Skip to main content
SpringerLink
Log in

Fractals and Surface Rroughness in EHL

  • Conference paper
IUTAM Symposium on Elastohydrodynamics and Micro-elastohydrodynamics

Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 134))

Abstract

Firstly, a brief introduction to fractals and similarity methods is given. Fractal models of rough surfaces are usually used when the spectral density function of surfaces has the power law character. It is argued that the main source for various misunderstandings in applications of fractals to mechanics is the lack of precise definitions and non-critical repetition of common statements about fractal geometry. Some key papers concerning fractal models of roughness and papers connecting EHL and fractals are reviewed. Two classes of fractal surfaces introduced by the author, namely the Cantor profile models and the parametric-homogeneous (PH) surfaces, are discussed. The well-known Weierstrass-Mandelbrot (W-M) profile is a particular case of PH-profiles. It is shown that only physical fractals (prefractals) should be attributed to real surfaces. It is argued that the Cantor profile is simple for analytical analysis. However, it has a minor drawback: all asperities of the profile have one-level character, while, as Archard showed, real roughness has a hierarchical structure. Finally, it is suggested to model rough surfaces by a multilevel prefractal model introduced by Borodich and Onishchenko.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Lubrecht, A.A. and Venner, C.H. (1999) Elastohydrodynamic lubrication of rough surfaces. Proc. Instn. Mech. Engrs. Part J. J. Eng. Trib., 213, 397–404.

    Google Scholar 

  2. Zhuravlev, V.A. (1940) On question of theoretical justification of the Amontons-Coulomb law for friction of unlubricated surfaces. Zh. Tekh. Fiz., 10, 1447–1452.

    Google Scholar 

  3. Greenwood, J.A. and Williamson, J.B.P. (1966) Contact of nominally flat surfaces. Phil. Trans. Roy. Soc. Lond., A295, 300–319.

    Google Scholar 

  4. Archard, J.F. (1957) Elastic deformation and the laws of friction. Proc. Roy. Soc. Lond., A243, 190–205.

    Google Scholar 

  5. Borodich, F.M. and Mosolov, A.B. (1992) Fractal roughness in contact problems. J. Appl. Math. Mech., 56, 786–795.

    Article  MATH  MathSciNet  Google Scholar 

  6. Whitehouse, D.J. and Archard, J.F. (1970) The properties of random surfaces of significance in their contact. Proc. Roy. Soc. Lond., A316, 97–121.

    Google Scholar 

  7. Brown, S.R. (1995) Simple mathematical model of rough fracture. J. Geophys. Res., 100(B4), 5941–5952.

    Article  Google Scholar 

  8. Sayles, R.S. and Thomas, T.R. (1978) Surface topography as a nonstationary random process. Nature, 271, 431–434.

    Article  Google Scholar 

  9. Berry, M.V. and Hannay, J.H. (1978) Topography of random surfaces. Nature, 273, 573.

    Article  Google Scholar 

  10. Falconer, K.J. (1990) Fractal Geometry: Mathematical Foundations an Applications, John Wiley, Chichester.

    MATH  Google Scholar 

  11. Vilenkin, N.Ya. (1968) Stories about Sets, Academic Press, New York.

    MATH  Google Scholar 

  12. Mandelbrot, B.B. (1975) Les Objects Fractals: Forme, Hasard et Dimension, Flammarion, Paris.

    Google Scholar 

  13. Liu, S.H., Kaplan T. and Gray, L.J. (1986) Theory of the AC response of rough interfaces. In: Fractals in Physics, L. Pietronero and E. Tosatti (eds), North Holland, Amsterdam, pp. 383–392.

    Google Scholar 

  14. Borodich, F.M. (1998) Parametric homogeneity and non-classical self-similarity. II. Some applications. Acta Mechanica, 131, 47–67.

    Article  MathSciNet  Google Scholar 

  15. Borodich, F.M. and Mosolov, A.B. (1991) Fractal contact of solids. Sov. Phys.-Tech. Phys., 61, 50–54.

    Google Scholar 

  16. Argatov, I.I. and Dmitriev, N.N. (2003) Fundamentals of Theory of Elastic Discrete Contact, Politekhnika Press, St. Petersburg.

    Google Scholar 

  17. Borodich, F.M. and Onishchenko, D.A. (1993)Fractal roughness for problem of contact and friction (the simplest models). J. Frict. Wear, 14, 14–19.

    Google Scholar 

  18. Borodich, F.M. and Onishchenko, D.A. (1999) Similarity and fractality in the modelling of roughness by multilevel profile with hierarchical structure. Int. J. Solids Struct., 36, 2585–2612.

    Article  MATH  MathSciNet  Google Scholar 

  19. Warren, T.L. and Krajcinovic, D. (1996) Random Cantor set models for the elastic perfectly plastic contact of rough surfaces. Wear, 196, 1–15.

    Article  Google Scholar 

  20. Plesha, M.E. and Ni, D. (2001) Scaling of geological discontinuity normal load-deformation response using fractal geometry. Int. J. Num. Anal. Meth. Geomech., 25, 741–756.

    Article  MATH  Google Scholar 

  21. Warren, T.L., Majumdar, A. and Krajcinovic, D. (1996) A fractal model for the rigid-perfectly plastic contact of rough surfaces. J. Appl. Mech., 63, 47–54.

    MATH  MathSciNet  Google Scholar 

  22. Schmittbuhl, J., Vilotte, J.-P. and Roux, S. (1996) Velocity weakening friction: A renormalization approach. J. Geophys. Res., 101(B6), 13911–13917.

    Article  Google Scholar 

  23. Sugimura, J. (1995) Fractal surfaces and hydrodynamic lubrication. J. Japan Soc. Trib., 40, 549–554 (in Japanese).

    Google Scholar 

  24. Kennedy, F.E., Brown, C.A., Kolodny, J. and Sheldon, B.M. (1999) Fractal analysis of hard disk surface roughness and correlation with static and lowspeed friction. Trans. ASME J. Tribology, 121, 968–974.

    Google Scholar 

  25. Jhon, M.S. and Choi, H.J. (2001) Lubricants in future data storage technology. J. Ind. Eng. Chem., 7, 263–275.

    Google Scholar 

  26. Plouraboue, F., Prat, M. and Letalleur, N. (2001) Sliding lubricated anisotropic rough surfaces. Phys. Rev. E, 64, Art. No. 011202.

    Google Scholar 

  27. Whitehouse, D.J. (2001) Fractal of fiction. Wear, 249, 345–353.

    Article  Google Scholar 

  28. Greenwood, J.A. and Wu, J.J. (2001) Surface roughness and contact: An apology. Meccanica, 36, 617–630.

    Article  MATH  Google Scholar 

  29. Bhushan, B. (2001) Modern Tribology Handbook, CRC Press, Boca Raton, p. 1430.

    Google Scholar 

  30. Borodich, F.M. (1999) Fractals and fractal scaling in fracture mechanics. Int. J. Fracture, 95, 239–259.

    Article  Google Scholar 

  31. Davies, S. and Hall, P. (1999) Fractal analysis of surface roughness by using spatial data. J. R. Statist. Soc. B, 61, 3–37.

    Article  MATH  MathSciNet  Google Scholar 

  32. Borodich, F.M. (1993) Similarity properties of discrete contact between a fractal punch and an elastic medium. C.R. Acad. Sci. (Paris), Ser. 2, 316, 281–286.

    MATH  Google Scholar 

  33. Berry, M.V. and Lewis, Z.V. (1980) On the Weierstrass-Mandelbrot fractal functions. Proc. Roy. Soc. Lond., A370, 459–484.

    MathSciNet  Google Scholar 

  34. Roques-Carmes, C., Wehbi, D., Quiniou, J.F. and Tricot, C. (1988) Modelling engineering surfaces and evaluating their non-integer dimension for application in material science. Surface Topography, 1, 435–443.

    Google Scholar 

  35. Sun, X. and Jaggard, D.L. (1990) Wave scattering from non-random fractal surfaces. Optics Communications, 78, 20–24.

    Article  Google Scholar 

  36. Majumdar, A. and Bhushan, B. (1990) Role of fractal geometry in roughness characterization and contact mechanics of surfaces. J. Tribology, 112, 205–216.

    Article  Google Scholar 

  37. Moreira, J.G., Kamphorst Leal da Silva, J. and Oliffson Kamphorst, S. (1994) On the fractal dimension of self-affine profiles. J. Phys. A: Math. Gen., 27, 8079–8089.

    Article  MATH  Google Scholar 

  38. Lopez, J., Hansali, G., Zahouani, Le Bosse, J.C. and Mathia, T. (1995) 3-D fractal-based characterization for engineered surface topography. Int. J. Mach. Tools Manufacture, 35, 211–217.

    Article  Google Scholar 

  39. Blackmore, D. and Zhou, J.G. (1996) A general fractal distribution function for rough surface profiles. SIAM J. Appl. Math., 56, 1694–1719.

    Article  MATH  MathSciNet  Google Scholar 

  40. Bhushan, B. (1995) A fractal theory of the temperature distribution at elastic contacts of fast sliding surfaces. Discussion. J. Tribology, 117, 214–215.

    Google Scholar 

  41. Malcai, O., Lidar, D.A., Biham, O. and Avnir, D., (1997) Scaling range and cutoffs in empirical fractals. Phys. Rev. E, 56, 2817–2828.

    Article  Google Scholar 

  42. Mandelbrot, B.B. (1985) Self-affine fractals and fractal dimension. Phys. Scripta., 32, 257–260.

    MATH  MathSciNet  Google Scholar 

  43. Hansen, A., Hinrichsen, E.L. and Roux, S. (1991) Roughness of crack interfaces. Phys. Rev. Lett., 66, 2476–2479.

    Article  Google Scholar 

  44. Måløy, K.J., Hansen, A., Hinrichsen, E.L. and Roux, S. (1992) Experimental measurements of the roughness of brittle cracks. Phys. Rev. Lett., 68, 2266–2269.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Engineering, Cardiff University, Cardiff, CF24 3AA, UK

    F. M. Borodich

Authors
  1. F. M. Borodich
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Cardiff, UK

    R. W. Snidle  & H. P. Evans  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer

About this paper

Cite this paper

Borodich, F.M. (2006). Fractals and Surface Rroughness in EHL. In: Snidle, R.W., Evans, H.P. (eds) IUTAM Symposium on Elastohydrodynamics and Micro-elastohydrodynamics. Solid Mechanics and Its Applications, vol 134. Springer, Dordrecht . https://doi.org/10.1007/1-4020-4533-6_29

Download citation

  • DOI: https://doi.org/10.1007/1-4020-4533-6_29

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-1-4020-4532-5

  • Online ISBN: 978-1-4020-4533-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation