Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. II: The vigdergauz microstructure

Yury Grabovsky, Robert Kohn

Research output: Contribution to journalArticle

Abstract

For modeling coherent phase transformations, and for applications to structural optimization, it is of interest to identify microstructures with minimal energy or maximal stiffness. The existence of a particularly simple microstructure with extremal elastic behavior, in the context of two-phase composites made from isotropic components in two space dimensions, has previously been shown. This "Vigdergauz microstructure" consists of a periodic array of appropriately shaped inclusions. We provide an alternative discussion of this microstructure and its properties. Our treatment includes an explicit formula for the shape of the inclusion, and an analysis of various limits. We also discuss the significance of this microstructure (i) for minimizing the maximum stress in a composite, and (ii) as a large volume fraction analog of Michell trusses in the theory of structural optimization.

Original languageEnglish (US)
Pages (from-to)949-972
Number of pages24
JournalJournal of the Mechanics and Physics of Solids
Volume43
Issue number6
DOIs
StatePublished - 1995

Fingerprint

microstructure
Microstructure
composite materials
Composite materials
Structural optimization
trusses
energy
inclusions
Trusses
optimization
phase transformations
Volume fraction
stiffness
Phase transitions
Stiffness
analogs

ASJC Scopus subject areas

  • Mechanical Engineering
  • Mechanics of Materials
  • Condensed Matter Physics

Cite this

@article{75ca0783a9ae43e79c5575ef40c48490,
title = "Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. II: The vigdergauz microstructure",
abstract = "For modeling coherent phase transformations, and for applications to structural optimization, it is of interest to identify microstructures with minimal energy or maximal stiffness. The existence of a particularly simple microstructure with extremal elastic behavior, in the context of two-phase composites made from isotropic components in two space dimensions, has previously been shown. This {"}Vigdergauz microstructure{"} consists of a periodic array of appropriately shaped inclusions. We provide an alternative discussion of this microstructure and its properties. Our treatment includes an explicit formula for the shape of the inclusion, and an analysis of various limits. We also discuss the significance of this microstructure (i) for minimizing the maximum stress in a composite, and (ii) as a large volume fraction analog of Michell trusses in the theory of structural optimization.",
author = "Yury Grabovsky and Robert Kohn",
year = "1995",
doi = "10.1016/0022-5096(95)00017-D",
language = "English (US)",
volume = "43",
pages = "949--972",
journal = "Journal of the Mechanics and Physics of Solids",
issn = "0022-5096",
publisher = "Elsevier Limited",
number = "6",

}

TY - JOUR

T1 - Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. II

T2 - The vigdergauz microstructure

AU - Grabovsky, Yury

AU - Kohn, Robert

PY - 1995

Y1 - 1995

N2 - For modeling coherent phase transformations, and for applications to structural optimization, it is of interest to identify microstructures with minimal energy or maximal stiffness. The existence of a particularly simple microstructure with extremal elastic behavior, in the context of two-phase composites made from isotropic components in two space dimensions, has previously been shown. This "Vigdergauz microstructure" consists of a periodic array of appropriately shaped inclusions. We provide an alternative discussion of this microstructure and its properties. Our treatment includes an explicit formula for the shape of the inclusion, and an analysis of various limits. We also discuss the significance of this microstructure (i) for minimizing the maximum stress in a composite, and (ii) as a large volume fraction analog of Michell trusses in the theory of structural optimization.

AB - For modeling coherent phase transformations, and for applications to structural optimization, it is of interest to identify microstructures with minimal energy or maximal stiffness. The existence of a particularly simple microstructure with extremal elastic behavior, in the context of two-phase composites made from isotropic components in two space dimensions, has previously been shown. This "Vigdergauz microstructure" consists of a periodic array of appropriately shaped inclusions. We provide an alternative discussion of this microstructure and its properties. Our treatment includes an explicit formula for the shape of the inclusion, and an analysis of various limits. We also discuss the significance of this microstructure (i) for minimizing the maximum stress in a composite, and (ii) as a large volume fraction analog of Michell trusses in the theory of structural optimization.

UR - http://www.scopus.com/inward/record.url?scp=0029326671&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0029326671&partnerID=8YFLogxK

U2 - 10.1016/0022-5096(95)00017-D

DO - 10.1016/0022-5096(95)00017-D

M3 - Article

AN - SCOPUS:0029326671

VL - 43

SP - 949

EP - 972

JO - Journal of the Mechanics and Physics of Solids

JF - Journal of the Mechanics and Physics of Solids

SN - 0022-5096

IS - 6

ER -