Method of calculation of elastic effective properties of two-phase polydisperse media using multipoint statistical descriptors and the integral equations technique

The aim of this study is to develop a new analytical approach to calculation of effective properties of elastic heterogeneous media based on multipoint approximations of solutions of stochastic boundary value problems. Prediction of macroscopic properties of heterogeneous media is associated with the need for a reliable description of their microstructural behavior, including the interaction between individual components. A number of analytical and numerical approaches have been developed to evaluate the effective properties of structurally inhomogeneous media. However none of them makes it possible to calculate the effective properties of such media with an absolute accuracy. One of the main limitations is imposed by taking into account the features of the microstructure of the medium, such as orientation, size, shape and distribution of inclusions, as well as the features of influence of the matrix on inclusions. In this paper, multipoint approximations of solutions of boundary value problems are used to calculate the effective characteristics, in which it is necessary to employ higher-order correlation functions, which allows to take into account the multiparticle interaction of microstructural elements with a higher extent. Analytical expressions for the calculation of the effective properties of structurally inhomogeneous media using multipoint higher-order approximations of solutions of stochastic boundary value problems in elastic formulations are obtained. A numerical comparison of the calculation results of relative effective characteristics of porous inhomogeneous polydisperse media with spherical inclusions of different volume fractions is performed. For the numerical solution of the integral-differential equations, a global adaptive strategy is applied in conjunction with the multidimensional integration rule and the IMT variable transformation rule to handle the singularity of the Green’s function. Some conclusions are made on the effectiveness and limitations of the proposed approach.