A mode-III Yoffe-type crack on the interface of a piezoelectric bi-layer with the strip-like electrical saturation and mechanical yielding zones

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.