Structural strength scaling law for fracture of plastic-hardening metals and testing of fracture properties

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

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


The small-scale yielding fracture of plastic-hardening metals is a well-understood theory, essentially conceived by Hutchinson, Rice and Rosengren (hence the name HRR theory). However, even though specimens of rather different sizes have been tested to verify the small-scale yielding theory, an analytical scaling law for the size effect transition from elastic–plastic behavior through small- and large-scale yielding to fracture process zone has apparently not been formulated. Such a scaling law would be useful for the design as well as measurement of mode-I ductile fracture properties of metals, and is the aim of this study. Unlike the fracture of quasibrittle materials such as concrete or composites, the modeling of plastic-hardening materials is complicated by a millimeter scale singular yielding zone that forms between the micrometer-scale fracture process zone (FPZ) and the elastic (unloading) material on the outside. Essential for the large-scale transitional size effect is the effective yielding zone size, which is here calculated from the equivalence of the virtual works within the plastic-hardening zone and elastic singular stress fields within the transition zone, and is shown to depend on the crack-parallel T-stress. The size effect analysis requires taking into account not only the dissipation in the FPZ delivered by the J-integral flux of energy through the yielding zone, but also the energies released from the structure and from the unloaded band of plasticized material trailing the advancing yielding zone. Equating the rates of energy releases and energy dissipation leads to an approximate energetic size effect (scaling) law that matches the calculated small- and large-size asymptotic behaviors, when the crack ligament contains the yielding zone. The law is similar to that for quasibrittle fracture but its coefficients depend on the fracture energy and the yielding zone size in a different way. This law, reducible to linear regression, can be exploited for size effect testing of fracture energy (or critical J-integral) and effective size rp of the yielding zone. An effect of high crack-parallel stress T on rp is likely but is relegated to future study, as it would not affect the scaling law derived. For testing of the transition from the small-size range (large-scale yielding) to the large-size range (small-scale yielding), a modified size effect method, requiring nonlinear optimization, is developed. The size effect law is verified by scaled tests of notched specimens of aluminum.

Original languageEnglish (US)
Article number101141
JournalExtreme Mechanics Letters
StatePublished - Feb 2021


  • Asymptotic matching
  • Ductile fracture
  • Energy dissipation
  • Failure load scaling
  • Fracture energy release
  • J-integral
  • Plastic-hardening metals
  • Size effect on structure strength
  • Small-scale yielding fracture
  • Testing of aluminum

ASJC Scopus subject areas

  • Bioengineering
  • Chemical Engineering (miscellaneous)
  • Engineering (miscellaneous)
  • Mechanics of Materials
  • Mechanical Engineering


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