We formulate and solve the mathematical problem for antiplane cracks in a pre-stressed and pre-polarized piezoelectric material with static initial fields, assuming the initially deformed configuration of the body is locally stable. Using the boundary conditions of antiplane cracks, we get the Riemann-Hilbert problems. Nonhomogeneous linear complex differential equations having the unknown complex potential are obtained. For constant value of the applied incremental forces can be obtained the complex potentials, incremental displacement and stress fields corresponding to the third mode of the classical fracture. The problem of interaction of two collinear, unequal cracks in a pre-stressed and pre-polarized piezoelectric material, is also studied.

中文翻译：

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预应力和预极化压电材料中的裂纹相互作用

假设物体的初始变形构型是局部稳定的，我们将制定并解决具有静态初始场的预应力和预极化压电材料中反平面裂纹的数学问题。利用反平面裂纹的边界条件，我们得到了黎曼-希尔伯特问题。获得具有未知复数电位的非齐次线性复数微分方程。对于所施加的增量力的恒定值，可以获得与经典裂缝的第三种模式相对应的复势，增量位移和应力场。还研究了在预应力和预极化的压电材料中两个共线不等裂纹的相互作用问题。