Moduli Of Two-phase Composites Research Articles

Two-parameter integral representation formula for the effective elastic moduli of two-phase composites

In this article, the integral representation formula (IRF) for the family of composites described in (1) is derived and the relation between moments of positive measures and the microstructure are established explicitly. This family represents composites consisting of one phase with fixed isotropic elasticity tensor and the other phase with varying elasticity tensor, e.g. viscoelastic materials whose complex-valued moduli change with frequency. Every choice of complex bulk modulus and shear modulus for constituent in Ω2 corresponds to a unique choice of parameters ξ1 and ξ2, the contrast between the bulk moduli and shear moduli, respectively. All choices with Im(ξ1) · Im(ξ2) > 0 have well-defined effective elasticity tensor C*(ξ1, ξ2). For each component of C* that can be obtained by ε 0 : C* : ε 0 with some real symmetric matrix ε 0, the Borel measure in the IRF is independent of the choice of (ξ1, ξ2); it is determined by the microstructure and the reference medium C (1). This implies that information of microstructure of the composites and the contrast of constituent properties are separated. Hence it potentially provides a way for retrieving microstructural information from measurement of C* of different (ξ1, ξ2) and C (1). The relation between the moments of the Borel measure and statistical information of the microstructure is established by first deriving the Taylor series expansion of C* near the homogeneous case ξ1 = ξ2 = 0 and then analytically extending it by using the IRF, which is valid for every choice of (ξ1, ξ2) ∈ U 2 or (−U)2.

Double-Inclusion Model and Overall Moduli of Multi-Phase Composites

The double-inclusion model consists of an ellipsoidal inclusion of arbitrary elasticity, containing another ellipsoidal heterogeneity of arbitrary elasticity, size, and orientation, which are embedded in an infinitely extended homogeneous domain of yet another arbitrary elasticity. Average field quantities for the double inclusion are obtained analytically, and used to estimate the overall moduli of two-phase composites. The technique includes the self-consistent and other related methods as special cases. Furthermore, exact bounds for the overall moduli are obtained on the basis of the double-inclusion model. The double-inclusion model has been generalized (Nemat-Nasser and Hori, 1993) to a multi-inclusion model, where, again, all the average field quantities are estimated analytically. The application of the multiinclusion model includes a composite containing inclusions with multi-layer coatings.

Double-inclusion model and overall moduli of multi-phase composites

The double-inclusion model consists of an ellipsoidal inclusion which contains an ellipsoidal heterogeneity and is embedded in an infinitely extended homogeneous domain. The elasticity of the inclusion, its heterogeneity, and that of the infinite domain may be distinct and arbitrary. The ellipsoidal heterogeneity may include other inclusions, or it may have variable elasticity. Average field quantities for the double inclusion are estimated analytically with the aid of a theorem which generalizes the Tanaka-Mori observation (1972; J. Elast. 2, 199–200). It is shown that the averaging scheme based on the double-inclusion model produces the overall moduli of two-phase composites with greater flexibility and hence effectiveness than the self-consistent and the Mori-Tanaka (1973; Acta Metall. 21, 571–574) methods, and, indeed, includes as special cases these methods, providing alternative interpretations for them. The double-inclusion model is then generalized to multi-inclusion models where, again, all the average field quantities are estimated analytically. As examples of the application of the multi-inclusion model, a composite containing inclusions with multilayer coatings and a composite consisting of several distinct materials are considered, and their overall moduli are analytically estimated. In addition, for a set of nested ellipsoidal regions of arbitrary aspect ratios and relative locations, which is embedded in an infinitely extended homogeneous elastic solid of arbitrary elasticity, and which undergoes transformations with uniform but distinct transformation strains within each annulus, it is shown that the resulting strain field averaged over each annulus can be computed exactly and in closed form; the transformation strains in the innermost region need not be uniform. Explicit results are presented for an embedded double inclusion, as well as a nested set of n inclusions.

A Possible Use of Bounds on Effective Moduli of Composite Materials

We advocate the use of bounds on the effective moduli of two-phase com posites to deduce possible ranges of microstructural parameters from ex perimental measurements of the effective moduli. In particular we give ex plicit expressions for the bounds on the volume fractions a third-order cor relation parameter. The bounds on the volume fractions are accurate to second-order terms in δx = x2 — x1 and δμ = μ2 — μ2, where x2 and μ2 (i = 1,2) are the bulk and shear moduli of the phases. When the moduli ratios are of the order 10 or less, the second-order bounds on the volume fractions are tight enough to recommend their use in evaluating end products.