Strain-softening bar and beam: Exact non-local solution

Zdeněk P. Bažant*, Aleksander Zubelewicz

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

17 Scopus citations


Using the recently developed imbricate non-local continuum approach, zones of strain softening (distributed microcracking) which have a finite size can be modeled. A differential approximation of the averaging integrals for the non-local continuum makes it possible to obtain exact analytical solutions for uniaxiul softening in a bar or for flexural softening in a beam. The differential equations of the problem along with the essential and natural boundary conditions and the conditions at the interface between the softening and non-softening regions are derived by a variational procedure based on the principle of virtual work. The failure due to strain softening is analyzed as a stability problem. In contrast to the blunt crack band model, the size of the strain-softening region is treated as an unknown to be solved by stability analysis. Numerical results show that the size of the strain-softening region is approximately constant, and so is the energy dissipated due to failure. Ductility diagrams, giving the strain at failure as a function of beam size and support stiffness are also calculated and are found to he quite similar to those obtained previously by local analysis with an assumed size of the softening region. These conclusions lend further support to the use of a blunt crack band model for localized cracking.

Original languageEnglish (US)
Pages (from-to)659-673
Number of pages15
JournalInternational Journal of Solids and Structures
Issue number7
StatePublished - 1988

ASJC Scopus subject areas

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

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