In this paper, we derive the second-order crack opening displacement tensor for an arbitrarily oriented elliptical crack in an elliptically orthotropic (EO) matrix. This result is obtained in explicit closed form. The approach is based on the Saint-Venant’s idea of linear transformation between boundary value problems for elliptically orthotropic and isotropic bodies. The solution utilizes the classical representation of an ellipsoid crack where the smallest aspect ratio approaches zero and the transformation of the Taylor expansion of the corresponding Hill tensor. It is shown, in particular, that transformed cracks have neither the same in-plane aspect ratio nor the same vanishing aspect ratio. It requires a correction factor in the crack opening displacement tensor. Some specific relative orientations of the crack with respect to the symmetry planes of the EO matrix are considered in detail and effective properties are calculated in the case of randomly distributed cracks. The result is also extended to the case of a cylindrical (plane strain) crack.

在本文中，我们推导了椭圆正交各向异性（EO）矩阵中任意取向的椭圆形裂纹的二阶裂纹张开位移张量。以显式封闭形式获得此结果。该方法基于Saint-Venant关于椭圆形正交各向异性和各向同性物体的边值问题之间的线性变换的思想。该解决方案利用了椭圆形裂纹的经典表示形式，其中最小纵横比接近零，并且转换了相应的Hill张量的Taylor展开。尤其显示出，相变的裂纹既不具有相同的面内纵横比也不具有相同的消失纵横比。它需要在裂纹开口位移张量中具有校正因子。详细考虑了裂纹相对于EO矩阵对称面的某些特定相对方向，并在随机分布的裂纹情况下计算了有效特性。结果还扩展到圆柱形（平面应变）裂纹的情况。