Scaling of quasibrittle fracture: Hypotheses of invasive and lacunar fractality, their critique and Weibull connection

Zdeněk P. Bažant*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

81 Scopus citations


Considerable progress has been achieved in fractal characterization of the properties of crack surfaces in quasibrittle materials such as concrete, rock, ice, ceramics and composites. Recently, fractality of cracks or microcracks was proposed as the explanation of the observed size effect on the nominal strength of structures. This explanation, though, has rested merely on intuitive analogy and geometric reasoning, and did not take into account the mechanics of crack propagation. In this paper, the energy-based asymptotic analysis of scaling presented in the preceding companion paper in this issue [1] is extended to the effect of fractality on scaling. First, attention is focused on the propagation of fractal crack curves (invasive fractals). The modifications of the scaling law caused by crack fractality are derived, both for quasibrittle failures after large stable crack growth and for failures at the initiation of a fractal crack in the boundary layer near the surface. Second, attention is focused on discrete fractal distribution of microcracks (lacunar fractals), which is shown to lead to an analogy with Weibull's statistical theory of size effect due to material strength randomness. The predictions ensuing from the fractal hypothesis, either invasive or lacunar, disagree with the experimentally confirmed asymptotic characteristics of the size effect in quasibrittle structures. It is also pointed out that considering the crack curve as a self-similar fractal conflicts with kinematics. This can be remedied by considering the crack to be an affine fractal. It is concluded that the fractal characteristics of either the fracture surface or the microcracking at the fracture front cannot have a significant influence on the law of scaling of failure loads, although they can affect the fracture characteristics.

Original languageEnglish (US)
Pages (from-to)41-65
Number of pages25
JournalInternational Journal of Fracture
Issue number1
StatePublished - 1997


  • Asymptotic analysis
  • Fractal cracks
  • Fracture energy release
  • Fracture mechanics
  • Invasive fractals
  • Lacunar fractals
  • Quasibrittle materials
  • Scaling
  • Size effect

ASJC Scopus subject areas

  • Computational Mechanics
  • Modeling and Simulation
  • Mechanics of Materials


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