International Journal of Fracture

, Volume 184, Issue 1–2, pp 137–149 | Cite as

Statistics of ductile fracture surfaces: the effect of material parameters

  • Laurent Ponson
  • Yuanyuan Cao
  • Elisabeth Bouchaud
  • Viggo Tvergaard
  • Alan Needleman


The effect of material parameters on the statistics of fracture surfaces is analyzed under small scale yielding conditions. Three dimensional calculations of ductile crack growth under mode I plane strain, small scale yielding conditions are carried out using an elastic-viscoplastic constitutive relation for a progressively cavitating plastic solid with two populations of void nucleating second phase particles represented. Large particles that result in void nucleation at an early stage are modeled discretely while small particles that require large strains to nucleate are homogeneously distributed. The three dimensional analysis permits modeling of a three dimensional material microstructure and of the resulting three dimensional stress and deformation states that develop in the fracture process region. Material parameters characterizing void nucleation are varied and the statistics of the resulting fracture surfaces is investigated. All the fracture surfaces are found to be self-affine over a size range of about two orders of magnitude with a very similar roughness exponent of \(0.56\,\pm \,0.03\). In contrast, the full statistics of the fracture surfaces is found to be more sensitive to the material microscopic fracture properties: height fluctuations are shown to crossover from a Student’s distribution with power law tails at small scales to a Gaussian behavior at large scales, but this transition occurs at a material dependent length scale. Using the family of Student’s distributions, this transition can be described introducing an additional exponent \(\mu = 0.15\,\pm \,0.02\), the value of which compares well with recent experimental findings. The description of the roughness distribution used here gives a more complete quantitative characterization of the fracture surface morphology which allows a better comparison with experimental data and an easier interpretation of the roughness properties in terms of microscopic failure mechanisms.


Fracture surfaces Roughness statistics Ductile fracture Crack growth Scaling behavior Finite elements 



The financial support provided by the U.S. National Science Foundation through Grant CMMI-1200203 and by the European Union through the ToughBridge Marie Curie Grant (LP) is gratefully acknowledged. We also thank Jean-Philippe Bouchaud and Stéphane Vernède for fruitful discussions


  1. Alava MJ, Nukala PKVV, Zapperi S (2006) Statististical models of fracture. Adv Phys 55:349–476CrossRefGoogle Scholar
  2. Belytschko T, Chiapetta RL, Bartel HD (1976) Efficient large scale non-linear transient analysis by finite elements. Int J Numer Methods Eng 10:579–596CrossRefGoogle Scholar
  3. Bonamy D, Ponson L, Prades S, Bouchaud E, Guillot C (2006) Scaling exponents for fracture surfaces in homogeneous glass and glassy ceramics. Phys Rev Lett 97:135504CrossRefGoogle Scholar
  4. Bonamy D, Bouchaud E (2011) Failure of heterogeneous materials: a dynamic phase transition? Phys Rep 498:1–44CrossRefGoogle Scholar
  5. Bouchaud E, Lapasset G, Planès J (1990) Fractal dimension of fractured surfaces: a universal value? Europhys Lett 13:73–79CrossRefGoogle Scholar
  6. Bouchaud E (2003) The morphology of fracture surfaces: a tool for understanding crack propagation in complex materials. Surf Rev Lett 10:797–814CrossRefGoogle Scholar
  7. Bouchbinder E, Mathiesen J, Procaccia I (2004) Roughening of fracture surfaces: the role of plastic deformation. Phys Rev Lett 92:245505CrossRefGoogle Scholar
  8. Cherepanov GP, Balankin AS, Ivanova VS (1995) Fractal fracture mechanics–a review. Eng Fract Mech 51:997–1033CrossRefGoogle Scholar
  9. Chu CC, Needleman A (1980) Void nucleation effects in biaxially stretched sheets. J Eng Mater Technol 102:249–256CrossRefGoogle Scholar
  10. Gurson AL (1975) Plastic flow and fracture behavior of ductile materials incorporating void nucleation, growth and interaction. Ph.D. Thesis, Brown University.Google Scholar
  11. Måløy KJ, Hansen A, Hinrichsen EL, Roux S (1992) Experimental measurements of the roughness of brittle cracks. Phys Rev Lett 68:213–215CrossRefGoogle Scholar
  12. Mandelbrot BB, Passoja DE, Paullay AJ (1984) Fractal character of fracture surfaces of metals. Nature 308:721–722CrossRefGoogle Scholar
  13. Morel S, Bonamy D, Ponson L, Bouchaud E (2008) Transient damage spreading and anomalous scaling in mortar crack surfaces. Phys Rev E 78:016112CrossRefGoogle Scholar
  14. Nakano A, Kalia RK, Vashista P (1995) Dynamics and morphology of brittle cracks: a molecular-dynamics study of silicon nitride. Phys Rev Lett 75:3138–3141CrossRefGoogle Scholar
  15. Needleman A, Tvergaard V, Bouchaud E (2012) Prediction of ductile fracture surface roughness scaling. J Appl Mech 79:031015Google Scholar
  16. Peirce D, Shih CF, Needleman A (1984) A tangent modulus method for rate dependent solids. Comput Struct 18:875–887CrossRefGoogle Scholar
  17. Ponson L, Auradou H, Vié P, Hulin JP (2006) Low self-affine exponents of fractured glass ceramics surfaces. Phys Rev Lett 97:125501CrossRefGoogle Scholar
  18. Ponson L, Bonamy D, Bouchaud E (2006) Two dimensional scaling properties of experimental fracture surfaces. Phys Rev Lett 96:035506 Google Scholar
  19. Ponson L, Bonamy D, Auradou H, Mourot G, Morel S, Bouchaud E, Guillot C, Hulin J-P (2006) Anisotropic self-affine properties of experimental fracture surfaces. Int Fract 140:27–36Google Scholar
  20. Ponson L (2007) Crack propagation in disordered materials: how to decipher fracture surfaces. Ann Phys 32:1–120Google Scholar
  21. Ponson L, Auradou H, Pessel M, Lazarus V, Hulin J-P (2007) Failure mechanisms and surface roughness statistics of fractured Fontainebleau sandstone. Phys Rev E 76:036108CrossRefGoogle Scholar
  22. Ramanathan S, Ertas D, Fisher DS (1997) Quasistatic crack propagation in heterogeneous media. Phys Rev Lett 79:873–876CrossRefGoogle Scholar
  23. Santucci S, Måløy KJ, Delaplace A, Mathiesen J, Hansen A, Bakke J, Schmittbuhl J, Vanel L, Ray P (2007) Statistics of fracture surfaces. Phys Rev E 75:016104CrossRefGoogle Scholar
  24. Tvergaard V (1981) Influence of voids on shear band instabilities under plane strain conditions. Int J Fract 17:389–407CrossRefGoogle Scholar
  25. Tvergaard V (1982a) On localization in ductile materials containing spherical voids. Int J Fract 18:237–252Google Scholar
  26. Tvergaard V (1982b) Influence of void nucleation on ductile shear fracture at a free surface. J Mech Phys Solids 30:399–425Google Scholar
  27. Tvergaard V, Needleman A (1984) Analysis of the cup-cone fracture in a round tensile bar. Acta Metall 32:157–169CrossRefGoogle Scholar
  28. Vernède S, Cao Y, Bouchaud JP, Ponson L (2013) Extreme events and non-Gaussian statistics of experimental fracture surfaces (submitted for publication)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Laurent Ponson
    • 1
  • Yuanyuan Cao
    • 1
  • Elisabeth Bouchaud
    • 2
  • Viggo Tvergaard
    • 3
  • Alan Needleman
    • 4
  1. 1.Institut Jean le Rond d’Alembert (UMR 7190), CNRSUniversité Pierre et Marie CurieParisFrance
  2. 2.CEA-Saclay and ESPCIParis TechParisFrance
  3. 3.Department of Mechanical EngineeringThe Technical University of DenmarkLyngbyDenmark
  4. 4.Department of Materials Science and EngineeringUniversity of North TexasDentonUSA

Personalised recommendations