Size effect in Paris law and fatigue lifetimes for quasibrittle materials: Modified theory, experiments and micro-modeling

Kedar Kirane, Zdeněk P. Bažant*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

76 Scopus citations


Unlike brittle materials, quasibrittle materials exhibit a structure size effect on the fatigue crack growth rate, particularly on the Paris law coefficient (prefactor). This size effect is strong for specimens not much larger than the dominant material inhomogeneities (or aggregate sizes in concrete), and vanishes for very large structures. It can be quantified by a size adjustment of Paris law which is similar to the size effect law for monotonic loading. But the transitional size D0c at which the transition is centered is not the same. Previous experiments aimed at quantitative analysis of this size effect involved only one or a few specimens per size. Thus the huge scatter, inevitable in fatigue tests, distorted the estimates of D0c and, thereby, also of the size of the cyclic fracture process zone (FPZ), to which D0c is proportional. Here, more reliable estimates of D0c and the cyclic FPZ size are obtained by conducting, on concrete, multiple fatigue tests per size and taking the average. Furthermore, these length characteristics are also estimated numerically using the latest version of the microplane constitutive damage model for concrete (model M7), extended to quasibrittle fatigue. It is conclusively shown that the cyclic FPZ is smaller than the monotonic FPZ. Further, the numerical simulations of the cyclic deformations within the FPZ reveal that the D0c obtained from the previous form of the size-adjusted Paris law is not proportional to the cyclic FPZ size. A new form is proposed and verified by updated dimensional analysis. It involves the transitional sizes for both the monotonic and cyclic size effects and is seen to yield values of D0c that are proportional to the cyclic FPZ size. The ensuing size effect on fatigue lifetimes is simulated using both the previous and new forms. Both forms are found to predict a non-monotonic size effect on the lifetimes, initially decreasing and, after a minimum, eventually increasing with increasing size. It is also shown that, similar to Paris law for brittle fatigue, an extension to the fatigue threshold at nearly vanishing amplitude is impossible because of a transition to Charles-Evans law for static fatigue. Finally, a possible ramification to fatigue of micrometer-scale metallic devices is pointed out.

Original languageEnglish (US)
Pages (from-to)209-220
Number of pages12
JournalInternational Journal of Fatigue
StatePublished - Feb 2016


  • Cyclic fracture process zone
  • Dimensional analysis
  • Fatigue lifetime
  • Quasibrittle materials
  • Scaling of fatigue

ASJC Scopus subject areas

  • Modeling and Simulation
  • General Materials Science
  • Mechanics of Materials
  • Mechanical Engineering
  • Industrial and Manufacturing Engineering


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