Tribology Letters

, Volume 35, Issue 3, pp 191–203 | Cite as

Improved Elastic Contact Model Accounting for Asperity and Bulk Substrate Deformation

  • Chang-Dong Yeo
  • Raja R. Katta
  • Andreas A. Polycarpou
Original Paper


An improved elastic contact model for a single asperity system is proposed accounting for both the effects of bulk substrate and asperity deformations. The asperity contact stiffness is based on the Hertzian solution for spherical contact, and the bulk substrate stiffness on the solution of Hertzian pressure on a circular region of the elastic half-space. Depending on the magnitude of the applied load, as well as the geometrical and physical properties of the asperity and bulk materials, the bulk substrate could have considerable contribution to the overall contact stiffness. The proposed single asperity model is generalized using two parameters based on physical and geometrical properties, and is also verified using finite element analysis. A parametric study for a practical range of geometric and physical parameters is performed using finite element analysis to determine the range of validity of the proposed model and also to compare it with the Hertz contact model. The single asperity model is extended to rough surfaces in contact and the contact stiffness from the proposed model and the simpler Greenwood–Williamson asperity model are compared to experimental measurements.


Contact mechanics Surface roughness Magnetic data storage Thin solid film Finite element analysis 

List of symbols


Nominal contact area


Real contact area (rough surface contact)


Separation based on asperity heights (rough surface contact)


Effective elastic modulus between asperity and bulk substrate


Asperity effective elastic modulus


Asperity elastic modulus


Bulk substrate effective elastic modulus


Bulk substrate elastic modulus


Contact force


Contact force computed based on Hertz model


Generalized function




Asperity height


Separation based on surface heights (rough surface contact)


Hardness coefficient


Composite spring constant between asperity and bulk substrate


Asperity spring constant


Bulk substrate spring constant


Contact stiffness in rough surface contact


Contact force (rough surface contact)


Maximum contact pressure


Asperity radius


Radial distance from the center of contact


Radius of circular contact region on top of bulk substrate


Normal displacement of the contact surface


Asperity height measured from the mean of asperity heights


Normal applied displacement (or surface approach)


Asperity deformation


Bulk substrate deformation


Areal density of asperities


Physical index parameter


Asperity Poisson’s ratio


Bulk substrate Poisson’s ratio


Geometrical index parameter


Standard deviation of asperity heights


Distribution function of asperity heights


Surface interference


Critical interference at the inception of plastic deformation



This research was supported by grants from the National Science Foundation under Grant number CAREER CMS-0239232 and the Information Storage Industry Consortium (INSIC) EHDR Program. The roughness measurements were performed at the Center for Microanalysis of Materials at the University of Illinois, which is supported by the U.S. Department of Energy under Grant DEFG02-96-366 ER45439.


  1. 1.
    Patton, S.T., Zabinski, J.S.: Failure mechanisms of capacitive MEMS RF switch contacts. Tribol. Lett. 19(4), 265–272 (2005)CrossRefGoogle Scholar
  2. 2.
    Suh, A.Y., Polycarpou, A.A.: Adhesion and pull-off forces for polysilicon MEMS surfaces using the sub-boundary lubrication model. ASME J. Tribol. 125, 193–199 (2003)CrossRefGoogle Scholar
  3. 3.
    Lee, S.-C., Polycarpou, A.A.: Microtribodynamics of pseudo-contacting head-disk interfaces intended for 1 Tbit/in2. IEEE Trans. Mag. 41(2), 812–818 (2005)CrossRefADSGoogle Scholar
  4. 4.
    Barber, J.R., Ciavarella, M.: Contact mechanics. Int. J. Solids Struct. 37, 29–43 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Greenwood, J.A., Williamson, B.P.: Contact of nominally flat surfaces. Proc. R. Soc. Lond. A 295, 300–319 (1966)CrossRefADSGoogle Scholar
  6. 6.
    Chang, W.R., Etsion, I., Bogy, D.B.: An elastic-plastic model for the contact of rough surfaces. ASME J. Tribol. 109, 257–263 (1987)CrossRefGoogle Scholar
  7. 7.
    Kogut, L., Etsion, I.: Elastic-plastic contact analysis of a sphere and a rigid flat. ASME J. Appl. Mech. 69, 657–662 (2002)zbMATHCrossRefGoogle Scholar
  8. 8.
    Greenwood, J.A., Wu, J.J.: Surface roughness and contact: an apology. Meccanica 36, 617–630 (2001)zbMATHCrossRefGoogle Scholar
  9. 9.
    McCool, J.I.: Comparison of models for the contact of rough surfaces. Wear 107, 37–60 (1986)CrossRefGoogle Scholar
  10. 10.
    Shi, X., Polycarpou, A.A.: Measurement and modeling of normal contact stiffness and contact damping at the meso scale. ASME J Vib. Acoust. 127, 52–60 (2005)CrossRefGoogle Scholar
  11. 11.
    Suh, A.Y., Mate, C.M., Payne, R.N., Polycarpou, A.A.: Experimental and theoretical evaluation of friction at contacting magnetic storage slider-disk interface. Tribol. Lett. 23–3, 177–190 (2006)CrossRefGoogle Scholar
  12. 12.
    Archard, J.F.: Elastic deformation and the laws of friction. Proc. R. Soc. Lond. A 243, 190–205 (1957)CrossRefADSGoogle Scholar
  13. 13.
    Majumdar, A., Bhushan, B.: Fractal model of elastic–plastic contact between rough surfaces. ASME J. Tribol. 113, 1–11 (1991)CrossRefGoogle Scholar
  14. 14.
    Sahoo, P., Roy Chowdhury, S.K.: A fractal analysis of adhesive friction between rough solids in gentle sliding. Proc. Inst. Mech. Eng. Part J: J Eng. Tribol. 214, 583–595 (2000)CrossRefGoogle Scholar
  15. 15.
    Komvopoulos, K., Yan, W.: Three-dimensional elastic–plastic fractal analysis of surface adhesion in microelectromechanical systems. ASME J. Tribol. 120, 808–813 (1998)CrossRefGoogle Scholar
  16. 16.
    Suh, A.Y., Polycarpou, A.A., Conry, T.F.: Detailed surface roughness characterization of engineering surfaces undergoing tribological testing leading to scuffing. Wear 255, 556–568 (2003)CrossRefGoogle Scholar
  17. 17.
    Mbise, G.W., Niklassont, G.A., Granqvist, C.G.: Scaling of surface roughness in evaporated calcium fluoride films. Solid State Commun. 97, 965–969 (1996)CrossRefADSGoogle Scholar
  18. 18.
    Siegert, M., Plischke, M.: Slope selection and coarsening in molecular beam epitaxy. Phys. Rev. Lett. 73, 1517–1520 (1994)PubMedCrossRefADSGoogle Scholar
  19. 19.
    Hui, C.Y., Lin, Y.Y., Baney, J.M., Kramer, E.J.: The mechanics of contact and adhesion of periodically rough surfaces. J. Polym. Sci. Part B: Polym. Phys. 39, 1195–1214 (2001)CrossRefADSGoogle Scholar
  20. 20.
    Jeng, Y.-R., Aoh, J.-H., Wang, C.-M.: Thermosonic wire bonding of gold wire onto copper pad using the saturated interfacial phenomena. J. Phys. D Appl. Phys. 34, 3515–3521 (2001)CrossRefADSGoogle Scholar
  21. 21.
    Suh, A.Y., Lee, S.-C., Polycarpou, A.A.: Adhesion and friction evaluation of textured slider surfaces in ultra-low flying head-disk interfaces. Tribol. Lett. 17(4), 739–749 (2004)CrossRefGoogle Scholar
  22. 22.
    Iida, K., Ono, K.: Design consideration of contact/near-contact sliders based on a rough surface contact model. ASME J. Tribol. 125, 562–570 (2003)CrossRefGoogle Scholar
  23. 23.
    Shi, X., Polycarpou, A.A.: Investigation of contact stiffness and contact damping for magnetic storage head-disk interfaces. ASME J. Tribol. 130, 021901-1-9 (2008)Google Scholar
  24. 24.
    Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1985)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Chang-Dong Yeo
    • 1
    • 2
  • Raja R. Katta
    • 1
  • Andreas A. Polycarpou
    • 1
  1. 1.Department of Mechanical Science and EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Seagate Technology LLCShakopeeUSA

Personalised recommendations