Many natural composite materials such as carbonate rocks contain systems of oriented or partially oriented thin inclusions (microcracks) filled with a soft elastic cement. In this paper we have studied the influence of the inclusion orientation, shape, and elastic properties on the effective elastic properties of micro-inhomogeneous materials. We have calculated the components of the compliance tensor of such materials as a function of crack density. The results were obtained for thin ellipsoidal inclusions with elastic compliance much larger than the elastic compliance of the matrix. To calculate the effective compliance tensor, we have used the “non-interaction approximation” (NIA). The application of the NIA allows us to evaluate the influence of peculiarities in the spatial distribution of inclusions on the effective properties of the medium. To simplify the calculations, we have used the special tensorial basis (T-basis). We have obtained the explicit expressions for the effective elastic compliance tensor of inhomogeneous materials. To verify the analytical expressions, we have compared our results for elastic moduli to the results obtained by numerical modeling of a material containing crack-like inclusions. The results obtained by using the two methods are in very good agreement, thus indicating that our analytic method can be applied even at sufficiently large values of crack density.

许多天然复合材料（例如碳酸盐岩）包含填充有软弹性水泥的定向或部分定向的薄包裹体（微裂纹）系统。在本文中，我们研究了夹杂物的取向、形状和弹性性能对微观非均匀材料有效弹性性能的影响。我们计算了此类材料的柔量张量的分量作为裂纹密度的函数。结果是针对弹性柔量远大于基体弹性柔量的薄椭圆体夹杂物获得的。为了计算有效柔量张量，我们使用了“非交互近似”（NIA）。NIA 的应用使我们能够评估夹杂物空间分布的特性对介质有效特性的影响。为了简化计算，我们使用了特殊的张量基（T-basis）。我们得到了非均匀材料有效弹性柔量张量的显式表达式。为了验证解析表达式，我们将弹性模量的结果与通过包含裂纹状夹杂物的材料的数值模拟获得的结果进行了比较。使用两种方法获得的结果非常一致，因此表明我们的分析方法甚至可以在裂纹密度y 值足够大的情况下应用。