## Abstract

In this paper, the problem of two collinear cracks in functionally graded piezoelectric materials (FGPMs) under in-plane electromechanical loads is examined. The elastic, piezoelectric and dielectric constants of the FGPMs are assumed to vary continuously in space. The theoretical formulations are derived by using Fourier transforms and the resulting singular integral equations are solved with Chebyshev polynomials. A dielectric crack model with deformation-dependent electric boundary condition is adopted in the fracture analysis of FGPMs. Numerical simulations are made to show the effect of the dielectric medium, the material gradient and the geometry of interacting cracks upon the fracture parameters at crack tips. A critical state for applied electromechanical loading is identified, which determines whether the traditionally impermeable (or permeable) crack model serves as the upper or lower bound of the current dielectric crack model.

## 1. Introduction

Recently, the concept of functionally graded materials (FGMs) with continuous change of composition and material property has been extended to piezoelectric materials. These new kinds of non-homogeneous piezoelectric materials are called functionally graded piezoelectric materials (FGPMs; Wu *et al*. 1996; Hudnut *et al*. 2000; Zhu *et al*. 2000), which are expected to possess the advantages of both FGMs and piezoelectric materials and to improve the reliability and lifetime of piezoelectric structures and devices. For example, Takagi *et al*. (2002) have shown that the piezoelectric structures made of FGPMs with optimized composition profile can perform better than those homogeneous piezoelectric ones. However, the commonly used piezoelectric materials, with stronger electromechanical coupling effects, are generally brittle and have the tendency to develop multiple cracks during manufacturing and service processes. As a result, the fracture analysis of interacting cracks in FGPMs becomes significant in the design and applications of these new materials.

One important issue involved in studying the fracture behaviour of piezoelectric materials is the electric boundary condition along crack surfaces. There are two typical crack models using different electric boundary conditions. One is the electrically permeable model (Parton 1976) with perfect contact of crack surfaces and continuous electric potential and normal component of electric displacement across the crack surfaces. The other is the electrically impermeable model (Deeg 1980), in which the electric induction of the medium filling the crack is neglected. These two models represent two limiting cases, where dielectric permittivity of the medium filling the crack is assumed to be infinite and zero, respectively. In most engineering situations, cracks in piezoelectric materials are filled with dielectric medium, such as air or vacuum. When the piezoelectric medium is subjected to a tensile stress, the crack will open up (mode I crack) and both the dielectric crack filling and crack surface separation will play a crucial role in the fracture performance of this material. To evaluate the effect of dielectric medium filling the crack upon the electric boundary condition, an elliptical crack model has been developed to study the fracture behaviour of cracked piezoelectric materials (McMeeking 1989; Sosa 1991; Dunn 1994; Zhang & Tong 1996; Zhang *et al*. 1998). Their investigation indicated that the electrically permeable condition may underestimate the effect of electric field on the crack propagation and the impermeable one may overestimate this effect. Another intermediate crack model (Parton & Kudryavtsev 1988; Dascalu & Homentcovschi 2002), in which the crack is represented by a dielectric thin layer with pre-assumed thickness, has also been proposed to study this dielectric medium effect. These two models can be reduced to the traditionally permeable and impermeable ones depending on the pre-assumed crack surface separation. However, it should be mentioned that for both elliptical and intermediate crack models the thickness of the crack must be pre-assumed, which is the original profile of the crack and is independent of the crack deformation. For a slit crack without initial crack opening, the electric boundary condition will be very sensitive to the crack opening due to the applied electromechanical loads. As a result, a deformation-dependent electric boundary condition should be considered. Recently, a dielectric crack model has been used to study the fracture behaviour of piezoelectric materials (Hao & Shen 1994; Xu & Rajapakse 2001; Wang & Jiang 2002, 2004). Such a model avoids any pre-assumed thickness of the crack and considers the ‘real’ electric boundary condition along the crack surfaces, which is expected to predict the electromechanical behaviour of piezoelectric materials more accurately.

The problems of interacting cracks in homogeneous piezoelectric materials have been extensively studied (Meguid & Wang 1998; Gao & Wang 1999; Chen & Worswick 2000; Hao 2001; Wang & Jiang 2005; Zhou *et al*. 2007). Since FGPMs are newly developed materials, there exist relatively fewer studies on the fracture analysis of these kinds of materials. Li & Weng (2002*a*,*b*) first considered the anti-plane problem of a finite crack in a strip of FGPM with elastic stiffness, piezoelectric constant and dielectric permittivity of the FGPM varying continuously along the thickness of the strip. It is found that the material gradient has significant effect upon the fracture properties of FGPMs. Owing to the fact that brittle FGPMs are susceptible to developing multiple cracks, the interacting crack problems for FGPMs have drawn the interests of research communities recently (Ma *et al*. 2004; Liang 2006; Zhou & Wu 2006; Zhang *et al*. 2008). However, it should be noted that most of these existing studies have been limited to using the traditionally impermeable and permeable crack models without considering the effect of dielectric medium filling the crack. Since the dielectric constant of piezoelectric ceramics is usually 10^{3} higher than that of the dielectric medium filling the crack, it is essential to study the effect of dielectric medium filling the crack and crack deformation upon the fracture property of multiple crack problems in FGPMs, especially when cracks open up. To consider this effect, Zhou & Chen (2008) used a dielectric crack model to study the interaction of two parallel cracks with material property varying along crack line direction. To the authors' knowledge, there are no existing studies on the interacting collinear cracks in FGPMs considering the dielectric medium effect.

Therefore, the objective of the current work is to provide a comprehensive theoretical study of the fracture behaviour of collinear dielectric cracks in FGPMs. The problem is formulated in §2. Section 3 gives a general solution for a single crack developed by using Fourier transforms. Then the solution for two collinear dielectric cracks is derived in §4. Numerical simulations are made and the results are discussed in §5. Finally, §6 concludes our work.

## 2. Formulation of the problem

The problem envisaged is a plane strain problem of two collinear cracks with lengths 2*a*_{I} and 2*a*_{II} located in an infinite FGPM medium, which is subjected to mechanical loads and an electric displacement . Two local Cartesian coordinate systems (*x*_{I}, *y*_{I}) and (*x*_{II}, *y*_{II}) are employed to describe these two collinear cracks as shown in figure 1. The poling direction of the medium is along the global coordinate *y*-direction and the distance between two cracks is *d*. Owing to the crack surface separation caused by the applied electromechanical loading, there exists electric potential drop across crack surfaces.

In the absence of body force and free charges, the basic equations governing the electromechanical behaviour of transversely isotropic FGPMs are given in the global Cartesian coordinate system (*x*, *y*) as(2.1)(2.2)(2.3)(2.4)(2.5)(2.6)where *σ*_{ij} (*i*, *j*=1, 2) and *D*_{i} (*i*=1, 2) are stress and electric displacement components, respectively; *u*, *v* and *Φ* are displacements and electric potential; *c*_{11}, *c*_{12}, *c*_{22} and *c*_{33} are elastic constants; *e*_{12}, *e*_{22} and *e*_{31} are piezoelectric constants; and *ϵ*_{11} and *ϵ*_{22} are dielectric constants.

For FGPMs, the variation of material property may exhibit an arbitrary format. However, to make the problem more mathematically tractable, a certain specific form is generated with all the material constants having the same exponential distribution with gradient *α* as(2.7)

Substituting equations (2.2)–(2.7) into equation (2.1) results in the following governing equations:(2.8)(2.9)(2.10)where(2.11)

To evaluate the effect of dielectric crack filling, a dielectric crack model is used in the current work. Considering the perturbation field only, the mechanical and electric boundary conditions along the crack surface (|*x*_{I}|<*a*_{I}, |*x*_{II}|<*a*_{II}) are given as(2.12)(2.13)where *ϵ*_{0}=8.85×10^{−12} C(V m)^{−1} is the dielectric permittivity of air (or vacuum) filling the crack and *v*^{k+}−*v*^{k−} is the crack-opening displacement (COD) caused by the applied loading. When *ϵ*_{0}=0 or ∞, this dielectric crack model reduces to the traditionally impermeable and permeable crack models. For each individual crack, outside the crack surfaces along the crack line, the following continuity conditions should be satisfied:(2.14)where ** t**={

*σ*

_{21},

*σ*

_{22},

*D*

_{2}}

^{T},

**={**

*d**u*,

*v*,

*Φ*}

^{T}.

## 3. Single crack solution

Following the same procedure of solving the problem of a cracked FGM (Konda & Erdogan 1994) and applying Fourier transforms with respect to *x* to equations (2.8)–(2.10), the solutions of *u*^{*}, *v*^{*} and *Φ*^{*} (the Fourier transform of *u*, *v* and *Φ*) satisfying the regularity conditions at infinity can be written as(3.1)which is equivalent to the method developed by Ding *et al*. (1996). *C*_{j} (*j*=1, 2, …, 6) are unknown functions of *s* to be determined from boundary conditions, and *λ*_{j} (*j*=1, 2, …, 6) are the roots of the following equation:(3.2)with *X*_{i} (*i*=1, 2, …, 7) being given in the electronic supplementary material. For the material constants and gradient considered in the current work, the roots of equation (3.2) can be calculated numerically with three roots *λ*_{1}, *λ*_{3}, *λ*_{5} having positive real parts, while three of them *λ*_{2}, *λ*_{4}, *λ*_{6} have negative real parts. When the material gradient *α*=0, these roots can be reduced to the same format as discussed by Suo *et al*. (1992). In equation (3.1), *a*_{j} and *b*_{j} (*j*=1, 2, …, 6) are(3.3)where *X*_{aj}, *X*_{bj}, *Y*_{j} (*j*=1, 2, …, 6) are given in the electronic supplementary material.

As a mathematical model, a crack can be modelled as distributed dislocations (Bilby & Eshelby 1968). The dislocation density functions are defined as(3.4)where *d*_{i} are elements of displacement and electric potential field vector as defined following equation (2.14). Applying Fourier transform to equation (3.4) and using the continuity condition for stress and electric displacement fields in equation (2.14), the unknown functions *C*_{1}, *C*_{2}, *C*_{3}, *C*_{4}, *C*_{5} and *C*_{6} in equation (3.1) can be expressed in terms of the dislocation density functions as(3.5)(3.6)where *f*_{j}, *g*_{j}(*s*) (*j*=1, 2, …, 9) and Δ(*s*) are given in the electronic supplementary material.

By considering the continuity condition for displacement and electric potential in equation (2.14) and substituting equations (3.1), (3.5) and (3.6) into the constitutive equations, the stress and electric displacement fields ** t**={

*σ*

_{21},

*σ*

_{22},

*D*

_{2}}

^{T}at

*y*=0 can then be expressed as(3.7)where

*a*is the half-length of crack and with

*h*

_{ij}(

*i*,

*j*=1, 2, 3) being given in the electronic supplementary material.

Detailed asymptotic analysis of *h*_{ij} (*s*, 0) indicates that *h*_{11}, *h*_{22}, *h*_{33}, *h*_{23} and *h*_{32} are odd functions and approach to constants when *s* tends to infinity, while the others are even functions and approach to zero with increasing *s*, i.e.(3.8)The asymptotic behaviour of *h*_{ij} governs the singular solution of the problem. Separating the singular parts of the kernels in equation (3.7), the stress and electric displacement fields in the Cartesian coordinate system (*x*, *y*) attached to the crack centre can be expressed as(3.9)(3.10)

(3.11)

## 4. Solution of interacting dielectric cracks

The stresses and electric displacement along crack I surfaces caused by the existence of crack II are defined as , while those quantities along crack II surfaces caused by the existence of crack I are defined as . After using the superposition technique, the total stress and electric displacement fields along each individual crack surface should satisfy the mechanical and electric boundary conditions given in equations (2.12) and (2.13), i.e.(4.1)(4.2)Substituting equations (3.9)–(3.11) into equations (4.1) and (4.2) results in singular integral equations for each crack *k* (*k*=I, II) in its local Cartesian coordinate systems (*x*_{k}, *y*_{k}),(4.3)(4.4)(4.5)where *p*=I, *q*=II and *X*_{k}=−*a*_{I}−*a*_{II}−*d* when *k*=I, but *p*=II, *q*=I and *X*_{k}=*a*_{I}+*a*_{II}+*d* when *k*=II. The integral equations (4.3)–(4.5) are characterized by the square root singularity; therefore, the general solutions can be determined by expanding the dislocation density functions , and as(4.6)where *T*_{l} are the Chebyshev polynomials of the first kind and (*k*=I, II) are unknown coefficients. The orthogonality condition of Chebyshev polynomials and the continuity condition for displacement and electric potential (2.14) result in (*k*=I, II). Substituting (4.6) into (4.3)–(4.5) and truncating the Chebyshev polynomials to the *N*th term, the following algebraic equations can be obtained by using the properties of Chebyshev polynomials:(4.7)(4.8)(4.9)where *k*=I, II represents cracks I and II, respectively. *p*=I, *q*=II, *r*_{k}=0 and *X*_{k}=−*a*_{I}−*a*_{II}−*d* when *k*=I, but *p*=II, *q*=I, *r*_{k}=1 and *X*_{k}=*a*_{I}+*a*_{II}+*d* when *k*=II. *J*_{l} is the Bessel function of the first kind with *l*th order. To solve equations (4.7)–(4.9), these equations are assumed to be satisfied at *N* collocation points along the surfaces of each crack. The unknown coefficients , , (*k*=I, II) can then be obtained, which will be used to determine the electromechanical fields of the FGPMs with collinear dielectric cracks.

After the stress and electric displacement fields are obtained, the fracture parameters, such as stress and electric displacement intensity factors at the left and right tips of crack *k* (*k*=I, II), can be determined as(4.10)

In addition to these traditional fracture parameters, a COD intensity factor *K*_{COD} (Wang & Jiang 2002), which can be used to describe the opening deformation of the crack surfaces, is also introduced to evaluate the fracture behaviour(4.11)

## 5. Results and discussion

The current work will consider only the cases where a normal tensile stress and an electric displacement are applied to the medium. In this situation, the crack will open up and the dielectric medium inside the crack will play a crucial role in the fracture behaviour of cracked FGPMs as argued by Chiang & Weng (2007). In numerical simulations, the material constants given in equation (2.7) along crack lines are taken as those of lead-zirconate-titanate-4 piezoceramics.

Firstly, we restrict our attention on the effect of crack dimension upon crack interaction for FGPMs with collinear cracks. For the case when the medium is subjected to a normal tensile stress and an electric displacement *D*_{2}=1×10^{−3} C m^{−2}, figure 2 shows the variation of normalized stress intensity factor at the right (inner) tip of crack I with different crack length ratios. The superscript ‘S’ represents the mode I intensity factor for single crack problem in a homogeneous medium, and length of crack I and the distance between two cracks are fixed as *a*_{I}=1 mm and *d*=0.5*a*_{I}, respectively. An amplification effect is observed with all being greater than 1. For homogeneous (*αa*_{I}=0) medium, increases with the increase in *a*_{II}/*a*_{I}, which is consistent with the result from Wang & Jiang (2005). The effect of material gradient upon this fracture parameter can also be observed. For example, increases with material gradient when the length ratio *a*_{II}/*a*_{I} is small. However, when *a*_{II}/*a*_{I} is relatively large, increases first with material gradient, then decreases. This phenomenon is attributed to the combined effects of material gradient and crack interaction, which means that the material gradient may reduce the amplification effect. For the left (inner) tip of crack II, the variation of with *a*_{II}/*a*_{I} is plotted in figure 3. A similar amplification effect is observed. It shows that this normalized stress intensity factor decreases with the increase in *a*_{II}/*a*_{I} for a homogeneous piezoelectric medium (*αa*_{I}=0) as expected. However, for the graded materials, the tendency for the variation of this normalized stress intensity factor depends on both the material gradient and the crack length ratios. decreases and then increases with *a*_{II}/*a*_{I} when the material gradient *αa*_{I} is small, while increases with the increase in *a*_{II}/*a*_{I} when *a*_{I} approaches a bigger value, e.g. *αa*_{I}=1.2. The effects of crack dimension and material gradient upon electric displacement intensity factor have a similar tendency as mode I stress intensity factor and thus the results are not present here. Owing to the non-symmetry of material properties about the crack line, the mode II intensity factor does not vanish. Figure 4 presents the variation of normalized with *a*_{II}/*a*_{I}. It can be seen from this figure that for non-homogeneous medium with different material gradients, decrease with the increase in *a*_{II}/*a*_{I} first, then increase but change the sign. It means that the coupling effect caused by the material gradient may be enhanced or impeded by the crack interaction depending on how strong the interaction is. The normalized with *a*_{II}/*a*_{I} for the left (inner) tip of crack II is shown in figure 5. This mode II stress intensity factor becomes 0 for some specific combination of material gradient and crack length ratio. From these two figures, we can conclude that for FGPMs with specific gradient, the interaction between collinear cracks with appropriate crack length ratio may make this coupling effect between modes I and II disappear. The dependence of normalized stress intensity factor on crack size has also been observed by Zhou & Chen (2008) for the problem of parallel dielectric cracks in FGPMs.

To compare the interaction effects upon two collinear cracks, both modes I and II stress intensity factors at the inner tips of these cracks are plotted in figure 6 for different material gradients. To see the pure crack interaction, these intensity factors are normalized by their counterparts for a single crack in the same graded material. Owing to the symmetry of the problem when these two cracks have the same length, the curves for both *k*_{I} and *k*_{II} at the inner tips of cracks intersect when *a*_{II}/*a*_{I}=1. It is observed that at crack inner tips when *a*_{I}>*a*_{II}, but when *a*_{II}>*a*_{I}. Therefore, crack II overtakes crack I with for a range of length ratio *a*_{II}/*a*_{I}>1 according to the interaction effect, which is consistent with the results of homogeneous piezoelectric materials. It should be mentioned that material gradient has a significant effect upon *k*_{I}. For this collinear crack problem, the coupling of modes I and II is caused by the non-symmetry of material properties about the crack line. However, it is observed in this figure that this normal and shear coupling (*k*_{II}) is also influenced significantly by crack interaction. Therefore, mode I and mode II coupling in non-homogeneous materials are more complicated due to the combined effect of material gradient and crack interaction. The interacting effects of the collinear cracks upon inner and outer crack tips are compared by *k*_{I} in figure 7 for crack I; it is clearly demonstrated that the interacting effect is stronger at the inner tips than the outer tips.

The effect of crack position upon crack interaction is also investigated for the case where cracks I and II are assumed to have the same length 2*a*_{1}=2*a*_{2}=2*a*=2 mm. The loading condition for the medium is the same as that described in figure 2. Since the mode I stress intensity factor and the electric displacement intensity factor have the same variation tendency, we will show only the results for the electric displacement intensity factor. Figures 8 and 9 present the variation of at the inner and outer crack tips with *d*/*a* for different material gradients. The amplification effect is clearly demonstrated in these two figures when the crack interaction exists, and *k*_{D} tends to be constant when two cracks are far away. It is also indicated in these figures that material gradient has a significant effect upon the electric displacement intensity factor. Figure 10 shows the comparison for the normalized electric displacement intensity factors *k*_{D} at the inner and outer crack tips with the changing crack distance *d*/*a*. As the crack distance increases, the electric displacement intensity factor *k*_{D} decreases gradually. For small *d*/*a*, the values of *k*_{D} at the inner tips are larger than those at the outer tips. When *d*/*a* is sufficiently large, *k*_{D} at both inner and outer crack tips approaches the value for the single crack problem (Jiang 2008). For example, and are obtained when *d*/*a*=5, which is approaching the single crack result in FGPM with gradient *αa*=1.0. The variation of *k*_{II} with *d*/*a* is plotted in figure 11 for different material gradients, in which the sign for has been changed. For inner crack tips, *k*_{II} increases with *d*/*a* and then decreases, while this trend is reversed for the outer crack tips. When *d*/*a* is small (i.e. the crack interaction is strong), the value of *k*_{II} at inner crack tips is smaller than that at outer crack tips. This trend will change with the increase in *d*/*a* but will depend on the material gradient. When *d*/*a* is sufficiently large, these values for the inner and outer crack tips tend to be the same as the single crack problem. Noda & Wang (2002) obtained similar results in their study on the collinear cracks in traditional FGMs with material constants obeying a power-law type relationship.

By using the dielectric crack model with crack boundary conditions given in equations (2.12) and (2.13), the effect of dielectric medium upon the fracture parameters will be studied. This effect will play a crucial role for the fracture analysis of FGPMs as argued by Chiang & Weng (2007), especially for the mode I crack in the current study. To see this effect, the variation of electric displacement intensity factors *K*_{D} at the inner and outer tips of two collinear cracks with applied stress for different crack models are depicted in figure 12 when subjected to an applied electric displacement . These two collinear cracks are assumed to have the same length 2*a*_{1}=2*a*_{2}=2*a*=2 mm and the distance between these two cracks is *d*=0.5*a*. It can be seen that the *K*_{D} of the impermeable and permeable models are the upper and lower bounds when for both inner and outer crack tips, and the value of is related to material properties and applied electric displacement. However, when , the situation is completely reversed. is called the critical state for the applied electromechanical loading first defined by Chiang & Weng (2007) for the single crack problem in homogeneous piezoelectric materials. This critical state for electromechanical loading can also be observed from the other fracture parameter COD intensity *K*_{COD} as shown in figure 13. From these two figures, we can see that the results of the current dielectric crack model are always between those of the two traditionally impermeable and permeable crack models. Therefore, the dielectric crack model may be more accurate to predict the fracture behaviour of interacting cracks in FGPMs.

## 6. Concluding remarks

This work provides a theoretical analysis for the in-plane problem of two collinear cracks in infinite FGPMs under electromechanical loading by using the dielectric crack model. Numerical analysis was performed to evaluate the influence of crack geometry, material gradient and dielectric medium upon the fracture parameters for the cracked FGPMs. It is concluded that the crack geometry and material gradient have a great effect upon the fracture behaviour of FGPMs. A critical state for the applied electromechanical loading is also observed for the interacting crack problem of FGPMs, which determines when the traditionally impermeable/permeable crack model serves as the upper or lower bound for the dielectric crack model. By considering the effect of crack surface deformation and the dielectric medium filling the crack, this real crack model is expected to provide more accurate prediction on the fracture behaviour of FGPMs.

## Acknowledgments

This work was supported by Natural Sciences and Engineering Research Council of Canada (NSERC) and ADF small grants of UWO.

## Footnotes

- Received December 16, 2008.
- Accepted February 10, 2009.

- © 2009 The Royal Society