Critical comparison of the boundary effect model with cohesive crack model and size effect law

Christian Carloni, Gianluca Cusatis, Marco Salviato*, Jia Liang Le, Christian G. Hoover, Zdeněk P. Bažant

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

34 Scopus citations


For several decades it has been clear that the size effect on structural strength, exhibiting a major non-statistical component, is a quintessential property of all quasibrittle materials. However, progress in design codes and practice for these materials has been retarded by protracted controversies about the proper mathematical form and justification of the size effect law (SEL). A fresh exception is the American Concrete Institute which, in 2019, becomes the first concrete code-making society to adopt the SEL based on quasibrittle fracture mechanics. This article begins by discussing several long-running controversies that have recently abated, and then focuses critically on the so-called Boundary Effect Model (BEM), promoted for concrete relentlessly for two decades, in ever-changing versions, by Xiaozhi Hu et al. The BEM is here compared to the quasibrittle SEL based on asymptotic matching. Its errors, weaknesses and inconsistencies are identified—including incorrect large- and small-size asymptotic size effects, conflicts with broad-range comprehensive test data and with the cohesive crack model, incorrect aggregate-size dependence of strength, illogical dependence on ligament stress profile, inability to capture the statistical part of size effect at large sizes, simplistic effect of boundary proximity, and lack of distinction between Type 1 and 2 size effects. In contrast to the SEL, the BEM is not applicable to mixed and shear fracture modes and to complex geometries of engineering structures, and is not transplantable from concrete to other quasibrittle materials. The purpose of this critique is to help crystallize a consensus about the proper size effect formulation, not only for concrete structures but also, and mainly, for other quasibrittle materials and structures, including airframes made of fiber composites, ceramic components and micrometer-scale devices, and for failure assessments of sea ice, rock, stiff soils, bone, and various bio- or bio-mimetic materials, for all of which the non-statistical size effect is yet to be widely accepted in practice.

Original languageEnglish (US)
Pages (from-to)193-210
Number of pages18
JournalEngineering Fracture Mechanics
StatePublished - Jun 15 2019


  • Analysis of experimental data
  • Concrete structures
  • Design codes
  • Energetic size effect
  • Fracture mechanics
  • Quasibrittle materials
  • Size effect justification
  • Size effect law
  • Statistical size effect

ASJC Scopus subject areas

  • Materials Science(all)
  • Mechanics of Materials
  • Mechanical Engineering


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