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# Introduction

In previous papers [1,2], an algorithm combining digital-image and spring network techniques was developed and applied to study the effective moduli of 2-D random isotropic composites. Three limitations were: 1) the digital resolution required to represent the desired microstructure, 2) the Poisson's ratios of each phase were required to be greater than or equal to 1/3, and 3) the geometry of the digital representation was hexagonal pixels arranged on a triangular lattice. The first limitation is of course inherent to any numerical digitization scheme, while the second and third limitations were a result of the spring lattice technique used.

The specific case of two-phase composites, where each phase had the same Poisson's ratio but different Young's modulus, was studied using the above algorithm [1]. It was found numerically that when the phase Poisson's ratios, 1 = 2, were above a critical value * = 1/3, the composite or effective Poisson's ratio was always greater than 1/3 and less than 1 = 2. Effective medium theory [3] was shown to describe the results [1,2] rather accurately. The validated effective medium theory, in a parameter range not accessible to the algorithm, was then used to show that when 1 = 2 < 1/3, the value of was always less than 1/3 and greater than 1 = 2. When 1 = 2 = 1/3, = 1/3 as well, within a few percent accuracy. The effective medium theory predicted that * = 1/3 exactly, with this value being a fixed point for any area fraction of the two phases.

These results were obtained for microstructures composed of circular inclusions of one phase distributed randomly in a matrix of the other phase, with either freely overlapping circles or hard circles that were not allowed to overlap. Using the same effective medium theory, it was shown that the use of elliptical inclusions with aspect ratios up to 10 only slightly changed the overall behavior of . The value * = 1/3 was no longer a fixed point, but instead there was a small range around 1 = 2 = 1/3, in which was S-shaped, dipping above and below 1/3 as the volume fraction varied. Above this range, 1/3 < < 1 = 2, and below this range, 1/3 > > 1 = 2.

In the present work, we describe an algorithm that can compute the effective moduli of a composite in 2-D or 3-D, for arbitrary values of Poisson's ratio and Young's modulus. It can also handle anisotropic elastic stiffness tensors, for any number of phases. The microstructure of the composite, as long as it can be adequately represented by an ordinary digital image, can be completely general. In particular, the composite microstructure is definitely not limited to the case usually considered in analytic treatment of composites, that of inclusions with a simple geometry randomly placed in a matrix. This algorithm was developed to operate on model and real 3-D digital images of materials with complex microstructures, as part of a general program studying the various physical properties of such materials [4]. This program is being carried out in order to develop quantitative theoretical microstructure-property relationships in heterogeneous materials complex enough to require such models, like cement-based materials [4] and sintered ceramic materials [5].

To illustrate the operation of the elastic algorithm, and to extend our previous 2-D work [1] to 3-D, we use a general non-particle-based microstructure introduced recently [6,7] to study the composite Poisson's ratio for the case of equal phase Poisson's ratios in 3-D, and in 2-D for cases that were inacessible to our previous algorithm. We first describe the algorithms used to solve the elastic equations and generate the complex composite microstructures.

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