## Abstract

A critical state for electromechanical loads that determines when the traditional impermeable (or permeable) crack model serves as the upper or the lower bound of the dielectric crack model, first proposed for homogeneous piezoelectric materials, is studied further for functionally graded piezoelectric materials (FGPMs) in the current work. The analytical formulations of a single crack and two interacting cracks in the FGPMs are derived by using Fourier transforms, and the resulting integral equations are solved with Chebyshev polynomials. Numerical simulations are conducted to show the effect of crack length, positions of two interacting cracks and material gradient of FGPMs at this critical state. Interesting results show that the combination of the material gradient and the crack length *α**a* plays an important role in determining this critical state. Our solutions also reveal there may exist several critical states for two interacting cracks in the FGPMs.

## 1. Introduction

Piezoelectric materials, both homogeneous and inhomogeneous (e.g. functionally graded piezoelectric materials (FGPMs) with varying spatial material properties), are widely used as electromechanical devices in smart structures because of their electromechanical coupling effect. However, the commonly used piezoelectric ceramics are brittle in nature and have the tendency to develop cracks during manufacturing and service processes. The existence of cracks may affect the mechanical integrity of these structures and prohibit their potential applications. Therefore, it is essential to better understand the fracture behaviour of this class of materials.

An important issue involved in the fracture analysis of piezoelectric materials is the electric boundary condition along crack surfaces. Most of the existing studies have been focused on using permeable (Parton 1976) and impermeable (Deeg 1980) crack models to predict the fracture behaviour of the homogeneous piezoelectric materials (Mikhailov & Parton 1990; Pak 1990; Shindo *et al.* 1990; Suo *et al.* 1992; Park & Sun 1995; Meguid & Wang 1998) and the FGPMs (Li & Weng 2002*a*,*b*; Ma *et al.* 2004; Zhou & Wu 2006; Zhang *et al.* 2008). These traditional crack models neglect the effect of dielectric medium filling the crack and have physical limitations by assuming the dielectric permittivity as infinite and zero, respectively. The work by some researchers (McMeeking 1989; Dunn 1994; Sosa & Khutoryansky 1996; Zhang & Tong 1996; Zhang *et al.* 1998) has revealed that these two traditional crack models may lead to some unrealistic results in certain cases. It should be mentioned that the simplicity of these models is one of the reasons that they are adopted in many studies, especially for mode II and mode III cracks where the crack thickness is assumed as zero. However, for a crack under a tensile loading (mode I), the crack will open up and the dielectric permittivity plays a crucial role as argued by Chiang & Weng (2007). The traditional crack models without considering the dielectric medium effect may not be accurate in this situation. Therefore, a dielectric crack model accounting for the effects of both crack filling and crack deformation, even though cumbersome, has been widely used to study the fracture behaviour of piezoelectric materials (Hao & Shen 1994; Xu & Rajapakse 2001; Wang & Jiang 2002; Zhou & Chen 2008; Yan & Jiang 2009). This dielectric crack model is expected to provide more accurate results than the traditional crack models under certain conditions, and a transition among different crack models will occur according to the applied loads. This model reduces to an intermediate crack model (Parton & Kudryavtsev 1988; Dascalu & Homentcovschi 2002) when the crack is represented by a dielectric thin layer with unchanged thickness.

To study the transition among different crack models, it is interesting to mention a phenomenon observed by Chiang & Weng (2007) when they studied the nonlinear behaviour of a penny-shaped dielectric crack in a homogeneous piezoelectric solid. In their work, by examining the effective electric displacement of the crack medium, they found a critical state under an applied tensile stress and a positive electric displacement , where is the critical value for . When , the response of an impermeable crack serves as the upper bound of the dielectric crack model, whereas the permeable one serves as the lower bound, and when the situation is exactly reversed. This critical state provides a defining point, at which the nonlinear response of the dielectric crack undergoes a transition. A similar phenomenon was observed by Yan & Jiang (2009) for the cracks in the FGPMs. The introduction of material gradient in the FGPMs will make crack interaction more complicated. The combination of the material gradient and the crack interaction may have a significant effect on this critical state, but has not been studied in detail in the literature.

Therefore, the objective of this work is to provide a systematic study of the critical state of dielectric cracks in the FGPMs. Attention will be focused on the effects of the material gradient, the crack geometries and the crack interaction at this critical state. The general formulations are developed in §2. Section 3 provides a general solution for a single crack. The method for solving two interacting cracks is introduced in §4. Numerical simulations are made and the results are discussed in §5. Finally, §6 concludes this work.

## 2. Formulation of the problem

The plane problem envisaged is an infinite FGPM medium containing two parallel cracks of lengths 2*a*_{I} and 2*a*_{II} subjected to mechanical loads (*i*=*x*,*y*) and an electric displacement . The poling direction of the medium is along the *y*-direction in the global Cartesian coordinate system (*x*,*y*) as shown in figure 1. Two local Cartesian coordinate systems (*x*_{k},*y*_{k}) with *k*=I and II are attached to the centre of each crack to describe the crack position, and the relative crack positions are measured by *d* and *h*. Due to the crack deformation caused by the applied loads, there exists an electric potential jump *Φ*^{k+}−*Φ*^{k−}(*k*=I,II) across the crack surfaces. To account for the effect of both the crack filling and the crack deformation, 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*_{k}|<*a*_{k},*k*=I,II) for each crack are given in its local coordinate systems as
2.1
and
2.2
where superscript + and − symbols represent the upper and the lower surfaces of the crack, and *κ* is the dielectric permittivity of the medium filling the crack, *κ*=*κ*_{0}=8.85×10^{−12} C (Vm)^{−1} for air (or vacuum), for example. Here *v*^{k}(*x*_{k},0^{+})−*v*^{k}(*x*_{k},0^{−}) is the crack opening displacement caused by the applied loading. It is interesting to mention that the influence of the dielectric medium is to introduce an extra electric displacement Δ*D*_{y}=*κ*(*Φ*^{k}(*x*_{k},0^{+})−*Φ*^{k}(*x*_{k},0^{−}))/(*v*^{k}(*x*_{k},0^{+})−*v*^{k}(*x*_{k},0^{−})) into the effective electric displacement imposing on the crack surfaces, which is caused by the crack deformation. Thus the electric boundary condition is deformation-dependent (or loading-dependent), which will cause the nonlinear fracture behaviour of FGPMs, e.g. the fracture parameters of FGPMs will vary nonlinearly with the applied electromechanical loads (Hao & Shen 1994; Xu & Rajapakse 2001; Wang & Jiang 2002; Chiang & Weng 2007). When *κ*=0 or , this dielectric crack model reduces to the traditionally impermeable or permeable crack models.

In the absence of body force and free charge, the equilibrium and charge equations for the plane problem of piezoelectric materials are expressed in the global Cartesian coordinates (*x*,*y*) as
2.3
where *σ*_{xx},*σ*_{xy} and *σ*_{yy} are the stress components, and *D*_{x} and *D*_{y} are the electric displacement components. For a transversely isotropic piezoelectric material with the poling direction along the *y*-axis, the constitutive relations are
2.4
and
2.5
where *c*_{11},*c*_{13},*c*_{33} and *c*_{44} are elastic constants, *e*_{31},*e*_{33} and *e*_{15} are piezoelectric constants and ε_{11} and ε_{33} are dielectric constants. Strain ϵ_{xx},ϵ_{yy} and ϵ_{xy}, and electric field *E*_{x} and *E*_{y} are given by
2.6
with *u*, *v* and *Φ* being displacements and electric potential.

For inhomogeneous piezoelectric materials, such as FGPMs, the individual elastic constants, piezoelectric constants and dielectric constants may vary independently. To make the problem more mathematically tractable, all these material constants are assumed to vary in a certain specific form with the same exponential distribution along the poling direction, i.e.
2.7
where *α* is the material gradient and and are the material constants along *y*=0.

Substituting equations (2.4)–(2.7) into equation (2.3), the governing equations for the FGPM can be expressed as
2.8
2.9
and
2.10
where coefficients *β*_{1} to *β*_{14} can be expressed in terms of the material constants, which are given in the electronic supplementary material.

To solve the governing equations (2.8)–(2.10), Fourier transform with respect to *x* is applied to these equations. Following the same procedure of solving the problem of a cracked functionally graded material (FGM) (Konda & Erdogan 1994), the general expressions of *u*,*v* and *Φ* in the Fourier transform domain are given as
2.11
2.12
and
2.13
which is equivalent to the method developed by Ding *et al.* (1996). The coefficients and *f*_{i} (*i*=1,2,…,9) are given in the electronic supplementary material. *C*_{j} are unknown functions of *s* to be determined from the boundary conditions, and are the roots of the equation
2.14
with *X*_{i}(*i*=1,2,…,7) being given in the electronic supplementary material. For the material constants and the gradient considered in the current work, the roots of equation (2.14) can be calculated numerically with three roots *λ*_{1},*λ*_{3} and *λ*_{5} having positive real parts and three roots *λ*_{2},*λ*_{4} and *λ*_{6} having 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).

## 3. Solution of a single crack

As a mathematical model, a crack can be modelled as distributed dislocations. For each individual crack *k* (*k*=I,II), the generalized dislocation density functions for piezoelectric materials are defined in the local coordinate system as
3.1
where *d*_{i} are components of *d*={*u*,*v*,*Φ*}^{T}, and the continuity condition along the crack line is
3.2
which results in
3.3
Applying Fourier transform to equation (3.1) with respect to *x*_{k}, *C*_{2},*C*_{4} and *C*_{6} in equations (2.11)–(2.13) can then be expressed in terms of , which is the Fourier transform of ψ_{j},
3.4
where Δ(*s*) and are given in the electronic supplementary material.

By considering equations (3.3) and (3.4) and substituting equations (2.6) and (2.11)–(2.13) into the constitutive equations (2.4) and (2.5), the stress and electric displacement fields *t*={*σ*_{yx},*σ*_{yy},*D*_{y}}^{T} caused by the existence of this single crack can be expressed in the local coordinate system (*x*_{k},*y*_{k}) as
3.5
with *K**_{ij} being given as
3.6
where *h*_{ij},*h*^{+}_{ij} and *h*^{−}_{ij}(*i*,*j*=1,2,3) are given in the electronic supplementary material.

Detailed asymptotic analysis of *h**_{ij} (including *h*_{ij},*h*^{+}_{ij} and *h*^{−}_{ij} derived in the local coordinate system) indicates that *h**_{11}, *h**_{22}, *h**_{33}, *h**_{23} and *h**_{32} are odd functions of *s*, whereas the others are even functions. *h*_{11},*h*_{22},*h*_{33},*h*_{23} and *h*_{32} are related to the material gradient and the position of each individual crack. When the position of crack *k* (*k*=I,II) is fixed at *y*=*h*_{k} according to the global coordinate system (*x*,*y*), they approach to constants when *s* tends to infinity, i.e.
3.7
whereas the others approach to zero with increasing *s*. This asymptotic behaviour of *h*_{ij} governs the singular solution of the problem.

## 4. Solution of two interacting cracks

By using the superposition technique, the boundary conditions (2.1) and (2.2) for each crack can be expressed in its own local coordinate system as
4.1
where and are stresses and electric displacement caused by the *k*th (*k*=I,II) crack itself; and are the stresses and electric displacement along the *k*th crack surfaces caused by the *j*th crack.

Separating the singular parts of the kernels in equation (3.5) and substituting them into equation (4.1), the following singular integral equations in the local Cartesian coordinate systems (*x*_{k},*y*_{k})(*k*=I,II) can be obtained as
4.2
4.3
and
4.4
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. Here *d* is positive when the left tip of crack II is at the right-hand side of the right tip of crack I, otherwise *d* is negative. *h* is the total vertical separation distance between the two cracks. The integral equations (4.2)–(4.4) are characterized by the square root singularity, and therefore the general solutions can be determined by expanding the dislocation density functions , and as
4.5
where *T*_{l} are Chebyshev polynomials of the first kind and are unknown coefficients. The orthogonality condition of Chebyshev polynomials and the continuity condition for the displacement and electric potential in equation (3.3) result in . Substituting equation (4.5) into equations (4.2)–(4.4) and truncating the Chebyshev polynomials to the *N*th term, the following algebraic equations can be obtained by using the properties of Chebyshev polynomials:
4.6
4.7
and
4.8
where *J*_{l} is the Bessel function of the first kind with the *l**t**h* order. To solve equations (4.6)–(4.8), these equations are assumed to be satisfied at *N* collocation points along the surfaces of each crack. The unknown coefficients and can then be obtained, which will be used to determine the electromechanical fields of the FGPMs with two parallel dielectric cracks. Consequently, the fracture parameters, such as stress and electric displacement intensity factors, at the left and the right tips of crack *k* (*k*=I,II) can be determined. Here we will give only the effective electric displacement along the crack surfaces and the electric displacement intensity factor
4.9
and
4.10
These parameters will be used to study the critical state of the dielectric cracks in the piezoelectric materials.

## 5. Results and discussion

For numerical simulations, the piezoelectric medium is assumed to subject to a normal tensile stress and an electric displacement C/m^{−2}, and the material properties at *y*=0 are taken as those of commercial PZT-4 ceramics. To study the effect of crack interaction on the critical state of interacting dielectric cracks, we will consider only the cases of two collinear and centre-aligned parallel cracks to clarify the problem in a simpler way.

First, attention is focused on the problem where the piezoelectric medium contains only one single crack. To show what is a critical state for the dielectric crack, we will start with a single-crack problem in a homogeneous medium, which was studied by Chiang & Weng (2007). For PZT-4 ceramics, the variation of with the applied mechanical load is plotted in figure 2 with different dielectric permittivity of the crack medium. It is seen that when *κ*=0, is a constant as expected. When , is linearly proportional to . For a finite value of *κ*, exhibits nonlinearity and a transition from the impermeable model to the permeable model occurs with the increase of *κ*. Regardless of the value of *κ*, all curves pass through a critical point () with MPa. The existence of this critical state is attributed to that the applied critical stress will ensure the additional term Δ*D*_{y}=*κ*(*Φ*^{k}(*x*_{k},0^{+})−*Φ*^{k}(*x*_{k},0^{−})/(*v*^{k}(*x*_{k},0^{+})−*v*^{k}(*x*_{k},0^{−}))=0 in the effective electric displacement in equation (2.2). For the homogeneous medium, we know that the effective electric displacement distributes uniformly along the crack surfaces under any applied electromechanical loads, and this distribution does not change with the crack length. Due to this reason, this critical state for the dielectric crack will not change with the crack length as shown in this figure, i.e. the distribution of along the crack with length 2*a* (*a*=1 mm) is identical with that along the crack with length 2*a* (*a*=2 mm). This critical state can also be observed from the fracture parameter such as the electric displacement intensity factor *K*_{D} in figure 3 for the crack with different crack lengths. It is also seen in this figure that *K*_{D} varies nonlinearly with the applied stress due to the nonlinear electric boundary condition, unlike the straight lines for the linear permeable and impermeable crack models.

Strictly speaking, the distribution of the effective electric displacement along the crack surfaces in FGPMs is not uniform, especially when the material gradient is at a higher level. Figure 4 shows the distribution of along the crack surface when MPa. It is indicated in this figure that varies with the change of the crack length from *a*=1 to *a*=4 mm for a fixed material gradient (*α*=0.2 mm^{−1}, for example). However, if we fix a variable combining the crack length and the material gradient as *α**a*=0.8, i.e. *α*=0.2 mm^{−1} and *a*=4 mm, or *α*=0.4 mm^{−1} and *a*=2 mm, or any other combinations, it is seen in this figure that is identical for these cases. Therefore, it is concluded that the critical state for the dielectric crack in the FGPMs, if it exists, is dependent on the crack length unlike in the homogeneous medium, but controlled by *α**a*. A similar phenomenon that the fracture property depends on *α**a* was found for the crack problem of traditional FGMs (Delale & Erdogan 1983; Konda & Erdogan 1994). Figure 5 plots the electric displacement intensity factor *K*_{D} versus the applied tensile stress with different combinations of *α* and *a*, but with a fixed *α**a*=0.8. It is clearly indicated in this figure that the critical state for the dielectric crack in the FGPMs does exist and is controlled by *α**a*. Obviously, the critical applied stress cannot ensure the additional term Δ*D*_{y}=*κ*(*Φ*^{k}(*x*_{k},0^{+})−*Φ*^{k}(*x*_{k},0^{−}))/(*v*^{k}(*x*_{k},0^{+})−*v*^{k}(*x*_{k},0^{−}))=0 at every point along the crack surface as for the homogeneous medium (Chiang & Weng 2007). However, the average contribution of this additional term along the crack surface is zero, which may contribute to the existence of such a critical state observed from the fracture parameters. The variation of the critical stress with the crack length is depicted in figure 6 when *α*=0.2 mm^{−1}, indicating that the critical stress decreases with the increase of the crack length.

To consider the critical state of interacting cracks in piezoelectric materials, we take some specific cases as examples. For two collinear interacting cracks in the homogeneous piezoelectric medium (*α*=0), it is found that the effective electric displacement along the crack surface is uniformly distributed as the single-crack problem. It is expected that this kind of crack interaction has a trivial effect upon this critical state for the interacting dielectric cracks in the homogeneous medium and the critical value for is approximately 20 MPa as shown in figure 7. Again, this critical state is independent of the crack length. However, this uniform distribution of the effective electric displacement no longer exists for the same problem in the inhomogeneous medium. For the case when the material gradient *α*=0.4 mm^{−1} and MPa, the variation of along the crack surface for two collinear cracks is plotted in figure 8. It is seen from this figure that when these two cracks have the same crack length (*a*_{I}=*a*_{II}=2 mm, for example), the distribution of for each crack from the inner tip to the outer tip is identical, whereas this symmetric distribution is broken up when the cracks have different crack lengths (*a*_{II}=2*a*_{I}, for example). Therefore, for two collinear cracks with the same crack length, there exists one critical state () as shown in figure 9. It is also indicated that this critical state is dependent on the variable *α**a*. However, for two collinear cracks with different crack lengths, there exist two critical states () and () as shown in figure . In the inhomogeneous medium, the crack interaction is complicated by the material gradient, and the fracture property of the cracked FGPMs is determined by the combined effect of the material gradient and the crack interaction. Therefore, the crack interaction effect on this critical state is non-trivial and cannot be ignored. As a result, these two critical states in figure 10 are not coincident with the corresponding value of the single-crack problem in the FGPMs with the same *α**a*.

For two centre-aligned parallel cracks, the effective electric displacement along the crack surface no longer distributed uniformly even for the homogeneous medium as shown in figure 11 when MPa, *a*_{II}=4*a*_{I}(*a*_{I}=1 mm) and *h*=2*a*_{I}. It is also seen from this figure that the distribution of the effective electric displacement for these two parallel cracks with different crack lengths is not the same, indicating the possibility of different critical states for each individual crack. Figure 12 plots the variation of the electric displacement intensity factor *K*_{D} with the applied tensile stress for different crack models. It is indicated that there exists a critical state () for crack I, when , the impermeable crack model is the upper bound, whereas the permeable crack model is the lower bound. When , this situation is completely reversed. For crack II, when the impermeable (or permeable) serves as the upper (or lower) bound is controlled by the critical state (). For two parallel cracks in the inhomogeneous piezoelectric medium, the combined effect of the material gradient and the crack interaction will make the problem more complicated. However, it is found that the critical state is still dependent on *α**a* for the cracks with the same crack length. For example, figure 13 shows the variation of *K*_{D} for crack II versus the applied tensile stress. It is obvious that for any fixed *α**a* (0.8, for example), the critical values for the two cases are the same. Figure 14 plots the distribution along the surfaces of these two parallel cracks for *α**a*=0.8. It is obviously demonstrated in this figure that the effective electric displacement for these two cracks is different. This phenomenon is due to the vanishing symmetry of these two cracks caused by the inhomogeneity of the material properties. Thus, the critical states for crack I and crack II do not coincide by comparing figures 13 and 15 even for the two parallel cracks with the same crack length. It is concluded that for the two interacting cracks, the effective electric displacement distribution governs the critical states.

## 6. Concluding remarks

This study provides a systematic study of the critical state of the dielectric crack in piezoelectric materials for both single crack and two interacting crack problems, which determines when the impermeable (or permeable) crack serves as the upper or the lower bound for the dielectric crack. Numerical simulations are performed to indicate that the crack length, the material gradient and the crack interaction have a significant effect on this critical value. For FGPMs, the effective electric displacement distribution along the crack surfaces becomes more complicated, which may result in different critical states for each individual crack. It is also found that a new feature of the combination of the material gradient and the crack length *α**a* will play a crucial role in determining this critical state. The results calculated from the dielectric model are always between those of the permeable and impermeable crack models, and the nonlinear response of the dielectric crack undergoes a transition according to this critical state.

## Acknowledgements

This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

## Footnotes

- Received April 10, 2009.
- Accepted June 29, 2009.

- © 2009 The Royal Society