Analysis of cracks in 3D piezoelectric media with various electrical boundary conditions

A weakly singular, symmetric Galerkin boundary element method capable of solving problems of isolated cracks in three-dimensional, linear anisotropic piezoelectric, infinite media with various types of crack-face boundary conditions including impermeable, permeable, semi-permeable, and the energetically consistent boundary condition introduced by Landis (Int J Solids Struct 41:6291–6315, 2004) is established. The key governing boundary integral equation used in the formulation possesses several crucial features including its desirable symmetric weak-form, weakly singular nature, and ability to treat general material anisotropy, arbitrary crack configurations and any type of boundary condition on the crack surface. The positive consequence of utilizing the singularity-reduced integral equations in the modeling, is that all involved singular integrals can be interpreted in the sense of Riemann and their validity requires only continuous crack-face data allowing C0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {C}^{0}$$\end{document}-interpolation functions to be employed everywhere in the numerical discretization. Special crack-tip elements with appropriate square-root functions are adopted in a local region along the crack front to accurately approximate the relative crack-face displacement and electric potential. With use of these crack-tip elements, the stress and electric intensity factors can be extracted directly in terms of crack-front nodal data. A system of nonlinear algebraic equations resulting from semi-permeable and energetically consistent boundary conditions is solved by standard Newton–Raphson iterative scheme. Various numerical examples of both planar and non-planar cracks under different types of electrical boundary conditions are considered and the proposed technique is found promising and computationally robust. In addition, it was determined that using crack-tip elements along the crack front significantly enhances the computational performance and that the stress and electric intensity factors can be obtained accurately using relatively coarse meshes.