Research Papers

Optimal Percolation Thresholds of Two- and Three-Dimensional Engineering Composites

[+] Author and Article Information
X. Frank Xu

Department of Civil, Environmental and Ocean Engineering,  Stevens Institute of Technology, Hoboken, NJ 07030

J. Eng. Mater. Technol 134(3), 031008 (May 07, 2012) (7 pages) doi:10.1115/1.4006503 History: Received September 29, 2011; Revised February 13, 2012; Published May 04, 2012; Online May 07, 2012

Theoretical prediction of percolation thresholds universally applicable for various composites remains a major theoretical challenge. In the work done by Xu (2011, “Ellipsoidal Bounds and Percolation Thresholds of Transport Properties of Composites,” Acta Mech., 223 , pp. 765–774), a variational method is developed to predict optimal percolation thresholds for transport properties of three dimensional composites subjected to full dispersion of fillers. In this paper, simplified formulae are provided for engineering applications of 3D composites. New formulae are derived for optimal percolation thresholds of 2D composites, i.e., laminates and thin films, and for composites containing a combination of fillers with different aspect ratios. The effects of dimensionality and waviness are especially discussed.

Copyright © 2012 by American Society of Mechanical Engineers
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Figure 1

Comparison of the simplified formulae Eq. 7 (dashed line on left) and Eq. 8 (dashed line on the right) with Eq. 6 (solid lines)

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Figure 2

The absolute difference (error) of the simplified formulae Eq. 7 (left) and Eq. 8 (right) from Eq. 6 (dashed—n = 1010 ; solid—n = 107 )

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Figure 3

The coordinate system of the laminate/film indicated with Euler angles

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Figure 4

Comparison of percolation thresholds for laminates/films characterized with normalized thickness t̃=100 (solid lines) and t̃=1 (dashed lines)

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Figure 5

Percolation thresholds of composites with different aspect ratios of fillers: left—η = 10, 20, 50, 100, 200 top-down with n→∞; right—η = 0.1, 0.05, 0.02, 0.01, 0.005 top-down with n=0

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Figure 6

Percolation threshold of 2D composites containing long parallel fibrous fillers with an elliptic cross-section

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Figure 7

Percolation threshold vs. percentage of nanotubes with η1  = 1000, for a 3D composite containing nantotubes with aspect ratios 1000 and 100 (continous—Eq. 26, dash—Eq. 25 using the mean aspect ratio)

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Figure 8

Histograms of the aspect ratios, based on 1 × 106 samples generated for each of the length and the diameter characterized with the uniform distribution (left), and the truncated normal distribution with further restriction to an interval (10, 500) (right)

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Figure 9

Determination of the effective length for a high aspect ratio filler with waviness



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