The paper presents the non-local stress and electric displacement solution of a 3D semi-permeable rectangular crack in infinite orthotropic piezoelectric materials (OPMs). With the help of the generalized Almansi's theorem and 2D Fourier transform, the boundary problem is formulated by three pairs of dual integral equations, and the displacement jumps across the crack surfaces are defined. The Schmidt method is used to solve the dual integral equations. The non-local stress field (NSF) and the non-local electric displacement field (NEDF) along the crack edges are deduced. Numerical results are reported to explain the influence of the size of the rectangular crack, the lattice parameter and the electric permittivity of the air inside the crack on the NSF and the NEDF at the crack edges in OPMs in detail. The present non-local solutions exhibit no stress and electric displacement singularities along the crack edges, and may be benefit future works.

中文翻译：

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无限正交各向异性压电材料中 3D 半透矩形裂纹的非局部应力和电位移解

本文提出了无限正交各向异性压电材料 (OPM) 中 3D 半渗透矩形裂纹的非局部应力和电位移解。借助广义阿尔曼西定理和二维傅里叶变换，边界问题由三对对偶积分方程表示，并定义了裂纹表面的位移跳跃。Schmidt 方法用于求解对偶积分方程。推导出沿裂纹边缘的非局部应力场（NSF）和非局部电位移场（NEDF）。数值结果详细解释了矩形裂纹尺寸、裂纹内部空气的晶格参数和介电常数对OPMs裂纹边缘处的NSF和NEDF的影响。