Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. II: The vigdergauz microstructure

Yury Grabovsky, Robert V. Kohn

Research output: Contribution to journalArticle


For modeling coherent phase transformations, and for applications to structural optimization, it is of interest to identify microstructures with minimal energy or maximal stiffness. The existence of a particularly simple microstructure with extremal elastic behavior, in the context of two-phase composites made from isotropic components in two space dimensions, has previously been shown. This "Vigdergauz microstructure" consists of a periodic array of appropriately shaped inclusions. We provide an alternative discussion of this microstructure and its properties. Our treatment includes an explicit formula for the shape of the inclusion, and an analysis of various limits. We also discuss the significance of this microstructure (i) for minimizing the maximum stress in a composite, and (ii) as a large volume fraction analog of Michell trusses in the theory of structural optimization.

Original languageEnglish (US)
Pages (from-to)949-972
Number of pages24
JournalJournal of the Mechanics and Physics of Solids
Issue number6
StatePublished - Jun 1995


ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering

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