Periodic inclusion - Matrix microstructures with constant field inclusions

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

doi:https://doi.org/10.1007/s11661-006-9019-z

Periodic inclusion - Matrix microstructures with constant field inclusions

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

Research output: Contribution to journal › Article › peer-review

29 Scopus citations

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	American English
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	https://doi.org/10.1007/s11661-006-9019-z
State	Published - Apr 2007
Externally published	Yes

ASJC Scopus subject areas

Condensed Matter Physics
Mechanics of Materials
Metals and Alloys

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title = "Periodic inclusion - Matrix microstructures with constant field inclusions",

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.",

author = "Liping Liu and James, {Richard D.} and Leo, {Perry H.}",

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.",

year = "2007",

month = apr,

doi = "https://doi.org/10.1007/s11661-006-9019-z",

language = "American English",

volume = "38",

pages = "781--787",

journal = "Metallurgical transactions. A, Physical metallurgy and materials science",

issn = "0360-2133",

publisher = "Springer Boston",

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T1 - Periodic inclusion - Matrix microstructures with constant field inclusions

AU - Liu, Liping

AU - James, Richard D.

AU - Leo, Perry H.

N1 - 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.

PY - 2007/4

Y1 - 2007/4

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.

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U2 - https://doi.org/10.1007/s11661-006-9019-z

DO - https://doi.org/10.1007/s11661-006-9019-z

M3 - Article

SN - 0360-2133

VL - 38

SP - 781

EP - 787

JO - Metallurgical transactions. A, Physical metallurgy and materials science

JF - Metallurgical transactions. A, Physical metallurgy and materials science

IS - 4

ER -

Periodic inclusion - Matrix microstructures with constant field inclusions

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