Application of Function of Complex Variable and MATLAB to Analysis of Piezoelectric Body Stress and Strain State with Crack

Abstract

In this paper, applying Linear Elastic Fracture Mechanics (LEFM) and linear theory of elasticity, spatial stress and strain states, and electric fields states in the piezoelectric material at the points of the vicinity of the crack tip are analyzed for the case of material shear along crack plane. Method of analytical functions of a complex variable in polar-cylindrical coordinates is used. A stress function and electrical potential function are introduced, which are expressed by polar coordinates and trigonometric series. The analysis shows the characteristics of coordinates of the mechanical stress tensor, vector of piezoelectric field of displacement, and tensor of relative mechanical deformations. Components of the analyzed tensors and vectors of the coupled fields of piezoelectric material have singular values at the tip of the semi-infinite crack. A numerical example is given for a piezoelectric material (PZT-4) through MATLAB. A visualization is presented for the spatial states of coupled fields of stress, deformation, and electrical displacement, and of the influence of every parameter on crack propagation process in piezoelectric material.