The behavior of two parallel symmetry permeable interface cracks in a piezoelectric layer bonded to two half piezoelectric materials planes

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Abstract

In this paper, the behavior of two parallel symmetry permeable interface cracks in a piezoelectric layer bonded to two half piezoelectric materials planes subjected to an anti-plane shear loading is investigated by using Schmidt method. 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. This process is quite different from that adopted previously. The normalized stress and electrical displacement intensity factors are determined for different geometric and property parameters for permeable crack surface conditions. Numerical examples are provided to show the effect of the geometry of the interacting cracks, the thickness and the materials constants of the piezoelectric layer upon the stress and the electric displacement intensity factor of the cracks. Contrary to the impermeable crack surface condition solution, it is found that the electric displacement intensity factors for the permeable crack surface conditions are much smaller than the results for the impermeable crack surface conditions.

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 defects, e.g. cracks, holes, etc. arising during their manufacture process. Therefore, it is of great importance to study the electro-elastic interaction and fracture behavior of piezoelectric materials. Moreover, it is known that the failure of solids results from the cracks, and in most cases, the unstable growth of the crack is brought about by the external loads. So, the study of the fracture mechanics of piezoelectric materials is much important in recent research, especially when multiple interface cracks are involved.

In the theoretical studies of crack problems, several different electric boundary conditions at the crack surfaces have been proposed by numerous researchers. For example, for the sake of analytical simplification, the assumption that the crack surfaces are impermeable to electric fields was adopted by Deeg (1980), Pak, 1990, Pak, 1992, Sosa and Pak (1990), Sosa, 1991, Sosa, 1992, Suo et al. (1992), and Gao et al. (1997), etc. In this model, the assumption of the impermeable cracks refers to the fact that the crack surfaces are free of surface charge and thus the electric displacement vanish insides the crack. In fact, cracks in piezoelectric materials consist of vacuum, air or some other gas. This requires that the electric fields can propagate through the crack, so the electric displacement component perpendicular to the crack surfaces should be continuous across the crack surfaces. Along this line, Zhank and Hack (1992) analyzed crack problems in piezoelectric materials. In addition, usually the conducting cracks which are filled with conducting gas or liquid are also applied to be a kind of simplified cracks models in piezoelectric materials by many researchers, such as McMeeking (1989) and Suo (1993). Dunn (1994), Zhang and Tong (1996), and Sosa and Khutoryansky (1999) avoided the common assumption of electric impermeability and utilized more accurate electric boundary conditions at the rim of an elliptical flaw to deal with anti-plane problems in piezoelectricity. They analyzed the effects of electric boundary conditions at the crack surfaces on the fracture mechanics of piezoelectric materials.

Layered materials can be used to manufacture high performance structures in order to achieve a high strength-to-weight ratio. Therefore, the analysis of laminated piezoelectric composite structures has attracted the attention of many researchers in recent years, such as Shen et al., 1999a, Shen et al., 1999b and Shen et al. (2000). Kim and Jones (1996) have studied the behavior of brittle fracture at the interface between two dissimilar piezoelectric materials. Beom and Atluri (1996) derived the complete form of stress and electric displacement fields of an interfacial crack between two dissimilar anisotropic piezoelectric media. The plane problem of a crack terminating at the interface of a bimaterial piezoelectric was treated by Qin and Yu (1997). In particular, control of laminated structures including piezoelectric devices was the subject of research by Tauchert (1996), Lee and Jiang (1996), Batra and Liang (1997), and Heyliger (1997). Many piezoelectric devices comprise both piezoelectric and structural layers, and an understanding of the fracture process of piezoelectric structural systems is of great importance in order to ensure the structural integrity of piezoelectric devices (Shindo et al., 1998; Narita et al., 1999; Chen et al., 1998). Recently, Soh et al. (2000) have investigated the behavior of a bi-piezoelectric ceramic layer with an interfacial crack by using the dislocation density function and the singular integral equation method. To our knowledge, the electro-elastic behavior of a piezoelectric ceramic with two parallel interface cracks subjected to an anti-plane shear loading has not been studied despite the fact that many piezoelectric devices are constructed in a laminated form by using the Schmidt method.

In the present paper, we consider the electro-elastic behavior of two parallel symmetry permeable interface cracks in a piezoelectric layer bonded to two same half piezoelectric materials planes subjected to an anti-plane shear is investigated using the Schmidt method (Morse and Feshbach, 1958). It is a simple and convenient method for solving this problem. 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 gaps of two crack surface displacement are expanded in a series of Jacobi polynomials. This process is quite different from that adopted in previous works (Han and Wang, 1999; Deeg, 1980; Pak, 1992; Sosa, 1992; Suo et al., 1992; Park and Sun, 1995; Zhang and Tong, 1996; Gao et al., 1997; Wang, 1992; Narita et al., 1999; Chen et al., 1998; Shen et al., 1999a, Shen et al., 1999b; Shen et al., 2000; Kim and Jones, 1996; Beom and Atluri, 1996; Qin and Yu, 1997; Soh et al., 2000). The form of solution is easy to understand. Numerical solutions are obtained for the stress and electric displacement intensity factors for permeable crack surface conditions. Note that the conducting crack condition is a special case of the permeable crack considered by other researchers (Parton, 1976; Zhank and Hack, 1992). Another main objective of the present study is to investigate the effect of the layer thickness, the distance between two cracks and the material constants of the two dissimilar materials on the fracture behavior.

Section snippets

Formulation of the problem

Fig. 1 shows a layered structure made by bonding together two same half piezoelectric materials plane. The piezoelectric materials layers are layer 2 and layer 3 of thickness h, with two parallel interface cracks of length 2l created. A Cartesian coordinate system (x, y, z) is positioned with its origin at the center between two parallel interface cracks for reference purposes. Note that the z-axis is oriented in the poling direction of the piezoelectric materials, and the xy plane is the

Solution

The solutions of , can be written asw(1)(x,y)=2π0A1(s)e−sycos(sx)ds,φ(1)(x,y)=e15(1)ε11(1)w(1)(x,y)+2π0B1(s)e−sycos(sx)ds,y⩾hw(2)(x,y)=2π0[A2(s)e−sy+B2(s)esy]cos(sx)ds,φ(2)(x,y)=e15(2)ε11(2)w(2)(x,y)+2π0[C2(s)e−sy+D2(s)esy]cos(sx)ds,h⩾y⩾0w(3)(x,y)=2π0[A3(s)esy+B3(s)e−sy]cos(sx)ds,φ(3)(x,y)=e15(2)ε11(2)w(3)(x,y)+2π0[C3(s)esy+D3(s)e−sy]cos(sx)ds,0⩾y⩾−hw(4)(x,y)=2π0A4(s)esycos(sx)ds,φ(4)(x,y)=e15(1)ε11(1)w(4)(x,y)+2π0B4(s)esycos(sx)ds,y⩽−hwhere Ak(s), Bk(s) (k=1, 2, 3, 4), Cj(s)

Solution of the dual integral equation

The Schmidt method (Morse and Feshbach, 1958) is used to solve the dual integral equations. The gap function of the crack surface displacement is represented by the following series:f1(x)=f2(x)=∑n=1anP2n−21/2,1/2xl1−x2l21/2,for−l⩽x⩽l,y=0f1(x)=f2(x)=w(1)(x,h+)−w(2)(x,h)=0,for|x|>l,y=0where an are unknown coefficients to be determined and Pn(1/2,1/2)(x) is a Jacobi polynomial (Gradshteyn and Ryzhik, 1980). The Fourier transform of , is (Erdelyi, 1954)f̄1(s)=∑n=1anGn1sJ2n−1(sl),Gn=2π(−1)n−1Γ2n−

Intensity factors

The coefficients an are known, so that the entire perturbation stress field and the perturbation electric displacement can be obtained. However, in fracture mechanics, it is of importance to determine the perturbation stress σyz and the perturbation electric displacement Dy in the vicinity of the crack's tips. σyz(1), σyz(2), σyz(3), σyz(4), Dy(1), Dy(2), Dy(3) and Dy(4) along the crack line can be expressed respectively asσyz(1)(x,h)=σyz(2)(x,h)=σyz(3)(x,−h)=σyz(4)(x,−h)=σyz=2πn=1anGn0

Numerical calculations and discussion

This section presents numerical results of several representative problems. Adopting the first 10 terms in the infinite series (51), we followed the Schmidt procedure. From the literatures (see e.g. Itou, 1978; Zhou et al., 1999a, Zhou et al., 1999b), it can be seen that the Schmidt method performs satisfactorily if the first 10 terms of the infinite series (51) are retained. The solution does not change with an increase of the number of terms in (51) beyond 10. The precision of present

Acknowledgements

The authors are grateful for financial support from the Post Doctoral Science Foundation of Hei Long Jiang Province, the Natural Science Foundation of Hei Long Jiang Province, the National Science Foundation with the Excellent Young Investigator Award and the Scientific Research Foundation of Harbin Institute of Technology (HIT.2000.30).

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