Second-rank equilibrium and transport properties of fibrous composites: Effective predictions and bounds

A direct approach to the evaluation of the effective permeability, permittivity and transport properties of an arbitrary composite is presented in terms of gradient and flux-density concentrators. A set of requirements are presented, which are imposed on the effective properties and ultimately result in conditions of admissibility for the concentrators. In the scope of bi-constituent, poly-phase composites, two approximate choices for the concentrators are discussed in detail: the Hatta-Taya theory and the poly-inclusion theory. The Hatta-Taya formulation is shown, in general, to yield an effective property which is unsymmetric and which depends on the matrix properties at unitary volume fraction of the embedded material. The poly-inclusion theory is here applied for the first time to second-rank properties. Regardless of the constitution, morphology and texture of the inhomogeneities, the poly-inclusion approach is shown to satisfy all admissibility requirements with the exception of consistency here defined to indicate form identity of mutually inverse properties. A proof is presented which infers relationships between the symmetry group of the orientation distribution function and the symmetry group of the effective properties for special classes of composites. Bounds of order n are presented for macroscopically homogeneous and isotropic composites comprised of an arbitrary number of anisotropic constituents. The case of n = 2 corresponds to the Hashin-Shtrikman bounds which to date appear to have only been calculated for composites with isotropic constituents. Application of the effective properties to the analysis of functionally graded materials (FGMs) is addressed.