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

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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.