Abstract—

Problems of plane cracks of normal fracture (mathematical cuts) in the middle plane of a transversely isotropic elastic layer, the outer faces of which are under conditions of a sliding support, are considered. Isotropic planes are parallel or perpendicular to layer faces. Using the Fourier integral transform, the problems are reduced to integro-differential equations for crack opening, from which one can obtain the known equations of the corresponding problems for a transversely isotropic space and an isotropic layer by passing to the limit. A regular asymptotic method is applied for elliptical cracks; this method is effective for a relatively thick layer. It is shown the applicability domain of the method narrows with increasing anisotropy that is characterized by the roots of the characteristic equation (for an isotropic material, all roots are equal to unity). For strip-like cracks, closed solutions are obtained based on special approximations of the kernel symbols of integral equations, the relative errors of which decrease with increasing anisotropy. Calculations are made for known transversely isotropic materials.

中文翻译：

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各向同性层中的平面裂纹

摘要-

考虑了横向各向同性弹性层的中间平面中的正常断裂（数学切口）的平面裂纹的问题，该横观各向同性的弹性层的外表面处于滑动支撑的条件下。各向同性平面平行于或垂直于层面。使用傅立叶积分变换，将问题简化为用于开裂的积分微分方程，从中可以通过求解极限，获得横向各向同性空间和各向同性层的相应问题的已知方程。对于椭圆形裂纹，采用常规渐近法。该方法对于较厚的层是有效的。结果表明，该方法的适用范围随着各向异性的增加而缩小，其特征在于特征方程的根（对于各向同性材料，所有根都等于1）。对于条状裂纹，基于积分方程内核符号的特殊近似来获得封闭解，其相对误差随各向异性的增加而减小。对已知的横向各向同性材料进行计算。