Microplane model M5 with kinematic and static constraints for concrete fracture and anelasticity. I: Theory

Zdenek P Bazant*, Ferhun C. Caner

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

67 Scopus citations


Presented is a new microplane model for concrete, labeled M5, which improves the representation of tensile cohesive fracture by eliminating spurious excessive lateral strains and stress locking for far postpeak tensile strains. To achieve improvement, a kinematically constrained microplane system simulating hardening nonlinear behavior (nearly identical to previous Model M4 stripped of tensile softening) is coupled in series with a statically constrained microplane system simulating solely the cohesive tensile fracture. This coupling is made possible by developing a new iterative algorithm and by proving the conditions of its convergence. The special aspect of this algorithm (contrasting with the classical return mapping algorithm for hardening plasticity) is that the cohesive softening stiffness matrix (which is not positive definite) is used as the predictor and the hardening stiffness matrix as the corrector. The softening cohesive stiffness for fracturing is related to the fracture energy of concrete and the effective crack spacing. The postpeak softening slopes on the microplanes can be adjusted according to the element size in the sense of the crack band model. Finally, an incremental thermodynamic potential for the coupling of statically and kinematically constrained microplane systems is formulated. The data fitting and experimental calibration for tensile strain softening are relegated to a subsequent paper in this issue, while all the nonlinear triaxial response in compression remains the same as for Model M4.

Original languageEnglish (US)
Pages (from-to)31-40
Number of pages10
JournalJournal of Engineering Mechanics
Issue number1
StatePublished - Jan 1 2005


  • Concrete
  • Damage
  • Finite element method
  • Fracture
  • Inelastic action
  • Numerical models
  • Softening

ASJC Scopus subject areas

  • Mechanics of Materials
  • Mechanical Engineering


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