Analysis of fractures in 3D linear piezoelectric media

Abstract

This paper presents a numerical technique called a weakly singular, symmetric Galerkin boundary element method (SGBEM) that can be used to analyze cracks in three-dimensional, generally anisotropic, linear piezoelectric infinite and finite media under various types of electrical boundary conditions including impermeable, permeable, semi-permeable and Landis-type conditions. One of advantage features of the present technique is that the governing equations are in symmetric forms, which are obtained by using two sets of boundary integral equations: the first one is the integral equation for the generalized displacement (i.e. displacement and electric potential) and the other one is the integral equation for the generalized traction (i.e. traction and surface electric charge). Since such pair of integral equations contain only weakly singular kernels of O(1/r), the standard ℃-interpolation functions can be employed to approximate the solutions by using the Galerkin scheme. Another positive feature is the use of special crack tip elements along the local region of the crack front to accurately model the near tip field.As a result, the stress and electric intensity factors can therefore be obtained accurately by using relatively coarse meshes. The weakly singular SGBEM is validated through various numerical experiments for both infinite and finite boundary value problems under several types of electrical boundary conditions. It was found that obtained results for a penny-shaped crack are in excellent agreement with the exact solutions. Subsequently, more complex problems are treated to demonstrate the versatility of the current technique to model cracks and bodies of various geometries under various loading conditions. Finally, the influence of electrical boundary conditions on the stress intensity factors and the electric intensity factor for piezoelectric infinite and finite media is thoroughly investigated.