About this Journal Submit a Manuscript Table of Contents
Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 652306, 22 pages
http://dx.doi.org/10.1155/2010/652306
Research Article

Fractal Geometry-Based Hypergeometric Time Series Solution to the Hereditary Thermal Creep Model for the Contact of Rough Surfaces Using the Kelvin-Voigt Medium

1Mechanical Engineering Department, The University of Jordan, Amman 11942, Jordan
2Computer Engineering Department, The University of Jordan, Amman 11942, Jordan

Received 28 January 2010; Accepted 23 May 2010

Academic Editor: Ming Li

Copyright © 2010 Osama M. Abuzeid et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. 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, Boston, Mass, USA, 1992.
  3. 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.
  4. 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. View at Publisher · View at Google Scholar
  5. M. Li and W. Zhao, “Representation of a stochastic traffic bound,” to appear in IEEE Transactions on Parallel and Distributed Systems, IEEE Computer Society Digital Library, IEEE Computer Society, http://doi.ieeecomputersociety. org/10.1109/TPDS.2009.162. View at Publisher · View at Google Scholar
  6. M. Li and S. C. Lim, “Modeling network traffic using generalized Cauchy process,” Physica A, vol. 387, no. 11, pp. 2584–2594, 2008. View at Publisher · View at Google Scholar
  7. M. Li, “Generation of teletraffic of generalized Cauchy type,” Physica Scripta, vol. 82, no. 2, Article ID 025007, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. K. L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, UK, 1985.
  9. J. A. Greenwood and J. B. P. Williamson, “Contact of nominally flat surfaces,” Proceedings of the Royal Society of London. Series A, vol. 295, no. 1442, pp. 300–319, 1966.
  10. A. Majumdar and B. Bhushan, “Fractal model of elastic-plastic contact between rough surfaces,” Journal of Tribology, vol. 113, pp. 1–11, 1991.
  11. F. M. Borodich and A. B. Mosolov, “Fractal roughness in contact problems,” Journal of Applied Mathematics and Mechanics, vol. 56, no. 5, pp. 786–795, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. 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.
  13. T. L. Warren, A. Majumdar, and D. Krajcinovic, “A fractal model for the rigid-perfectly plastic contact of rough surfaces,” Journal of Applied Mechanics, vol. 63, no. 1, pp. 47–54, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. O. Abuzeid, “Linear viscoelastic creep model for the contact of nominal flat surfaces based on fractal geometry: Maxwell type medium,” Dirasat-Engineering Sciences, The University of Jordan, vol. 30, no. 1, pp. 22–36, 2003.
  15. 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. View at Publisher · View at Google Scholar
  16. 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. View at Publisher · View at Google Scholar
  17. F. Borodich, “Fractals and surface roughness in EHL,” in IUTAM Symposium on Elastohydrodynamics and Micro-Elastohydrodynamics, R. Snidle and H. Evans, Eds., vol. 134 of Solid Mechanics and Its Applications, pp. 397–408, Springer, Dordrecht, The Netherlands, 2006. View at Publisher · View at Google Scholar
  18. B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, San Francisco, Calif, USA, 1982. View at MathSciNet
  19. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. D. Wójcik, I. Białynicki-Birula, and K. Zyczkowski, “Time evolution of quantum fractals,” Physical Review Letters, vol. 85, no. 24, pp. 5022–5025, 2000.
  21. A. N. Al-Rabadi, Reversible Logic Synthesis: From Fundamentals to Quantum Computing, Springer, Berlin, Germany, 2004. View at MathSciNet
  22. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. 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, 13 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. G. Toma, “Specific differential equations for generating pulse sequences,” Mathematical Problems in Engineering, vol. 2010, Article ID 324818, 11 pages, 2010. View at MathSciNet
  25. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. S. Y. Chen, Y. F. Li, and J. Zhang, “Vision processing for realtime 3-D data acquisition based on coded structured light,” IEEE Transactions on Image Processing, vol. 17, no. 2, pp. 167–176, 2008. View at Publisher · View at Google Scholar · View at PubMed · View at MathSciNet
  27. S. Y. Chen, Y. F. Li, Q. Guan, and G. Xiao, “Real-time three-dimensional surface measurement by color encoded light projection,” Applied Physics Letters, vol. 89, no. 11, Article ID 111108, 2006. View at Publisher · View at Google Scholar
  28. K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Chichester, UK, 1990. View at MathSciNet
  29. P. S. Modenov and A. S. Parkhomenko, Geometric Transformations. Vol. 1: Euclidean and Affine Transformations, Academic Press, New York, NY, USA, 1965. View at MathSciNet
  30. R. F. Voss, “Random fractal forgeries,” in Fundamental Algorithms in Computer Graphics, R. A. Earnshaw, Ed., pp. 805–835, Springer, Berlin, Germany, 1985.
  31. A. Majumdar and C. L. Tien, “Fractal characterization and simulation of rough surfaces,” Wear, vol. 136, no. 2, pp. 313–327, 1990.
  32. 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.
  33. M. Li, “Fractal time series—a tutorial review,” Mathematical Problems in Engineering, vol. 2010, Article ID 157264, 26 pages, 2010. View at MathSciNet
  34. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  35. R. S. Sayles and T. R. Thomas, “Surface topography as a nonstationary random process,” Nature, vol. 271, no. 5644, pp. 431–434, 1978. View at Publisher · View at Google Scholar
  36. S. R. Brown, “Simple mathematical model of a rough fracture,” Journal of Geophysical Research, vol. 100, no. 4, pp. 5941–5952, 1995. View at Publisher · View at Google Scholar
  37. M. V. Berry and Z. V. Lewis, “On the Weierstrass-Mandelbrot fractal function,” Proceedings of the Royal Society of London. Series A, vol. 370, no. 1743, pp. 459–484, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. E. H. Lee and J. R. M. Radok, “The contact problem for viscoelastic bodies,” Journal of Applied Mechanics, vol. 27, pp. 438–444, 1960. View at Zentralblatt MATH · View at MathSciNet
  39. 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.
  40. T. C. T. Ting, “Contact problems in the linear theory of viscoelasticity,” Journal of Applied Mechanics, vol. 35, pp. 248–254, 1968.
  41. G. R. Nghieh, H. Rahnejat, and Z. M. Jin, “Contact mechanics of viscoelastic layered surface,” in Contact Mechanics III, M. H. Aliabadi and A. Samartin, Eds., pp. 59–68, Computational Mechanics Publications, Boston, Mass, USA, 1997.
  42. 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.
  43. D. J. Whitehouse and J. F. Archard, “The properties of random surfaces of significance in their contact,” Proceedings of the Royal Society of London. Series A, vol. 316, pp. 97–121, 1970.
  44. J. R. M. Radok, “Visco-elastic stress analysis,” Quarterly of Applied Mathematics, vol. 15, pp. 198–202, 1957. View at Zentralblatt MATH · View at MathSciNet
  45. 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.
  46. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  47. T. Junisbekov, V. Kestelman, and N. Malinin, Stress Relaxation in Viscoelastic Materials, Science Publishers, Enfield, NH, USA, 2nd edition, 2003.
  48. 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. View at Publisher · View at Google Scholar
  49. J. Boyle and J. Spencer, Stress Analysis for Creep, Butterworths-Heinemann, London, UK, 1st edition, 1983.
  50. I. H. Shames and F. A. Cozzarelli, Elastic and Inelastic Stress Analysis, Prentice-Hall International, Englewood Cliffs, NJ, USA, 1992.
  51. G. E. Roberts and H. Kaufman, Table of Laplace Transforms, W. B. Saunders, Philadelphia, Pa, USA, 1966. View at MathSciNet
  52. L. J. Slater, Confluent Hypergeometric Functions, Cambridge University Press, New York, NY, USA, 1960. View at MathSciNet
  53. W. Nowacki, Thermoelasticity, Pergamon Press, Oxford, UK, 2nd edition, 1986. View at MathSciNet
  54. 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.