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TECHNICAL PAPERS

# The Eshelby Tensors in a Finite Spherical Domain—Part II: Applications to Homogenization

[+] Author and Article Information
Shaofan Li1

Department of Civil and Environmental Engineering, University of California, Berkeley, CA 94720li@ce.berkeley.edu

Gang Wang, Roger A. Sauer

Department of Civil and Environmental Engineering, University of California, Berkeley, CA 94720

1

Corresponding author.

J. Appl. Mech 74(4), 784-797 (Jun 14, 2006) (14 pages) doi:10.1115/1.2711228 History: Received April 06, 2006; Revised June 14, 2006

## Abstract

In this part of the work, the Eshelby tensors of a finite spherical domain are applied to various homogenization procedures estimating the effective material properties of multiphase composites. The Eshelby tensors of a finite domain can capture the boundary effect of a representative volume element as well as the size effect of the different phases. Therefore their application to homogenization does not only improve the accuracy of classical homogenization methods, but also leads to some novel homogenization theories. This paper highlights a few of them: a refined dilute suspension method and a modified Mori–Tanaka method, the exterior eigenstrain method, the dual-eigenstrain method, which is a generalized self-consistency method, a shell model, and new variational bounds depending on the different boundary conditions. To the best of the authors’ knowledge, this is the first time that a multishell model is used to evaluate the Hashin–Shtrikman bounds for a multiple phase composite $(n⩾3)$, which can distinguish some of the subtleties of different microstructures.

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## Figures

Figure 1

Average Eshelby tensor coefficients s1I, s1E(i=1) and s2I, s2E(i=2)

Figure 10

Variational bounds for a three-phase composite material: (a) bounds for bulk modulus; and (b) bounds for shear modulus.

Figure 11

Influence of phase position on three-phase variational bounds

Figure 4

Illustration of interior and exterior eigenstrain method

Figure 5

Mori–Tanaka homogenization for the interior and exterior eigenstrain methods

Figure 6

Effective shear modulus for: (a) C̃=aCI+(1−a)CE, and (b) C̃=C¯

Figure 7

A three-layer shell model

Figure 8

Influence of β on the effective shear modulus

Figure 9

Improved Hashin–Shtrikman bounds for the effective bulk and shear moduli

Figure 2

Effective moduli κ¯, μ¯ (or κeff and μeff) obtained by using the dilute suspension method

Figure 3

Effective moduli κ¯, μ¯ (or κeff and μeff) obtained by using the Mori–Tanaka method

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