# Asymptotics of the effective conductivity of composites with closely spaced inclusions of optimal shape

Research output: Contribution to journalArticle

6 Citations (Scopus)

### Abstract

We consider a mathematical model of a high contrast two phase composite material with inclusions (fibres) close to touching in two space dimensions. The inclusions form a periodic array and have an optimal shape which is a curvilinear square with rounded-off angles ('nearly square' shape) described by a flattening parameter m. We derive an asymptotic formula for the effective conductivity Âδ of the composite when the interparticle distance δ goes to zero. This formula captures the dependence of Âδ on the parameter m. We provide a rigorous justification for this asymptotic formula by a variational duality approach.

Original language English (US) 83-106 24 Quarterly Journal of Mechanics and Applied Mathematics 58 1 https://doi.org/10.1093/qjmamj/hbh022 Published - Feb 1 2005

### Fingerprint

Effective Conductivity
Optimal Shape
Asymptotic Formula
Inclusion
Composite
inclusions
conductivity
composite materials
Composite materials
Composite Materials
Justification
Duality
Fiber
flattening
Mathematical Model
Mathematical models
Angle
Fibers
mathematical models
Zero

### All Science Journal Classification (ASJC) codes

• Condensed Matter Physics
• Mechanics of Materials
• Mechanical Engineering
• Applied Mathematics

### Cite this

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title = "Asymptotics of the effective conductivity of composites with closely spaced inclusions of optimal shape",
abstract = "We consider a mathematical model of a high contrast two phase composite material with inclusions (fibres) close to touching in two space dimensions. The inclusions form a periodic array and have an optimal shape which is a curvilinear square with rounded-off angles ('nearly square' shape) described by a flattening parameter m. We derive an asymptotic formula for the effective conductivity {\^A}δ of the composite when the interparticle distance δ goes to zero. This formula captures the dependence of {\^A}δ on the parameter m. We provide a rigorous justification for this asymptotic formula by a variational duality approach.",
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year = "2005",
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language = "English (US)",
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journal = "Quarterly Journal of Mechanics and Applied Mathematics",
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In: Quarterly Journal of Mechanics and Applied Mathematics, Vol. 58, No. 1, 01.02.2005, p. 83-106.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Asymptotics of the effective conductivity of composites with closely spaced inclusions of optimal shape

AU - Gorb, Y.

AU - Berlyand, Leonid V.

PY - 2005/2/1

Y1 - 2005/2/1

N2 - We consider a mathematical model of a high contrast two phase composite material with inclusions (fibres) close to touching in two space dimensions. The inclusions form a periodic array and have an optimal shape which is a curvilinear square with rounded-off angles ('nearly square' shape) described by a flattening parameter m. We derive an asymptotic formula for the effective conductivity Âδ of the composite when the interparticle distance δ goes to zero. This formula captures the dependence of Âδ on the parameter m. We provide a rigorous justification for this asymptotic formula by a variational duality approach.

AB - We consider a mathematical model of a high contrast two phase composite material with inclusions (fibres) close to touching in two space dimensions. The inclusions form a periodic array and have an optimal shape which is a curvilinear square with rounded-off angles ('nearly square' shape) described by a flattening parameter m. We derive an asymptotic formula for the effective conductivity Âδ of the composite when the interparticle distance δ goes to zero. This formula captures the dependence of Âδ on the parameter m. We provide a rigorous justification for this asymptotic formula by a variational duality approach.

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U2 - 10.1093/qjmamj/hbh022

DO - 10.1093/qjmamj/hbh022

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JO - Quarterly Journal of Mechanics and Applied Mathematics

JF - Quarterly Journal of Mechanics and Applied Mathematics

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