Tribology Letters

, 66:146 | Cite as

Theoretical and Finite Element Analysis of Static Friction Between Multi-Scale Rough Surfaces

  • Xianzhang WangEmail author
  • Yang Xu
  • Robert L. Jackson
Original Paper


The current work considers the multi-scale nature of roughness in a new model that predicts the static friction coefficient. This work is based upon a previous rough surface contact model, which used stacked elastic–plastic 3-D sinusoids to model the asperities at multiple scales of roughness. A deterministic model of a three-dimensional deformable rough surface pressed against a rigid flat surface is also carried out using the finite element method (FEM). The accuracy of the deterministic FEM model is also considered. At the beginning of contact, which is surface-point contact, the asperities or peaks are isolated, sharp, and the contact areas consist of an inadequate number of elements and sources of error. In this range of contact, the results are not presented as real or accurate. As the normal load increases, the number of the contact elements become larger, and thus, the results become more accurate. That is, the deterministic FEM results are most accurate at high loads. Spectral interpolation is used to smooth the geometry in between the original measured nodes. The effects of normal load and plasticity index on static friction are then analyzed. The results predicted by the theoretical model are also compared to other existing rough surface friction contact models and the FEM results. They are in a good qualitative agreement, especially for higher loads and higher plasticity indices. The FEM model also has significant error, but it is more accurate at higher loads where the proposed multi-scale static friction model and FEM model are in better agreement.


Contact mechanics Static friction Surface roughness analysis and models 




Area of contact


Contact area under normal preload only


Nominal contact area of the surface


Real contact area


Contact area at sliding inception


Critical yield stress coefficient


Elastic modulus


\({E \mathord{\left/ {\vphantom {E {\left( {1 - {v^2}} \right)}}} \right. \kern-0pt} {\left( {1 - {v^2}} \right)}}\)


Spatial frequency (reciprocal of wavelength)


Contact force for single asperity


Friction force


Normal preload


Dimensionless normal preload


Tangential load


Scan length


Number of nodes


Number of elements


Average pressure to cause complete contact (Elastic)


Average pressure to cause complete contact (Elastic–plastic)

\(\bar {p}\)

Average pressure over the entire surface


Yield strength


Displacement in the x direction

Greek symbols


Asperity wavelength


Asperity amplitude

\({\Delta _c}\)

Critical asperity amplitude


Sinusoidal asperity parameter


Plasticity index


Standard derivation on the surface heights

\({\sigma _s}\)

Standard derivation on the asperity heights


Poisson’s ratio

\({\tau _c}\)

Critical interfacial shear strength

\({\omega _0}\)

Interference under normal preload

\({\omega _c}\)

Critical interference (full stick condition)

\({\omega _{cs}}\)

Critical interference (perfect slip condition)

\({\mu _s}\)

Static friction coefficient

\({\mu _s}\)

Static friction coefficient



Critical value at the onset of plastic deformation (full stick condition)


Average value


Maximum value




From model by Johnson, Greenwood, and Higgson [1]


In the x direction


  1. 1.
    Kogut, L., Etsion, I.: A static friction model for elastic-plastic contacting rough surfaces. J. Tribol. 126(1), 34–40 (2004). CrossRefGoogle Scholar
  2. 2.
    Bowden, F.P., Bowden, T.D.: Friction and lubrication of solids. Clarendon, Oxford, (1954)Google Scholar
  3. 3.
    Kogut, L., Etsion, I.: A semi-analytical solution for the sliding inception of a spherical contact. J. Tribol. 125(3), 499–506 (2003). CrossRefGoogle Scholar
  4. 4.
    Brizmer, V., Kligerman, Y., Etsion, I.: Elastic–plastic spherical contact under combined normal and tangential loading in full stick. Tribol. Lett. 25(1), 61–70 (2007)CrossRefGoogle Scholar
  5. 5.
    Eriten, M., Polycarpou, A.A., Bergman, L.A.: Physics-based modeling for partial slip behavior of spherical contacts. Int. J. Solids Struct. 47(18–19), 2554–2567 (2010). CrossRefGoogle Scholar
  6. 6.
    Wu, A.Z., Shi, X., Polycarpou, A.A.: An Elastic–plastic spherical contact model under combined normal and tangential loading. J. Appl. Mech. 79(5) (2012). CrossRefGoogle Scholar
  7. 7.
    Wang, X., Xu, Y., Jackson, R.L.: Elastic–Plastic sinusoidal waviness contact under combined normal and tangential loading. Tribol Lett 65(2), 45 (2017)CrossRefGoogle Scholar
  8. 8.
    Chang, W.R., Etsion, I., Bogy, D.B.: Static friction coefficient model for metallic rough surfaces. J. Tribol. 110(1), 57–63 (1988)CrossRefGoogle Scholar
  9. 9.
    Kogut, L., Etsion, I.: A finite element based elastic-plastic model for the contact of rough surfaces. Tribol. T. 46(3), 383–390 (2003). doi: CrossRefGoogle Scholar
  10. 10.
    Cohen, D., Kligerman, Y., Etsion, I.: A model for contact and static friction of nominally flat rough surfaces under full stick contact condition. J. Tribol. 130(3) (2008). CrossRefGoogle Scholar
  11. 11.
    Li, L., Etsion, I., Talke, F.E.: Contact area and static friction of rough surfaces with high plasticity index. J. Tribol. 132(3) (2010). CrossRefGoogle Scholar
  12. 12.
    Greenwood, J., Williamson, J.P.: Contact of nominally flat surfaces. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 1966, pp. 300–319. The Royal SocietyGoogle Scholar
  13. 13.
    Etsion, I., Amit, M.: The effect of small normal loads on the static friction coefficient for very smooth surfaces. J. Tribol. 115(3), 406–410 (1993). doi: CrossRefGoogle Scholar
  14. 14.
    Cohen, D., Kligerman, Y., Etsion, I.: The effect of surface roughness on static friction and junction growth of an elastic-plastic spherical contact. J. Tribol. 131(2) (2009). CrossRefGoogle Scholar
  15. 15.
    Mccool, J.I.: Relating profile instrument measurements to the functional performance of rough surfaces. J. Tribol. 109(2), 264–270 (1987)CrossRefGoogle Scholar
  16. 16.
    Jackson, R.L., Green, I.: A finite element study of elasto-plastic hemispherical contact against a rigid flat. J. Tribol. 127(2), 343–354 (2005). CrossRefGoogle Scholar
  17. 17.
    Kogut, L., Jackson, R.L.: A comparison of contact modeling utilizing statistical and fractal approaches. J. Tribol. 128(1), 213–217 (2006). doi: CrossRefGoogle Scholar
  18. 18.
    Jackson, R.L., Green, I.: On the modeling of elastic contact between rough surfaces. Trib. Trans. 54(2), 300–314 (2011)CrossRefGoogle Scholar
  19. 19.
    Tayebi, N., Polycarpou, A.A.: Modeling the effect of skewness and kurtosis on the static friction coefficient of rough surfaces. Tribol. Int. 37(6), 491–505 (2004). CrossRefGoogle Scholar
  20. 20.
    Lee, C.H., Polycarpou, A.A.: Static friction experiments and verification of an improved elastic-plastic model including roughness effects. J. Tribol. 129(4), 754–760 (2007). CrossRefGoogle Scholar
  21. 21.
    Lee, C.H., Eriten, M., Polycarpou, A.A.: Application of elastic-plastic static friction models to rough surfaces with asymmetric asperity distribution. J. Tribol. 132(3), 031602 (2010). CrossRefGoogle Scholar
  22. 22.
    Dickey, R.D.I., Jackson, R.L., Flowers, G.T.: Measurements of the static friction coefficient between tin surfaces and comparison to a theoretical model. J. Tribol. 133(3), 031408 (2011). CrossRefGoogle Scholar
  23. 23.
    Archard, J.: Elastic deformation and the laws of friction. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 1957, pp. 190–205. The Royal Society (1957)Google Scholar
  24. 24.
    Ciavarella, M., Demelio, G., Barber, J.R., Jang, Y.H.: Linear elastic contact of the Weierstrass profile. Proc. Royal Soc. A 456, 387–405 (2000). CrossRefGoogle Scholar
  25. 25.
    Westergaard, H.M.: Bearing prssures and cracks. Journal of Applied Mechanics-Transactions of the Asme 6, 49–53 (1939)Google Scholar
  26. 26.
    Jackson, R.L., Streator, J.L.: A multi-scale model for contact between rough surfaces. Wear. 261(11–12), 1337–1347 (2006). CrossRefGoogle Scholar
  27. 27.
    Gao, Y.F., Bower, A.F.: Elastic-plastic contact of a rough surface with Weierstrass profile. Proc. Royal Soc. A 462(2065), 319–348 (2006). CrossRefGoogle Scholar
  28. 28.
    Wilson, W.E., Angadi, S.V., Jackson, R.L.: Surface separation and contact resistance considering sinusoidal elastic–plastic multi-scale rough surface contact. Wear 268(1), 190–201 (2010)CrossRefGoogle Scholar
  29. 29.
    Pastewka, L., Robbins, M.O.: Contact between rough surfaces and a criterion for macroscopic adhesion. Proc. Natl. Acad. Sci. USA. 111(9), 3298–3303 (2014). CrossRefGoogle Scholar
  30. 30.
    Polonsky, I.A., Keer, L.M.: Fast methods for solving rough contact problems: a comparative study. J. Tribol. 122(1), 36–41 (2000). doi: CrossRefGoogle Scholar
  31. 31.
    Yang, C., Persson, B.N.J., Israelachvili, J., Rosenberg, K.: Contact mechanics with adhesion: interfacial separation and contact area. Europhys. Lett. 84(4) (2008). CrossRefGoogle Scholar
  32. 32.
    Persson, B.N.J., Scaraggi, M.: Theory of adhesion: role of surface roughness. J. Chem. Phys. 141(12) (2014). CrossRefGoogle Scholar
  33. 33.
    Wu, J.J.: Numerical analyses on elliptical adhesive contact. J. Phys. D 39(9), 1899–1907 (2006). CrossRefGoogle Scholar
  34. 34.
    Ilincic, S., Vernes, A., Vorlaufer, G., Hunger, H., Dorr, N., Franek, F.: Numerical estimation of wear in reciprocating tribological experiments. Proc. Inst. Mech. Eng. J 227(J5), 510–519 (2013). CrossRefGoogle Scholar
  35. 35.
    Solhjoo, S., Vakis, A.I.: Continuum mechanics at the atomic scale: Insights into non-adhesive contacts using molecular dynamics simulations. J. Appl. Phys. 120(21) (2016). CrossRefGoogle Scholar
  36. 36.
    Rostami, A., Jackson, R.L.: Predictions of the average surface separation and stiffness between contacting elastic and elastic-plastic sinusoidal surfaces. Proc. Inst. Mech. Eng. J 227(12), 1376–1385 (2013). CrossRefGoogle Scholar
  37. 37.
    Rostami, A., Streator, J.L.: Study of liquid-mediated adhesion between 3D rough surfaces: a spectral approach. Tribol. Int. 84, 36–47 (2015)CrossRefGoogle Scholar
  38. 38.
    Medina, S., Dini, D.: A numerical model for the deterministic analysis of adhesive rough contacts down to the nano-scale. Int. J. Solids Struct. 51(14), 2620–2632 (2014). CrossRefGoogle Scholar
  39. 39.
    Putignano, C., Afferrante, L., Carbone, G., Demelio, G.: A new efficient numerical method for contact mechanics of rough surfaces. Int. J. Solids Struct. 49(2), 338–343 (2012). CrossRefGoogle Scholar
  40. 40.
    Afferrante, L., Carbone, G., Demelio, G.: Interacting and coalescing Hertzian asperities: a new multiasperity contact model. Wear. 278, 28–33 (2012). CrossRefGoogle Scholar
  41. 41.
    Muser, M.H., Dapp, W.B., Bugnicourt, R., Sainsot, P., Lesaffre, N., Lubrecht, T.A., Persson, B.N.J., Harris, K., Bennett, A., Schulze, K., Rohde, S., Ifju, P., Sawyer, W.G., Angelini, T., Esfahani, H.A., Kadkhodaei, M., Akbarzadeh, S., Wu, J.J., Vorlaufer, G., Vernes, A., Solhjoo, S., Vakis, A.I., Jackson, R.L., Xu, Y., Streator, J., Rostami, A., Dini, D., Medina, S., Carbone, G., Bottiglione, F., Afferrante, L., Monti, J., Pastewka, L., Robbins, M.O., Greenwood, J.A.: Meeting the contact-mechanics challenge. Tribol. Lett. 65(4) (2017).
  42. 42.
    Ning, Y., Polycarpou, A.A.: Extracting summit roughness parameters from random Gaussian surfaces accounting for asymmetry of the summit heights. J. Tribol. 126(4), 761–766 (2004). CrossRefGoogle Scholar
  43. 43.
    Yu, N., Polycarpou, A.A.: Combining and contacting of two rough surfaces with asymmetric distribution of asperity heights. J. Tribol. 126(2), 225–232 (2004). CrossRefGoogle Scholar
  44. 44.
    Xu, Y.: An analysis of elastic rough contact models. (2012)Google Scholar
  45. 45.
    Jackson, R.L., Bhavnani, S.H., Ferguson, T.P.: A multi-scale model of thermal contact resistance between rough surfaces. ASME J. Heat Transfer 130, 081301 (2008)CrossRefGoogle Scholar
  46. 46.
    Jackson, R.L., Ghaednia, H., Elkady, Y.A., Bhavnani, S.H., Knight, R.W.: A closed-form multiscale thermal contact resistance model. IEEE Trans. Compon. Packag. Manuf. Technol. 2(7), 1158–1171 (2012)CrossRefGoogle Scholar
  47. 47.
    Almeida, L., Ramadoss, R., Jackson, R., Ishikawa, K., Yu, Q.: Laterally actuated multicontact MEMS relay fabricated using MetalMUMPS process: experimental characterization and multiscale contact modeling. J. Micro/Nanolith. MEMS MOEMS 6(2), 023009 (2007)CrossRefGoogle Scholar
  48. 48.
    Johnson, K.L., Greenwood, J.A., Higginson, J.G.: The contact of elastic regular wavy surfaces. Int. J. Mech. Sci. 27(6), 383 (1985). doi: CrossRefGoogle Scholar
  49. 49.
    Jackson, R.L., Krithivasan, V., Wilson, W.E.: The pressure to cause complete contact between elastic-plastic sinusoidal surfaces. Proc. Inst. Mech. Eng. J 222(J7), 857–863 (2008). CrossRefGoogle Scholar
  50. 50.
    Krithivasan, V., Jackson, R.L.: An analysis of three-dimensional elasto-plastic sinusoidal contact. Tribol. Lett. 27(1), 31–43 (2007). CrossRefGoogle Scholar
  51. 51.
    Ghaednia, H., Wang, X., Saha, S., Jackson, R.L., Xu, Y., Sharma, A.: A Review of elastic-plastic contact mechanics. Appl. Mech. Rev. 69(6), 060804 (2017)CrossRefGoogle Scholar
  52. 52.
    Jackson, R.L.: An analytical solution to an archard-type fractal rough surface contact model. Tribol. T. 53(4), 543–553 (2010). CrossRefGoogle Scholar
  53. 53.
    Sahli, R., Pallares, G., Ducottet, C., Ali, B., Al Akhrass, I.E., Guibert, S., Scheibert, M.: J.: Evolution of real contact area under shear and the value of static friction of soft materials. Proc. Natl. Acad. Sci. USA. 115(3), 471–476 (2018). CrossRefGoogle Scholar
  54. 54.
    Ovcharenko, A., Halperin, G., Etsion, I., Varenberg, M.: A novel test rig for in situ and real time optical measurement of the contact area evolution during pre-sliding of a spherical contact. Tribol. Lett. 23(1), 55–63 (2006). CrossRefGoogle Scholar
  55. 55.
    Brizmer, V., Kligerman, Y., Etsion, I.: A model for junction growth of a spherical contact under full stick condition. J. Tribol. 129(4), 783–790 (2007). CrossRefGoogle Scholar
  56. 56.
    Ovcharenko, A., Halperin, G., Etsion, I.: In situ and real-time optical investigation of junction growth in spherical elastic-plastic contact. Wear. 264(11–12), 1043–1050 (2008). CrossRefGoogle Scholar
  57. 57.
    Zolotarevskiy, V., Kligerman, Y., Etsion, I.: The evolution of static friction for elastic–plastic spherical contact in pre-sliding. J. Tribol. 133(3) (2011). CrossRefGoogle Scholar
  58. 58.
    Jackson, R.L., Green, I.: A statistical model of elasto-plastic asperity contact between rough surfaces. Tribol. Int. 39(9), 906–914 (2006). CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentAuburn UniversityAuburnUSA

Personalised recommendations