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Research Papers

Ellipsoidal Bounds of Elastic Composites

[+] Author and Article Information
X. Frank Xu

Department of Civil, Environmental and Ocean Engineering,  Stevens Institute of Technology, Hoboken, NJ 07030xxu1@stevens.edu

J. Appl. Mech 79(2), 021016 (Feb 24, 2012) (8 pages) doi:10.1115/1.4005586 History: Received August 26, 2011; Revised December 12, 2011; Posted February 01, 2012; Published February 13, 2012; Online February 24, 2012

The formulation of rigorous bounds for the physical properties of composites constitutes one of the most fundamental parts of applied mechanics. In this work, the so-called ellipsoidal bounds, as a generalization of the Hashin-Shtrikman spherical bounds, are formulated for elastic moduli of multiphase composites. Explicit formulas are derived to estimate the bounds for the elastic moduli of isotropic composites. Asymptotic analyses are conducted for composites containing needlelike and disklike fillers with aspect ratios approaching infinity and zero, respectively. The new bounds and estimates are expected to be useful for polycrystals and composites containing fillers, especially with large or small aspect ratios, such as nanowires, nanotubes, and nanoplatelets.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic of a three-phase composite containing (a) spherical, and (b) ellipsoidal fillers

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Figure 2

Effective Young’s modulus predicted by the ellipsoidal and HS spherical bounds for (a) a 3-phase solid containing two types of oblate cavities, and (b) for a 2-phase nanocomposites containing carbon nanotubes

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Figure 3

Lower bound of bulk modulus versus volume fraction of needlelike fillers with κ1  = 100κ0 . (a) ν0  = 0.3 and ν1  = 0, 0.1, 0.2, 0.3, 0.4, and 0.499 for the curves from left to right; (b) ν1  = 0.4 and ν0  = 0, 0.1, 0.2, 0.3, 0.4, and 0.499 for the curves from right to left.

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Figure 4

Lower bound of shear modulus versus volume fraction of needlelike fillers with μ1  = 100μ0 . (a) ν0  = 0.3 and ν1  = 0, 0.1, 0.2, 0.3, 0.4, and 0.499 for the curves from right to left; (b) ν1  = 0.4 and ν0  = 0, 0.1, 0.2, 0.3, 0.4, and 0.499 for the curves from left to right.

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Figure 5

Percolation threshold versus the contrast ratio κ1  = 100κ0 for the bulk mode of composites containing needlelike fillers. (a) ν0  = 0.3 and ν1  = 0, 0.1, 0.2, 0.3, 0.4, and 0.499 for the curves ordered by the arrow; (b) ν1  = 0.4 and ν0  = 0, 0.1, 0.2, 0.3, 0.4, and 0.499 for the curves ordered by the arrow.

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Figure 6

Percolation threshold versus the contrast ratio μ1=100μ0 for the shear mode of composites containing needlelike fillers. (a) ν0  = 0.3 and ν1  = 0, 0.1, 0.2, 0.3, 0.4, and 0.499 for the curves ordered by the arrow; (b) ν1  = 0.4 and ν0  = 0, 0.1, 0.2, 0.3, 0.4, and 0.499 for the curves ordered by the arrow.

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Figure 7

(a) Lower bound of bulk modulus versus volume fraction of disklike fillers with ν0  = 0, 0.1, 0.2, 0.3, 0.4, and 0.499 for the curves from left to right; (b) Lower bound of shear modulus versus volume fraction of disklike fillers with ν0  = 0, 0.1, 0.2, 0.3, 0.4, and 0.499 for the curves from right to left

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Figure 8

Percolation threshold versus Poisson’s ratio of the matrix containing disklike reinforcing fillers

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