Précis of gap test results requiring reappraisal of line crack and phase-field models of fracture mechanics

Zdeněk P. Bažant*, A. Abdullah Dönmez, Hoang T. Nguyen

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


This paper presents a brief review of the recent advances in fracture mechanics at Northwestern University and Istanbul Technical University, prompted by the recent discovery of the gap test—a test that makes it easy and unambiguous to determine the effects of crack-parallel stresses on the mode-I fracture energy and, in consequence, on the nominal strength of structures of different sizes (aka, the size effect). The standard fracture specimens cannot reveal these effects since they have zero or negligible crack-parallel stresses. In addition, these effects cannot be reproduced by the standard, widely used, fracture models including the linear elastic fracture mechanics (LEFM), the cohesive crack model (CCM), as well as the popular computational models such as the extended finite element (XFEM) and the phase-field models (PFM). Therefore, it will be necessary to adopt fracture models that can reflect the tensorial damage behavior in the fracture process zone (FPZ), which is governed by at least two characteristic lengths, one for the FPZ length and one for the FPZ width. The modeling of elasto-plastic metals is even more complicated since the FPZ of micrometer-scale width is surrounded by a millimeter-scale plastic-hardening (yielding) zone. This role of the yielding zone has been understood well since the 1980s except for the scaling laws which are helpful for determining the effect of crack-parallel stresses more accurately. As a general conclusion, the line crack and phase field models cannot be used for practical problems with significant crack-parallel stress components (σxxzzxz). However, thanks to the finite width of its fracture front, the crack band model can be used, provided that its tensorial damage law is realistic. A new challenge for the nonlocal and gradient models is that they, too, will need to distinguish two independent material characteristic lengths, one for the direction of damage band and one transverse to it.

Original languageEnglish (US)
Article number113285
JournalEngineering Structures
StatePublished - Jan 1 2022


  • Cohesive crack model, Crack band model, Nonlocal models
  • Concrete
  • Fracture mechanics, Crack-parallel stresses
  • Fracture process zone
  • Fracture testing
  • Material characteristic length
  • Phase-field models
  • Quasibrittle behavior
  • T-stress
  • XFEM
  • Yielding zone

ASJC Scopus subject areas

  • Civil and Structural Engineering


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