J. Tribol. 123, 595–602 (2001)CrossRefGoogle Scholar
  15. 15.
    Dickrell, D.J., Dugger, M.T., Hamilton, M.A., Sawyer, W.G.: Direct contact-area computation for MEMS using real topographic surface data. J. Microelectromech. Syst. 16, 1263–1268 (2007)CrossRefGoogle Scholar
  16. 16.
    Delrio, F.W., De Boer, M.P., Knapp, J.A., Reedy, E.D., Clews, P.J., Dunn, M.L.: The role of van der Waals forces in adhesion of micromachined surfaces. Nat. Mater. 4, 629–634 (2005)CrossRefGoogle Scholar
  17. 17.
    Cook, R.D., Malkus, D.S., Plesha, M.E., Witt, R.J.: Concepts and Applications of Finite Elements Analysis, 4th edn. Wiley, New York (2001)Google Scholar
  18. 18.
    Plesha, M.E., Cook, R.D., Malkus, D.S.: FEMCOD—Program Description and User Guide. University of Wisconsin-Madison, Madison (1988)Google Scholar
  19. 19.
    Young, W.C.: Roark’s Formulas for Stress & Strain, 6th edn. McGraw-Hill, New York (1989)Google Scholar
  20. 20.
    Borodachev, N.M.: Impression of a punch with a flat square base into an elastic half-space. Int. Appl. Mech. 35, 989–994 (1999)CrossRefGoogle Scholar
  21. 21.
    De Boer, M.P., Luck, D.L., Ashurst, W.R., Maboudian, R., Corwin, A.D., Walraven, J.A., Redmond, J.M.: High-performance surface-micromachined inchworm actuator. J. Microelectromech. Syst. 13, 63–74 (2004)CrossRefGoogle Scholar
  22. 22.
    Zhuravlev, V.A.: On question of theoretical justification of the Amontons–Coulomb law for friction of unlubricated surfaces. Zh. Tekh. Fiz. 10, 1447–1452 (1940)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Can K. Bora
    • 1
  • Michael E. Plesha
    • 1
  • Robert W. Carpick
    • 2
  1. 1.Department of Engineering PhysicsUniversity of Wisconsin-MadisonMadisonUSA
  2. 2.Department of Mechanical Engineering and Applied MechanicsUniversity of PennsylvaniaPhiladelphiaUSA

Personalised recommendations