The notion of “randomness” in the mathematical theory of composites has typically been used abstractly within measure theory, making practical applications difficult. In contrast, engineering sciences often discuss randomness too loosely, lacking a theoretical foundation. This paper aims to bridge the gap between theory and applications, focusing on the effective properties of two-dimensional conducting composites with non-overlapping circular inclusions. It is shown that there is no universal minimum number of inclusions per cell in simulations of random composites. Even minor changes to Random Sequential Addition algorithms lead to different formulas for the effective constants. Application of the analytical representative volume element (aRVE) theory methodologically and practically addresses the diversity issue of random composites based on homogenization principles. In particular, it examines how the spatial arrangement of inclusions impacts the overall composite properties. The proposed method can be applied to a large number of inclusions and to symbolically given geometric and physical parameters relevant to optimal design problems. The method leverages structural sums and enables a more refined classification of different classes of composites, which was unattainable using previous approaches. The obtained results demonstrate a diversity of apparently similar composites. This paper outlines the investigation strategy and provides a detailed description of each step.

The approximate calculation of integrals is a common part of research in various fields of mathematics, primarily in the theory of approximations and compu

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We study approximate calculation of integrals of products of two functions by the method of tensor products of functionals. Under the assumption that one of them belongs to the Sobolev class with the...