Microplane model M7 for plain concrete. I: Formulation

Ferhun C. Caner, Zdeněk P. Bažant*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

118 Scopus citations


Mathematical modeling of the nonlinear triaxial behavior and damage of such a complexmaterial as concrete has been a long-standing challenge in which progress has been made only in gradual increments. The goal of this study is a realistic and robust material model for explicit finite-element programs for concrete structures that computes the stress tensor from the given strain tensor and some history variables. Themicroplanemodels, which use a constitutive equation in a vectorial rather than tensorial form and are semimultiscale by virtue of capturing interactions among phenomena of different orientation, can serve this goal effectively. This paper presents a new concrete microplane model, M7, which achieves this goal much better than the previous versions M1-M6 developed at Northwestern University since 1985. The basic mathematical structure of M7 is logically correlated to thermodynamic potentials for the elastic regime, the tensile and compressive damage regimes, and the frictional slip regime. Given that the volumetric-deviatoric (V-D) split of strains is inevitable for distinguishing between compression failures at low and high confinement, the key idea is to apply the V-Dsplit only to the microplane compressive stress-strain boundaries (or strain-dependent yield limits), the sumof which is compared with the total normal stress from the microplane constitutive relation. This avoids the use of the V-D split of the elastic strains and of the tensile stress-strain boundary, which caused various troubles in M3-M6 such as excessive lateral strains and stress locking in far postpeak uniaxial extension, poor representation of unloading and loading cycles, and inability to represent high dilatancy under postpeak compression in lower-strength concretes. Moreover, the differences between high hydrostatic compression and compressive uniaxial strain are accurately captured by considering the compressive volumetric boundary as dependent on the principal strain difference. The model is verified extensively in the companion paper.

Original languageEnglish (US)
Pages (from-to)1714-1723
Number of pages10
JournalJournal of Engineering Mechanics
Issue number12
StatePublished - 2013


  • Concrete
  • Constitutive modeling
  • Cracking damage
  • Inelastic behavior
  • Numerical algorithm
  • Softening
  • Thermodynamic potentials
  • Unloading
  • Yield limits

ASJC Scopus subject areas

  • Mechanics of Materials
  • Mechanical Engineering


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