Multi-scale micromorphic theory for hierarchical materials

Franck Vernerey*, Wing Kam Liu, Brian Moran

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

159 Scopus citations


For the design of materials, it is important to faithfully model macroscopic materials response together with mechanisms and interactions occurring at the microstructural scales. While brute-force modeling of all the details of the microstructure is too costly, many of the current homogenized continuum models suffer from their inability to capture the correct underlying deformation mechanisms-especially when localization and failure are concerned. To overcome this limitation, a multi-scale continuum theory is proposed so that kinematic variables representing the deformation at various scales are incorporated. The method of virtual power is then used to derive a system of coupled governing equations, each representing a particular scale and its interactions with the macro-scale. A constitutive relation is then introduced to preserve the underlying physics associated with each scale. The inelastic behavior is represented by multiple yield functions, each representing a particular scale of microstructure, but collectively coupled through the same set of internal variables. The theory is illustrated by two applications. First, a one-dimensional example of a three-scale material is presented. After the onset of softening, the model shows that the localization zone is distributed according to two distinct length scale determined by the model. Second, a two-scale continuum model is introduced for the failure of porous metals. By comparing the theory to a direct numerical simulation (DNS) of the microstructure for a specimen in tension, we show that the model capture the main physics, and at the same time, remains computationally affordable.

Original languageEnglish (US)
Pages (from-to)2603-2651
Number of pages49
JournalJournal of the Mechanics and Physics of Solids
Issue number12
StatePublished - Dec 2007
Externally publishedYes


  • Finite elements
  • Inhomogeneous material
  • Microstructures
  • Multi-scale micromorphic theory
  • Plastic collapse

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering


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