About this Journal Submit a Manuscript Table of Contents
Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 706326, 10 pages
http://dx.doi.org/10.1155/2012/706326
Research Article

Study of the Fractal and Multifractal Scaling Intervening in the Description of Fracture Experimental Data Reported by the Classical Work: Nature 308, 721–722(1984)

1Physics Faculty, University of Bucharest, P.O. Box MG-11, 077125 Bucharest, Romania
2Physics Department, University “Politehnica” of Bucharest, Splaiul Independenţei, 060042 Bucharest, Romania

Received 9 September 2011; Accepted 4 October 2011

Academic Editor: Cristian Toma

Copyright © 2012 Liliana Violeta Constantin and Dan Alexandru Iordache. 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. Santa Fe Institute for Studies in the Sciences of Complexity: (i) Bulletin, 2-3 issues/year, 1987-present, (ii) Lectures, 1989-present, (iii) Proceedings, 1986-present.
  2. R. Dobrescu and D. A. Iordache, Complexity Modeling, Politehnica Press, Bucharest, Romania, 2007.
  3. R. Dobrescu and D. A. Iordache, Complexity and Information, Romanian Academy Printing House, Bucharest, Romania, 2010.
  4. P. W. Anderson, “More is different,” Science, vol. 177, no. 4047, pp. 393–396, 1972. View at Publisher · View at Google Scholar · View at PubMed
  5. P. W. Anderson, “Physics: the opening to complexity,” Proceedings of the National Academy of Sciences of the United States of America, vol. 92, no. 15, pp. 6653–6654, 1995. View at Publisher · View at Google Scholar · View at Scopus
  6. K. G. Wilson, “Renormalization group and critical phenomena,” Physical Review B, vol. 4, no. 9, pp. 3174–3183, 1971. View at Publisher · View at Google Scholar · View at Scopus
  7. I. Prigogine and G. Nicolis, Self-Organization in Non-Equilibrium Systems: From Dissipative Structures to Order through Fluctuations, Wiley, New York, NY, USA, 1977.
  8. S. Solomon and E. Shir, “Complexity; a science at 30,” Europhysics News, vol. 34, no. 2, pp. 54–57, 2003. View at Publisher · View at Google Scholar
  9. S. Solomon, “Evolving uniform and nonuniform cellular automata networks,” in Annual Reviews of Computational Physics, D. Stauffer , Ed., pp. 243–294, World Scientific, River Edge, NJ, USA, 1995.
  10. A. L. Barabási and R. Albert, “Emergence of scaling in random networks,” Science, vol. 286, no. 5439, pp. 509–512, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. R. Albert, H. Jeong, and A. L. Barabási, “Diameter of the world-wide web,” Nature, vol. 401, no. 6749, pp. 130–131, 1999. View at Publisher · View at Google Scholar · View at Scopus
  12. A. L. Barabási, “Complex networks: from the cell to human diseases,” in Proceedings of the International Workshop on Quantitative Biology, Bucharest, Romania, May 2007.
  13. K. Bhattacharya, G. Mukherjee, and S. S. Manna, “The international trade network,” in Econo-Physics of Markets and Business Networks, A. Chatterjee and B. K. Chakrabarti, Eds., New Economic Windows Series, p. 139, Springer, New York, NY, USA, 2008.
  14. S. Y. Chen, Wei Huang, and C. Cattani, “Traffic dynamics on complex networks: a survey,” Mathematical Problems in Engineering, vol. 2012, Article ID 732698, 23 pages, 2012. View at Publisher · View at Google Scholar
  15. E. G. Backhoum and C. Toma, “Dynamical aspects of macroscopic andquantum 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
  16. E. G. Backhoum and C. Toma, “Specific mathematical aspects of dynamics generated by coherence functions,” Mathematical Problems in Engineering, vol. 2011, Article ID 436198, 10 pages, 2011. View at Publisher · View at Google Scholar
  17. C. Cattani and P. Sterian, “Modelling the hyperboloid elastic shell,” Scientific Bulletin of University “Politehnica” of Bucharest, Series A, vol. 71, no. 4, pp. 37–44, 2009.
  18. A. A. Gukhman, Introduction to the Theory of Similarity, Academic Press, New York, NY, USA, 1965.
  19. G. I. Barenblatt, Dimensional Analysis, Gordon and Breach, New York, NY, USA, 1987.
  20. G. I. Barenblatt, Scaling, Self-Similarity and Intermediate Asymptotics, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, UK, 1996.
  21. C. Shannon, “The mathematical theory of communication,” Bell System Technical Journal, vol. 27, no. 3, pp. 379–423, 1948.
  22. C. Shannon, “The mathematical theory of communication,” Bell System Technical Journal, vol. 27, no. 4, pp. 623–656, 1948.
  23. C. Shannon, “Prediction and entropy of printed English,” Bell System Technical Journal, vol. 30, no. 1, pp. 50–64, 1951.
  24. A. Carpinteri and B. Chiaia, “Multifractal nature of concrete fracture surfaces and size effects on nominal fracture energy,” Materials and Structures, vol. 28, no. 8, pp. 435–443, 1995. View at Publisher · View at Google Scholar · View at Scopus
  25. A. Carpinteri and B. Chiaia, “Multifractal scaling laws in the breaking behaviour of disordered materials,” Chaos, Solitons and Fractals, vol. 8, no. 2, pp. 135–150, 1997. View at Scopus
  26. D. A. Iordache, P. P. Delsanto, Şt. Puşcă, and V. Iordache, “Complexity, similitude, and fractals in physics,” in Proceedings of the 2nd International Symposium on “Interdisciplinary Applications of Fractal Analysis” (IAFA '05), vol. 3, pp. 7–12, Politehnica Press Publishing House, Bucharest, Romania, May 2005.
  27. P. P. Delsanto, D. A. Iordache, and Şt. Puşcă, “Study of the correlations between different effective fractal dimensions used for fracture parameters descriptions,” in Interdisciplinary Applications of Fractal and Chaos, R. Dobrescu and C. Vasilescu, Eds., pp. 136–153, Romanian Academy Printing House, Bucharest, Romania, 2004.
  28. P. P. Delsanto, A. S. Gliozzi, C. L. E. Bruno, N. Pugno, and A. Carpinteri, “Scaling laws and fractality in the framework of a phenomenological approach,” Chaos, Solitons and Fractals, vol. 41, no. 5, pp. 2782–2786, 2009. View at Publisher · View at Google Scholar · View at Scopus
  29. 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 · View at Scopus
  30. Y. Ma and T. G. Langdon, “Observations on the use of a fractal model to predict superplastic ductility,” Scripta Metallurgica et Materiala, vol. 28, no. 2, pp. 241–246, 1993.
  31. D. A. Iordache, Şt. Puşcă, G. Toma, V. Păun, A. Sterian, and C. Morarescu, “Analysis of compatibility with experimental data of Fractal descriptions of the fracture parameters,” Lecture Notes in Computer Science, vol. 3980, pp. 804–813, 2006. View at Publisher · View at Google Scholar
  32. E. E. Underwood and K. Banerji, “Fractals in fractography,” Materials Science and Engineering, vol. 80, no. 1, pp. 1–14, 1986. View at Scopus
  33. C. S. Pande, L. E. Richards, N. Louat, B. D. Dempsey, and A. J. Schwoeble, “Fractal characterization of fractured surfaces,” Acta Metallurgica, vol. 35, no. 7, pp. 1633–1637, 1987. View at Scopus
  34. C. W. Lung and Z. Q. Mu, “Fractal dimension measured with perimeter-area relation and toughness of materials,” Physical Review B, vol. 38, no. 16, pp. 11781–11784, 1988. View at Publisher · View at Google Scholar · View at Scopus
  35. Z. H. Huang, J. F. Tian, and Z. G. Wang, “Study of the slit island analysis as a method for measuring fractal dimension of fractured surface,” Scripta Metallurgica et Materialia, vol. 24, no. 6, pp. 962–972, 1990. View at Scopus
  36. R. E. Williford, “Multifractal fracture,” Scripta Metallurgica, vol. 22, no. 11, pp. 1749–1754, 1988. View at Scopus
  37. Z.Q. Mu and C. W. Lung, “Studies on the fractal dimension and fracture toughness of steel,” Journal of Physics D, vol. 21, no. 5, p. 848, 1988. View at Publisher · View at Google Scholar
  38. H. Su, Y. Zhang, and Z. Yan, Acta Metallurgica Sinica, vol. 3, p. 226, 1990.
  39. Şt. Puşcă, V. Solcan, and D. A. Iordache, “Procedure of extraction of numerical data from experimental plots. Application to the numerical analysis of some fractal studies,” in Proceedings of the 5th General Conference of the Balkan Physical Union, pp. 259–264, Vrnjacka Banja, Serbia, August 2003.
  40. J. F. Gouyet, Physique et Structures Fractales, Masson, Paris, France, 1992.
  41. D. L. Davidson, “Fracture surface roughness as a gauge of fracture toughness: aluminium-particulate SiC composites,” Journal of Materials Science, vol. 24, no. 2, pp. 681–687, 1989. View at Publisher · View at Google Scholar · View at Scopus
  42. K. Levenberg, “A method for the solution of certain problems in least squares,” Quarterly of Applied Mathematics, vol. 2, pp. 164–168, 1944.
  43. D. W. Marquardt, “An algorithm fot least-squares estimation of nonlinear parameters,” Journal of the Society for Industrial and Applied Mathematics, vol. 11, no. 2, pp. 431–441, 1963.
  44. E. Bodegom, D. W. McClure, P. P. Delsanto et al., Computational Physics Guide, Politehnica Press, Bucharest, Romania, 2009.
  45. W.T. Eadie, D. Drijard, F. E. James, M. Roos, and B. Sadoulet, Statistical Methods in Experimental Physics, North-Holland, Amsterdam, The Netherlands, 1982.
  46. W. Ledermann, Ed., Handbook of Applied Mathematics, vol. 6 of Statistics, John Wiley & Sons, New York, NY, USA, 1984.
  47. P. W. M. John, Statistical Methods in Engineering and Quality Assurance, John Wiley & Sons, New York, NY, USA, 1990.
  48. D. A. Iordache and V. Iordache, “Compatibility of multi-fractal and similitude descriptions of the fracture parameters relative to the experimental data for concrete specimens,” in Proceedings of the 1st South-East European Symposium on Interdisciplinary Approaches in Fractal Analysis, pp. 55–60, Bucharest, Romania, May 2003.
  49. A. Carpinteri and G. Ferro, “Scaling behaviour amd dual renormalization of experimental tensile softening responses,” Materials and Structures, vol. 31, no. 5, pp. 303–309, 1998. View at Publisher · View at Google Scholar
  50. M. R. A. van Vliet, Size effects in tensile fracture of concrete and rock, Ph.D. thesis, University of Delft, 2000.