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Research Papers

# On the Effective Elastic Properties of Macroscopically Isotropic Media Containing Randomly Dispersed Spherical Particles

[+] Author and Article Information
D. Cojocaru

Department of Mechanical Engineering, University of Delaware, Newark, DE 19716

A. M. Karlsson1

Department of Mechanical Engineering, University of Delaware, Newark, DE 19716karlsson@udel.edu

However, this method can be used effectively for area fractions up to 50% for 2D problems with circular inclusions (i.e., disks) being inserted into a square. We also note that $Vfp=0.275$ leads to a nearest-neighbor distance of $H1=1.06d$(19), which in practical terms is a relatively closed packed system.

Special numerical integration rules might be used for the elements crossed by the theoretical interface between the materials to alleviate this problem. For example, a particular $5×5×5$, the integration rule was used by Zohdi and co-worker (5-6,26) for the elements crossed by the interface. However, the application of these higher order quadrature rules requires additional programming effort, and is not always possible to implement for a commercial FE software (depending on the software used).

For example, for a 3D sample containing 100 spherical particles at 25% volume fraction, the statistically equivalent area fraction is 0.271 and the mean of the intersected number of particles is 18.2 (Table 1). Therefore, the 2D-FE model will contain 18 circular inclusions of the same-size accounting for 27.1% of the sample area. Thus, the radius of the inclusions is the radius of one inclusion representing 27.1% of a sample of unit area. The size of the 2D rectangular sample is calculated to accommodate 18 circular inclusions representing 27.1% of the sample area.

1

Corresponding author.

J. Eng. Mater. Technol 132(2), 021011 (Feb 19, 2010) (11 pages) doi:10.1115/1.4000229 History: Received April 07, 2009; Revised August 19, 2009; Published February 19, 2010; Online February 19, 2010

## Abstract

A computational scheme for estimating the effective elastic properties of a particle reinforced matrix is investigated. The randomly distributed same-sized spherical particles are assumed to result in a composite material that is macroscopically isotropic. The scheme results in a computational efficient method to establish the correct bulk and shear moduli by representing the three-dimensional (3D) structure in a two-dimensional configuration. To this end, the statistically equivalent area fraction is defined in this work, which depends on two parameters: the particle volume fraction and the number of particles in the 3D volume element. We suggest that using the statistically equivalent area fraction, introduced and defined in this work, is an efficient way to obtain the effective elastic properties of an isotropic media containing randomly dispersed same-size spherical particles.

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Copyright © 2010 by American Society of Mechanical Engineers
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## Figures

Figure 1

A configuration with 1000 randomly generated nonintersecting spheres representing 25% volume fraction

Figure 2

Equivalent area fraction as a function of the number of configurations investigated, obtained by cross-sectioning the 3D-RVE simultaneously with three orthogonal planes for (a) N=20 particles and Vfp=0.25, (b) N=1000 particles and Vfp=0.25, (c) N=100 particles and Vfp=0.05, and (d) N=100 particles and Vfp=0.25

Figure 3

Distribution of the equivalent area fraction obtained for Vf=0.22 and N=1000 particles by cross-sectioning the 3D cubic RVE with the three orthogonal planes. Histograms computed using 50,000 random configurations.

Figure 4

Equivalent area fraction as the function of the number of particles N obtained by cutting the cubic RVE with the XY plane, for various volume fractions: (a) the mean value and (b) the standard deviation. (Each point is based on 50,000 randomly generated configurations.)

Figure 5

Number of the particles intersected by plane XY as the function of the number of particles N for various volume fractions: (a) the mean value and (b) the standard deviation

Figure 6

Modeling inclusions using (a) unaligned mesh and (b) aligned mesh

Figure 7

Two-dimensional-FE modeling approaches: (a) a 3D configuration containing 100 randomly distributed spherical particles (Vf=0.25), (b) the 2D-FE model obtained by cross-sectioning the 3D specimen with the plane XY, and (c) the equivalent 2D model containing 18 same-size circular particles (Af=0.271)

Figure 8

Effective elastic properties as a function of volume fraction Vf predicted by the 2D-FE models using the two modeling approaches, compared with Hashin–Shtrikman bounds and the self-consistent method: (a) the bulk modulus Ka and (b) the shear modulus Ga. The vertical bars indicate ±1 standard deviation.

Figure 9

Three-dimensional-FE models of RVEs containing random configurations of 32 particles at 25% volume fraction: (a) aligned tetrahedral mesh, (b) unaligned hexahedral mesh, and (c) cross section along the plane XY. The contour plots show Mises stress

Figure 10

Effective elastic properties as a function of the volume fraction Vf predicted by the 3D-FE models using aligned and unaligned meshes compared with Hashin–Shtrikman bounds and the self-consistent method: (a) bulk modulus Ka and (b) shear modulus Ga

Figure 11

Effective properties as a function of the volume fraction Vf predicted by 2D- (direct generation) and 3D-FE models with aligned meshes: (a) bulk modulus Ka and (b) shear modulus Ga. The vertical bars indicate ±1 standard deviation.

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