Universal size-shape effect law based on comprehensive concrete fracture tests

Christian G. Hoover, Zdeněk P. Bažant*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

125 Scopus citations


The universal size-shape effect law is a law that describes the dependence of nominal strength of specimen or structure on both its size and the crack (or notch) length, over the entire range of interest, and exhibits the correct small-size and large-size asymptotic properties as required by the cohesive crackmodel (or crack bandmodel). Themain difficulty has been the transition of crack length from0, inwhich case the size effect is Type 1, to deep cracks (or notches), in which case the size effect is Type 2 and is fundamentally different fromType 1, with different asymptotes. In this transition, the problem is not linearizable because the notch is not much larger than the fracture process zone. The previously proposed universal lawcould not be verified experimentally for the Type 1-Type 2 transition because sufficient test datawere lacking. The current study is based on recently obtained comprehensive fracture test data for three-point bend beams cast from one batch of the same concrete and cured and tested under identical conditions. The test data reveal that the Type 1-Type 2 transition in the previous universal law has insufficient accuracy and cannot be captured by Taylor series expansion of the energy release rate function of linear elastic fracturemechanics. Instead, the size effect for a zero notch and for the transitional range is nowcharacterized in terms of the strain gradient at the specimen surface,which is themain variable determining the degree of stress redistribution by the boundary layer of cracking. The new universal law is shown to fit the comprehensive data quite well, with a coefficient of variation of only 2.3%.

Original languageEnglish (US)
Pages (from-to)473-479
Number of pages7
JournalJournal of Engineering Mechanics
Issue number3
StatePublished - 2014


  • Asymptotics of fracture
  • Cohesive crack
  • Failure of structures
  • Quasi-brittle fracture
  • Scaling
  • Statistics of experimental data

ASJC Scopus subject areas

  • Mechanics of Materials
  • Mechanical Engineering


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