MSC: 74Q20 74Q15 Keywords: Hashin-Shtrikman bounds Three-phase composite Transport problem a b s t r a c t In this work we analyse the influence of the spatial distribution function, introduced by Ponte Castañeda and Willis (1995), on the Hashin-Shtrikman bounds on the effective transport properties of a transversely isotropic (TI) three-phase particulate composite, i.e. when two distinct materials are embedded in a matrix medium. We provide a straightforward mechanism to construct associated

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... unds, independently accounting for the shape, size and spatial distribution of the respective phases, and assuming ellipsoidal symmetry. The main novelty in the present scheme resides in the consideration of more than a single inclusion phase type. Indeed, unlike the two-phase case, a two-point correlation function is necessary to characterize the spatial distribution of the inclusion phases in order to avoid overlap between different phase types. Moreover, once the interaction between two different phases is described, the scheme developed can straightforwardly be extended to multiphase composites. The uniform expression for the associated Hill tensors and the use of a proper tensor basis set, leads to an explicit set of equations for the bounds. This permits its application to a wide variety of phenomena governed by Laplace's operator. Some numerical implementations are provided to validate the effectiveness of the scheme by comparing the predictions with available experimental data and other theoretical results.