Microplane model M4 for concrete I: Formulation with work-conjugate deviatoric stress

Zdenek P. Bazant*, Ferhun C. Caner, Ignacio Carol, Mark D. Adley, Stephen A. Akers

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

290 Scopus citations


The first part of this two-part study presents a new improved microplane constitutive model for concrete, representing the fourth version in the line of microplane models developed at Northwestern University. The constitutive law is characterized as a relation between the normal, volumetric, deviatoric, and shear stresses and strains on planes of various orientations, called the microplanes. The strain components on the microplanes are the projections of the continuum strain tensor, and the continuum stresses are obtained from the microplane stress components according to the principle of virtual work. The improvements include (1) a work-conjugate volumetric deviatoric split-the main improvement, facilitating physical interpretation of stress components; (2) additional horizontal boundaries (yield limits) for the normal and deviatoric microplane stress components, making it possible to control the curvature at the peaks of stress-strain curves; (3) an improved nonlinear frictional yield surface with plasticity asymptote; (4) a simpler and more effective fitting procedure with sequential identification of material parameters; (5) a method to control the steepness and tail length of postpeak softening; and (6) damage modeling with a reduction of unloading stiffness and crack-closing boundary. The second part of this study, by Caner and Bazant, will present an algorithm for implementing the model in structural analysis programs and provide experimental verification and calibration by test data.

Original languageEnglish (US)
Pages (from-to)944-953
Number of pages10
JournalJournal of Engineering Mechanics
Issue number9
StatePublished - Sep 2000

ASJC Scopus subject areas

  • Mechanics of Materials
  • Mechanical Engineering


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