Generalized derivative and its use for analysis of the microstructure of a heterogeneous medium

An analytical account of the influence of the heterogeneous medium internal boundaries on the propagation of an elastic stress field through it is carried out. The introduced mathematical concept aimed at displaying the microstructure of a heterogeneous system is a derivative in a generalized sense. Based on the formalism of the generalized derivative, the operator is modified in the used initial model of the linear theory of elasticity. Green's function (built on the transformed operator) displays the microstructural features of the system. To obtain the effective coefficients of elasticity included in the averaged equations and describing the elastic properties of a heterogeneous medium, the method of conditional moments is used. Carrying out operations within the framework of this approach leads to integrals containing the modified averaged Green's function and the correlation function of the geometry of the structure. Based on these terms the microstructure of the system is integrally taken into account in the final effective elasticity coefficients.