The effective field method is applied to solution of the homogenization problem for anisotropic media containing random sets of thin inclusions of low conductivity (crack-like inclusions) or cracks. The derived expression for the tensor of effective conductivity of cracked media on the one hand, takes into account peculiarities of shapes and conductivity of thin inclusions, and on the other hand, reflect statistical properties of the inclusion distributions in the host medium. The crucial part of realization of the method is solution of the so-called one-particle problem that is the conductivity problem for an isolated inclusion embedded into an anisotropic host medium and subjected to a constant external field. This problem is reduced to solution of the integral equation for the potential jump on the middle surface of a thin inclusion. An efficient numerical method of solution of this equation is proposed. The integral equation is discretized by Gaussian approximating functions and reduced to a linear algebraic system for the coefficients of the approximation (the discretized problem). For Gaussian functions, the elements of the matrix of the discretized problem are calculated in explicit analytical forms (for isotropic host media) or they reduce to a standard 1D-integral (for arbitrary anisotropic host media) that can be tabulated. As a result, the matrix of the discretized problem is calculated fast. For inclusions with planar middle surfaces and regular grids of approximated nodes, this matrix has Toeplitz’ structure, and fast Fourier transform algorithm can be used for iterative solution of the discretized problem. The cases of a strongly anisotropic medium with circular, annular and square cracks are considered. Examples of solution of the homogenization problems for the medium containing cracks of various shapes are presented.

有效场法适用于各向异性介质中含有随机组的低电导率薄夹杂物（类裂纹夹杂物）或裂纹的均匀化问题的求解。裂纹介质有效电导率张量的导出表达式一方面考虑了薄夹杂物的形状和电导率的特点，另一方面反映了夹杂物在基质中分布的统计特性。实现该方法的关键部分是解决所谓的单粒子问题，即嵌入各向异性主体介质并受到恒定外场作用的孤立夹杂物的电导率问题。该问题简化为薄夹杂物中间表面电位跃变积分方程的求解。提出了求解该方程的有效数值方法。积分方程由高斯逼近函数离散化，并简化为逼近系数的线性代数系统（离散化问题）。对于高斯函数，离散化问题的矩阵元素以显式解析形式计算（对于各向同性宿主介质），或者它们简化为可以制成表格的标准一维积分（对于任意各向异性宿主介质）。因此，离散化问题的矩阵计算速度很快。对于具有平面中间面和近似节点规则网格的夹杂物，该矩阵具有Toeplitz'结构，可以使用快速傅里叶变换算法迭代求解离散化问题。具有圆形的强各向异性介质的情况，考虑环形和方形裂纹。给出了含有各种形状裂纹的介质的均匀化问题的求解实例。