Recent advancements in fractal geometric-based nonlinear time series solutions to the micro-quasistatic thermoviscoelastic creep for rough surfaces in contact. (English) Zbl 1213.74307

Summary: To understand the tripological contact phenomena, both mathematical and experimental models are needed. In this work, fractal mathematical models are used to model the experimental results obtained from literature. Fractal geometry, using a deterministic Cantor structure, is used to model the surface topography, where recent advancements in thermoviscoelastic creep contact of rough surfaces are introduced. Various viscoelastic idealizations are used to model the surface materials, for example, Maxwell, Kelvin-Voigt, Standard Linear Solid and Jeffrey media. Such media are modelled as arrangements of elastic springs and viscous dashpots in parallel and/or in series. Asymptotic power laws, through hypergeometric series, were used to express the surface creep as a function of remote forces, body temperatures and time. The introduced models are valid only when the creep approach of the contact surfaces is in the order of the size of the surface roughness. The obtained results using such models, which admit closed-form solutions, are displayed graphically for selected values of the systems’ parameters; the fractal surface roughness and various material properties. Results obtained showed good agreement with published experimental results, where the utilized methodology can be further extended to the utilization for the contact of surfaces within micro- and nano-electronic devices, circuits and systems.


74S30 Other numerical methods in solid mechanics (MSC2010)
74M15 Contact in solid mechanics
Full Text: DOI EuDML


[1] T. A. Alabed, O. M. Abuzeid, and M. Barghash, “A linear viscoelastic relaxation-contact model of a flat fractal surface: a Maxwell-type medium,” International Journal of Advanced Manufacturing Technology, vol. 39, no. 5-6, pp. 423-430, 2008.
[2] F. M. Borodich and D. A. Onishchenko, “Similarity and fractality in the modelling of roughness by a multilevel profile with hierarchical structure,” International Journal of Solids and Structures, vol. 36, no. 17, pp. 2585-2612, 1999. · Zbl 0939.74004
[3] J. A. Greenwood, “Problems with surface roughness,” in Fundamentals of Friction: Macroscopic and Microscopic Processes, I. L. Singer and H. M. Pollock, , Eds., pp. 57-76, Kluwer Academic Publishers, Boston, Mass, USA, 1992.
[4] A. Majumdar and B. Bhushan, “Role of fractal geometry in roughness characterization and contact mechanics of surfaces,” Journal of Tribology, vol. 112, no. 2, pp. 205-216, 1990.
[5] B. B. Mandelbrot, D. E. Passoja, and A. J. Paullay, “Fractal character of fracture surfaces of metals,” Nature, vol. 308, no. 5961, pp. 721-722, 1984.
[6] M. Li and W. Zhao, “Representation of a Stochastic Traffic Bound,” IEEE Transactions on Parallel and Distributed Systems, vol. 21, no. 9, pp. 1368-1372, 2010.
[7] M. Li and S. C. Lim, “Modeling network traffic using generalized Cauchy process,” Physica A, vol. 387, no. 11, pp. 2584-2594, 2008.
[8] M. Li, “Generation of teletraffic of generalized Cauchy type,” Physica Scripta, vol. 81, no. 2, Article ID 025007, 2010. · Zbl 1191.90013
[9] B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, San Francisco, Calif, USA, 1982. · Zbl 0504.28001
[10] J. Feder, Fractals, Physics of Solids and Liquids, Plenum Press, New York, NY, USA, 1988. · Zbl 0648.28006
[11] K. L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, UK, 1985. · Zbl 0599.73108
[12] J. A. Greenwood and J. B. P. Williamson, “Contact of nominally flat surfaces,” Proceedings of the Royal Society A, vol. 370, pp. 300-319, 1966.
[13] A. Majumdar and B. Bhushan, “Fractal model of elastic-plastic contact between rough surfaces,” Journal of Tribology, vol. 113, no. 1, pp. 1-11, 1991.
[14] F. M. Borodich and A. B. Mosolov, “Fractal roughness in the contact problems,” Journal of Applied Mathematics and Mechanics, vol. 56, no. 5, pp. 786-795, 1992. · Zbl 0789.73062
[15] F. Borodich, “Fractals and surface roughness in EHL,” in Proceedings of the Iutam Symposium on Elastohydrodynamics and Micro-elastohydrodynamics, R. Snidle and H. Evans, Eds., pp. 397-408, 2006.
[16] T. L. Warren and D. Krajcinovic, “Fractal models of elastic-perfectly plastic contact of rough surfaces based on the Cantor set,” International Journal of Solids and Structures, vol. 32, no. 19, pp. 2907-2922, 1995. · Zbl 0869.73066
[17] T. L. Warren, A. Majumdar, and D. Krajcinovic, “A fractal model for the rigid-perfectly plastic contact of rough surfaces,” ASME Journal of Applied Mechanics, Transactions, vol. 63, no. 1, pp. 47-54, 1996. · Zbl 0894.73142
[18] O. Abuzeid, “Linear viscoelastic creep model for the contact of nominal flat surfaces based on fractal geometry: Maxwell type medium,” Dirasat, vol. 30, no. 1, pp. 22-36, 2003.
[19] O. M. Abuzeid, “A linear viscoelastic creep-contact model of a flat fractal surface: Kelvin-Voigt medium,” Industrial Lubrication and Tribology, vol. 56, no. 6, pp. 334-340, 2004.
[20] O. Abuzeid, “A viscoelastic creep model for the contact of rough fractal surfaces: Jeffreys’ type material,” in Proceedings of the 7th International Conference Production Engineering and Design for Development, Cairo, Egypt, 2006.
[21] O. M. Abuzeid and P. Eberhard, “Linear viscoelastic creep model for the contact of nominal flat surfaces based on fractal geometry: standard linear solid (SLS) material,” Journal of Tribology, vol. 129, no. 3, pp. 461-466, 2007.
[22] O. M. Abuzeid and T. A. Alabed, “Mathematical modeling of the thermal relaxation of nominally flat surfaces in contact using fractal geometry: Maxwell type medium,” Tribology International, vol. 42, no. 2, pp. 206-212, 2009.
[23] O. M. Abuzeid, A. N. Al-Rabadi, and H. S. Alkhaldi, “Fractal geometry-based hypergeometric time series solution to the hereditary thermal creep model for the contact of rough surfaces using the Kelvin-Voigt medium,” Mathematical Problems in Engineering, vol. 2010, Article ID 652306, 22 pages, 2010. · Zbl 1425.74335
[24] O. Abuzeid, “Thermal creep model of rough fractal surfaces in contact: viscoelastic standard linear solid,” Industrial Lubrication and Tribology. In press.
[25] R. D. Mauldin and S. C. Williams, “On the Hausdorff dimension of some graphs,” Transactions of the American Mathematical Society, vol. 298, no. 2, pp. 793-803, 1986. · Zbl 0603.28003
[26] D. Wójcik, I. Białynicki-Birula, and K. Zyczkowski, “Time evolution of quantum fractals,” Physical Review Letters, vol. 85, no. 24, pp. 5022-5025, 2000.
[27] A. N. Al-Rabadi, Reversible Logic Synthesis, Springer, Berlin, Germany, 2004. · Zbl 1140.94022
[28] C. Cattani and A. Kudreyko, “Application of periodized harmonic wavelets towards solution of eigenvalue problems for integral equations,” Mathematical Problems in Engineering, vol. 2010, Article ID 570136, 8 pages, 2010. · Zbl 1191.65175
[29] E. G. Bakhoum and C. Toma, “Dynamical aspects of macroscopic and quantum transitions due to coherence function and time series events,” Mathematical Problems in Engineering, vol. 2010, Article ID 428903, 2010. · Zbl 1191.35219
[30] G. Toma, “Specific differential equations for generating pulse sequences,” Mathematical Problems in Engineering, vol. 2010, Article ID 324818, 11 pages, 2010. · Zbl 1191.37052
[31] G. Mattioli, M. Scalia, and C. Cattani, “Analysis of large-amplitude pulses in short time intervals: application to neuron interactions,” Mathematical Problems in Engineering, vol. 2010, Article ID 895785, 15 pages, 2010. · Zbl 1189.37099
[32] S. Y. Chen and Y. F. Li, “Determination of stripe edge blurring for depth sensing,” IEEE Sensors Journal, vol. 11, no. 2, pp. 389-390, 2011.
[33] S. Y. Chen and Q. Guan, “Parametric shape representation by a deformable NURBS model for cardiac functional measurements,” IEEE Transactions on Biomedical Engineering. In press.
[34] K. Falconer, Fractal Geometry, John Wiley &; Sons, Chichester, UK, 1990.
[35] P. S. Modenov and A. S. Parkhomenko, Geometric Transformations, vol. 1 of Euclidean and Affine Transformations, Academic Press, New York, NY, USA, 1965. · Zbl 0192.26701
[36] R. F. Voss, “Random fractal forgeries,” in Fundamental Algorithms in Computer Graphics, R. A. Earnshaw, Ed., pp. 805-835, Springer, Berlin, Germany, 1985.
[37] A. Majumdar and C. L. Tien, “Fractal characterization and simulation of rough surfaces,” Wear, vol. 136, no. 2, pp. 313-327, 1990.
[38] J. Lopez, G. Hansali, H. Zahouani, J. C. Le Bosse, and T. Mathia, “3D fractal-based characterisation for engineered surface topography,” International Journal of Machine Tools and Manufacture, vol. 35, no. 2, pp. 211-217, 1995.
[39] M. Li, “Fractal time series-a tutorial review,” Mathematical Problems in Engineering, vol. 2010, Article ID 157264, 26 pages, 2010. · Zbl 1191.37002
[40] M. Li and J.-Y. Li, “On the predictability of long-range dependent series,” Mathematical Problems in Engineering, vol. 2010, Article ID 397454, 9 pages, 2010. · Zbl 1191.62160
[41] R. S. Sayles and T. R. Thomas, “Surface topography as a nonstationary random process,” Nature, vol. 271, no. 5644, pp. 431-434, 1978.
[42] S. R. Brown, “Simple mathematical model of a rough fracture,” Journal of Geophysical Research, vol. 100, no. 4, pp. 5941-5952, 1995.
[43] M. V. Berry and Z. V. Lewis, “On the Weierstrass-Mandelbrot fractal function,” Proceedings of the Royal Society London Series A, vol. 370, no. 1743, pp. 459-484, 1980. · Zbl 0435.28008
[44] E. H. Lee and J. R. M. Radok, “The contact problems for viscoelastic bodies,” Journal of Applied Mechanics, vol. 27, pp. 438-444, 1960. · Zbl 0094.37503
[45] T. C. T. Ting, “The contact stress between a rigid indenter and a viscoelastic half-space,” Journal of Applied Mechanics, vol. 33, pp. 845-854, 1966. · Zbl 0154.22502
[46] T. C. T. Ting, “Contact problems in the linear theory of viscoelasticity,” Journal of Applied Mechanics, vol. 35, pp. 248-254, 1968. · Zbl 0167.24403
[47] G. R. Nghieh, H. Rahnejat, and Z. M. Jin, “Contact mechanics of viscoelastic layered surface,” in Contact Mechanics. III, M. H. Aliabadi and A. Samarti, Eds., pp. 59-68, Computational Mechanics Publications, Boston, Mass, USA, 1997.
[48] K. J. Wahl, S. V. Stepnowski, and W. N. Unertl, “Viscoelastic effects in nanometer-scale contacts under shear,” Tribology Letters, vol. 5, no. 1, pp. 103-107, 1998.
[49] D. J. Whitehouse and J. F. Archard, “The properties of random surfaces of significance in their contact,” Proceedings of the Royal Society A, vol. 316, pp. 97-121, 1970.
[50] A. Lumbantobing, L. Kogut, and K. Komvopoulos, “Electrical contact resistance as a diagnostic tool for MEMS contact interfaces,” Journal of Microelectromechanical Systems, vol. 13, no. 6, pp. 977-987, 2004.
[51] V. S. Radchik, B. Ben-Nissan, and W. H. Müller, “Theoretical modeling of surface asperity depression into an elastic foundation under static loading,” Journal of Tribology, vol. 124, no. 4, pp. 852-856, 2002.
[52] Y. Zhao and L. Chang, “A model of asperity interactions in elastic-plastic contact of rough surfaces,” Journal of Tribology, vol. 123, no. 4, pp. 857-864, 2001.
[53] A. Signorini, “Sopra alcune questioni di elastostatica,” Atti della Societa Italiana per il Progresso delle Sceienze, 1933. · JFM 59.1413.02
[54] P. E. D’yachenko, N. N. Tolkacheva, G. A. Andreev, and T. M. Karpova, The Actual Contact Area between Touching Surfaces, Consultant Bureau, New York, NY, USA, 1964.
[55] J. R. M. Radok, “Visco-elastic stress analysis,” Quarterly of Applied Mathematics, vol. 15, pp. 198-202, 1957. · Zbl 0125.13505
[56] N. J. Distefano and K. S. Pister, “On the identification problem for thermorheologically simple materials,” Acta Mechanica, vol. 13, no. 3-4, pp. 179-190, 1972. · Zbl 0239.73006
[57] T. Junisbekov, V. Kestelman, and N. Malinin, Stress Relaxation in Viscoelastic Materials, Science Publishers, Enfield, UK, 2nd edition, 2003.
[58] J. Boyle and J. Spencer, Stress Analysis for Creep, Butterworths-Heinemann, London, UK, 1st edition, 1983.
[59] W. S. Lee and C. Y. Liu, “The effects of temperature and strain rate on the dynamic flow behaviour of different steels,” Materials Science and Engineering A, vol. 426, no. 1-2, pp. 101-113, 2006.
[60] Z. Handzel-Powierza, T. Klimczak, and A. Polijaniuk, “On the experimental verification of the Greenwood-Williamson model for the contact of rough surfaces,” Wear, vol. 154, no. 1, pp. 115-124, 1992.
[61] W. Nowacki, Thermoelasticity, Pergamon Press, Oxford, UK, 2nd edition, 1986. · Zbl 0227.73009
[62] A. Hashem, “Study on reloading stress relaxation behavior for high temperature bolted steel,” Journal of Advanced Performance Materials, vol. 6, no. 2, pp. 129-140, 1999.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.