Abstract
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 bounds, 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.
Highlights
The prediction of the effective transport and elastic properties of multiphase materials has attracted the attention of scientists and engineers over many years
= 1, 2, i.e. θ(dk ),α(k ) = θ(d k),α( k) and p(k| ) = p( |k), something that is contradictory by definition. This situation cannot be realized from Theorem 3.5. This explains past results for three-phase composites, which illustrated that the Mori-Tanaka method produced results residing outside the Hashin-Shtrikman bounds [25], unless all inclusions are spherical, in which case the conditions for microstructural construction are met
The Hashin-Shtrikman bounds are routinely employed to bound the effective properties of composites
Summary
The prediction of the effective transport and elastic properties of multiphase materials has attracted the attention of scientists and engineers over many years now. It should be noted that the general form for the HS bounds applicable to arbitrarily anisotropic composites can be written down in some cases, works concerning the construction of such bounds from first principles (volume fractions, elastic or physical properties, shapes of phases of the composite and their spatial distribution) of a given material are not found, if they exist at all, for the three phase case For all these reasons, using a tensor-basis for transverse isotropy and exploiting the uniformity of the so-called Eshelby and Hill tensors [22], we construct explicit expressions for the HS bounds for the effective quasi-static transport properties (thermal or electrical conductivity, electrical resistivity or magnetic permeability) for transversely isotropic (TI) three phase media, incorporating information as regards the shape, relative size and distribution of the two filler phases. We illustrate the implementation of the scheme with several examples where comparisons with other theoretical predictions confirm that the present model can predict and bound the effective transport properties with accuracy
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