### 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) |
---|---|

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 1 2007 |

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**Periodic inclusion - Matrix microstructures with constant field inclusions.** / Liu, Liping; James, Richard D.; Leo, Perry H.

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TY - JOUR

T1 - Periodic inclusion - Matrix microstructures with constant field inclusions

AU - Liu, Liping

AU - James, Richard D.

AU - Leo, Perry H.

PY - 2007/4/1

Y1 - 2007/4/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=34347391686&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34347391686&partnerID=8YFLogxK

U2 - 10.1007/s11661-006-9019-z

DO - 10.1007/s11661-006-9019-z

M3 - Article

AN - SCOPUS:34347391686

VL - 38

SP - 781

EP - 787

JO - Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science

JF - Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science

SN - 1073-5623

IS - 4

ER -