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Review Articles

Asymptotic Homogenization of Composite Materials and Structures

[+] Author and Article Information
Alexander L. Kalamkarov1

Department of Mechanical Engineering, Dalhousie University, P.O. Box 1000, Halifax, NS, B3J 2X4, Canadaalex.kalamkarov@dal.ca

Igor V. Andrianov

Institute of General Mechanics, Rheinisch-Westfälische Technische Hochschule (Technical University of Aachen), Templergraben 64, Aachen D-52062, Germany

Vladyslav V. Danishevs’kyy

 Prydniprov’ska State Academy of Civil Engineering and Architecture, Chernishevs’kogo 24a, Dnipropetrovsk 49600, Ukraine

1

Corresponding author.

Appl. Mech. Rev 62(3), 030802 (Mar 31, 2009) (20 pages) doi:10.1115/1.3090830 History: Received July 30, 2008; Revised December 01, 2008; Published March 31, 2009

The present paper provides details on the new trends in application of asymptotic homogenization techniques to the analysis of composite materials and thin-walled composite structures and their effective properties. The problems under consideration are important from both fundamental and applied points of view. We review a state-of-the-art in asymptotic homogenization of composites by presenting the variety of existing methods, by pointing out their advantages and shortcomings, and by discussing their applications. In addition to the review of existing results, some new original approaches are also introduced. In particular, we analyze a possibility of analytical solution of the unit cell problems obtained as a result of the homogenization procedure. Asymptotic homogenization of 3D thin-walled composite reinforced structures is considered, and the general homogenization model for a composite shell is introduced. In particular, analytical formulas for the effective stiffness moduli of wafer-reinforced shell and sandwich composite shell with a honeycomb filler are presented. We also consider random composites; use of two-point Padé approximants and asymptotically equivalent functions; correlation between conductivity and elastic properties of composites; and strength, damage, and boundary effects in composites. This article is based on a review of 205 references.

Copyright © 2009 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Composite material with hexagonal array of cylindrical fibers

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Figure 2

(a) 3D periodic composite and (b) unit cell Y

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Figure 3

(a) Fiber-reinforced composite with fiber volume fraction close to maximum and (b) asymptotic model

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Figure 4

Unit cell of the regular square lattice of cylindrical fibers

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Figure 5

Effective conductivity ⟨k⟩ in the perfectly regular case: solid curves—the present solution 313; circles—data from Ref. 64

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Figure 6

Asymptotic behavior of ⟨k⟩ in the perfectly regular case at kf/km→∞, c→cmax: solid curves—the present solution; dashed curves—the asymptotic formula from Ref. 65

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Figure 7

The sequences of [M/M]0, [M/M]1, and [M/M]2, M=2,4,6,12,18 uniformly converging to the effective conductivity λe(h)(h=λ2/λ1) of the square array of cylinders. Curves [M/M]2 are indistinguishable (solid line—(a)). The bounds [18/18]1 and [18/18]2 are very restrictive.

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Figure 8

The TPPA upper and lower bounds on the effective conductivity for a square array of densely packed highly conducting cylinders. For φ=0.785 the bounds coincide. For φ=0.7853, 0.78539 bounds are very restrictive. For higher volume fractions φ≥0.78539816 the difference between lower and upper bounds rapidly increases.

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Figure 9

Effective conductivity ⟨k⟩/km of the SC array versus volume fraction of inclusions c

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Figure 10

Effective conductivity ⟨k⟩/km of the BCC array versus volume fraction of inclusions c

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Figure 11

Effective conductivity ⟨k⟩/km of the FCC array versus volume fraction of inclusions c

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Figure 12

General view of the shaking-geometry composite material

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Figure 13

Deviation of the fibers about the regular square lattice

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Figure 14

Bounds on ⟨k⟩ in the nonregular case. Solid curves—the present solution: the lower bound 61 at δ=0 and the upper bound 62 for different values of δ. Dashed curves—the Hashin–Shtrikman bounds 63,64. (a) Dilute composite: c=0.2 and (b) densely packed composite: c=0.7.

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Figure 15

Composite material with square inclusions

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Figure 16

(a) Curvilinear thin 3D reinforced composite layer and (b) unit cell Ωδ

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Figure 17

(a) Wafer-reinforced shell and (b) unit cell

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Figure 18

Sandwich composite shell with a honeycomb filler

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