Meeting the Contact-Mechanics Challenge


This paper summarizes the submissions to a recently announced contact-mechanics modeling challenge. The task was to solve a typical, albeit mathematically fully defined problem on the adhesion between nominally flat surfaces. The surface topography of the rough, rigid substrate, the elastic properties of the indenter, as well as the short-range adhesion between indenter and substrate, were specified so that diverse quantities of interest, e.g., the distribution of interfacial stresses at a given load or the mean gap as a function of load, could be computed and compared to a reference solution. Many different solution strategies were pursued, ranging from traditional asperity-based models via Persson theory and brute-force computational approaches, to real-laboratory experiments and all-atom molecular dynamics simulations of a model, in which the original assignment was scaled down to the atomistic scale. While each submission contained satisfying answers for at least a subset of the posed questions, efficiency, versatility, and accuracy differed between methods, the more precise methods being, in general, computationally more complex. The aim of this paper is to provide both theorists and experimentalists with benchmarks to decide which method is the most appropriate for a particular application and to gauge the errors associated with each one.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12


  1. 1.

    Greenwood, J.A., Williamson, J.B.P.: Contact of nominally flat surfaces. In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 295(1442), pp. 300–319 (1966)

  2. 2.

    Archard, J.F.: Elastic deformation and the laws of friction. In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 243(1233), pp. 190–205 (1957)

  3. 3.

    Whitehouse, D.J., Archard, J.F.: The properties of random surfaces of significance in their contact. In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 316(1524), pp. 97–121 (1970)

  4. 4.

    Bush, A.W., Gibson, R.D., Thomas, T.R.: The elastic contact of a rough surface. Wear 35(1), 87–111 (1975)

    Article  Google Scholar 

  5. 5.

    Persson, B.N.J.: Theory of rubber friction and contact mechanics. J. Chem. Phys. 115(8), 3840 (2001)

    Article  Google Scholar 

  6. 6.

    Hyun, S., Pei, L., Molinari, J.-F., Robbins, M.O.: Finite-element analysis of contact between elastic self-affine surfaces. Phys. Rev. E 70(2), 026117 (2004)

    Article  Google Scholar 

  7. 7.

    Prodanov, Nikolay, Dapp, Wolf B., Müser, Martin H.: On the contact area and mean gap of rough, elastic contacts: dimensional analysis, numerical corrections, and reference data. Tribol. Lett. 53(2), 433–448 (2013)

    Article  Google 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. (EPL) 77(3), 38005 (2007)

    Article  Google Scholar 

  9. 9.

    Carbone, G., Bottiglione, F.: Asperity contact theories: do they predict linearity between contact area and load? J. Mech. Phys. Solids 56(8), 2555–2572 (2008)

    Article  Google Scholar 

  10. 10.

    Ciavarella, M., Demelio, G., Barber, J.R., Jang, Y.H.: Linear elastic contact of the Weierstrass profile. In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 456(1994), pp. 387–405 (2000)

  11. 11.

    Paggi, M., Ciavarella, M.: The coefficient of proportionality \(\kappa\) between real contact area and load, with new asperity models. Wear 268(7–8), 1020–1029 (2010)

    Article  Google Scholar 

  12. 12.

    Greenwood, J.A., Wu, J.J.: Surface roughness: an apology. Meccanica 36(6), 617–630 (2001)

    Article  Google Scholar 

  13. 13.

    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(11), 2355–2369 (2011)

    Article  Google Scholar 

  14. 14.

    Pastewka, L., Prodanov, N., Lorenz, B., Müser, M.H., Robbins, Mark O., Persson, Bo N.J.: Finite-size scaling in the interfacial stiffness of rough elastic contacts. Phys. Rev. E 87(6), 062809 (2013)

    Article  Google Scholar 

  15. 15.

    Pohrt, R., Popov, V.L., Filippov, A.E.: Normal contact stiffness of elastic solids with fractal rough surfaces for one- and three-dimensional systems. Phys. Rev. E 86(2), 026710 (2012)

    Article  Google Scholar 

  16. 16.

    Campañá, C., Müser, M.H., Robbins, M.O.: Elastic contact between self-affine surfaces: comparison of numerical stress and contact correlation functions with analytic predictions. J. Phys. Condens. Matter 20(35), 354013 (2008)

    Article  Google Scholar 

  17. 17.

    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(31), 312001 (2008)

    Article  Google Scholar 

  18. 18.

    Johnson, K.L., Kendall, K., Roberts, A.D.: Surface energy and the contact of elastic solids. In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 324(1558), pp. 301–313 (1971)

  19. 19.

    Müser, M.H.: Single-asperity contact mechanics with positive and negative work of adhesion: influence of finite-range interactions and a continuum description for the squeeze-out of wetting fluids. Beilstein J. Nanotechnol. 5, 419–437 (2014)

    Article  Google Scholar 

  20. 20.

    Müser, M.H., Dapp, W.B.: The contact mechanics challenge: problem definition. ArXiv e-prints, December 2015. arXiv:1512.02403

  21. 21.

    Boyce, B.L., Kramer, S.L.B., Fang, H.E., Cordova, T.E., Neilsen, M.K., Dion, K., Kaczmarowski, A.K., Karasz, E., Xue, L., Gross, A.J., Ghahremaninezhad, A., Ravi-Chandar, K., Lin, S.-P., Chi, S.-W., Chen, J.S., Yreux, E., Rüter, M., Qian, D., Zhou, Z., Bhamare, S., O’Connor, D.T., Tang, S., Elkhodary, K.I., Zhao, J., Hochhalter, J.D., Cerrone, A.R., Ingraffea, A.R., Wawrzynek, P.A., Carter, B.J., Emery, J.M., Veilleux, M.G., Yang, P., Gan, Y., Zhang, X., Chen, Z., Madenci, E., Kilic, B., Zhang, T., Fang, E., Liu, P., Lua, J., Nahshon, K., Miraglia, M., Cruce, J., DeFrese, R., Moyer, E.T., Brinckmann, S., Quinkert, L., Pack, K., Luo, M., Wierzbicki, T.: The Sandia fracture challenge: blind round robin predictions of ductile tearing. Int. J. Fract. 186(1–2), 5–68 (2014)

    Article  Google Scholar 

  22. 22.

    Tysoe, W.T., Spencer, N.D.: Contact-mechanics challenge. Tribol. Lubr. Technol. 71, 96 (2015)

    Google Scholar 

  23. 23.

    Hurst, H.E.: Long-term storage capacity of reservoirs. T. Am. Soc. Civ. Eng. 116, 770–799 (1951)

    Google Scholar 

  24. 24.

    Majumdar, A., Tien, C.L.: Fractal characterization and simulation of rough surfaces. Wear 136(2), 313–327 (1990)

    Article  Google Scholar 

  25. 25.

    Persson, B.N.J.: On the fractal dimension of rough surfaces. Tribol. Lett. 54(1), 99–106 (2014)

    Article  Google Scholar 

  26. 26.

    Maugis, D.: Adhesion of spheres: the JKR–DMT transition using a Dugdale model. J. Colloid Interface Sci. 150(1), 243–269 (1992)

    Article  Google Scholar 

  27. 27.

    Pastewka, L., Robbins, M.O.: Contact between rough surfaces and a criterion for macroscopic adhesion. Proc. Natl. Acad. Sci. 111(9), 3298–3303 (2014)

    Article  Google Scholar 

  28. 28.

    Müser, M.H.: A dimensionless measure for adhesion and effects of the range of adhesion in contacts of nominally flat surfaces. Tribol. Int. 100, 41–47 (2016)

    Article  Google Scholar 

  29. 29.

    Campañá, C., Müser, M.H.: Practical Green’s function approach to the simulation of elastic semi-infinite solids. Phys. Rev. B 74(7), 075420 (2006)

    Article  Google Scholar 

  30. 30.

    Polonsky, I.A., Keer, L.M.: Fast methods for solving rough contact problems: a comparative study. J. Tribol. 122(1), 36 (2000)

    Article  Google Scholar 

  31. 31.

    Stanley, H.M., Kato, T.: An FFT-based method for rough surface contact. J. Tribol. 119(3), 481 (1997)

    Article  Google Scholar 

  32. 32.

    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), R1–R62 (2004)

    Article  Google Scholar 

  33. 33.

    Yang, C., Persson, B.N.J., Israelachvili, J., Rosenberg, K.: Contact mechanics with adhesion: interfacial separation and contact area. EPL (Europhys. Lett.) 84(4), 46004 (2008)

    Article  Google Scholar 

  34. 34.

    Persson, B.N.J., Scaraggi, M.: Theory of adhesion: role of surface roughness. J. Chem. Phys. 141(12), 124701 (2014)

    Article  Google Scholar 

  35. 35.

    Wang, A., Müser, M.H.: Gauging persson theory on adhesion. Tribol. Lett. 65, 103 (2017). doi:10.1007/s11249-017-0886-9

    Article  Google Scholar 

  36. 36.

    Bennett, A.I., Harris, K.L., Sawyer, W.G., Müser, M.H., Angelini, T.: Illuminating pressing problems in soft contacts. Tribol. Lett. (submitted)

  37. 37.

    Jiunn-Jong, Wu: Numerical analyses on elliptical adhesive contact. J. Phys. D Appl. Phys. 39(9), 1899–1907 (2006)

    Article  Google Scholar 

  38. 38.

    Ilincic, S., Vorlaufer, G., Fotiu, P.A., Vernes, A., Franek, F.: Combined finite element-boundary element method modelling of elastic multi-asperity contacts. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol. 223(5), 767–776 (2009)

    Article  Google Scholar 

  39. 39.

    Ilincic, S., Tungkunagorn, N., Vernes, A., Vorlaufer, G., Fotiu, P.A., Franek, F.: Finite and boundary element method contact mechanics on rough, artificial hip joints. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol. 225(11), 1081–1091 (2011)

    Article  Google Scholar 

  40. 40.

    Ilincic, S., Vernes, A., Vorlaufer, G., Hunger, H., Dorr, N., Franek, F.: Numerical estimation of wear in reciprocating tribological experiments. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol. 227(5), 510–519 (2013)

    Article  Google Scholar 

  41. 41.

    Daw, M.S., Baskes, M.I.: Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals. Phys. Rev. B 29(12), 6443–6453 (1984)

    Article  Google Scholar 

  42. 42.

    Sheng, H.W., Kramer, M.J., Cadien, A., Fujita, T., Chen, M.W.: Highly optimized embedded-atom-method potentials for fourteen fcc metals. Phys. Rev. B 83(13), 134118 (2011)

    Article  Google Scholar 

  43. 43.

    Jones, J.E.: On the determination of molecular fields. II. From the equation of state of a gas. in: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 106(738), pp. 463–477 (1924)

  44. 44.

    Zhen, S., Davies, G.J.: Calculation of the Lennard-Jones n–m potential energy parameters for metals. Phys. Status Solidi (a) 78(2), 595–605 (1983)

    Article  Google Scholar 

  45. 45.

    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), 215102 (2016)

    Article  Google Scholar 

  46. 46.

    Plimpton, S.: Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117(1), 1–19 (1995)

    Article  Google Scholar 

  47. 47.

    Michael Brown, W., Wang, P., Plimpton, S.J., Tharrington, A.N.: Implementing molecular dynamics on hybrid high performance computers—short range forces. Comput. Phys. Commun. 182(4), 898–911 (2011)

    Article  Google Scholar 

  48. 48.

    Michael Brown, W., Kohlmeyer, A., Plimpton, S.J., Tharrington, A.N.: Implementing molecular dynamics on hybrid high performance computers—particle-particle particle-mesh. Comput. Phys. Commun. 183(3), 449–459 (2012)

    Article  Google Scholar 

  49. 49.

    Stukowski, A.: Visualization and analysis of atomistic simulation data with OVITO-the open visualization tool. Model. Simul. Mater. Sci. Eng. 18(1), 015012 (2009)

    Article  Google Scholar 

  50. 50.

    Rasband, W.S.: ImageJ.

  51. 51.

    Jackson, R.L., Streator, J.L.: A multi-scale model for contact between rough surfaces. Wear 261(11–12), 1337–1347 (2006)

    Article  Google Scholar 

  52. 52.

    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. Part J J. Eng. Tribol. 227(12), 1376–1385 (2013)

    Article  Google Scholar 

  53. 53.

    Rostami, A., Streator, J.L.: Study of liquid-mediated adhesion between 3d rough surfaces: a spectral approach. Tribol. Int. 84, 36–47 (2015)

    Article  Google Scholar 

  54. 54.

    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)

    Article  Google Scholar 

  55. 55.

    Carbone, G.: A slightly corrected Greenwood and Williamson model predicts asymptotic linearity between contact area and load. J. Mech. Phys. Solids 57(7), 1093–1102 (2009)

    Article  Google Scholar 

  56. 56.

    Carbone, G., Scaraggi, M., Tartaglino, U.: Adhesive contact of rough surfaces: comparison between numerical calculations and analytical theories. Eur. Phys. J. E 30(1), 65–74 (2009)

    Article  Google Scholar 

  57. 57.

    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(5), 973–982 (2012)

    Article  Google Scholar 

  58. 58.

    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)

    Article  Google Scholar 

  59. 59.

    Afferrante, L., Carbone, G., Demelio, G.: Interacting and coalescing Hertzian asperities: a new multiasperity contact model. Wear 278–279, 28–33 (2012)

    Article  Google Scholar 

  60. 60.

    Greenwood, J.A.: Constriction resistance and the real area of contact. Br. J. Appl. Phys. 17(12), 1621–1632 (1966)

    Article  Google Scholar 

  61. 61.

    Barber, J.R.: Bounds on the electrical resistance between contacting elastic rough bodies. In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 459(2029), pp. 53–66 (2003)

  62. 62.

    Dapp, W.B., Lücke, A., Persson, B.N.J., Müser, M.H.: Self-affine elastic contacts: percolation and leakage. Phys. Rev. Lett. 108(24), 244301 (2012)

    Article  Google Scholar 

  63. 63.

    Campañá, C.: Using Green’s function molecular dynamics to rationalize the success of asperity models when describing the contact between self-affine surfaces. Phys. Rev. E 78(2), 026110 (2008)

    Article  Google Scholar 

  64. 64.

    Abbott, E.J., Firestone, F.A.: Specifying surface quality: a method based on accurate measurement and comparison. Mech. Eng. 55, 569 (1933)

    Google Scholar 

  65. 65.

    Feng, J.Q.: Adhesive contact of elastically deformable spheres: a computational study of pull-off force and contact radius. J. Colloid Interface Sci. 238(2), 318–323 (2001)

    Article  Google Scholar 

  66. 66.

    Pastewka, L., Robbins, M.O.: Contact area of rough spheres: large scale simulations and simple scaling laws. Appl. Phys. Lett. 108(22), 221601 (2016)

    Article  Google Scholar 

  67. 67.

    Pei, L., Hyun, S., Molinari, J., Robbins, M.O.: Finite element modeling of elasto-plastic contact between rough surfaces. J. Mech. Phys. Solids 53(11), 2385–2409 (2005)

    Article  Google Scholar 

  68. 68.

    Pérez-Ràfols, F., Roland, L., van Riet, E.J., Almqvist, A.: On the flow through plastically deformed surfaces under unloading: a spectral approach. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. (2017). doi:10.1177/0954406217690180

  69. 69.

    Luan, B., Robbins, M.O.: Hybrid atomistic/continuum study of contact and friction between rough solids. Tribol. Lett. 36(1), 1–16 (2009)

    Article  Google Scholar 

  70. 70.

    Persson, B.N.J.: Contact mechanics for randomly rough surfaces: on the validity of the method of reduction of dimensionality. Tribol. Lett. 58(1), 58 (2015)

    Article  Google Scholar 

  71. 71.

    Sneddon, I.N.: The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. Int. J. Eng. Sci. 3(1), 47–57 (1965)

    Article  Google Scholar 

  72. 72.

    Lyashenko, I.A., 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(2), 223–229 (2013)

    Article  Google Scholar 

Download references


MHM thanks Wilfred Tysoe and Nicholas Spencer for indispensible support in the execution and the write-up of the contact-mechanics challenge. MHM and WBD thank the Jülich Supercomputing Centre for computing time on JUQUEEN. The contribution of GV and AV was funded by the Austrian COMET-Program (Project XTribology, No. 849109), and the work was carried out at the “Excellence Centre of Tribology” (AC2T research GmbH). MOR was supported by the NSF through Grant 1411144.

Author information



Corresponding author

Correspondence to Martin H. Müser.

Additional information

This article is part of the Topical Collection on Special Issue: The Contact-Mechanics Challenge.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Müser, M.H., Dapp, W.B., Bugnicourt, R. et al. Meeting the Contact-Mechanics Challenge. Tribol Lett 65, 118 (2017).

Download citation


  • Contact mechanics
  • Adhesion
  • Modeling
  • Nominally flat surfaces