×

Fractal roughness in contact problems. (English. Russian original) Zbl 0789.73062

J. Appl. Math. Mech. 56, No. 5, 681-690 (1992); translation from Prikl. Mat. Mekh. 56, No. 5, 786-795 (1992).
The roughness of real polished bodies is shown to be fractal in character. A relation is found between the fractal dimension of a surface and its statistical properties. Models are constructed of the contact of fractal-rough punches and the smooth surface of a deformable half-space by a modelling Winkler medium and a rigidly plastic medium. Asymptotic power laws have been obtained which associate the force operating on the punch and the depth of indentation for different (both plastic and elastic) models of the deformed base. The relation between the power index and the fractal dimension of the surface and the print is determined.

MSC:

74A55 Theories of friction (tribology)
74M15 Contact in solid mechanics
28A80 Fractals
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Demkin, N. B., The True Area of Contact Between Solid Surfaces (1962), Izd-vo Akad. Nauk SSSR: Izd-vo Akad. Nauk SSSR Moscow
[2] Bowden, F. P.; Tabor, D., (Friction and Lubrication of Solids, Vols 1 and 2. (1950), Oxford University Press: Oxford University Press London) · Zbl 0987.74002
[3] Johnson, K. L., (Contact Mechanics (1985), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0599.73108
[4] Sviridenok, A. I.; Chizhik, S. A.; Petrokovets, M. I., Mechanics of Discrete Contact (1990), Navuka i Tekhnika: Navuka i Tekhnika Minsk
[5] Goryacheva, I. G.; Dobychin, M. N., Contact Problems in Tribology (1988), Mashinostroyeniye: Mashinostroyeniye Moscow
[6] Borodich, F. M., On the problem of the contact of two previously distorted half-spaces, Prikl. Mekh. Tekl. Fiz., 2, 160-162 (1984)
[7] Mandelbrot, B. B., The Fractal Geometry of Nature (1982), Freeman: Freeman San Francisco · Zbl 0504.28001
[8] Pfeifer, P., Fractal dimension as working tool for surface-roughness problems, Appl. Surf. Sci., 18, 146-164 (1984)
[9] Ling, F. F., Fractal engineering surfaces and tribology, Wear, 136, 141-156 (1990)
[10] Gagnepain, J. J.; Roques-Carmes, C., Fractal approach to two-dimensional and three-dimensional surface roughness, Wear, 109, 119-126 (1986)
[11] Mandelbrot, B. B.; Passoja, D. E.; Paullay, A. J., Fractal character of fracture surfaces of metals, Nature, 308, 721-722 (1984)
[12] Sayles, R. S.; Thomas, T. R., Surface topography as a non-stationary random process, Nature, 271, 431-434 (1978)
[13] Dubuc, B.; Zucker, S. W.; Tricot, C.; Quiniou, J. F.; Wehbi, D., Evaluating the fractal dimension of surfaces, (Proc. Roy. Soc. London, A 425 (1989)), 113-127 · Zbl 0703.28006
[14] Feder, J., Fractals (1989), Plenum Press: Plenum Press New York · Zbl 0648.28006
[15] Archard, J. F., Elastic deformation and the laws of friction, (Proc. Royal Soc. Ser., A 243 (1957)), 190-205
[16] Borodich, F. M., Hertz-type contact problems for an anisotropic physically non-linear elastic medium, Probl. Prochnosti, 12, 47-53 (1989)
[17] Ling, F. F., The possible role of fractal geometry in tribology, Tribology Trans., 32, 497-505 (1989)
[18] Majumdar, A.; Bhushan, B., Role of fractal geometry in roughness characterization and contact mechanics of surfaces, Trans ASME J. Tribology, 112, 205-216 (1990)
[19] Borodich, F. M., On the deforming properties of multilayer metal packets, (MTT, 4 (1987), Izv. Akad. Nauk SSSR), 105-112
[20] Barenblatt, G. I.; Botvina, L. R., Self-similarity of fatigue fracture, (MTT, 4 (1983), Izv. Akad. Nauk SSSR), 161-165 · Zbl 0785.73059
[21] Mosolov, A. B.; Dinariyevo, O. Yu., Self-similarity and the fractal geometry of fracture, Probl. Prochnosti, 1, 3-7 (1988)
[22] Gist, G. A., Ultrasonic attenuation and fractal surfaces in porous media, Phys. Rev., B 39, 7295-7298 (1989)
[23] Avnir, D.; Farin, D.; Pfeifer, P., Chemistry in noninteger dimensions between two and three. II. Fractal surface of adsorbents, J. Chem. Phys., 79, 3566-3571 (1983)
[24] Hohr, A.; Neumann, H.-B.; Schmidt, P. W.; Pfeifer, P.; Avnir, D., Fractal surface and cluster structure of controlled-pore glasses and Vycor porous glass as revealed by small-angle X-ray and neutron scattering, Phys. Rev., B 38, 1462-1467 (1988)
[25] Falconer, K. J., Geometry of Fractal Sets (1985), Cambridge University Press · Zbl 0587.28004
[26] (Peitgen, H.-O.; Saupe, D., The Science of Fractal Images (1988), Springer: Springer New York)
[27] Mandelbrot, B., Self-affine Fractal Sets, (Fractals in Physics (1988), Mir: Mir Moscow), 9-47, Parts I-III
[28] Liu, S.; Kaplan, T.; Gray, L., The response of rough surfaces to an alternating current, (Fractals in Physics (1988), Mir: Mir Moscow), 543-552
[29] Borodich, F. M.; Mosolov, A. B., Fractal contact of solids, Zh. Tekh. Fiz., 61, 50-54 (1991)
[30] Hill, R., Mathematical Theory of Plasticity (1950), Clarendon Press: Clarendon Press Oxford · Zbl 0041.10802
[31] Aleksandrov, V. M.; Mkhitaryan, S. M., Contact Problems for Bodies with Thin Coverings and Interlayers (1983), Nauka: Nauka Moscow
[32] Shtayerman, I. Ya., The Contact Problem of the Theory of Elasticity (1949), Gostekhizdat: Gostekhizdat Moscow
[33] Giugliarelli, G.; Martin, A.; Stella, A. L., Spectral properties of fractal surfaces, Europhys Letters, 11, 101-106 (1990)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.