Abstract—When creating inhomogeneous materials with required functional characteristics, theoretical methods for predicting their properties depending on the composition and structure are of great importance. This paper offers a review of methods for calculating the effective electrophysical characteristics of inhomogeneous media that can take into account such structural features of these media as the multicomponent nature, probability distributions of orientations and shapes of inclusions, and several levels of nesting. The Maxwell Garnett approximation and its generalizations for inhomogeneous matrix-type media are presented in detail: the Fricke and Bragg–Pippard formulas for the cases of ellipsoidal homogeneous inclusions, as well as my generalizations of the Maxwell Garnett approximation for the cases of ellipsoidal homogeneous anisotropic inclusions oriented according to a probabilistic law, and ellipsoidal anisotropic inclusions with anisotropic shell. Variants of the effective medium approximation for media with homogeneous inclusions are considered: the Bruggeman approximation; generalized effective medium approach of D. Stroud; generalized singular approximation. A differential approach is presented on the basis of the generalized singular approximation, with the help of which an equation with a parameter is obtained that allows one to vary the degree of differentiality of the approximation. A description of the my approach for media with inclusions in a multilayer shell is given: a generalized effective field approximation, which can be considered a generalization of the generalized singular approximation to media with several levels of nesting. Methods for taking into account the probability distributions of inclusion orientations (using the representation theory of the SO(3) group) and inclusion shapes are described. It is established that the generalized effective-field approximation for inhomogeneous media with inclusions in a multilayer shell has the greatest degree of generality.

摘要：在制造具有所需功能特性的非均质材料时，根据成分和结构预测其性能的理论方法非常重要。本文综述了计算非均匀介质有效电物理特性的方法，这些方法可以考虑这些介质的结构特征，如多组分性质、夹杂物的方向和形状的概率分布以及多层嵌套。详细介绍了麦克斯韦加内特近似及其对非均匀基体类型介质的推广：针对椭球均匀夹杂物情况的 Fricke 和 Bragg-Pippard 公式，以及我对椭球均匀各向异性情况的麦克斯韦加内特近似的推广根据概率定律定向的包裹体，以及具有各向异性壳的椭圆体各向异性包裹体。考虑具有均匀内含物的介质的有效介质近似的变体：布鲁格曼近似； D. Stroud 的广义有效媒介方法；广义奇异近似。在广义奇异近似的基础上提出了一种微分方法，借助该方法可以获得带有参数的方程，该方程允许人们改变近似的微分程度。给出了我对多层壳中包含夹杂物的介质的方法的描述：广义有效场近似，它可以被认为是对具有多个嵌套级别的介质的广义奇异近似的概括。描述了考虑夹杂物方向的概率分布（使用 SO(3) 族的表示理论）和夹杂物形状的方法。已确定多层壳中含有夹杂物的非均匀介质的广义有效场近似具有最大程度的通用性。