The scattering of harmonic elastic anti-plane shear waves by a Griffith crack in a piezoelectric material plane by using the non-local theory

Abstract

In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material plane under anti-plane shear waves is investigated by using the non-local theory for impermeable crack face conditions. For overcoming the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and the electric displacement near the crack tips. By using the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. Contrary to the classical elasticity solution, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the circular frequency of incident wave and the lattice parameter. For comparison results between the non-local theory and the local theory for this problem, the same problem in the piezoelectric materials is also solved by using local theory.

Introduction

It is well known that piezoelectric materials produce an electric field when deformed, and undergo deformation when subjected to an electric field. The coupling nature of piezoelectric materials has attracted wide applications in electric–mechanical and electric devices, such as electric–mechanical actuators, sensors and structures. When subjected to mechanical and electrical loads in service, these piezoelectric materials can fail prematurely due to their brittleness and presence of defects or flaws produced during their manufacturing process. Therefore, it is important to study the electro-elastic interaction and fracture behavior of piezoelectric materials.

Many studies have been made on the electro-elastic fracture mechanics based on the modeling and analyzing of one crack in the piezoelectric materials (see, for example, [3], [12], [18], [19], [22], [23], [25], [29], [30], [32], [33]). The problem of the interacting fields among multiple cracks in a piezoelectric material has been studied by Han [14]. In Han's paper, the crack is treated as continuously distributed dislocations with the density function to be determined according to the conditions of external loads and crack surface. Most recently, Chen and Karihaloo [2] considered an infinite piezoelectric ceramic with impermeable crack-face boundary condition under arbitrary electro-mechanical impact. Sosa and Khutoryansky [26] investigated the response of piezoelectric bodies disturbed by internal electric sources. The impermeable boundary condition on the crack surface was widely used in the works [2], [22], [23], [28], [29]. However, these solutions contain stress and electric displacement singularity. This is not reasonable according to the physical nature. For overcoming the stress singularity in the classical elastic theory, Eringen [6], [8], [9] used the non-local theory to discuss the state of stress near the tip of a sharp line crack in an elastic plane subject to uniform tension, shear and anti-plane shear. Zhou [34], [35], [36], [37] used the non-local theory to study the state of the dynamic stress near the tip of a line crack or two line cracks in an elastic plane. These solutions did not contain any stress singularity, thus resolving a fundamental problem that persisted over many years. This enables us to employ the maximum stress hypothesis to deal with fracture problems in a natural way.

In the present paper, the scattering of harmonic elastic anti-plane shear waves by a Griffith impermeable crack subjects to anti-plane shear in piezoelectric materials is investigated by using the non-local theory. The traditional concept of linear elastic fracture mechanics and the non-local theory are extended to include the piezoelectric effects. For overcoming the mathematical difficulties, one-dimensional non-local kernel function is used instead of two-dimensional kernel function for the anti-plane dynamic problem to obtain the stress and electric displacement occur at the crack tips. For obtaining the theoretical solution and discussing the probability of using the non-local theory to solve the dynamic fracture problem in the piezoelectric materials, one has to accept some assumptions as Nowinski's [20], [21]. Certainly, the assumption should be further investigated to satisfy the realistic condition. Fourier transform is applied and a mixed boundary value problem is reduced to two pairs of dual integral equations. In solving the dual integral equations, the crack surface displacement and electric potential are expanded in a series of Jacobi polynomials. This process is quite different from that adopted in previous works [3], [6], [8], [9], [12], [14], [22], [23], [25], [29], [30], [32], [33]. As expected, the solution in this paper does not contain the stress and electric displacement singularity at the crack tip, thus clearly indicating the physical nature of the problem, namely, in the vicinity of the geometrical discontinuities in the body, the non-local intermolecular forces are dominant. For such problems, therefore, one must resort to theories incorporating non-local effects, at least in the neighborhood of the discontinuities.