The Crack Tip Fields for Anti-plane Crack in Functionally Graded Piezoelectric Materials with Arbitrarily Gradient Direction

Abstract

The higher order crack tip fields for anti-plane crack in functionally graded piezoelectric materials(FGPMs) under mechanical and electrical loadings are investigated. Although the elastic stiffness, piezoelectric parameter, and dielectric permittivity of FGPMs are assumed to be exponential functions,they are along arbitrary gradient direction. As usual, the crack surfaces are assumed to be electrically impermeable. Similar to the Williams’ solution of homogeneous elastic materials, the high order crack tip stress and electric displacement fields are obtained by the method of eigen-expansion. Introduction To get rid of the possible interfacial failure, FGPMs have been introduced to reduce the stress concentration at the interface. In order to make FGPMs to have high strength, high reliability and long lifetime, the behaviors of cracks in brittle piezoelectric materials have to be analyzed. Therefore, fracture mechanics problems of a cracked body in FGPMs have received much attention in recent decades. Ma et al. [1] investigated the electro-elastic behavior of a Griffith’s crack in a functionally graded piezoelectric strip. Ou [2] studied the internal crack problem located within one functionally graded piezoelectric strip. The crack is normal to the edge of the strip and the material properties vary along the direction of crack length. Hus and Chue [3] studied mode III arbitrarily oriented crack in an FGPM strip bonded to a homogeneous piezoelectric half plane. Yeh and Chue [4] investigated the anti-plane crack problem of a FGPM cracked strip bonded to an FGPM cracked half-plane. It should be mentioned that existing studies in fracture analysis of FGPMs are focused on the singular part of crack tip fields. No attempts have been made in giving the higher order crack tip fields for FGPMs with arbitrarily gradient direction. The purpose of this paper is to present the higher order crack tip fields and explicit expression of intensity factors. Basic equations Consider a crack in FGPMs under anti-plane shear tractions and in-plane electric displacements, as shown in Fig.1. The FGPM is poled in the z direction and isotropic in the xoy plane. The present work employs the following exponential functions to describe the continuous variations of material properties, 1 2 1 2 1 2 44 440 15 150 11 110 , , x y x y x y c c e e e e e β β β β β β e e + + + = = = (1) where 440 c is the shear modulus, 150 e is the piezoelectric coefficient, 110 e is the dielectric parameter at 0, 0 x y = = . International Conference on Materials, Environmental and Biological Engineering (MEBE 2015) © 2015. The authors Published by Atlantis Press 1007 The governing equations can be written