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We consider generalizations of the classical symmetrical Bruggeman equation based on the concept of shape-distributed particle systems. The use of the Beta distribution for the particle shape is shown to result in some known as well as unknown equations of the effective medium theory. However, these equations yield no percolation threshold. On the other hand, the use of one- and two-dimensional steplike distributions of spheroidal (ellipsoidal) shapes yields a percolation threshold depending on the distribution parameters. The problem of finding the percolation threshold to fit the systems under consideration, as well as the applicability area of the generalized Bruggeman equation and its relation to the Bergman representation, are discussed.

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To calculate the effective dielectric response of a dilute composite, a generalization of the Maxwell Garnett theory for small nonspherical particles distributed in shape is proposed. Various types of distribution function are analysed and the applicability of the simplest (steplike) distribution is discussed. It is shown that the use of the steplike distribution is more valid for particles having a higher imaginary part of the permittivity in the actual region. Besides, an alternative approach to the problem based on the spectral representation is also considered. As an illustration, the effective dielectric response of a system of semiconductor (SiC) and metal (Al) ellipsoidal particles is calculated.

Using a simple phenomenological approach, we calculate the percolation threshold for Bruggeman composites having microgeometry of two kinds. Both kinds of composites consist of spheroids whose shape follows the Beta distribution. At the same time, the first one is a mixture of spheroids equally oriented along their revolution axis. In this case the percolation threshold is shown to be the same as for an assembly of equally oriented identical spheroids whose shape corresponds to the most probable shape of the distribution. For such composites the percolation threshold can vary between 0 and 1. The second one is a random mixture of the spheroids. In this case the percolation threshold is expressed in terms of the Gauss hypergeometric function; it is shown to vary between 0 and 1/3. The derived analytical results are supplemented with numerical calculations carried out for different values of the Beta distribution parameters.

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