## Abstract

In the present paper, the dynamic behaviour of a Griffith crack situated at the interface of two bonded dissimilar functionally graded piezoelectric materials (FGPMs) is considered. It is assumed that the elastic stiffness, piezoelectric constant, dielectric permittivity and mass density of the FGPMs vary continuously as an exponential function of the *x* and *y* coordinates, and that the FGPMs are under anti-plane mechanical loading and in-plane electrical loading. By using an integral transform technique the problem is reduced to four pairs of dual integral equations, which are transformed into four simultaneous Fredholm integral equations with four unknown functions. By solving the four simultaneous Fredholm integral equations numerically the effects of the material properties on the stress and electric displacement intensity factors are calculated and displayed graphically.

## 1. Introduction

It is well known that a piezoelectric material produces an electric field when deformed and that it undergoes a deformation when subjected to an electric field. Owing to this intrinsically coupled phenomenon, piezoelectric materials are widely used as sensors and actuators in intelligent advanced structure design. When subjected to mechanical and electric stresses in service, piezoelectric materials can fail owing to defects, such as cracks, holes, etc., which might arise during their manufacture. It is therefore important to study the electromechanical behaviour of piezoelectric materials with defects. Moreover, it is known that failures of solids generally result from the final propagation of a crack, and, in most cases, the unstable growth of a crack is brought about by external dynamic loads. The study of the dynamic fracture mechanics of piezoelectric materials is therefore of great interest in current research.

Nowadays piezoelectric materials have wide applications in the smart systems of the aerospace, automotive, medical and electronic fields owing to the intrinsic coupling characteristics between the electric and mechanical fields in the materials. The mechanical reliability and durability of these materials have become increasingly important, since the piezoelectric materials are being extensively used as actuators or transducers in these technologies. The disadvantage of those materials is that they crack at low temperatures and creep at high temperatures. The fracture of piezoelectric materials has therefore received significant attention. The development of functionally graded materials (FGMs) has demonstrated that they have the potential to reduce the stress concentration and to increase the fracture toughness, and FGMs can be extended to piezoelectric materials to improve their reliability.

The main object of this paper is to explore the fracture mechanics of functionally graded piezoelectric materials (FGPMs) when material properties change in both the *x*- and *y*-directions, and there is a large effect on the stress intensity factors of the crack in the *x*-direction (parallel to the crack) since it is the anti-plane direction. The equation of equilibrium has been solved analytically and we have considered a crack situated at the interface of two bonded dissimilar graded piezoelectric half-spaces under plausible assumptions.

In the present paper, the dynamic and electric displacement intensity factors are determined when the crack is situated in the interface of two similar FGPMs under the time-harmonic anti-plane mechanical loading and in-plane electrical loading. In this way, we have considered diffraction of torsional waves by the crack as the crack is subjected to normal torsional waves. The Fourier transform technique is used to reduce the mixed boundary-value problem into four pairs of dual integral equations, which are transformed to four simultaneous Fredholm integral equations of the second kind. Solving the four simultaneous Fredholm integral equations numerically, results for stress intensity factors are obtained and shown graphically, showing the influence of graded piezoelectric materials on the crack formation.

Connected to this paper, Chen *et al*. (1997, 1998) and Li *et al*. (2000) have considered anti-plane shear problems for a crack between two dissimilar homogeneous piezoelectric materials. In this paper, we have solved the equations of equilibrium by the method discussed by Dhaliwal & Singh (1978).

In some practical structures, one major concern has been the mechanical failure of the FGPMs. This is normally due to the discrepancies in mechanical and piezoelectric properties between component materials, which results in cracking at the interface influencing the strength of the whole structure.

To meet the demand of advanced piezoelectric materials in lifetime and reliability, and with the help of modern material processing technology, the concept of functionally gradient materials has recently been extended to piezoelectric materials (e.g. Wu *et al*. 1996; Lee & Yu 1998; Hu *et al*. 2002, 2003, 2005; Li & Weng 2002*b*; Wang 2003; Wang & Zhang 2004).

It is also worth mentioning that many dynamic fracture problems have been studied in the past (some of the references are Jin & Zhong 2002; Kwon *et al*. 2002; Li & Weng 2002*a*; Chen *et al*. 2003; Kwon 2003).

It should also be noted that there is recent research work (Häusler & Balke 2001; Häusler *et al*. 2004) on the interface cracks at the interface of two bonded piezoelectric materials. Connected to this paper, the scattering of harmonic anti-plane shear waves by a crack in functionally gradient piezoelectric materials is discussed by Li *et al*. (2004, 2005).

There have been a number of studies in books (Joffe *et al*. 1971; Ikeda 1996) devoted to the theoretical analysis and engineering application of piezoelectric materials, and more recent work in FGMs has been done in reference (Borrelli *et al*. 2004).

To our knowledge, there is no major work on the fracture mechanics of FGPMs in which the properties of the material vary in both directions parallel and perpendicular to the crack. In other words, elastic stiffness, piezoelectric constant and dielectric permittivity vary in *x*- and *y*-directions (parallel and perpendicular to the crack).

## 2. Solution of equilibrium equation, formulation of the crack problem and reducing the problem to the solution of a system of simultaneous Fredholm integral equations

We consider an infinitely long crack of width 2*a* located at the interface *y*=0 of two piezoelectric half-spaces, where the crack boundary is parallel to the *z*-axis as shown in figure 1. The Cartesian coordinates *x*, *y*, *z* are the principal axes of the material symmetry, while the *z*-axis is oriented in the poling direction of the two piezoelectric half-spaces.

Under anti-plane deformation, the constitutive equations are(2.1)(2.2)where *w* is the component of displacement in the *z*-direction; (*σ*_{xz}, *σ*_{yz}) are the shear stress components; (*D*_{x}, *D*_{y}) are components of the electric displacement vector; *ϕ* is the electric potential; *c*_{44}, *e*_{15} and *ϵ*_{11} are the shear modulus, piezoelectric constant and dielectric constant, respectively; and *ρ* is the density and(2.3)In the following, we use the notations(2.4)The equilibrium equations are(2.5)Using equations (2.1)–(2.4), we can write equation (2.5) in the following form:(2.6)

(2.7)

We assume that the shear modulus, the piezoelectric constant and the dielectric constant are in the following form:(2.8)where *α*_{1}, *α*_{2} and *α*_{3} are constants and(2.9)*ρ*_{0} is constant and does not depend on *x* and *y*.

Substituting equations (2.8) and (2.9) into equations (2.6) and (2.7), we find that(2.10)

(2.11)

By multiplying equation (2.10) by *α*_{3}/*α*_{2} and equation (2.11) by *α*_{2}/*α*_{1} and then adding the resulting equations, we obtain(2.12)where(2.13)Subtracting equation (2.11) from equation (2.10), we find that(2.14)

(2.15)

Equations (2.12) and (2.14) hold only if(2.16)Equations (2.12) and (2.14) are of the same form. Substituting (where *ω* is the wave frequency)(2.17)into equation (2.12) leads to(2.18)where *k*_{1}=*ω*/*c* denotes the wavenumber. Assuming(2.19)(2.20)(2.21)where *a*_{0} and *b*_{0} are the constants, which means that equation (2.18) can be written as(2.22)with as a variable separation constant. With(2.23)and equations (2.22), (2.17) and (2.19), we obtain(2.24)

Introducing a new function *Ψ* by(2.25)and substituting equation (2.25) into equation (2.11), leads to(2.26)If we assume that(2.27)and , then by comparing equation (2.27) with equation (2.4) we conclude that(2.28)

Since the time factor exp(i*ωt*) is a common factor in all of the equations, it is dropped in the equations that follow.

The physical quantities for the region *y*>0 can be written in the following form:(2.29)(2.30)(2.31)(2.32)(2.33)(2.34)(2.35)(2.36)where(2.37)(2.38)where *ρ*_{1} is density of the material and *μ*^{0} is defined by equation (2.13)_{2}.

If we assume that for the region *y*<0 the shear modulus, piezoelectric constant and dielectric constant are in the following form:(2.39)where *ρ*=*ρ*_{2}e^{αx+βy}, *a*_{1}, *a*_{2}, *a*_{3} are material constants and *α* and *γ* is the material property gradient then the analogous equations to those of the upper half-space are(2.40)(2.41)(2.42)(2.43)(2.44)(2.45)(2.46)(2.47)where(2.48)(2.49)and(2.50)and *ρ*_{2} is density of the material for *y*<0.

For simplicity, it is assumed in the present study that the longitudinal shear stress and electric displacement acting at the boundaries *y*→∞→−∞ are zero at the upper and lower half-spaces and that these are bounded. That is, the combined electrical and mechanical loading occurs at(2.51)The following boundary conditions for the problem can be written in the form:(2.52)

(2.53)

(2.54)

(2.55)

(2.56)

(2.57)

From the boundary conditions (2.52), (2.54), (2.56) and (2.57) we find that(2.58)(2.59)(2.60)(2.61)(2.62)where(2.63)

Comparing the even and odd functions of *x*, which are the symmetrical and antisymmetrical parts, with respect to *x*, we find that(2.64)(2.65)(2.66)(2.67)and(2.68)(2.69)(2.70)(2.71)To solve equations (2.64)–(2.67), we assume that(2.72)

(2.73)

Making use of the boundary conditions (2.52)–(2.55) and equations (2.72) and (2.73), we obtain(2.74)(2.75)where(2.76)

(2.77)

Substituting the values of *A*_{1} and *C*_{1} into equations (2.64) and (2.66), and using the equations (2.72) and (2.73), we can write the equations (2.64)–(2.67) as(2.78)(2.79)(2.80)(2.81)where(2.82)

(2.83)

(2.84)

(2.85)

(2.86)

The solution of the integral equations (2.78)–(2.81) can be written as(2.87)(2.88)where *J*_{ν}( ) is the Bessel function of the first kind and *ν*≥0 and *Φ*_{1}(*t*) and *Ψ*_{1}(*t*) are the unknown functions to be determined from the two dual integral equations. The solutions of the two dual integral equations (2.78)–(2.81) can be written in the form(2.89)(2.90)where *I*_{ν}(*x*) is the modified Bessel function of the first kind.

Equations (2.89) and (2.90) are simultaneous Fredholm integral equations and(2.91)

(2.92)

(2.93)

(2.94)

For solving the equations (2.68)–(2.71), we assume that(2.95)

(2.96)

From the boundary conditions (2.52)–(2.55) and the equations (2.95) and (2.96), it follows that(2.97)

(2.98)

Substituting the values of *B*_{1}(*ξ*) and *D*_{1}(*ξ*) into equations (2.68)–(2.71) and using equations (2.95) and (2.96), we can write the equations (2.68)–(2.71) as(2.99)

(2.100)

(2.101)

(2.102)

The solution of the integral equations (2.99)–(2.102) can be written in the following form:(2.103)(2.104)where *Φ*_{2}(*t*) and *Ψ*_{2}(*t*) satisfies the following simultaneous Fredholm integral equations:(2.105)(2.106)where(2.107)

(2.108)

(2.109)

(2.110)

Solving equations (2.89), (2.90), (2.105) and (2.106) numerically, we get values for *Φ*_{1}(*t*), *Ψ*_{1}(*t*), *Φ*_{2}(*t*) and *Ψ*_{2}(*t*).

## 3. Expressions for stress intensity factors

From equations (2.31), (2.74), (2.75), (2.87), (2.88), (2.97), (2.98), (2.103) and (2.104), the stress intensity factors at *x*=*a* and −*a* are found to be(3.1)

(3.2)

The electric displacement intensity factors are(3.3)

(3.4)

## 4. Numerical results and discussion

Solving the simultaneous Fredholm integral equations (2.89), (2.90), (2.105) and (2.106) numerically, we have obtained values for *ϕ*_{1}(*t*), *ψ*_{1}(*t*), *ϕ*_{2}(*t*) and *ψ*_{1}(*t*) and from equations (3.1) and (3.2) the numerical results for normalized stress intensity factors and are obtained. These results are shown graphically in figures 2–7. In figures 2–4, the results at *x*=*a* and in figures 5–7 the results at *x*=−*a* are presented.

Figure 2 shows the variation of the dynamic normalized stress intensity factors with normalized wavenumbers for different values of *β*. With increases of the *β* parameter, the normalized dynamic stress intensity factor at the crack tip increases. This phenomenon is caused by the gradient of the FGPMs. This figure corresponds to a vertical non-homogeneity of the material with the non-homogeneity *α* parallel to the crack assumed to be zero.

Figure 3 shows the variation of the dynamic normalized stress intensity factors with the normal wavenumbers for *α*=10, *β*=5, *γ*=1. In this case, when the normalized wavenumbers increase, the stress intensity factor increases.

Figure 4 shows that, when the vertical part of the non-homogeneity of the lower half-space of the crack increases, the stress intensity factor increases with *k*_{1}*a* as well.

Figure 5 shows that, when the lateral part of the non-homogeneity is zero, the normalized stress intensity factor at *x*=−*a* decreases with *β*, the vertical non-homogeneity. Also, the stress intensity factor increases as *k*_{1}*a* increases.

Figure 6 shows that, when the lateral part of the non-homogeneity increases, the normalized stress intensity factors decrease.

Figure 7 shows that, when the non-homogeneity of the lower half-space of the crack increases, the normalized stress intensity factor decreases.

In figures 2–7 we assume that

## 5. Conclusions

We have developed an electroelastic fracture mechanics theory to determine the singular stress and electric field by an interface crack between two bonded dissimilar FGPMs. In order to simplify the problem, we have assumed that the non-homogeneities in the *x*-direction in the upper and lower half-spaces are the same. The anti-plane electroelastic problem with the crack at the interface of the FGPMs has been analysed mathematically under dynamic loading. The numerical results show that the effect of the non-homogeneity, which varies in the *x*- and *y*-directions, has considerable effect on the dynamic stress intensity factors.

## Footnotes

↵† Deceased 10 October 2007.

- Received September 29, 2008.
- Accepted December 4, 2008.

- © 2009 The Royal Society