A framework for microplane models at large strain, with application to hyperelasticity

Ignacio Carol*, Milan Jirásek, Zdeněk P. Bažant

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

49 Scopus citations


A general framework is proposed for the formulation of microplane models at large strain. It is based on the thermodynamic approach to microplane formulation recently presented by the authors, which defines the macroscopic free energy of the material as an integral of a microplane free-energy potential over all possible orientations. By simple differentiation with respect to strain, it is possible to obtain the consistent definition of microplane stresses and integral expressions for evaluation of the macroscopic stress tensor. To apply this approach to large strains, new microplane strain measures need to be defined, including volume change, stretch of fibers, "thickening" of planes, deviatoric parts of the stretch and thickening, and distortion (shear) angles. Based on these, various microplane formulations are developed. Each formulation starts with the definition of microplane stresses and the derivation of the integral expressions which are valid for the general case of dissipative materials. Then, these expressions are particularized to specific forms of hyperelastic potentials leading to various hyperelastic models. The simplest model, with a quadratic microplane potential in terms of the fiber stretch, corresponds to the classical Gaussian statistical theory of long-chain molecules and leads to the neo-Hookean type of macroscopic free-energy potential. Many other, more complex forms of the microplane potential are investigated and their relation to existing models for rubber elasticity is analyzed. It is shown that, in the small-strain limit, they collapse into well-known small-strain microplane formulations, either with restricted or with unrestricted values of Poisson's ratio.

Original languageEnglish (US)
Pages (from-to)511-557
Number of pages47
JournalInternational Journal of Solids and Structures
Issue number2
StatePublished - Jan 2004


  • Anisotropy
  • Constitutive model
  • Finite elasticity
  • Finite strain
  • Hyperelasticity
  • Large deformations
  • Large strain
  • Microplane model
  • Rubber elasticity

ASJC Scopus subject areas

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


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