In this paper, we propose a self-consistent scheme for the calculation of the components of the effective electrical conductivity tensor. The calculations were fulfilled for a microinhomogeneous material, the components of which have the Hall effect. The presence of the Hall effect leads to appearance of asymmetry of the components of the conductivity tensor and to dependence of these components on the magnitude of the magnetic field applied to the material. Our approach is based on the Generalized Differential Effective Medium (GDEM) method. This method generalizes the classical differential scheme (DEM) for the case of several inclusion types instead of one. In this case, the GDEM scheme leads to a system of matrix differential equations that were solved numerically. This solution was obtained for materials containing spherical or cylindrical inclusions (3D and 2D-problems). In the case of cylindrical inclusions, the results were obtained for inclusions with the symmetry axes orthogonal to the magnetic field. The application of the GDEM method allows us to consider the percolation effect for 2D and 3D-microheterogeneous materials. The results obtained are compared to the experimental data and the calculation results obtained by other self-consistent schemes.

中文翻译：

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具有霍尔效应的渗透微不均匀材料有效电导率的广义微分方案


在本文中，我们提出了一种自洽的方案来计算有效电导率张量的分量。对微不均匀材料进行了计算，其组件具有霍尔效应。霍尔效应的存在导致电导率张量分量出现不对称性，并导致这些分量依赖于施加到材料上的磁场的大小。我们的方法基于广义差分有效介质 （GDEM） 方法。此方法将经典微分方案 （DEM） 推广到多个包含类型而不是一个包含类型的情况下。在这种情况下，GDEM 方案导致一个矩阵微分方程组，该方程组以数值方式求解。对于包含球形或圆柱形夹杂物的材料（3D 和 2D 问题），可以获得此解。在圆柱形夹杂物的情况下，对于对称轴与磁场正交的夹杂物，可以获得结果。GDEM 方法的应用使我们能够考虑 2D 和 3D 微异质材料的渗流效应。将所得结果与实验数据和其他自洽方案得到的计算结果进行了比较。