Nonlocal microplane model with strain-softening yield limits

Zdeněk P. Bažant*, Giovanni Di Luzio

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

76 Scopus citations


The paper deals with the problem of nonlocal generalization of constitutive models such as microplane model M4 for concrete, in which the yield limits, called stress-strain boundaries, are softening functions of the total strain. Such constitutive models call for a different nonlocal generalization than those for continuum damage mechanics, in which the total strain is reversible, or for plasticity, in which there is no memory of the initial state. In the proposed nonlocal formulation, the softening yield limit is a function of the spatially averaged nonlocal strains rather than the local strains, while the elastic strains are local. It is demonstrated analytically as well numerically that, with the proposed nonlocal model, the tensile stress across the strain localization band at very large strain does soften to zero and the cracking band retains a finite width even at very large tensile strain across the band only if one adopts an "over-nonlocal" generalization of the type proposed by Vermeer and Brinkgreve [In: Chambon, R., Desrues, J., Vardoulakis, I. (Eds.), Localisation and Bifurcation Theory for Soils and Rocks, Balkema, Rotterdam, 1994, p. 89] (and also used by Planas et al. [Basic issue of nonlocal models: uniaxial modeling, Tecnical Report 96-jp03, Departamento de Ciencia de Materiales, Universidad Politecnica de Madrid, Madrid, Spain, 1996], and by Strömberg and Ristinmaa [Comput. Meth. Appl. Mech. Eng. 136 (1996) 127]). Numerical finite element studies document the avoidance of spurious mesh sensitivity and mesh orientation bias, and demonstrate objectivity and size effect.

Original languageEnglish (US)
Pages (from-to)7209-7240
Number of pages32
JournalInternational Journal of Solids and Structures
Issue number24-25
StatePublished - Dec 2004


  • Concrete
  • Damage
  • Finite elements
  • Fracture
  • Microplane model
  • Nonlocal continuum
  • Numerical methods
  • Quasibrittle materials
  • Softening

ASJC Scopus subject areas

  • Modeling and Simulation
  • Materials Science(all)
  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering
  • Applied Mathematics


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