## Abstract

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 language | English (US) |
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Pages (from-to) | 781-787 |

Number of pages | 7 |

Journal | Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science |

Volume | 38 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2007 |

### Bibliographical note

Funding Information:Two of the authors (LL and RDJ) acknowledge the financial support of the National Science Foundation through Grant No. DMS-0304326. One of the authors (PHL) acknowledges the support of the Department of Energy through Grant No. DE-FG02-99ER45770.