A complete characterization of the possible bulk and shear moduli of planar polycrystals

M. Avellaneda; A. V. Cherkaev; L. V. Gibiansky; G. W. Milton; M. Rudelson

doi:10.1016/0022-5096(96)00018-X

A complete characterization of the possible bulk and shear moduli of planar polycrystals

M. Avellaneda, A. V. Cherkaev, L. V. Gibiansky, G. W. Milton, M. Rudelson

Mathematics

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Abstract

This paper investigates the range of possible elastic moduli of two-dimensional isotropic polycrystals. The polycrystals are comprised of grains obtained from a single orthotropic material. The overall elastic properties are described by an effective bulk modulus Κ₀ and an effective shear modulus μ₀. The pair (Κ₀, μ₀) is shown to be confined to a rectangle in the (Κ, μ) plane. Microstructures are identified which correspond to every point within the rectangle, and in particular to the corner points. Optimal bounds on the effective Poisson's ratio and Young's modulus follow immediately. Under a certain constraint on the crystal moduli, the rectangle degenerates to a line segment: the effective shear modulus of such a polycrystal is microstructure independent. This extends earlier work that established microstructure independence for polycrystals constructed from square symmetric crystals.

Original language	English (US)
Pages (from-to)	1179-1218
Number of pages	40
Journal	Journal of the Mechanics and Physics of Solids
Volume	44
Issue number	7
DOIs	https://doi.org/10.1016/0022-5096(96)00018-X
State	Published - Jul 1996

ASJC Scopus subject areas

Condensed Matter Physics
Mechanics of Materials
Mechanical Engineering

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10.1016/0022-5096(96)00018-X

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title = "A complete characterization of the possible bulk and shear moduli of planar polycrystals",

abstract = "This paper investigates the range of possible elastic moduli of two-dimensional isotropic polycrystals. The polycrystals are comprised of grains obtained from a single orthotropic material. The overall elastic properties are described by an effective bulk modulus Κ0 and an effective shear modulus μ0. The pair (Κ0, μ0) is shown to be confined to a rectangle in the (Κ, μ) plane. Microstructures are identified which correspond to every point within the rectangle, and in particular to the corner points. Optimal bounds on the effective Poisson's ratio and Young's modulus follow immediately. Under a certain constraint on the crystal moduli, the rectangle degenerates to a line segment: the effective shear modulus of such a polycrystal is microstructure independent. This extends earlier work that established microstructure independence for polycrystals constructed from square symmetric crystals.",

author = "M. Avellaneda and Cherkaev, {A. V.} and Gibiansky, {L. V.} and Milton, {G. W.} and M. Rudelson",

note = "Funding Information: The financial supportf or this work comesf rom various sources: M. Avellanedai s thankful to theA rmy ResearchO fficea nd theN ational ScienceF oundation ; A. V. Cherkaevi s thankful to theN ational ScienceF oundation and the PackardF oundation ; L. V. Gibiansky is thankful to theA ir Force Officeo f ScientificR esearcht,o theD epartmenot f Energy,a nd Army Research Office and the Packard Foundation; G. W. Milton is thankful to the Army ResearchO ffice, the National ScienceF oundation, and the Packard Foundation. We are gratefult o J. Erker for his assistancein writing a computerp rogramf or computinge ffectivem oduli of millions of randomlyg eneratedm ultiple rank laminatesT. his work ultimatelyl eadto thei dentificationo f the microstructurecso rrespondingt o the corner points A and D.",

year = "1996",

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doi = "10.1016/0022-5096(96)00018-X",

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AU - Rudelson, M.

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PY - 1996/7

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N2 - This paper investigates the range of possible elastic moduli of two-dimensional isotropic polycrystals. The polycrystals are comprised of grains obtained from a single orthotropic material. The overall elastic properties are described by an effective bulk modulus Κ0 and an effective shear modulus μ0. The pair (Κ0, μ0) is shown to be confined to a rectangle in the (Κ, μ) plane. Microstructures are identified which correspond to every point within the rectangle, and in particular to the corner points. Optimal bounds on the effective Poisson's ratio and Young's modulus follow immediately. Under a certain constraint on the crystal moduli, the rectangle degenerates to a line segment: the effective shear modulus of such a polycrystal is microstructure independent. This extends earlier work that established microstructure independence for polycrystals constructed from square symmetric crystals.

AB - This paper investigates the range of possible elastic moduli of two-dimensional isotropic polycrystals. The polycrystals are comprised of grains obtained from a single orthotropic material. The overall elastic properties are described by an effective bulk modulus Κ0 and an effective shear modulus μ0. The pair (Κ0, μ0) is shown to be confined to a rectangle in the (Κ, μ) plane. Microstructures are identified which correspond to every point within the rectangle, and in particular to the corner points. Optimal bounds on the effective Poisson's ratio and Young's modulus follow immediately. Under a certain constraint on the crystal moduli, the rectangle degenerates to a line segment: the effective shear modulus of such a polycrystal is microstructure independent. This extends earlier work that established microstructure independence for polycrystals constructed from square symmetric crystals.

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A complete characterization of the possible bulk and shear moduli of planar polycrystals

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