A piezoelectric material with a periodic distribution of slant mode-III cracks

Abstract

By using the dislocation layer modelling, electroelastic analysis of a periodic distribution of slant cracks in a piezoelectric material is made under anti-plane shear and in-plane electric loading. The cracks are assumed to be either electrically permeable or impermeable. With the aid of the fundamental solution for a screw dislocation, the problem is reduced to solving a singular integral equation. In particular, closed-form solutions of electroelastic field are derived for both coplanar and parallel periodic cracks, respectively. By using a collocation technique and the quadrature formula for slant cracks, the resulting singular integral equation is solved approximately by a system of algebraic equations, and the field intensity factors as well as mechanical strain energy release rate are obtained. Numerical results show the effects of electric loading and geometric parameters on the normalized mechanical strain energy release rate.

Introduction

As well known, crack theory is quite important for the investigation of brittle fracture in elastic solids. For brittle piezoelectric materials, due to the intrinsic coupling characteristics between electric and elastic behaviors, the analysis of electroelastic field for a cracked piezoelectric material is more complicated than that of elastic field for a cracked purely elastic material. Because of the wide application of piezoelectric materials in micro-electromechanical system such as sensors, actuators, and transducers, their failure analysis has recently become the subject of intensive study. In engineering applications, for a transversely isotropic piezoelectric material, applied electric field is generally parallel to the poling axis, and in-plane deformations are coupled with applied electric field. However, when electric field is applied perpendicular to the poling axis, a piezoelectric actuator acts through thickness-shear mode, and in this case anti-plane shear deformation is coupled with applied electric field. Such piezoelectric actuators have many advantages over extension-mode actuators (Sun and Zhang, 1995; Mueller and Zhang, 1998). On the other hand, the presence of cracks in piezoelectric materials gives rise to electroelastic field concentration, and further alters the performance of piezoelectric materials. Therefore, the analysis of electroelastic behavior in cracked piezoelectric materials is a prerequisite. Considerable researches have been reported on the singular electromechanical behavior of cracked piezoelectric materials via various approaches.

In particular, the problem involving a mode-III straight crack or a circular arc-crack embedded in an infinite piezoelectric sheet subjected to uniform anti-plane shear and in-plane electric loading at infinity has been studied by Pak (1990a), and Zhong and Meguid (1997), respectively. Based on the exact electric boundary conditions at the rim of a hole, an infinite piezoelectric sheet containing an elliptical hole in a state of anti-plane shear deformation was studied by Dunn (1994), using the Esheby inclusion method, and by Zhang and Tong (1996), using conformal mapping technique. Later, Shindo et al., 1996, Shindo et al., 1997 analyzed the singular electroelastic behavior of the horizontal and vertical central mode-III cracks in a piezoelectric strip and obtained the numerical results for the field intensity factors by solving a resulting integral equation. This problem is further extended to an eccentric anti-plane shear crack by numerical method (Shin et al., 2000) and analytic approach (Li, 2002), respectively. A cracked rectangular piezoelectric body under anti-plane shear and in-plane electric loading has been studied by Kwon and Lee (2000). Due to the interaction of multiple cracks, the electroelastic field for doubly-periodic arrays of cracks of equal length in an infinite piezoelectric sheet has been analyzed by Pak and Goloubeva (1996), who considered the rectangular and diamond-shaped arrays of cracks loaded by uniform anti-plane shear and in-plane electric loading, and determined the intensity factors of the singular electroelastic field. Moreover, the obtained results can be used to predict the effective material properties of a damaged piezoelectric material from the view of micromechanics.

It is well known that dislocations have a significant effect on most behaviors of a material. Moreover, within the framework of continuum theory, cracks can be modelled by the pile-ups of infinitesimal dislocation layers (Lardner, 1974). Recently, a mode-III crack in a piezoelectric half-space was solved by Tupholme (1999), who used dislocation layer modelling combined with an image method to obtain a singular integral equation, and determined the singular electroelastic field in closed form when subjected to uniform anti-plane shear and in-plane electric displacement.

By use of the dislocation layer modelling, in this paper we consider a periodic distribution of slant mode-III cracks in an infinite piezoelectric material. The problem is reduced to solving a singular integral equation. For the piezoelectric material subjected to uniform anti-plane shear and constant in-plane electric loading, analytical expressions for the singular electroelastic fields are derived for both parallel and coplanar periodic cracks, and the corresponding field intensity factors are determined explicitly. Numerical results of the mechanical strain energy release rates are presented graphically to show the effects of electric loading and some geometric parameters.