Abstract This paper is concerned with the problem of a Yoffe-type moving crack along the interface of an infinitely long piezoelectric bi-layer with the distinct strip-like electrical saturation and mechanical yielding zones in front of the crack tips. In this paper, we simplify the mixed boundary value problem into coupling Fredholm integral equations of the second kind by applying the Fourier integral transformation technique and the Copson method. With appropriate formula derivation and numerical discretization, we successfully obtained the numerical solutions for any layer-thickness. For the Yoffe-type crack, by assuming the crack propagates sub-sonically along the interface, three different cases, i.e., the region of electrical saturation is longer, shorter than, or equal to the domain of mechanical yielding are respectively considered in this article. The influences of the electromechanical loadings, the layer-thicknesses and crack velocity on the length of the electrical/mechanical yielding zone as well as the energy release rate under small-scale mechanical and electrical yielding are studied. When the layer-thicknesses of the piezoelectric bi-material approach infinity, the relationship obtained in this paper between the length of the electrical/mechanical yielding zone and the loadings reduces to the analytical expression of the semi-infinite piezoelectric bi-material in literature. For the piezoelectric bi-material with finite layer-thicknesses, the numerical results show that the lengths of the strip-like yielding zones rely on both the loadings and the layer-thicknesses. The energy release rate under the small-scale yielding condition depends on the loadings, layer-thicknesses as well as crack velocity. The solution technique proposed in this article has certain reference value for solving the interfacial crack problem with different yielding zones in layered materials under the coupling multi-fields.

中文翻译：

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带状电饱和区和机械屈服区的压电双层界面上的 III 型 Yoffe 型裂纹

摘要 本文研究了沿无限长压电双层界面的Yoffe型移动裂纹问题，裂纹尖端前有明显的条状电饱和区和机械屈服区。在本文中，我们通过应用傅里叶积分变换技术和 Copson 方法将混合边值问题简化为耦合的第二类 Fredholm 积分方程。通过适当的公式推导和数值离散，我们成功地获得了任何层厚的数值解。对于 Yoffe 型裂纹，假设裂纹沿界面以亚音速传播，三种不同的情况，即电饱和区长于、短于、或等于机械屈服域在本文中分别考虑。研究了机电载荷、层厚和裂纹速度对电/机械屈服区长度以及小尺度机电屈服下能量释放率的影响。当压电双材料的层厚接近无穷大时，本文得到的电/机械屈服区长度与载荷之间的关系简化为文献中半无限压电双材料的解析表达式。对于有限层厚的压电双材料，数值结果表明条状屈服区的长度取决于载荷和层厚。小规模屈服条件下的能量释放率取决于载荷、层厚和裂纹速度。本文提出的求解技术对求解多场耦合下层状材料不同屈服区的界面裂纹问题具有一定的参考价值。