Universal Effective Toughness Distribution for Heterogeneous Brittle Materials

  • S. Roux
  • Y. Charles
  • F. Hild
  • D. Vandembroucq
Conference paper
Part of the Iutam Bookseries book series (IUTAMBOOK, volume 10)


A brittle material whose microstructure is heterogeneous and random will display at large enough scales a deterministic brittle character. Prior to this limit, the effective macroscopic toughness of the material has a statistical distribution whose shape is discussed in the present paper. It is proposed that it has a universal shape characterized by only two parameters, namely, an asymptotic toughness, and a size-dependent width. Predictions of this theoretical result expressed in terms of crack size distribution are tested against experimental indentation data.


Brittleness toughness size-effects 


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  1. 1.
    Alava M.J., Nukala P.K.V.V. and Zapperi S., “Statistical models of fracture”, Adv. Phys. Vol. 55, n°3–4, pp. 349–476, 2006.CrossRefGoogle Scholar
  2. 2.
    Bouchaud E., “Scaling properties of cracks”, J. Phys. Cond. Mat. Vol. 9, n°3, pp. 4319–4344, 1997.CrossRefGoogle Scholar
  3. 3.
    Brazovskii S. and Nattermann T., “Pinning and sliding of driven elastic systems: from domain walls to charge density waves”, Adv. Phys., Vol. 53, n°2, pp. 177–252, 2004.CrossRefGoogle Scholar
  4. 4.
    Charles Y. and Hild F., “Crack arrest in ceramic/steel assemblies”, Int. J. Fract. Vol. 15, n°3, pp. 251–272, 2002.CrossRefGoogle Scholar
  5. 5.
    Charles Y., Hild F. and Roux S., “The issue of crack arrest”, ASME J. Eng. Mech. Tech. Vol. 125 n°3, pp. 333–340, 2003.CrossRefGoogle Scholar
  6. 6.
    Charles Y., Hild F., Roux S. and Vandembroucq D., “Material independent crack arrest statistics: application to indentation experiments”, Int. J. Fract., Vol. 142, pp. 51–67, 2006. Arxiv: cond-mat/0604156.CrossRefGoogle Scholar
  7. 7.
    Fisher D.S., “Sliding charge-density waves as a dynamic critical phenomenon”, Phys. Rev. B. Vol. 31, pp. 1396–1427, 1985.CrossRefGoogle Scholar
  8. 8.
    Gao H. and Rice J. R., “A first order perturbation analysis of crack trapping by arrays of obstacles”, ASME J. Appl. Mech. Vol. 56, pp. 828–836, 1989.zbMATHGoogle Scholar
  9. 9.
    Hansen A. and Schmittbuhl J., “Origin of the universal roughness exponent of brittle fracture surfaces: Stress-weighted percolation in the damage zone”, Phys. Rev. Lett. Vol. 90, pp. 045504, 2003.CrossRefGoogle Scholar
  10. 10.
    Jeulin D., “Fracture statistics models and crack propagation in random media”, Appl. Mech. Rev, Vol. 47 n°1, pp. 141–150, 1994.CrossRefGoogle Scholar
  11. 11.
    Kardar M., “Nonequilibrium dynamics of interfaces and lines”, Phys.Rep. Vol. 301, pp. 85–112, 1998.CrossRefGoogle Scholar
  12. 12.
    Katzav E., Adda-Bedia M. and Derrida B., “Fracture surfaces of heterogeneous materials: a 2D solvable model”, EuroPhys. Lett. Vol. 78, 46006, 2007. arxiv:cond-mat/0610185.CrossRefGoogle Scholar
  13. 13.
    Moretti P., Miguel M.C., Zaiser M. and Zapperi S., “Depinning transition of dislocation assemblies: pileups and low-angle grain boundaries”, Phys. Rev. B. Vol. 69, n°21, pp. 214103, 2004.CrossRefGoogle Scholar
  14. 14.
    Ponton C. B. and Rawlings R. D., “Vickers indentation fracture toughness test – Part 1 – review of literature and formulation of standardized indentation toughness equations”, Mat. Sci. Tech. Vol. 5, pp. 865–872, 1989.Google Scholar
  15. 15.
    Ponton C.B. and Rawlings R.D., “Vickers indentation fracture toughness test – Part 2 – application and evaluation of standardized indentation toughness equations”, Mat. Sci. Tech. Vol. 5, pp. 961–976, 1989.Google Scholar
  16. 16.
    Ramanathan S. and Fisher D.S., “Quasi-static crack propagation in heterogeneous media”, Phys. Rev. Lett. Vol. 79, pp. 873–876, 1997.CrossRefGoogle Scholar
  17. 17.
    Rolley E., Guthmann C., Gombrowicz R. and Repain V., “Roughness of the contact line on a disordered substrate”, Phys. Rev. Lett. Vol. 80, n°13, pp. 2865–2868, 1998.CrossRefGoogle Scholar
  18. 18.
    Rosso A. and Krauth W., “Roughness at the depinning threshold of long range elastic string”, Phys. Rev. E. Vol. 65, pp. 025101, 2002.CrossRefGoogle Scholar
  19. 19.
    Roux S., Vandembroucq D. and Hild F., “Effective toughness of heterogeneous brittle materials”, Eur. J. Mech. A/Solids Vol. 22, n 5, pp. 743–749, 2003.Google Scholar
  20. 20.
    Schmittbuhl J., Roux S., Vilotte J.P. and Måløy K.J., “Interfacial crack pinning: effect of non local interaction”, Phys. Rev. Lett. Vol. 74, pp. 1787–1790, 1995.CrossRefGoogle Scholar
  21. 21.
    Schmittbuhl J. and Måløy K.J., “Direct observation of a self-affine crack propagation”, Phys. Rev. Lett. Vol. 78, pp. 3888–3891, 1997.CrossRefGoogle Scholar
  22. 22.
    Skoe R., Vandembroucq D. and Roux S., “Front propagation in random media: from extrenal to activated dynamics”, Int. J. Modern Physics C Vol. 13, pp. 751–757, 2002.CrossRefGoogle Scholar
  23. 23.
    Vandembroucq D. and Roux S., “Large scale simulations of ultrametric depinning”, Phys. Rev. E. Vol. 70, pp. 026103, 2004.CrossRefGoogle Scholar
  24. 24.
    Weibull W., “A statistical theory of the strength of materials”, Roy. Swed. Inst. Eng. Res., Vol. 151, 1939.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • S. Roux
    • 1
    • 2
  • Y. Charles
  • F. Hild
  • D. Vandembroucq
  1. 1.Laboratoire Surface du verre et InterfacesUnité Mixte de Recherche CNRS/Saint-GobainFrance
  2. 2.Laboratoire de Méecanique et TechnologieENS-Cachan, Université Paris VI and UMR CNRS 8535France

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