Advertisement

Tribology Letters

, 67:57 | Cite as

Elasticity Does Not Necessarily Break Down in Nanoscale Contacts

Comparing Stresses from Atomistic Simulations to Continuum Theory
  • Martin H. MüserEmail author
Original Paper
  • 13k Downloads

Abstract

Atomistic structures can have (sharp) features that are not accounted for in standard continuum theories. A prominent example is a Hertzian contact in which, however, the indenting tip is cut out of a crystal, whereby the tip acquires a discretized height profile. The microscopic stresses observed for such quantized indenters show sharp stress peaks at the edges of the height steps so that the stress profiles differ from those produced by smooth, parabolic indenters. Such deviations are frequently misinterpreted as the breakdown of continuum theory at the nanoscale. In this Letter, the stress peaks are confirmed to also occur in a continuum treatment containing steps. In addition, it is shown that analytical solutions for smooth tips can compare extremely well to those with steps if both stress fields are passed through the same (Gaussian) filter smearing out the features in real space with a resolution close to the broadest terrace of the quantized tip. Related statements are shown to also hold for the stress distribution function of randomly rough indenters with quantized height profiles.

Keywords

Contact mechanics Linear elasticity Modeling and theory 

Notes

Acknowledgements

MM thanks Mark O. Robbins for extensive and helpful feedback on the manuscript.

References

  1. 1.
    Luan, B., Robbins, M.O.: The breakdown of continuum models for mechanical contacts. Nature 435(7044), 929–932 (2005)CrossRefGoogle Scholar
  2. 2.
    Luan, B., Robbins, M.O.: Contact of single asperities with varying adhesion: comparing continuum mechanics to atomistic simulations. Phys. Rev. E 74(2), 026111 (2006)CrossRefGoogle Scholar
  3. 3.
    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
  4. 4.
    Persson, B.N.J.: Theory of rubber friction and contact mechanics. J. Chem. Phys. 115(8), 3840 (2001)CrossRefGoogle Scholar
  5. 5.
    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)CrossRefGoogle Scholar
  6. 6.
    Zhou, Y., Moseler, M., Müser, M.H.: Solution of boundary-element problems using the fast-inertial-relaxation-engine method. Phys. Rev. B (in print)Google Scholar
  7. 7.
    Müser, 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., Ashtari Esfahani, H., 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), 118 (2017)CrossRefGoogle Scholar
  8. 8.
    Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge, UK (1985)CrossRefGoogle Scholar
  9. 9.
    Pastewka, L., Sharp, T.A., Robbins, M.O.: Seamless elastic boundaries for atomistic calculations. Phys. Rev. B 86(7), 075459 (2012)CrossRefGoogle Scholar
  10. 10.
    Klemenz, A., Gola, A., Moseler, M., Pastewka, L.: Contact mechanics of graphene-covered metal surfaces. Appl. Phys. Lett. 112(6), 061601 (2018)CrossRefGoogle Scholar
  11. 11.
    Müser, M.H.: Internal, elastic stresses below randomly rough contacts. J. Mech. Phys. Solids 119, 73–82 (2018)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(2), 026117 (2004)CrossRefGoogle Scholar
  13. 13.
    Campañá, C., Müser, M.H.: Contact mechanics of real vs. randomly rough surfaces: a Green’s function molecular dynamics study. Europhys. Lett. 77(3), 38005 (2007)CrossRefGoogle Scholar
  14. 14.
    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)CrossRefGoogle Scholar
  15. 15.
    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(2), 433–448 (2014)CrossRefGoogle Scholar
  16. 16.
    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, 83–102 (2015)CrossRefGoogle Scholar
  17. 17.
    Majumdar, A., Tien, C.L.: Fractal characterization and simulation of rough surfaces. Wear 136(2), 313–327 (1990)CrossRefGoogle Scholar
  18. 18.
    Palasantzas, G.: Roughness spectrum and surface width of self-affine fractal surfaces via the k-correlation model. Phys. Rev. B 48(19), 14472–14478 (1993)CrossRefGoogle Scholar
  19. 19.
    Persson, B.N.J.: On the fractal dimension of rough surfaces. Tribol. Lett. 54(1), 99–106 (2014)CrossRefGoogle Scholar
  20. 20.
    Todd, B.D., Evans, D.J., Daivis, P.J.: Pressure tensor for inhomogeneous fluids. Phys. Rev. E 52(2), 1627–1638 (1995)CrossRefGoogle Scholar
  21. 21.
    Wang, A., Müser, M.H.: Gauging Persson theory on adhesion. Tribol. Lett. 65(3), 103 (2017)CrossRefGoogle Scholar
  22. 22.
    Braun, O.M., Kivshar, Y.S.: Nonlinear dynamics of the Frenkel–Kontorova model. Phys. Rep. 306(1–2), 1–108 (1998)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Materials Science and EngineeringSaarland UniversitySaarbrückenGermany

Personalised recommendations