Analytical formulas for effective conductivity of dispersed fibrous composites

This work is supported by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan AP23486576. The steady-state conductivity of fibrous composite materials is considered. The Hashin principle is used to calculate effective properties, as well as deterministic and stochastic approaches are applied. Structural sums are calculated, approximate analytical formulas for effective conductivity are constructed depending on the concentration of inclusions and the contrast parameter. Calculations are given for perfectly conducting fibers and insulators. The procedure for quantitative analysis of composite structures is considered based on the analytical theory of a representative volume element, which allows determining the parameters characterizing changes in the structure of a composite material that occurred as a result of the technological process. The steady-state thermal conductivity of two-dimensional (fibrous) two-phase dispersed composites with non-overlapping inclusions is presented, when a set of particles with fixed shapes and sizes is embedded in a matrix. It is shown that the effective constants of deterministic and random composites can be estimated in two ways, leading to the same result. The developed technique allows one to perform a quantitative analysis of the composite structure with an assessment of the effects resulting from the applied technological process both in the first approximation at the level of the reinforcing phase concentration and in higher orders of approximation, including effects at the micro- and macroscales. In this paper, we extend the analytical method of the representative volume element and calculate the high-order terms of the effective conductivity.