### 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) |
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Pages (from-to) | 83-106 |

Number of pages | 24 |

Journal | Quarterly Journal of Mechanics and Applied Mathematics |

Volume | 58 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2005 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

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**Asymptotics of the effective conductivity of composites with closely spaced inclusions of optimal shape.** / Gorb, Y.; Berlyand, Leonid V.

Research output: Contribution to journal › Article

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.

UR - http://www.scopus.com/inward/record.url?scp=19344362712&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=19344362712&partnerID=8YFLogxK

U2 - 10.1093/qjmamj/hbh022

DO - 10.1093/qjmamj/hbh022

M3 - Article

AN - SCOPUS:19344362712

VL - 58

SP - 83

EP - 106

JO - Quarterly Journal of Mechanics and Applied Mathematics

JF - Quarterly Journal of Mechanics and Applied Mathematics

SN - 0033-5614

IS - 1

ER -