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A New Fudge Factor for Persson’s Theory of Purely Normal Elastic Rough Surface Contact

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Abstract

After more than two decades of development, Persson’s theory has become one of the dominantly used theoretical tools to analyze the interaction between a nominally flat rough surface and a rigid flat. The original theory (Persson in J. Chem. Phys. 115(8):3840–3861, 2001) tends to overestimate and underestimate the relative contact area and strain energy, respectively, for linear elastic purely normal contact. Several fudge factors, obtained empirically based on numerical solutions, have been used to amend the strain energy associated with each wavenumber, resulting in scale history-dependent formulations for the relative contact area and probability density function (PDF) of contact pressure, which significantly complicate the solution process. We provide a new fudge factor that results in simple formulations of the relative contact area and PDF of contact pressure, which only relies on the present scale. Compared with the relative contact area and PDF of contact pressure solved by Green’s Function Molecular Dynamics (GFMD), Persson’s theory using the new fudge factor shows similar accuracy to other variants. Among all variants of Persson’s theory, the use of the new hybrid formulation of strain energy results in the best agreement with GFMD. Using the new fudge factor, various interfacial properties (e.g., the average interfacial gap) can be derived with a simple form, and solved more quickly, with acceptable accuracy.

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Data Availability

Matlab functions that implement various variants of Persson’s theory are publicly available on GitHub (https://github.com/xylcsi/PerssonModelFudgeFactor).

Notes

  1. \(P_0(p, {\zeta })\) is continuous at \(p = 0\), i.e., \(P_0(p \rightarrow 0^+, {\zeta }) = P_0(p = 0, {\zeta }) = 0\).

  2. This correction was initially proposed to achieve good agreement between the elastic strain energy predicted by Persson’s theory and numerical models.

References

  1. Greenwood, J. A., Williamson, J. P.: Contact of nominally flat surfaces. Proc. Royal Soc. London. Ser. A. Math. Phys. Sci. 295(1442), 300–319 (1966)

  2. Persson, B.N.: Theory of rubber friction and contact mechanics. J. Chem. Phys. 115(8), 3840–3861 (2001)

    Article  ADS  CAS  Google Scholar 

  3. Majumdar, A., Bhushan, B.: Fractal model of elastic-plastic contact between rough surfaces. J. Tribol. 113(1), 1–11 (1991)

    Article  Google Scholar 

  4. Scaraggi, M., Persson, B. N.: General contact mechanics theory for randomly rough surfaces with application to rubber friction. J. Chem. Phys. 143(22), 224111 (2015)

  5. Persson, B.: Elastoplastic contact between randomly rough surfaces. Phys. Rev. Lett. 87(11), 116101 (2001)

  6. Wang, A., Müser, M.H.: Gauging Persson theory on adhesion. Tribol. Lett. 65(3), 1–10 (2017)

    Article  ADS  Google Scholar 

  7. Almqvist, A., Campana, C., Prodanov, N., et al.: 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  ADS  MathSciNet  Google Scholar 

  8. Persson, B.: On the elastic energy and stress correlation in the contact between elastic solids with randomly rough surfaces. J. Phy.: Cond. Matter 20(31), 312001 (2008)

  9. Persson, B.: On the electric contact resistance. Tribol. Lett. 70(3), 88 (2022)

    Article  Google Scholar 

  10. Persson, B.: On the role of flexoelectricity in triboelectricity for randomly rough surfaces. Europhys. Lett. 129(1), 10006 (2020)

    Article  ADS  CAS  Google Scholar 

  11. Persson, B.: Leakage of metallic seals: role of plastic deformations. Tribol. Lett. 63, 1–6 (2016)

    Article  Google Scholar 

  12. 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)

    Article  Google Scholar 

  13. Afferrante, L., Bottiglione, F., Putignano, C., et al.: Elastic contact mechanics of randomly rough surfaces: an assessment of advanced asperity models and Persson’s theory. Tribol. Lett. 66(2), 1–13 (2018)

    Article  Google Scholar 

  14. Yang, C., Persson, B.: Contact mechanics: contact area and interfacial separation from small contact to full contact. J. Phys.: Cond. Matter 20(21), 215214 (2008)

  15. Manners, W., Greenwood, J.: Some observations on Persson’s diffusion theory of elastic contact. Wear 261(5–6), 600–610 (2006)

    Article  CAS  Google Scholar 

  16. Xu Y, Li X, Chen Q, et al. Persson’s theory of purely normal elastic rough surface contact: a tutorial based on stochastic process theory. Int. J. Solids Struct. (2024): 112684

  17. Dapp, W.B., Prodanov, N., Müser, M.H.: Systematic analysis of Persson’s contact mechanics theory of randomly rough elastic surfaces. J. Phys.: Cond. Matter 26(35), 355002 (2014)

  18. Yang, C., Tartaglino, U., Persson, B.: A multiscale molecular dynamics approach to contact mechanics. Eur. Phys. J. E 19(1), 47–58 (2006)

    Article  CAS  PubMed  Google Scholar 

  19. Persson, B.: Relation between interfacial separation and load: a general theory of contact mechanics. Phys. Rev. Lett. 99(12), 125502 (2007)

  20. Venugopalan, S., Irani, N., Nicola, L.: Plastic contact of self-affine surfaces: Persson’s theory versus discrete dislocation plasticity. J. Mech. Phys. Solids 132, 103676 (2019)

  21. Zhou, Y., Moseler, M., Müser, M.H.: Solution of boundary-element problems using the fast-inertial-relaxation-engine method. Phys. Rev. B 99(14), 144103 (2019)

  22. Zhou Y, Müser M H. Effect of structural parameters on the relative contact area for ideal, anisotropic, and correlated random roughness. Front. Mech. Eng. (2020): 59

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 52105179), Fundamental Research Funds for the Central Universities (No. JZ2023HGTB0252), and Natural Science Foundation of Jiangsu Province (No. BK20220555). YX would like to thank Dr. Anle Wang for kindly sharing the GFMD results used in Fig. 3d.

Funding

This work was supported by the National Natural Science Foundation of China (No. 52105179), Fundamental Research Funds for the Central Universities (No. JZ2023HGTB0252), and Natural Science Foundation of Jiangsu Province (No. BK20220555).

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Authors and Affiliations

  1. School of Mechanical Engineering, Hefei University of Technology, Hefei, 230009, China

    Yang Xu, Longan Zhu & Feiyun Xiao

  2. Anhui Province Key Laboratory of Digital Design and Manufacturing, Hefei, 230009, China

    Yang Xu

  3. Department of Civil Engineering, Yangzhou University, Yangzhou, 225127, China

    Yunong Zhou

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  1. Yang Xu
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  2. Longan Zhu
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Contributions

Y.X.: methodology, conceptualization, software, validation, formal analysis, visualization, investigation, writing—original draft, funding acquisition, supervision, writing—review and editing. L.Z.: visualization, software, validation, formal analysis, writing—review and editing. F.X.: methodology, writing—review and editing. Y.Z.: software, visualization, investigation, writing—review and editing.

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Correspondence to Yang Xu or Yunong Zhou.

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Xu, Y., Zhu, L., Xiao, F. et al. A New Fudge Factor for Persson’s Theory of Purely Normal Elastic Rough Surface Contact. Tribol Lett 72, 36 (2024). https://doi.org/10.1007/s11249-024-01838-4

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