Periodic inclusion - Matrix microstructures with constant field inclusions

Liping Liu, Richard D. James, Perry H. Leo

Research output: Contribution to journalArticlepeer-review

29 Scopus citations


We find a class of special microstructures consisting of a periodic array of inclusions, with the special property that constant magnetization (or eigenstrain) of the inclusion implies constant magnetic field (or strain) in the inclusion. The resulting inclusions, which we term E-inclusions, have the same property in a finite periodic domain as ellipsoids have in infinite space. The E-inclusions are found by mapping the magnetostatic or elasticity equations to a constrained minimization problem known as a free-boundary obstacle problem. By solving this minimization problem, we can construct families of E-inclusions with any prescribed volume fraction between zero and one. In two dimensions, our results coincide with the microstructures first introduced by Vigdergauz, [1,2] while in three dimensions, we introduce a numerical method to calculate E-inclusions. E-inclusions extend the important role of ellipsoids in calculations concerning phase transformations and composite materials.

Original languageAmerican English
Pages (from-to)781-787
Number of pages7
JournalMetallurgical and Materials Transactions A: Physical Metallurgy and Materials Science
Issue number4
StatePublished - Apr 2007
Externally publishedYes

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Metals and Alloys


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