Variational bounds on the effective moduli of anisotropic composites

Graeme W. Milton, Robert V. Kohn

Research output: Contribution to journalArticlepeer-review


The vritional inequalities of Hashin and Shtrikman are transformed to a simple and concise form. They are used to bound the effective conductivity tensor σ* of an anisotropic composite made from an arbitrary number of possibly anisotropic phases, and to bound the effective elasticity tensor C* of an anisotropic mixture of two well-ordered isotropic materials. The bounds depend on the conductivities and elastic moduli of the components and their respective volume fractions. When the components are isotropic the conductivity bounds, which constrain the eigenvalues of σ*, include those previously obtained by Hashin and Shtrikman, Murat and Tartar, and Lurie and Cherkaev. Our approach can also be used in the context of linear elasticity to derive bounds on C* for composites comprised of an arbitrary number of anisotropic phases. For two-component composites our bounds are tighter than those obtained by Kantor and Bergman and by Francfort and Murat, and are attained by sequentially layered laminate materials.

Original languageEnglish (US)
Pages (from-to)597-629
Number of pages33
JournalJournal of the Mechanics and Physics of Solids
Issue number6
StatePublished - 1988

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering


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