A thermodynamically consistent approach to microplane theory Part I. Free energy and consistent microplane stresses

Ignacio Carol*, Milan Jirásek, Zdeněk Bažant

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

98 Scopus citations


Microplane models are based on the assumption that the constitutive laws of the material may be established between normal and shear components of stress and strain on planes of generic orientation (so-called microplanes), rather than between tensor components or their invariants. In the kinematically constrained version of the model, it is assumed that the microplane strains are projections of the strain tensor, and the stress tensor is obtained by integrating stresses on microplanes of all orientations at a point. Traditionally, microplane variables were defined intuitively, and the integral relation for stresses was derived by application of the principle of virtual work. In this paper, a new thermodynamic framework is proposed. A free-energy potential is defined at the microplane level, such that its integral over all orientations gives the standard macroscopic free energy. From this simple assumption, it is possible to introduce consistent microplane stresses and their corresponding integral relation to the macroscopic stress tensor. Based on this, it is shown that, in spite of the excellent data fits achieved, many existing formulations of microplane model were not guaranteed to be fully thermodynamically compliant. A consequence is the lack of work conjugacy between some of the microplane stress and strain variables used, and the danger of spurious energy dissipation/generation under certain load cycles. The possibilities open by the new theoretical framework are developed further in Part II companion paper.

Original languageEnglish (US)
Pages (from-to)2921-2931
Number of pages11
JournalInternational Journal of Solids and Structures
Issue number17
StatePublished - Mar 7 2001


  • Anisotropy
  • Concrete
  • Constitutive model
  • Damage
  • Microplane model
  • Plasticity
  • Thermodynamics

ASJC Scopus subject areas

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


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