#### Vol. 6, No. 5, 2011

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Effective property estimates for heterogeneous materials with cocontinuous phases

### Patrick Franciosi, Renald Brenner and Abderrahim El Omri

Vol. 6 (2011), No. 5, 729–763
##### Abstract

This work concerns heterogeneous multiphase materials which may exhibit omnidirectional full or partial cocontinuity of several or all phases. The estimate of their effective (mechanical or physical) properties is not yet well handled as compared to those for well-defined aggregate or reinforced-matrix structures, especially in the context of homogenization methods. We propose in this framework a modeling scheme which aims at accounting for such phase cocontinuity features. In the mechanical application field, the modeling validity restricts to elastic properties of unloaded materials or in load situations as far as bending and torsion effects of possibly strut-like phase parts are not essential. For other physical properties (dielectric, magnetic, etc.), the modeling applications concern those for which homogenization approaches are relevant. The modeling is based on a material’s morphology description in terms of a generalization of so-called “fiber systems” that were introduced in early literature reports. Using parameters that describe the clustering characteristics of the individual phases and of their assemblage, we have considered these fiber systems both within a layer-based approach of the material structure and within an aggregate-like one. By these two routes, we have obtained two estimate forms that differ only slightly in definition. The presentation uses the elasticity formalism, in simple cases of isotropic mixtures of two-phase materials with isotropic phase behavior but the modeling extends to $n$-phase anisotropic materials as established separately. Our estimates are compared with basic variational bounds and homogenization estimates, with some literature data and with homogenization results obtained with the fast Fourier transform approach on numerical structures. All data are matched with different parameter sets corresponding to different types of phase organization. The two estimates remain nearly equal for all of the examined structures regardless of phase contrast and with only slight differences consistent with their definition difference.

##### Keywords
phase cocontinuity, effective properties, heterogeneous materials, clustering