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

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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 languageEnglish (US)
Pages (from-to)83-106
Number of pages24
JournalQuarterly Journal of Mechanics and Applied Mathematics
Volume58
Issue number1
DOIs
StatePublished - Feb 1 2005

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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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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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AU - Berlyand, Leonid V.

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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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