The effective medium and the average field approximations vis-à-vis the Hashin–Shtrikman bounds. I. The self-consistent scheme in matrix-based composites

Abstract

The effective medium approximation (EMA) and the average field approximation (AFA) are two classical micromechanics models for the determination of effective properties of heterogeneous media. They are also known in the literature as ‘self-consistent’ approximations. In the AFA, the basic idea is to estimate the actual average field existing in a phase through a configuration in which a typical particle of that phase is embedded in the homogenized medium. In the EMA, on the other hand, one or more representative microstructural elements of the composite is embedded in the homogenized effective medium subjected to a uniform field, and the demand is made that the dominant part of the far-field disturbance vanishes. Both parts of this study are concerned with two-phase, matrix-based, effectively isotropic composites with an inclusion phase consisting of randomly oriented particles of arbitrary shape in general, and ellipsoidal shape in particular. The constituent phases are assumed to be isotropic. It is shown that in those systems the AFA and EMA give different predictions, with the distinction between them becoming especially striking regarding their standing vis-à-vis the Hashin–Shtrikman (HS-bounds). While due to its realizability property the EMA will always obey the bounds, we show that there are circumstances in which the AFA may violate the bounds. In the AFA for two-phase matrix-based composites, the embedded inclusion is a particle of the inclusion phase. If the particle is directly embedded in the effective medium, the method is called here the self-consistent scheme–average field approximation (SCS-AFA), and will obey the HS-bounds for an inclusion shape that is simply connected. If the embedded entity is a matrix-coated particle, then the method is called the generalized self-consistent scheme–average field approximation (GSCS-AFA), and may violate the HS-bounds. On the other hand, in the EMA for matrix-based composites with well-separated inclusions, we indicate that in view of its premises the embedding with a matrix-coated particle generally becomes the appropriate one, and the method is thus called the generalized self-consistent scheme–effective medium approximation (GSCS-EMA). Part I of this study is concerned with SCS-AFA in dielectrics and elasticity, and Part II with the GSCS-AFA and GSCS-EMA in dielectrics.