Scaling of Effective Moduli of Generalised Continua

  • A.V. Dyskin
  • E. Pasternak
Conference paper
Part of the Iutam Bookseries book series (IUTAMBOOK, volume 10)


We model materials with self-similar structure by a set of continua, CH, each representing the original material with all structural elements of sizes smaller than H. We then continue this sequence in a self-similar manner to H→ 0. The intersection of these continua gives a fractal F where all fields scale according to the power law. By considering scaling of the energy in linear elastic Cosserat continuum we find that if the classical elastic moduli scale with an exponent α, the Cosserat moduli scale with exponents α and α+2. Using this rule, we determine the dispersion relations for travelling waves and find the full set of the exponents for the cases of material with cracks and pores.


Self-similar approximation fractal approximation Cosserat continuum effective characteristics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Moore CA, Gill J. “Fractals in Geomechanics”, Computer Methods and Advances in Geomechanics, Beer G, Booker JR, Carter JP (eds.), Balkema, Rotterdam, 1991, pp. 371–376.Google Scholar
  2. 2.
    Xie H, Sanderson DJ, Peacock DCP. “A fractal model and energy dissipation for en echelon fractures”, Engng Fract. Mech. vol. 48, no. 5, pp. 655–662, 1994.CrossRefGoogle Scholar
  3. 3.
    Panagiotopoulos PD, Panagouli O. “Mechanics of fractal bodies. Data compression using fractals”, Chaos Solitons Fractals, vol. 8, no. 2, pp. 253–267, 1997.zbMATHCrossRefGoogle Scholar
  4. 4.
    Strichartz R. “Analysis on fractals”, Not. AMS, vol. 12, no. 10, pp. 1199–1208, 1999.MathSciNetGoogle Scholar
  5. 5.
    Rupnowski P. “Calculations of J integrals around fractal defects in plates”, Int. J. Fract. vol.111, pp. 381–394, 2001.CrossRefGoogle Scholar
  6. 6.
    Carpinteri A, Cornetti P. “A fractional calculus approach to the description of stress and strain localization in fractal media”, Chaos Solitons Fractals, vol. 13, pp. 85–94, 2002.zbMATHCrossRefGoogle Scholar
  7. 7.
    Zosimov VV, Lyamishev LM. “Fractals in wave processes”, Physics-Uspekhi, vol. 38, no. 4, pp. 347–384, 1995.CrossRefGoogle Scholar
  8. 8.
    Makris N “Generalized differentiation and the complex memory of structures”, Fractals, vol. 2 no. 2, pp. 315–320, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Podubny I. Fractional Differential Equations, San-Diego, Boston, New York, London, Sydney, Tokyo, Toronto, Academic Press, 1999.Google Scholar
  10. 10.
    Gol’dstein RV, Mosolov AB. “Cracks with a fractal surface”, Sov. Phys. Dokl. vol. 36 no. 8, pp. 603–605, 1991.Google Scholar
  11. 11.
    Bazant ZP. “Scaling laws in mechanics of failure”, J. Engg Mech. vol. 119 no. 9, pp. 1828–1844, 1993.CrossRefGoogle Scholar
  12. 12.
    Carpinteri A. “Scaling laws and renormalization groups for strength and toughness of disordered materials”, Int J. Solids Struct. vol. 31, no. 3, pp. 291–302, 1994.zbMATHCrossRefGoogle Scholar
  13. 13.
    Cherepanov GP, Balankin AS, Ivanova VS. “Fractal fracture mechanics – a review”, Engg. Fract. Mech. vol. 51, no. 6, pp. 997–1033, 1995.CrossRefGoogle Scholar
  14. 14.
    Borodich FM. “Some fractal models of fracture”, J. Mech. Phys. Solids, vol. 45, no. 2, pp. 239–259, 1997.zbMATHCrossRefGoogle Scholar
  15. 15.
    Yavari A, Sarkani S, Moyer, Jr. ET. “The mechanics of self-similar and self-affine fractal cracks”, Int. J Fract. vol. 114, pp. 1–27, 2002.CrossRefGoogle Scholar
  16. 16.
    Dyskin AV. “Effective characteristics and stress concentrations in materials with self-similar microstructure”, Int. J. Solids Struct, vol. 42 no. 2, pp. 477–502, 2004.CrossRefGoogle Scholar
  17. 17.
    Borodich FM. “Fractals and fractal scaling in fracture mechanics”, Int. J Fract. vol. 95, pp. 239–259, 1999.CrossRefGoogle Scholar
  18. 18.
    Saouma VE, Fava G. “On fractals and size effects”, Int. J Fract. vol. 137, pp. 231–249, 2006.CrossRefGoogle Scholar
  19. 19.
    Dyskin AV. “Multifractal properties of self-similar stress distributions”, Philoso. Mag. vol. 86, no. 21–22, pp. 3117–3136, 2006.CrossRefGoogle Scholar
  20. 20.
    Tada H, Paris PC, Irwin GR. The Stress Analysis of Cracks. Handbook. Third edition. vol. II. New York: ASME Press, 1985.Google Scholar
  21. 21.
    Dyskin AV. “Continuum fractal mechanics of Earth’s crust”, Pure Appl. Geophys. vol. 161, pp. 1979–1989, 2004.CrossRefGoogle Scholar
  22. 22.
    Barenblatt GI, Botvina LR. “Application of the similarity method to damage calculation and fatigue crack growth studies”, Defects and Fracture Sih GC, Zorski H (eds.), Martinus Nijhoff Publishers 1980, pp. 71–79.Google Scholar
  23. 23.
    Achieser NI. Theory of Approximation. New York, Frederic Unger Publ. Co, 1956.zbMATHGoogle Scholar
  24. 24.
    Nowacki W. Theoria Spr¸zystości. Warszawa, Państwowe Wydawnictwo Naukowe, 1970.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • A.V. Dyskin
    • 1
  • E. Pasternak
  1. 1.School of Civil and Resource EngineeringThe University of Western AustraliaCrawleyAustralia

Personalised recommendations