fracture problems in antiplane piezoelectricity

In: Recent Trends in Fracture Mechanics Author: Yunan Prawoto

ISBN: 978-1-61470-615-1 ©2012 Nova Science Publishers, Inc.

Chapter 6

FRACTURE PROBLEMS IN ANTIPLANE PIEZOELECTRICITY Y. Eugene Pak*1, Dhaneshwar Mishra2 and Seung-Hyun Yoo2 Advanced Institutes of Convergence Technology, Seoul National University Suwon, Korea 1 Structural Mechanics Lab, Department of Mechanical Engineering, Ajou University Suwon, Korea2

ABSTRACT This article reviews extensive work that has been done to better understand the mechanics of piezoelectric materials in the presence defects such as cracks. In particular, antiplane problems are considered because they offer a considerable simplification in the analyses while revealing important physical phenomena without complicated mathematics that arise due to the anisotropic and electroelastic nature of piezoelectric materials. A simple crack, multi-crack, crack-defect interactions, elliptical defects and dynamic crack problems are reviewed in the context of linear piezoelectricity. Physically meaningful conclusions are drawn that can find their counterparts in more realistic inplane loading cases. Keywords: Piezoelectric Fracture, Mode III Crack, Antiplane Piezoelectricity, Energy Release Rate

6.1. INTRODUCTION It is well known that piezoelectric materials produce an electric field when deformed and undergo deformation when subjected to an electric field. Due to this intrinsic coupling phenomenon, piezoelectric materials are used widely in technology such as high-power sonar transducers, electromechanical actuators, and piezoelectric power supplies. When subjected to mechanical and electrical stresses in service, these piezoelectric materials, which are mostly *

E-mail address:genepak@ snu.ac.kr

2 Y. Eugene Pak, Dhaneshwar Mishra and Seung-Hyun Yoo brittle ceramics, can fail prematurely due to the propagation of flaws or defects produced during the manufacturing process. Therefore, it is important that the fracture processes in pizoelectric materials be understood and analyzed so that reliable service lifetime predictions of the components can be made. This article reviews extensive work that has been done to better understand the mechanics of piezoelectric materials in the presence defects such as cracks. Due to the anisotropic nature of piezoelectric materials and the coupling of an additional field variable in the governing equations, piezoelectric boundary value problems generally pose serious mathematical difficulty. When a defect such as a crack is introduced, the problem becomes even more difficult. The anitplane problems reviewed in this article offer a simplification that makes it possible to find a closed-form or analytical solutions readily. A simple crack, multi-crack, crack-defect interactions, elliptical defects and dynamic crack problems are reviewed in the context of linear antiplane piezoelectricity. These antiplane mode III piezoelectric crack problems will reveal important physical phenomena without complicated mathematics that can find their counterparts in more realistic inplane loading cases. Parton (1976) did a pioneering work—publishing perhaps the first archived paper on piezoelectric crack—on a finite crack at the interface of a piezoelectric material and a conductor subjected to a far-field uniform tension. Cherapanov (1977) did one of the earlier works on path-independent integrals for electromechanically coupled solids. A more general defect mechanics of piezoelectric material was considered by Deeg (1980) using Green’s function method and modeling the defect with a collinear dislocation and a charge dipole line. Ever since these pioneering works, there has been an extensive research in this field and some excellent fundamental works and review articles have been written by McMeecking (1989, 1999, 2001), Pak (1990a), Sosa and Pak (1990), Suo (1992), Sosa (1992), Zhang and Hack (1992), Park (1994), Park and Sun (1995a,b), Zhang and Tong (1996), Fulton and Gao (1998, 1999), Sih and Zuo (2000), Zuo and Sih (2000), Hall (2001), Soh et al. (2001), Qin (2001), Zhang et al. (2002), Zhang and Gao (2004), Li and Lee (2004a), Chen and Hasebe (2005) and Kuna (1999, 2010). In this review, we will consider only the antiplane fracture problems of linear piezoelecticity. We will first introduce the basic formulations and a simple crack problem which will set the foundation for more involving problems.

6.2. ANTIPLANE PIEZOELECTRIC FORMULATIONS 6.2.1. Governing Equations The linear piezoelectric formulations that are basic to most of the antiplane crack analyses reviewed in this article are introduced. The three-dimensional formulation of linear piezoelectricity that was presented by Tiersten (1969) is briefly summarized here. The variational formulation for a static linear piezoelectric material occupying region V bounded by surface S can be derived from the Hamilton’s principle as follows: (6.1)

3 Fracture Problems in Antiplane Piezoelectricity where H is the electric enthalpy density, Ti is the applied surface traction, ui is the displacement, q is the applied surface charge, and φ is the electric potential whose negative of the gradient is the electric field, i.e., (6.2) In this formulation, the displacement and the electric field are taken to be the independent field variables. For linear piezoelectric materials, the electric enthalpy density is expressed as

(6.3) where c ijk l are the elastic moduli measured in a constant electric field, are the dielectric constants measured at constant strain, e i k l are the piezoelectric constants, and ε i j is the strain tensor, (6.4)

The variational formulation for the piezoelectric continuum provides the following results: Governing field equations

(6.5) Boundary conditions

(6.6) Constitutive equations

(6.7) In the expressions above,  i j is the stress tensor, ni is the unit normal vector, and Di is the electric displacement vector.

4 Y. Eugene Pak, Dhaneshwar Mishra and Seung-Hyun Yoo

6.2.2. Constitutive Relations For a special case of a piezoelectric material representing a transversely isotropic medium belonging to a hexagonal crystal class 6mm, the constitutive relations (6.7) are of the form (Berlincourt et al., 1964)

 xx   yy  zz   zy  zx   xy

 c11 c12 c13 0    c12 c11 c11 0    c13 c13 c33 0 =     0 0 0 c44    0 0 0 0   0 0 0 0 

0 0 0 0 c44 0

    0   0   0    0   1 c11  c12  2  0

 xx   yy  zz   zy  zx   xy

   0 0 e31    0 0 e  Ex  31        - 0 0 e33   E y    0 e15 0   E   z  e 0 0 15     0 0 0 

(6.8)

(6.9) The piezoelectric boundary value problem simplifies considerably if we consider only the out-of-plane displacement and the in-plane electric fields such that (6.10) .

(6.11)

In this case the governing equations (6.5) simplify to (6.12) (6.13) where  2 is the two-dimensional Laplacian operator

5 Fracture Problems in Antiplane Piezoelectricity (6.14)

The constitutive relations (6 . 8 ) and (6 . 9 ) become

(6.15)

6.2.3. Electrical Boundary Conditions on Crack Faces In analyzing the piezoelectric crack problem, it can be assumed that the crack contains a small but finite volume that is filled with vacuum (or air). Of course, in mathematical idealization the crack is modeled as a slit with only a length parameter. However, proper physical boundary conditions must be prescribed at the crack faces taking into account the interactions that take place at the interface between the piezoelectric material and the void inside the crack. To be formal, one must solve a two-domain problem with the appropriate governing equations and the boundary conditions for each domain. Through boundary and continuity conditions across the crack face, Pak (1990a) has shown that

(6.16)

This expression shows the relative magnitude of the normal components of the electric field just inside and outside the piezoelectric crack face as a function of the relative magnitude of the material constants. In examining the actual values of these material constants used in the analysis, one can easily see that the ratio is of the same order of magnitude as the dielectric constant of the piezoelectric material , which in turn is three orders of magnitude greater than the dielectric constant of the vacuum, . Since the ratio of the material constants is very small, i.e., one can simplify the problem by assuming that the ratio of the electric fields,

, approaches zero at the piezoelectric-vacuum interface. (6.17)

It is interesting to note that, in examining the expression given in (6.16), the ratio of the fields is small, independent of the relative magnitude of the dielectric constants , for is also much greater than . This indicates that, for piezoelectric materials, there are two independent reasons for the electric field just inside the material to be small

6 Y. Eugene Pak, Dhaneshwar Mishra and Seung-Hyun Yoo at the boundary. Therefore, one can set or at the boundary of the piezoelectric material and decouple the domains by ignoring the field inside the crack. The validity of this simplification is subjected to the condition that the crack is filled with a vacuum or a non-conducting gas (i.e., the crack faces are not in perfect contact with each other). In most realistic cases, this assumption is valid especially when a far-field crackopening mechanical load is present. The assumptions made thus far constitute what in the literature is called the impermeable crack. There are other boundary conditions which are discussed extensively in the literature (Dunn, 1994; Hao and Sen, 1994; Kumar and Singh, 1997b; Ru, 1999; Wang and Jiang, 2002; Wang and Mai, 2003; Landis, 2004; Chen et al., 2005; Nam and Watanabe, 2007) and are well summarized by Kuna (2010) as follows: a) Permeable crack: the crack is fully penetrated by the electric field. Equal electric potentials are prescribed on both crack faces. This condition can occur if the crack remains closed (6.18) where the superscripts (+, -) indicate the upper and lower crack faces. b) Semi-permeable crack: this is a more realistic condition allowing limited permittivity and Maxwell stress traction on crack faces. According to Parton (1988) and Hao and Shen (1994), the opposite crack faces are considered as a set of parallel capacitors flushed by a vertical electric field Ey as shown in Fig. 6.2.1.

(6.19)

c) Conducting crack: this is valid when the crack interior is filled with a conducting medium such as an electrolyte fluid or moist air not allowing electric field to be built up inside. (6.20)

Figure. 6.2.1 The iterative capacitor analogy, ICA (Kuna, 2010).

7 Fracture Problems in Antiplane Piezoelectricity The permeable and conducting crack conditions are two far extremes of approximations and more realistic model would be the semi-permeable crack suggested by Landis (2004) as “energetically consistent boundary conditions.”

6.2.4. Path-Independent Integrals Conservation laws that lead to path-independent integrals can be derived by a direct procedure shown by Eshelby (1970) for a static elastic continuum. It involves the Lagrangian density only and consists of differentiating it with respect to the independent variables. The method can be demonstrated by taking electric enthalpy density to be the Lagrangian density and simply differentiating it with respect to the spatial coordinate, xk. This operation leads to

(6.21) where Fk denotes the kth component of the conservation integral and δij is Kronecker delta. Pak (1990a) has shown that this expression is physically meaningful in that its value, when evaluated on a surface enclosing a defect, is the total potential energy release rate perinfinitesimal translation of the defect in the xk direction. The x-component of this integral is the generalized J-integral for piezoelectric materials. Another path independent integral of elastostatics is the M-integral which is related to the energy release rate of a self-similarly expanding defect enclosed by the contour of the integral. The M-integral can be generalized to take piezoelectric effect into account

(6.22) Freund (1978) showed that for a slit crack, we can relate M to J through the relationship M = 2aJ, where 2a is the crack length. Using this relationship, Pak and Kim (2010) showed that the energy release rate for a crack under a various mechanical and thermal loading can be obtained by first evaluating the M-integral for an elliptical hole then performing a limiting process of the minor axis approaching zero in length.

6.3. SINGLE CRACK PROBLEMS In this section, we will present the status of single crack analyses in piezoelectric materials by considering various types and orientations of the crack, types of boundary conditions used on the crack faces, methods of analysis, failure criteria as well as the numerical results of the stress and electric field intensity factors and energy release rates. We will consider far-field antiplane mechanical and inplane electrical loads, and will consider both electrically permeable and impermeable types of cracks. Consider a simplest case of a mode III fracture problem in which a finite crack of length 2a is embedded in an infinite piezoelectric medium subjected to a far-field antiplane shear, σzy = τ∞, and an inplane electric field, Ey = E∞ as shown in Figure 6.3.1. The boundary conditions

8 Y. Eugene Pak, Dhaneshwar Mishra and Seung-Hyun Yoo on the upper and the lower surfaces of the crack are to be free of the surface traction and the surface charge for the given far-field loading conditions, i.e., (6.23) This would be what in the literature is called the “impermeable boundary condition.” Using complex function method, the crack tip solution obtained is

(6.24) where these crack-tip field intensity factors are defined as

(6.25)

It is shown that all the field variables have the same crack tip singular behavior as the classical mode III fracture problem. It is also shown that the stress and the electric fields decouple at the crack tip. Therefore, when there is an electrical load in addition to a mechanical load, it is not clear that the stress intensity factor alone can characterize the fracture behavior. Therefore, we will consider the energy release rate concept in characterizing defects subjected to more than one field loading. The path-independent integral derived earlier can now be evaluated to obtain the energy release rate for the mode III piezoelectric fracture problem under consideration. Denoting J to be the x-component of the conservation integral Fk, the path-independent integral takes the form

(6.26) Using the solution obtained previously, evaluating this integral on a vanishingly small contour at a crack tip results in (6.27)

9 Fracture Problems in Antiplane Piezoelectricity

Figure 6.3.1. Infinite piezoelectric material with an embedded finite crack subjected to far-field antiplane mechanical and inplane electrical loading (Pak, 1990a).

This result can also be obtained by considering the work done in closing the crack tip over an infinitesimal distance Δx, which can be calculated by

(6.28) The crack extension force, G, can then be calculated as the energy released in propagating the crack an infinitesimal distance, i.e., (6.29)

This result is the same as that obtained by evaluating the path-independent integral (6.26) on a contour enclosing a crack tip. Therefore, for linear piezoelectric materials, as in the purely elastic case, the value of J is identical to the crack extension force, G: (6.30)

10 Y. Eugene Pak, Dhaneshwar Mishra and Seung-Hyun Yoo As the expression shows, the negative sign in front of the electrical load term decreases the crack extension force in the presence of electrical loads regardless of the direction. Some . quantitative insight on the magnitude of this electrical effect can be obtained by substituting 8 2 6 the tensile strength of 10 N/m for τ and the poling voltage of 10 V/m for E in (6.30). We observe that the electrical term decreases the total crack extension force by about 8%. In the presence of a finite size macrocrack, the effect will be even more significant since the critical mechanical load will be lower. It seems that this effect is large enough to be observed from an experiment. Pak (2010) has shown that this crack arresting effect arises because the total electrical energy decreases when the overall dielectric constant increases (due to decrease in crack size) which is energetically favorable. He also showed that this effect has a physical origin in the interaction between the applied electric field and the induced surface charges. However, the indentation fracture test conducted by Tobin and Pak (1993) showed the crack arresting effect occurs only when the electric field is applied in the opposite direction of poling. For a permeable crack, 1 / singularity still exist for both stress and electric displacement along the plane of the crack (Park and Sun, 1994);

(6.31) The stress and electric displacement intensity factors can be given as (6.32) In the crack plane (θ = 0 ) ahead of the crack tip, the stresses and electric fields are not coupled, i.e., electrical field alone cannot produce mechanical stress. This is universal in all piezoelectric slit crack analyses. So when the stress intensity factor is used to determine crack instability, the electric field should have no effect which contradicts with some of the experimental findings. Thus, they proposed to use the mechanical energy release rate as a criterion to predict crack growth. They also argue that because fracture is a mechanical process there is no role of electrical loading on fracture and thus the mechanical energy release rate should be used as the fracture criterion in piezoelectric materials. Shindo et al., (1996) extended this work and presented the singular stress and electrical fields in a piezoelectric ceramic strip with a finite crack under a longitudinal shear by proposing to use the strain intensity factor in place of the stress intensity factor. A finite element simulation of crack propagation in piezoelectric ceramics (Fang et al., 1998) suggests that the crack arrests when  = 0 for a negative electric field, but it propagates at  = 84° on the basis of the maximum stress criterion. This result indicates a limitation of the mechanical strain energy density criterion proposed by Park and Sun (1995) when the ratio of electric field to the mechanical load is large. A crack problem in a piezoelectric material under a general mechanical and uniform electric loading was considered by Yang and Kao (1999) using exact boundary conditions. Here, the authors have explained that the electrically permeable and impermeable cracks are two extreme cases. They tried to impose the actual boundary condition at the crack face by

11 Fracture Problems in Antiplane Piezoelectricity introducing the mechanical load and the electrical boundary condition separately in two steps and then combine the results by superposing both results. To consider the electrical boundary condition on the crack face, the authors assumed that there is a thin layer (Fig. 6.3.2) of thickness 2h between the upper and lower semi-infinite piezoelectric material. The electrical boundary conditions at the interface are (6.33) where and are the components of electric field and electric displacement inside the crack, respectively.

Figure 6.3.2. Antiplane loading of two semi-infinite piezoelectric materials (Yang and Kao, 1999).

The mechanical loading on crack faces in the absence of electric field is imposed as (6.34) (6.35) In Yang and Kao’s work, the electrical field intensity factor derived by this method is zero as there is no charge inside the piezoelectric material and along the crack faces. This is different from the result given by Pak (1990a) and Zhang and Hack (1992). Two important cases have been considered (i) the mechanical loading on crack face is constant and (ii) the loading is a point force applied in the middle of the crack face. In the first case, where a constant mechanical loading is applied on the crack face, the field intensity factors come out to be the same as those given by Pak (1990a). When a point force applied on the middle of the crack face is considered, then for

(6.36)

12 Y. Eugene Pak, Dhaneshwar Mishra and Seung-Hyun Yoo where  is the delta function. The field intensity factor expressions can be written as

.

(6.37)

All field variables except for the electric field have square root singularity at the crack tip. A far-field electrical loading in the absence of the mechanical loading in antiplane deformation, the boundary condition can be written as (i) which corresponds to traction free boundary condition at infinity. ( i i ) A second set of boundary condition implies the strain or displacement of the piezoelectric material at infinity is zero. The field intensity factors give rise to

(6.38) These results indicate that the electric field is non-singular in front of the crack tip due to the presence of free charges inside the piezoelectric material and along the crack faces. The energy release rate for this case only depends upon the resultant stress distribution generated by the mechanical deformation and the electromechanical interaction. The crack driving force for the crack growth remains positive under the combination of the far-field electrical and mechanical loading which is different from Pak’s (1990a) result where a negative value of the energy release is presented. However, the theoretical results of Pak (1990a), Sosa and Pak (1990) and experimental result of Wang and Singh (1994) were confirmed by Kumar and Singh (1996a, b, 1997a, b) by finite element simulations. McMeecking (1999, 2004) pointed out that the energy release rate at the crack tip and the total energy release rate are different. He suggested that the energy release rate at the crack tip should be utilized to estimate the failure of piezoelectric materials where there are both mechanical and electrical loads acting simultaneously. A semi-infinite antiplane impermeable crack in a piezoelectric material was analyzed by the Mellin transform method and the field intensity factors and the mechanical energy release rate were derived (Li and Fan, 2000). These can serve as the fundamental solution from which purely elastic results can be extracted by removing the electromechanical coupling part. The analysis of a transversally isotropic piezoelectric material in the framework of linear piezoelectric fracture mechanics (Kwon and Lee, 2000) showed that the energy release rate is

13 Fracture Problems in Antiplane Piezoelectricity dependent on the electrical loading under a constant strain loading. However, it is independent of the electrical loading under a constant stress loading which is shown by using the electric displacement strip saturation model presented by Gao et al. (1997). For a finite crack in a finite orthotropic piezoelectric plate, Shindo et al. (1996a) showed that the stress intensity factors and the energy release rate increases when the ratio of the crack length to the plate width increases. The electrical nonlinear behavior of an antiplane shear crack in a piezoelectric ceramic layer constrained between two orthotropic layers (Kwon and Shin, 2006) was examined by electrical strip saturation model. This model is based on the electrically unified crack surface condition. A new term called electric crack condition parameter (ECCP) was introduced and the effects of electrical crack conditioning parameter (ECCP) along with the elliptical flaw shape parameters are studied taking into consideration of permittivity inside the crack, applied electric field, crack length, electromechanical coupling coefficient, crack position and adjacently-bonded elastic material properties. The solutions resulting from the unified crack boundary condition fall between impermeable and permeable crack boundary conditions as both permeable and impermeable crack conditions are two extreme cases (Zhang and Tong, 1995). A discontinuous crack model was proposed by Liu et al. (2008) which removed the restrictions of electric displacement strip saturation model so as to facilitate the evaluation of the electric yielding effect on the crack parameter. The stress intensity factors and energy release rate was shown to increase with the sample length under a constant strain loading. A mixed electrical boundary value problem for two dimensional piezoelectric crack was solved (Huang and Kuang, 2003) by the analytical continuum method to evaluate the field intensity factors at the crack tip. Antiplane shear behavior of a permeable Griffith crack in a piezoelectric material (Zhengong et al., 2003) by the use of a non-local theory suggests that it can be used to study any scale crack, either micro or macro scale, as it provides non-singular stresses at the crack tip. The minimum energy release rate exists with a positive value for various electrical loads. When the crack is at an arbitrary position (Li and Lee, 2004), all the energy release rates (ERR) are the same for permeable cracks, while for impermeable cracks, the ERR calculated from the internal energy can predict the crack growth better which increases with increase in the crack length. The ERR is independent of the electrical loading if a traction (stress) is prescribed at the boundaries parallel to the crack face. When strain is prescribed, the electrical loading has to be taken into consideration. The normalized ERR increases with increasing crack length. Furthermore, when eccentricities rise with reference to the vertical or horizontal directions, the normalized ERR increases. Mode III crack in a piezoelectric N-type semiconductor (Hu et al., 2007) was analyzed by Fourier transform method to reduce the mixed Boundary Value Problem (BVP) to a pair of dual integral equation; a numerical solution of these equations yields electromechanical fields near the crack tip. A half-plane piezoelectric solid with an embedded crack and a traction free boundary (Yang et al., 2007) has been analyzed and the singular integral equation has been formulated for materials with general anisotropic piezoelectric properties and cracks with arbitrary orientations. Kernel functions were developed for a general anisotropic piezoelectric material and the complex form was specialized to the case of a transversally isotropic piezoelectric material in real form. These Kernel functions can be reduced to purely elastic

14 Y. Eugene Pak, Dhaneshwar Mishra and Seung-Hyun Yoo terms when the electrical effects disappear. Generalized stress intensity factors for a transversely isotropic material were presented to show the coupling effect. The mechanical load has a significant effect in the electric displacement intensity factors, but there is a negligible inverse piezoelectricity effect on the mechanical stress intensity factors. The effect of elastic coating on fracture behavior of a piezoelectric fiber with a pennyshaped crack was investigated by the Hankel transform and potential function methods. The effects of the thickness and elastic material properties of the coating on the fracture behavior of piezoelectric fiber composite was studied by Qin et al. (2006). The mode III stress and electric field intensity factors were derived for edge-cracked circular piezoelectric shafts (Zhou et al., 2009) by Hamiltonian formalism and the singularities near the crack tip were represented in terms of the exponential series. The stress and electric intensity factors were obtained by the first pair coefficients of the expansion. Both stress and electric displacement factors do not increase monotonically due to the applied surface charges. Thus, it was shown that both parameters can be optimized by choosing appropriate surface charges and loading angles. A limited permeable crack in an interlayer between piezoelectric materials with different zones of electrical saturation and mechanical yielding was studied by Loboda et al. (2010). Here, two distinct zones—a zone of mechanical yielding and a zone of electrical saturation of unknown length were introduced as crack continuations. Outside of these zones, the semiinfinite spaces were assumed to be perfectly bonded. The unknown yield and saturated zone lengths were found from the conditions of the finiteness of stress and electrical displacement at the ends of these zones. Considered cases were for the electrically saturated zone being longer and shorter than the mechanical yielding zone. It is shown that the same equation for the Griffith crack model can be used to determine the electrical displacement in the crack region. The results from the proposed model were compared with the Griffith crack model for the crack-tip stress and electric fields and the energy release rate.

6.4. BLUNT ELLIPTICAL CRACK PROBLEMS In order to model a permeable crack with proper electrical interactions that take place on crack faces, an elliptical hole is modeled with the electric field inside the hole. Then, as a limiting case of the elliptical hole tending to a flat slit crack, a permeable crack solution can be obtained. A theoretical solution is provided by Meguid and Zhong (1997) for the elliptical inhomogeneity in a piezoelectric material under an antiplane shear and an inplane electric field by deriving the general complex potential expressions in closed form. In a more explicit form, an elliptical piezoelectric inclusion (Fig. 6.4.1) embedded in an infinite piezoelectric matrix is analyzed by Pak (1994, 2010) in the framework of linear piezoelectricity. Using the conformal mapping technique, a closed-form solution is obtained for the case of a far-field antiplane mechanical load and an inplane electrical load. The solution to a permeable elliptical hole problem is obtained as a limiting case of vanishing elastic modulus of the inclusion. This enables the study of the nature of crack tip electric field singularity which is shown to depend on the electrical boundary condition imposed on the crack faces. The energy release rate of a self-similarly expanding slender crack in the presence of an electric field is obtained by using the generalized M-integral. The energy release rate expression indicates that the electric field has a crack arresting influence.

15 Fracture Problems in Antiplane Piezoelectricity This effect is inferred to have a more fundamental physical origin in the interaction between the applied electric field and the induced surface charges on the crack faces. An explicit solution for the elliptical hole problem can be obtained by setting the elastic and the piezoelectric constants of the inclusion to be zero. This process enable us to model a slit crack more realistically by letting the crack faces be electrically permeable and electric fields to exit inside the crack. The elliptical hole solution thus obtained reveals that the stress and electric fields along the x-axis ahead of the elliptical notch in the matrix (M) has the form

(6.39)

(6.40)

Figure 6.4.1. Elliptical inclusion embedded in a piezoelectric medium (Pak, 2010).

16 Y. Eugene Pak, Dhaneshwar Mishra and Seung-Hyun Yoo These expressions show that the stress and electric fields are also decoupled for the case of an elliptical hole as in the antiplane slit crack case (Pak, 1990). Decoupling of the field variables corresponding to the far-field loads (i.e., stresses and electric fields) is due to the fact that the governing equations are decoupled from the outset which is a special characteristic of the antiplane piezoelectricity considered herein. Other field variables such as strains and electric displacements show coupling through the constitutive relations (6.15). In the case of a piezoelectric inclusion with non-zero elastic and piezoelecric constants, however, coupling is present. The coupling is also present in the plane strain case of an elliptical hole subjected to an inplane uniaxial tension and inplane electric field parallel to the poling direction as shown by Sosa (1991). It was also shown that the electrical and mechanical fields are coupled everywhere at all aspect ratios of the ellipse. However, in the limiting case of a slit crack, b → 0, on the crack plane, θ = 0, the stress and the electric fields decouple as was first revealed in Deeg’s (1980) analysis. This decoupling at the crack tip makes it difficult to characterize electric field-induced crack propagation based on the stress intensity factor criterion. This necessitated the use of the energy release rate criterion. The electric field expression shown above is similar to that found by McMeeking (1989) in studying a fracture problem for electrostrictive materials. He also found a region front of the crack away from the crack tip in which 1/ field behavior is observed. It was also found that this behavior depends on the relative values of (b/a) and . It is interesting to observe that in the limiting case of an elliptical cavity approaching a slit crack, (b/a) → 0, the ratio of the crack tip electric field to the applied field, , along the x-axis as shown in (6.41) approaches one, indicating that there is no disturbance of the uniform applied field, where as the crack tip stress exhibit classical mode III singularity behavior

(6.41) However, if we express the electric field at the tip of the ellipse at x = a first, and then perform the limiting process of approaching a slit crack, we obtain

(6.42) This shows an anomaly in the mathematical solution where an electric field concentration at the tip of a crack depends on the order of the limiting process. It should be noted that by modeling the crack faces to be permeable, the electric field becomes finite at the crack tip. Imposing the impermeable boundary condition is equivalent to assuming that = 0 (in light of the fact that >> ), which would make the crack tip field artificially singular as shown by Pak (1990a), i.e.,

(6.43) Although the impermeable boundary condition is a good approximation in modeling blunt cavities, it is shown here that it can lead to an erroneous characterization of electric field in

17 Fracture Problems in Antiplane Piezoelectricity the slit crack limit. However, in realistic cracks with small but finite b/a ratio, there would be an electric field concentration, which is bounded by the ratio of the dielectric constants and the piezoelectric coupling factor as shown in (6.42). This concentrated field can influence the fracture behavior either through directly influencing the crack tip stresses, which none of the slit crack models predict, or indirectly through influencing the energenics of the fracture process that may consist of surface charge interactions or domain switching effects. According to the crack extension force shown in (6.30), when a high electric field is applied to a piezoelectric material with a crack, the negative term containing the square of the applied field will dominate and the crack arresting behavior will ensue independent of the applied field direction. The Mode III crack extension force expression shown in (6.30), albeit simple, reveal most of the physics that macroscopic continuum models can reveal about the effects of electrical fields on the fracture behavior of slit cracks in piezoelectric materials. More complicated Mode I plane strain solution (Sosa and Khutoryansky, 1996) and the strength of materials solution (Suo et al., 1992) predict essentially the same behavior. However, the crack extension force expression shown in (6.30) has been evaluated using the solution obtained by assuming the impermeable boundary condition that induces artificial singularity in the electric fields at the crack tip. In realistic cracks with some bluntness (b  0), there would be an electric field concentration, not quite as singular as the impermeable case, that can still influence the energetics of crack propagation. To quantify this effect, the generalized M-integral (6.22) is evaluated on a contour coinciding with the rim of the elliptical hole. This relationship can be extended to a slender elliptical cavity to obtain the crack extension force for slightly opened cracks. This will at most introduce an error on the order of the aspect ratio, b/a, which for very slender cracks is negligible. The crack extension force for various cases of b/a is plotted in Fig. 6.4.2 for PZT-5 including the impermeable case showing the greatest crack-arresting effect. In more realistic permeable cases, the electric field influence is present but diminishes as the crack becomes progressively more slender, and vanishes altogether in the slit crack limit. This implies that it may be difficult to observe the crack arresting phenomenon in an actual experiment for a slit crack. It is clear that the impermeable boundary condition on the crack faces overestimates the crack-arresting influence of the applied electric field. Other authors have considered various elliptical inhomogeneity problems in antiplane piezoelectricity. It was shown that the electroelastic fields within the three-phase inhomogeneity (Sudok, 2002) are uniform provided that the interfaces are two confocal ellipses. It was also shown that the confocal elliptical interface curve has advantage in minimizing the electroelastic fields. Elliptical cavity under an antiplane mechanical and inplane electrical loads (Zhang and Tong, 1996) were formulated by the complex potential method, and the electrical field inside the cavity was shown to be uniform whose magnitude varied with the shape of the ellipse. When the cavity is reduced to a slit crack, the electric field strength inside the cavity is inversely proportional to the permitivity of the cavity. The energy release rate depends on the stress intensity factor as in the purely elastic case without coupling with the electric field. If the ratio of the major to minor axes of the ellipse is much smaller than the permittivity, then the electric field does not have any role on energy release rate or crack propagation. An elliptical rigid inclusion of elastic conductor (Chung and Ting, 1996) subjected to a line force, a torque or a line charge was studied to derive solution at the interface.


Figure 6.4.2 Effect of electric field on crack extension force for various ratios of b/a (Pak, 2010).

Stress concentration at an elliptical hole in a transversely isotropic piezoelectric solid (Dai et al., 2006) was studied and a comparison between the stress concentration factor of the elliptical hole and the circular hole was made. The electromechanical coupling effect was studied and it was found that it can be helpful in reducing the stress concentration. It was shown that the influence of the dielectric constant of the medium inside the hole on the stress concentration is weak for a wide range of dielectric constants.

6.5. MULTI-CRACK PROBLEMS Electroelastic properties of a cracked piezoelectric materials under a longitudinal shear and an inplane electrical load were considered for analysis to study influence of cracks on electroelastic properties of piezoelectric materials (Pak and Goloubeva, 1996). A single stack of cracks as well as doubly periodic array of cracks with both rectangular and diamond shaped cracks are taken into consideration as shown in Fig. 6.5.1. Multi-crack interaction in piezoelectric solid and their effect on fracture and electroelastic properties were studied. The crack-tip field intensity factor and the change in stored electroelastic energy due to presence of many micro cracks are analyzed. This study can help in predicting the effective elastic, piezoelectric and dielectric constants of a damaged piezoelectric material which can result in the presence of many micro cracks. For the single stack case, we arrive at the following expressions for the stress and the electric displacement intensity factors, respectively

(6.44)

Here, and are the stress and electric displacement intensity factors for a single crack. Cracks in a strained piezoelectric solid change the total electric enthalpy, H

19 Fracture Problems in Antiplane Piezoelectricity of the system. It is possible to find the change in the electric enthalpy due to the presence of single stack of cracks,

(6.45) where

(6.46) Here, Δ Hs c is the change of electric enthalpy in an infinite piezoelectric body with a single crack.

Figure 6.5.1. (a) Single stack, (b) rectangular and (c) diamond-shaped arrays of cracks in a piezo electric material (Pak and Goloubeva, 1996).

Electroelastic constants of a piezoelectric body can be determined from the electric enthalpy stored in the body when it is subjected to an electromechanical load. Given the same loading, the stored electric enthalpy of a body with cracks are different from that of a homogeneous one. For the body with a rectangular array of cracks, one would expect the material to exhibit orthotropic behavior in the otherwise isotropic x-y plane. Therefore, we can characterize the behavior of a piezoelectric material containing a rectangular array of cracks with the effective orthotropic elastic , dielectric , and piezoelectric , constants. The electroelastic constants can be evaluated by equating the electric enthalpy density of the homogeneous orthotropic piezoelectric body, H*, to that of the isotropic piezoelectric body with cracks, H. We obtain the following effective orthotropic electroelastic constants of the body with rectangular array of cracks:

(6.47)

20 Y. Eugene Pak, Dhaneshwar Mishra and Seung-Hyun Yoo where (6.48) These are the effective material constants in the y-direction, perpendicular to the crack faces. The material behavior in the x-direction (parallel to the crack faces) would be the same as in the undamaged material. Expressions (6.47) show that the values of the effective electroelastic moduli are determined by material properties of the body without cracks, as well as by the length and the density of cracks. Diamond-shaped array of cracks show a similar behavior. This is clearly observed from Figures 6.5.2 and 6.5.3, which show the normalized effective electroelastic constants as functions of vertical and horizontal crack spacings for both rectangular and diamond shaped cracks. One can see that for the antiplane piezoelectric problem considered herein, the electroelastic behavior of a piezoelectric material is identical to the elastic behavior for the purely elastic case (Delameter et al., 1977), and the change in the elastic, dielectric and piezoelectric moduli are identical to each other:

(6.49)

Figure 6.5.2 Effective electroelastic constants for the rectangular array of cracks (Pak and Goloubeva, 1996).

Figure 6.5.3 Normalized effective electroelastic constants as functions of vertical and horizontal crack spacings for both rectangular and diamond shaped cracks (Pak and Goloubeva, 1996).

21 Fracture Problems in Antiplane Piezoelectricity This indicates that the changes in the piezoelectric and dielectric properties due to the presence of cracks are the same as the change in the elastic property. This reveals an interesting possibility of characterizing the mechanical damage in piezoelectric materials by simply measuring the change in the dielectric constant, i.e., the capacitance, which is much easier to measure than the changes in the elastic or the piezoelectric constants. The problem of two collinear permeable cracks lying at the mid-plane of a piezoelectric strip subjected to an antiplane shear loading was studied by Zhong and Li (2005). They used a Fourier transform technique to obtain an explicit closed form solution for the electroelastic field intensity factors and energy release rate at the inner and the outer crack tips. It was found that the electric field was uniform, while the crack-tip stress, strain and electric displacement became singular. Using the energy release rate as a failure criterion, it can be concluded that the effect of electric field on the crack growth is strongly dependent on the applied elastic displacement. The normalized energy release rate of the inner crack tip was always greater than that of the outer crack tip and thus the inner crack is more likely to grow. The interaction of two cracks intensified with decreased distance between the two. Unequally sized doubly-periodic cracks were analyzed and a closed form solution for the crack-tip field intensity factors and the effective electroelastic moduli of such cracked piezoelectric materials were derived (Tong et al., 2006). Dimensionless field intensity factor, K*, has been plotted against a vertical dimensionless spacing, ω2/a, for various values of dimensionless horizontal spacing, ω1/a in Fig. 6.5.4. They compared their result for the multistack solutions with the single stack solution given by Pak and Goloubeva (1996) for the case when ω1/a is sufficiently large for any values of ω2/a. When ω1/a and ω2/a tends to infinity, the dimensionless field intensity factor, K*, becomes one. It was shown that for rectangular array of cracks, the amplifying effect of the cracks in a row predominates. For a diamond shaped array of equidistant cracks, the study showed a complex dependence of the dimensionless field intensity factor on ω1/a. While discussing the shielding or amplifying effects of micro cracks, the authors observed that the stack-interleaving micro cracks shield the macro cracks while row interleaving and diamond shaped interleaving micro cracks amplify the field intensities at the macro crack tip. The magnification effect of the rowinterleaving micro cracks is more acute than that of the diamond shaped interleaving micro cracks. Small cracks tend to increase the electroelastic damage of the material and the most acute is in the case of row-interleaving micro cracks.

Figure 6.5.4 Comparison of Solution for a single stack of cracks (Tong et al., 2006).

22 Y. Eugene Pak, Dhaneshwar Mishra and Seung-Hyun Yoo A piezoelectric material with a periodic distribution of slant mode-III cracks (Li and Lee, 2005) were studied by the dislocation layer modeling. Both permeable and impermeable cracks were studied by developing a fundamental solution for a screw dislocation. The effect of electrical loading and geometrical parameters on the mechanical strain energy release rate were studied which showed a strong influence of the geometric parameters on mechanical strain energy release rate (MSERR). The relationship between the electric and the stress fields near a crack tip was obtained for three non-symmetric permeable mode III cracks by Zhou et al. (2009b). They showed that the stress and displacement intensity factors at the crack tip depend on the lengths and spacing of the cracks (Fig. 6.5.5). They discussed about the crack shielding effects explaining it in terms of the stress intensity factor and the crack orientation angle. The effect of geometrical parameters and applied mechanical loading on field intensity factors and energy release rate were also discussed. Interactions of multiple, parallel, symmetric, and permeable cracks in a piezoelectric material subjected to an antiplane shear and an inplane electrical load were studied by Zhou et al. (2009c). They used the Schmidt method to develop a relationship between the electric and the stress fields near a crack tip.

Figure 6.5.5 Geometry and coordinate system for three parallel non symmetric cracks (Zhou et al., 2009b).

The relationship between the displacement/electric potentials and the stress/electric displacements have been expressed in terms of the stress/electric field intensity factors along with the crack opening displacement and the energy release rate. A periodic crack problem in bonded piezoelectric materials was investigated by Shenghu and Xing (2007). They discussed about the variation of the stress intensity factor as a function of the crack periodicity for different values of the material inhomogeneity. When we discuss the numerical methods used for multi-crack problems in antiplane piezoelectric fracture, the following works provide a useful insight. A mixed boundary value problem was formulated to model periodic cracks in piezoelectric materials in terms of the surface displacement and the electrical potentials (Han and Wang, 2005) resulting in a system of hypersingular integral equations that provide analytical solution to predict crack spacing effect. Denda and Mansukh (2005) formulated a numerical Green’s function approach with an embedded crack opening displacement (COD) in the boundary integral method (BEM) to solve for multi-crack solutions. This method does not need any further post processing to

23 Fracture Problems in Antiplane Piezoelectricity determine the stress and the electric displacement intensity factors. The problem of an infinite sequence of parallel cracks in an infinitely extended piezoelectric solid was considered by Yang et al. (2008). This work is an extension of the authors work for the single crack problem using the kernel function method. They discussed about the crack spacing effect and the electromechanical coupling along with the evaluation of the fracture parameters such as stress intensity factors. The arbitrarily oriented multiple cracks (Athanasius et al., 2010) were analyzed by a Green’s function method, and hypersingular integral equations were developed to evaluate the stress and electric displacement intensity factors. Electromechanical fracture analysis for corners and cracks (Hwu and Ikeda, 2008) was carried out by introducing multiple wedges to study cracks and interface cracks in piezoelectric materials. In this study, the corners and cracks were studied together and the stress intensity factors were evaluated. They extended the stress intensity factor expressions developed for cracks to general corners.

6.6. INTERFACE CRACK PROBLEMS Materials with interfaces are common in piezoelectric devices and often times that is where failure occurs. Therefore, studying fracture behavior of piezoelectric materials with interface cracks would be important in studying failure characteristics of piezoelectric devices. There are various studies on interface cracks in piezoelectric materials under antiplane mechanical and inplane electrical loading. The near tip singular fields of a piezoelectric interface crack at the interface can be expressed as r -1/2 ±iε and r -1/2 ±κ in place of the square root singularity (Suo, 1992), where  and κ are material constants. When the upper and lower materials are the same, then it reduces to a usual square root singularity. These material constants have a strong influence on the near-tip field intensity factors. Near-tip fields and intensity factors for interfacial cracks in dissimilar anisotropic piezoelectric media were considered by Boem and Atluri (1996) and presented the field intensity factors in real terms which include bi-material constants.

Figure 6.6.1 Schematic of interfacial crack: definition of geometry and loading (Soh et al., 2000).

An interface crack between two orthotropic piezoelectric materials has been discussed theoretically (Narita and Shindo, 1998; 1999) and its effect on fracture parameters such as stress intensity factors and energy release rate has been presented. The behavior of a bilayer piezoelectric ceramic with a central interfacial crack (Fig. 6.6.1) subjected to an antiplane shear and an inplane electrical loading has been studied by a dislocation density functions

24 Y. Eugene Pak, Dhaneshwar Mishra and Seung-Hyun Yoo method (Soh et al., 2000) to estimate the field intensity factors and energy release rate. In this study, the effect of crack length to material layer thickness on energy release rate was discussed. The results for both conducting and non-conducting cracks were presented. The influence of a permeable electrical boundary condition on interface crack in a bi-material compound (Govorukha et al., 2006) under a pure electrical loading was studied to see the effects on fracture parameters of interface cracks. The validity of this model decreases as the mechanical load starts to increase. Theoretical and numerical analyses of a permeable interfacial crack (Kwon and Lee, 2001; Ou and Chen, 2004; Qun and Yiheng, 2007) at the bimaterial interface show two oscillatory singularities in addition to the usual inverse square root singularity. They found that the crack tip stress intensity factor and the energy release rate depend upon the far-field mechanical loading only and thus no effect of electrical loading has been shown. Fracture analysis for a crack at the interface between a dielectric and a piezoelectric layer in a multi-stack cylindrical piezoelectric sensor polarized along its axis was studied by Li and Lee (2009b). They showed the effects of the geometry, boundary conditions and material properties of dielectric and piezoelectric layers. It was suggested that a specific geometric ratio of piezoelectric to dielectric layer and its material gradient is needed to make the device robust. A single crack and two collinear cracks between two dissimilar piezoelectric material layers have been studied (Wang and Sun, 2004) and the energy release rate and field intensity factors have been derived and plotted for both permeable and impermeable cracks. The antiplane problem of an interfacial crack ahead of an elliptical inhomogeneity in a piezoelectric material was investigated by Fang et al. (2011). For the case of an interfacial crack between two dissimilar functionally graded piezoelectric materials (Singh et al., 2008), it has been found that the stress intensity factors depend upon the lateral variation of the material property. A numerical study on an interfacial crack in a three-dimensional piezoelectric media (Zhao et al., 2004) has been carried out by a boundary integral method (BEM) and various parameters that affect fracture strength have been identified. The field intensity factors, energy release rates and generalized strain energy density at the crack tip were provided.

6.7. CRACKS EMANATING FROM A HOLE It is well known that stress concentration occurs at the rim of a hole from which cracks can emanate and grow eventually causing a catastrophic failure. Therefore, studying the fracture behavior of piezoelectric materials with cracks emanating from a hole can provide insight about its failure. Mode III impermeable cracks originating from the rim of a circular hole in a piezoelectric solid (Fig. 6.7.1) were studied by Wang and Gao (2008) through complex variable analysis and conformal mapping. Their solution gave stress intensity factor and the energy release rate as (6.50) where

can be determined from


(6.51)

and R is the radius of the hole,  is a real parameter such that 0 ≤  ≤1,

(6.52)

in which κ is number of cracks emanating from the hole, and

(6.53)

Figure 6.7.1 Cracks emanating from a hole (Wang and Gao, 2008).

When the coefficients of field intensity factors (Fig. 6.7.2) are plotted against the crack length, we can see that the field intensity factors for a single crack are slightly greater than for double cracks. For the energy release rate, it increases with increase in both positive electrical and mechanical loads, but starts decreasing when the electrical field becomes negative in both the single and double crack cases. When we consider the non-symmetric permeable cracks emanating from an elliptical hole (Guo et al., 2009), the stress intensity factor and the energy release rate increase with the increase in crack length. This behavior is similar to the circular hole case and is irrespective of which side the crack length increases. The energy release rate for two non-symmetrical radial cracks at the edge of a circular hole is slightly larger than that from an elliptical hole. This method can be used to study cracks emanating from a circular hole, T-shaped cracks,

26 Y. Eugene Pak, Dhaneshwar Mishra and Seung-Hyun Yoo cross shaped cracks and edge cracks. A complex potential method was used to analyze multiple cracks emanating from a circular hole in a piezoelectric material (Guo et al., 2010) providing field intensity factors and energy release rate. This general analytical solution can be transformed for specific cases of a single crack or double cracks emanating from a hole. The results show that the piezoelectric material containing three radial cracks spaced equally at 1200 apart originating from a circular hole is the easiest to fail for all cases of cracks originating from a circular hole. Moreover, if there exist multiple cracks (n  3) originating from a circular hole, an increase in the number of cracks can enhance the reliability of these materials.

Figure 6.7.2 Variation of g () as a function of L/R (Wang and gao, 2008).

A semi-infinite crack penetrating to the center of a piezoelectric circular inhomogeneity bonded to an infinite piezoelectric matrix through a linear viscous interface was studied by Wang et al. (2009). The stress and the electric displacement intensity factors were shown to be

(6.54)

(6.55) where superscript (2) refers to the circular inhomogeneity,  is the eigenvalue of the governing equations and Hij are the material parameters. It is observed from the above expression that as t , the viscous interface becomes free-sliding and does not sustain shear loads resulting in .


6.8. CRACK-DEFECT INTERACTION 6.8.1. Crack-Dislocation Interaction Dislocations are often generated at the crack tip and their interaction with the crack can influence the fracture behavior of the material. Therefore, it would be interesting to study the interaction between a dislocation and an antiplane crack in piezoelectric materials. Pak (1 990b) first considered the force acting on a piezoelectric screw dislocation in the framework of linear piezoelectricity by considering a straight dislocation with the Berger vector normal to the isotropic Basel plane. The dislocation causes discontinuous displacement and electric potential across the slip and they are subjected to a line force and a line charge at the core. Generalized expression for piezoelectric interaction energy is expressed as a sum of the stored interaction electric enthalpy and potential energy of the external body and surface forces which helps to calculate the forces acting on a piezoelectric screw dislocation subjected to a line force and line charge at the core. The generalized expression for the force acting on a dislocation in piezoelectric crystal with a line load and a line charge attached to it is (6.56)

where Fk is the generalized Peach-Kohler force including piezoelectric effect and body force. The force acting on any piezoelectric singularity can be evaluated by the path-independent integral (6.21). Lee et al. (2000) considered the interaction between a semi-infinite crack and a screw dislocation with a line force, p, and a line charge, q, in a piezoelectric material with an impermeable boundary condition (Fig. 6.8.1). They found the force acting on the dislocation by a semi-infinite crack to be

(6.57)


(6.58)

Figure 6.8.1 Screw dislocation with a line force near a sime-infinite crack in piezoelectric material (Lee et al., 2000).

At the same time, the force acting on the semi-finite crack by a dislocation and a far-field electromechanical loading is found to be

(6.59)

29 Fracture Problems in Antiplane Piezoelectricity Table 6.1 Crack-tip shielding effect induced by a line-force, a line-charge, and a screw dislocation when b,  , p, q, are positive(+) ( : sheilding, :antisheilding). Position Intensity Factors/variables

− ππ q b   

The crack-tip shielding effect due to the presence of a screw dislocation with a line-force and a line-charge as a function of the dislocation position is summarized in Table 1. A piezoelectric screw dislocation interacting with a finite crack in an infinite piezoelectric medium under a remote uniform loading was considered by Chen et al. (2005) as shown in Fig 6.8.2. They studied how two different electrical boundary conditions along the crack surfaces affect the mechanical and electrical fields produced by a line screw dislocation with a line force and a line charge attached at the core. It is shown here that the electrical loading always impedes the crack propagation for electrically impermeable crack while it does not affect crack propagation for the electrically permeable crack. It has been also shown that the line charge does not contribute to the energy release rate for the electrically permeable crack. At the same time, the image force (force acting on the dislocation due to the presence of a crack under a far-field loading) is not affected by the remote electric field for electrically permeable crack. It is positive for electrically impermeable crack if the dislocation has a nonzero electric potential dislocation, φ, or a line charge, q. For an impermeable crack, the image force becomes negative for a non-zero mechanical dislocation, b, or a line force, p. For an electrically permeable crack, all image forces are negative. This indicates that for an electrically permeable crack, the dislocation is always attracted towards the defect. The remote applied electric field also influences the magnitude of the image force. It contributes a positive value when the applied electric field is positive while for a negative electric field, the image force is reduced. Unified electrical boundary condition for a crack interacting with a dislocation in piezoelectric media has been studied by Chen et al. (2005). They used the electric crack condition parameter proposed by Gao et al. (1997) for single and multiple cracks in homogeneous and functionally graded piezoelectric materials. The authors discussed variety of electrical boundary condition at the crack faces considering permeable and impermeable crack boundary conditions as two extreme cases. A screw dislocation interacting with both the interface and the collinear interfacial cracks between two bonded dissimilar piezoelectric media has been studied (Wu et al., 2003) and the fracture parameters have been derived. It was shown that the electroelastic field near the interfacial crack tip exhibits square root singularity without oscillations. A piezoelectric screw dislocation interacting with the interfacial collinear rigid lines (anti cracks) in a piezoelectric bi-material was studied by Xiao et al. (2007, 2008). The rigid lines considered are either conducting or dielectric lines and the dislocation core is subjected to a line-force and a line-charge. A square root singularity of the field variables near the tip of the interfacial rigid line was observed.


Figure 6. 8.2 A charged screw dislocation around a finite crack of length 2a in an infinite piezoelectric medium under remote loads (Chen et al., 2005).

The strain energy density intensity factor (SEDIF) at the tip of the rigid line can be expressed as (6.60) where C(1) is a material property matrix and J(x) is a measure of the generalized stress jump. The rigid line extension force acting on the tip can be evaluated as

(6.61) where Δty(r) is the generalized stress jump behind the tip and u(r) denotes the generalized displacement ahead of the tip. The effect on rigid line extension force and the generalized strain energy density due to a Burger vector and the line load position has been discussed. The electroelastic interaction between a piezoelectric dislocation and collinear rigid lines (anti cracks) shows that there isn’t any effect of glide dislocation on antiplane deformation, thus, no effect on the stress intensity factor (Chen et al., 2007a). The climb dislocations are not sensitive to the shielding effects. The rigid dielectric line always repels the mechanical dislocation in the radial direction, while it has little effect on the electrical dislocation. The rigid conducting line always repels the dislocation in radial direction and attracts it in tangential direction. The study of electroelastic interaction between a piezoelectric screw dislocation and an elliptical inhomogeneity containing a confocal blunt crack under a far-field longitudinal shear and an inplane electric field discusses the effect of dislocation position on the image force (Yu et al., 2010). The elliptical blunt crack strongly influences the interaction between the screw dislocation and the inhomogeneity. It strengthens the repulsion to the dislocation when the matrix is hard which will be in an unstable equilibrium position. The shape of the inhomogeneity and the blunt crack was shown to have a strong influence on the interaction between the screw dislocation and the inhomogeneity. The screw dislocation could retard the crack initiation when it is located in the region -1050 < θ < 1050. The retardation area in piezoelectric materials is different from that of elastic materials. The energy release rate, G, can be positive or negative under combined mechanical and electrical load, while strain energy density is always positive. The increase of inhomogeneity curvature and the decrease

31 Fracture Problems in Antiplane Piezoelectricity of relative shear modulus value can both lead to the decline of the energy release rate. The existence of a dislocation makes G decrease and consequently inhibits the crack growth. It also makes strain energy density decrease for a positive electrical load and increase for a negative electrical load.

6.8.2. Crack-Interface Interaction Micro crack initiation and its growth in materials can cause abrupt failure when it reaches a critical size. This is also true for piezoelectric materials subjected to a coupled electromechanical loading. When the material system is made up of interfaces, as is often the case for piezoelectric actuators, the chances of failure increases much more. Thus, analyzing crackinterface interaction in piezoelectric materials can give a better insight about its failure near an interface boundary. There are quite a number of studies that have been carried out in this area.

Figure 6.8.3 Two bonded dissimilar piezoelectric ceramics with a crack perpendicular to and terminating at the interface. (Li and Wang, 2007).

An antiplane shear crack normal to and terminating at the interface (Fig. 6.8.3) of two bonded piezoelectric ceramics was studied by Li and Wang (2007) to provide an insight into the effects of various material constants on field intensity factors. The field intensity factors in terms of a material mismatch parameter, α, crack length a, and material constants derived by a Fourier integral transform for permeable cracks can be written as

(6.62) When two piezoelectric ceramics are identical, i.e., α = 1/2, it becomes (6.63) Other field intensity factors are


(6.64)

(6.65)

(6.66) The singularity at the crack tip does not obey the inverse square root form, rather it is governed by the singularity of the power law α (0 < α < 1). This matches with the result of an interface crack terminating at the interface in purely elastic materials. The material constants affect the order of singularity. If both sides of the piezoelectric material are the same, the singularity at the crack tip reduces to the usual inverse square root. It is also shown that the material property on both sides of the interface changes, the order of singularity also changes. The material constants also strongly affect the field intensity factors. The electric field near the interface becomes singular unlike the cases for permeable cracks in homogeneous piezoelectric materials. Electroelastic analysis of a cracked piezoelectric ceramic strip sandwiched between two elastic dielectrics with the crack at the center of the piezoelectric material and perpendicular to the interface (Li, 2005) showed the effect of permeable and impermeable electrical boundary conditions at the interface. It is found that when an impermeable boundary condition is considered, the electric loading has significant effect on the crack growth, while for permeable case, this effect disappears. Figure 6.8.4 plots strain intensity factors against the applied electrical loading with various values of , where  denotes the electrical boundary condition on crack surfaces with zero value denoting the impermeable crack and one denoting the permeable crack. The electrical loading aids or inhibits crack growth depending on the direction and the magnitude of the loading when  < 1. When  = 1, i.e., a permeable crack, the electrical loading does not have any effect on the strain intensity factor. Mode III crack problems for two bonded functionally graded piezoelectric materials have been studied by Cheu et al. (2005). The authors have considered the crack to be normal to the interface of bonded two half planes made up of functionally graded piezoelectric materials. The properties of two materials, such as elastic modulus, piezoelectric constant and dielectric constant, are assumed to be in exponential forms and to vary along the crack direction. The singular integral equations for impermeable and permeable cracks are derived and solved by using the Gauss chebyshev integration technique. It shows that the stresses and electrical displacements around the crack tips have the conventional square root singularity. The stress intensity and electric displacement intensity factors are highly affected by the material inhomogeneity parameters. A sub-interface crack in an anisotropic piezoelectric bi-material was studied (Yang et al., 2008) by developing the singular integral equations with kernel functions expressed in

33 Fracture Problems in Antiplane Piezoelectricity complex form for a transversely isotropic piezoelectric bi-material. The field intensity factors for a coupled mechanical and electrical loading can be evaluated by evaluating the decoupled mechanical stress intensity factor and the electric displacement intensity factor when special properties of one of the bi-material are considered. The shielding effect of the imperfect interface on a mode III permeable crack (Li and Lee, 2009a) in a layered piezoelectric sensor (permeable crack parallel to imperfect interface) showed that the imperfect interface develops shielding effect. For an electrically permeable crack, a mechanical imperfection has a greater shielding effect than the dielectric imperfection.

Figure 6.8.4. Variation of strain intensity factor against applied electric loading for various values of  when 0= 8 MPa (Li, 2005).

6.9. CRACKS IN FUNCTIONALLY GRADIENT PIEZOELECTRIC MATERIAL Functionally graded materials are widely used in modern engineering applications. Piezoelectric materials can be engineered to have functionally graded electroelastic properties, thus, it is worthwhile to investigate the fracture properties of such materials. Most of the studies in the literature have considered the exponential gradation of the material properties. We will study the effect of material gradation on field intensity factors that govern the fracture behavior. A antiplane crack problem in a functionally graded piezoelectric material revealed that the form of the stress and electric displacement singularities at a crack tip are the same as in the homogeneous piezoelectric materials (Li and Weng, 2002a). However, the magnitudes of the field intensity factors are dependent upon the material gradient, and it was shown that the material gradation causes reduction in stress intensity factors. Fracture analysis of a functionally graded piezoelectric strip (Ma et al., 2005) with a Griffith crack (Fig.6. 9.1) showed the stress and electric displacement intensity factors as

(6.67)


Figure 6.9.1 Geometry of the crack problem in a functionally graded piezoelectric strip (Ma et al, 2005).

(6.68)

where an and bn are unknown coefficients that depend on the properties of functionally graded piezoelectric materials, and Γ(n) is the gamma function. The numerical study showed that the gradation of the material properties has a considerable effect on field intensity factors. As the strip thickness to crack length ratio increases, the stress intensity factors decrease. Two parallel symmetric permeable cracks in a functionally gradient piezoelectric/piezomagnetic material under an antiplane shear loading (Zhou and Wang, 2004; Zhou et al., 2005) showed the same behavior as a single crack in a functionally gradient piezoelectric material. In addition, the stress intensity factors increase with the increase in the distance between the parallel cracks. The electric displacement factors behave the same as the stress intensity factors. The problem of periodic cracks in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric material with an exponential variation of the elastic, piezoelectric and dielectric constants for a permeable and an impermeable crack was considered for analysis (Ueda, 2003; Wang, 2003; Ding and Li, 2008 a, b; Li et al., 2010; Dai et al., 2010), and the field intensity factors along with the energy release rate have been derived. In these studies, the effects of periodic crack spacing, material constants and geometrical parameters on fracture behavior have been discussed. The magnitude of field intensity factors can increase or decrease with the increasing material gradient, depending on the type of material property distribution. It has also been shown that the crack-tip stress and electric displacement intensity factors are heavily dependent upon the material inhomogeneity. The problem of a finite crack in a semi-infinite strip of a graded piezoelectric material under an electric loading was considered by Ueda (2006). In this work, he discussed the effect of various parameters such as crack size, crack location, material inhomogeneity on stress and electric displacement intensity factors.

35 Fracture Problems in Antiplane Piezoelectricity A mode III eccentric crack in a functionally graded piezoelectric material was considered for analysis by Ou et al. (2006) for the following boundary conditions, namely the strip is (i) mechanically clamped and electrically closed, (ii) mechanically opened (traction free) and electrically opened (iii) mechanically clamped and electrically closed, and (iv) mechanically and electrically opened. The authors found that the edge boundary condition and the material inhomogeneity have a significant control on the stress and electric displacement intensity factors. An investigation of the antiplane shear behavior of a permeable Griffith crack in functionally graded piezoelectric materials has been carried out by Zhou et al. (2007). They used the non-local theory and presented the effect of material inhomogeneity on the stress and electric displacement intensity factors. A study of a functionally graded piezoelectric strip bonded between two dissimilar piezoelectric half-planes with a mode III crack (Hu et al., 2005; Yong and Zhou, 2007; Torshizian and Kargarnovin, 2010) concluded that the electric displacement loads have no effect on the stress intensity factors for permeable cracks. This finding is in agreement with the homogeneous piezoelectric materials with permeable cracks. Electrically nonlinear antiplane shear crack in a functionally graded piezoelectric strip (Kwon, 2003) utilizes the parameter called electric crack condition parameter (ECCP) developed by the author for single crack problems in homogeneous piezoelectric materials. The ECCP can describe all the electrical crack boundary conditions in accordance with the aspect ratio of the ellipsoidal crack and the permittivity inside the crack. This includes both traditional permeable and impermeable crack boundary conditions. When the size of the saturated strip is very small, the field intensity factors and the energy release rate can be expressed as (6.69)

(6.70) where 1 and 2 are auxiliary functions that govern the singular behavior of the stress and electric fields, κ0 defines the strength of the electromechanical coupling and 0 is the normalized electrical load. The electric crack condition parameter (ECCP) can be determined for an elliptical flaw of the form (6.71)

where α is the aspect ratio of the ellipsoid. For different electrical crack conditions, the expressions of energy release rates are

for saturated unified crack

(6.72)

36 Y. Eugene Pak, Dhaneshwar Mishra and Seung-Hyun Yoo for saturated impermeable crack

(6.73)

for traditional impermeable crack (6.74)

for saturated permeable crack

(6.75)

for traditional permeable crack

(6.76)

where (6.77)

The energy release rates based on the unified crack boundary condition is always positive and falls between those obtained from the impermeable permeable cracks (Fig.6.9.2). It has also been found that the energy release rates are dependent on the electric crack conditioning parameter with two ellipsoidal crack parameters, the direction and magnitude of electrical loads, the material gradation, the crack length, the electro-mechanical coupling coefficient and the crack location. An angled crack in a bonded functionally gradient piezoelectric material under an antiplane shear was considered by Chue and Yeh (2010). They showed that the energy density factor at the crack tip increases when it is located within the softer material. They also discussed about using the strain energy density theory to study the failure that can occur at a finite distance away from the crack tip.

Figure 6.9.2 Normalized ERR ( ) versus the electrical crack condition parameter (Dr ) in a center cracked piezoelectric strip (Kwon, 2003).


6.10. DYNAMIC CRACK PROBLEMS 6.10.1. Electroelastic Wave Scattering Piezoelectric materials have a wide area of applications where they can be subjected to dynamics loads. Under a continuous dynamic or cyclic loading, microcracks can form and grow in piezoelectric devices that can eventually lead to a catastrophic failure. The study on dynamic crack problems in piezoelectric materials can be broadly divided into three major areas, namely, wave scattering, moving crack and transient crack analyses. We will review the numerical and theoretical studies of piezoelectric materials subjected to dynamic antiplane mechanical and inplane electrical loading resulting in electroelastic wave scattering and dynamic crack propagation. The basic formulation for an infinite piezoelectric ceramic for dynamic crack analysis can be written as (6.78) (6.79) where (ux, uy, uz) and (Ex, Ey, Ez) are components of the displacement and the electric field vectors. The electric field components can be written in terms of the electric potential (x, y, t), (6.80) The governing equations are obtained as (6.81) (6.82) where  is the mass density and 2 is the Laplacian operator in the variables x and y. Solving these equations for a single finite crack in an infinite piezoelectric material subjected to a farfield harmonic antiplane shear wave using the integral transform method, the dynamic energy release rate and dynamic stress intensity factors were determined (Narita and Shindo, 1998). Figure 6.10.1 presents the variation of the normalized dynamic stress intensity factor as a function of the normalized frequency. The dynamic stress intensity factor shows a similar behavior as the purely elastic case (Loeber and Sih, 1968), which drops rapidly beyond the first maximum and exhibits oscillations of approximately constant period as the frequency increases. The scattering patterns in piezoelectric media with a conducting crack were somewhat different than in elastic media (Li et al., 2005). The amplitudes of electric displacement and electric field phase function decrease with increase in the incident angles. When the incident angle increases, the incident wave is more focused on opening the crack. The scattering pattern in piezoelectric media does not generate shadow zone behind the half-plane slit which

38 Y. Eugene Pak, Dhaneshwar Mishra and Seung-Hyun Yoo is in contrast with the purely elastic media. The crack is somewhat transparent to incident waves. This is because in a piezoelectric medium the incident electro-acoustic waves interact with the crack to produce the electric head wave that can penetrate the crack surface. Also, in the piezoelectric materials, different headway exists in many different scenarios which is completely different from the purely elastic medium. The amplitudes for both phase functions are larger for the incident acoustic source than for the electric source (Wang et al., 2000). Thus, a large electromechanical coupling increases the electric displacement and electric field intensities. Studies on the diffraction and scattering of electro-acoustic waves by an interfacial crack between two dissimilar piezoelectric half spaces (Gu et al., 2006; To et al., 2005) showed that the interfacial crack is not completely opaque to the electro-acoustic waves, i.e., the electroacoustic wave can penetrate and transmit to the other side of the interfacial slit. These studies also confirm that the interaction between electric waves and acoustic waves will provide multiple electro-acoustic head waves. This result is in agreement with the findings of wave interaction for cracked piezoelectric medium. It was found that the ratio of the crack length to the layer width has significant influence on the field intensity factors when the scattering of antiplane shear waves by a finite crack in piezoelectric laminates was considered (Narita and Shindo, 1999). The scattering of antiplane shear waves by a circular piezoelectric inclusion embedded in a piezoelectric medium subjected to a steady-state electrical load was formulated using the wave function expansion method (Shindo et al., 2002a). They evaluated the dynamic stress intensity factors and electric displacement intensity factors. Analysis of antiplane shear wave scattering from two curved interface cracks between a piezoelectric fiber and an elastic matrix was conducted by Shindo et al. (2002b). They considered scattering of horizontally polarized shear waves from a single piezoelectric fiber partially bonded to an elastic matrix. The effects of frequency, crack angle, piezoelectric material constant on dynamic stress intensity factors have been investigated and presented. The scattering of electroelastic waves by an ellipsoidal inclusion in a piezoelectric medium based on the polarization method was studied by Ma and Wang (2005) and obtained various scattering cross sections. Study of dynamic interaction of antiplane shear waves with an arc shaped interfacial crack (Du and Wang, 2006) showed that the piezoelectric coefficient has a significant effect on crack opening displacement. The scattering of the harmonic antiplane shear waves by a crack in functionally graded piezoelectric materials (FGPM) with a Griffith crack was investigated with the assumption that the elastic stiffness, piezoelectric constant, dielectric permittivity and mass density of the FGPM vary continuously as an exponential function (Ma et al., 2005). It has been found that the material gradient property has considerable effect on the fracture behavior of FGPM. The dynamic stress intensity factor can be as much as 20% to 35% above the static value depending on the geometry and piezoelectric material properties. Therefore, the dynamic effects should always be taken into account to insure the integrity of materials subjected to dynamics loads (Narita and Shindo, 1997). Study of antiplane shear waves scattered by two collinear cracks in a piezoelectric material revealed the dependence of the dynamic stress intensity factors on shear stress wave velocity, crack length, electric loading and frequency of the incident wave (Zhou et al., 2002). As in the static case, the distance between collinear cracks has a significant effect on the


Figure 6.10.1. Dynamic stress intensity factor verses frequency (Narita and Shindo, 1998).

dynamic stress intensity factors. It decreases with increase in the distance between the cracks. The stress intensity factors increase with the increased electrical loading because of increased electromechanical coupling. The dynamic response of the electric field is independent of the mechanical load and thus dynamic elastic field promotes the propagation of the crack at different stages of the loading process. The investigation was carried out for the scattering of harmonic elastic antiplane shear waves by a Griffith crack in a piezoelectric material by using a non-local theory with an impermeable crack condition (Zhou and Wang, 2001). The non-local theory gives a finite stress at the crack tip. When a harmonic shear wave scattering problem in a piezoelectric/piezomagnetic functionally graded material with a crack was considered for analysis (Jun, 2007), it was found that the stress intensity factor increased with increasing circular frequency of the wave. The antiplane vibration of cracked piezoelectric materials was investigated by Chen and Yu (1998) using the integral transform methodology and presented the variations in dynamic field intensity factors with various geometric and loading parameters. A numerical study of the scattering problem in two-dimensional piezoelectric solids has been carried out by employing the hypersingular boundary element method (Saez et al., 2006), and the dynamic stress intensity factor was evaluated and compared with the theoretical results. A crack propagation study of a conductive mode III interfacial crack along a conductive interface between two piezoelectric materials was done by To et al. (2006). They showed that the crack propagation excites the electro-acoustic surface waves which are different from those in purely elastic media. They also presented that when the electromechanical coupling increases, the normalized dynamic stress intensity factor and energy release rate decrease.


6.10.2. Moving Crack Analysis Piezoelectric materials are extensively used in intelligent systems as sensors and actuators to monitor and control the dynamic behavior of structures. These piezoelectric devices are subjected to mechanical and electrical disturbances that can cause catastrophic failure wherein a crack can propagate at high speeds. Thus, studying dynamic crack analysis in a piezoelectric material would be helpful in understanding the behavior of a propagating crack in these materials. In this section, we will consider first the steady state moving crack problems followed by the transient response of dynamic crack analysis in the next section. In order to solve for the dynamic field intensity factors when a crack is moving steadily, the governing equations (6.81, 6.82) are attached to the free moving coordinate system x = X - vt,

y = Y,

and

z = Z.

(6.83)

The Bleustein-Gulyaev wave function, which is used to express the dynamic field intensity factors and energy release rates, derived from the governing equation attached to the free moving coordinate system can be written as (6.84) where α( ) and ( ) are coefficient functions expressed in terms of crack speed (v), speed of bulk shear wave (c), non-dimensional parameter (s), auxiliary petrubation parameter (ε), and kv is the electromechanical coupling coefficient. The vacuum abutted surface wave propagates at the speed of (Bleustein, 1968; Ikeda, 1990) (6.85)

The crack speed affects both the number and location of roots of the Bleustein-Gulyaev wave function. Three crack propagation speeds considered are sub-critical speed (v < cbg), transonic speed (cbg < v < c) and supersonic speed (v > c). The crack face traction in an elastic medium cannot drive the crack at supersonic speed unless there is a load acting at the crack tip.

Figure 6.10.2. Running Crack abutted to permeable vacuum at the crack interface (Li and Mataga, 1996 b).

1

41 Fracture Problems in Antiplane Piezoelectricity When the crack speed reaches the surface wave speed, both the stress intensity and electric displacement intensity factors change drastically. The expression for field intensity factors for mixed mode loading can be written as

(6.86) (6.87)

where Q0 is a load vector. These field intensity factors can be bifurcated in two terms, the self induced field intensity factors and cross over field intensity factors. The energy release rate expression per unit of the crack tip in the x-direction for a dynamic crack can be expressed as

(6.88)

where P and Q are load pairs and hTT, hDT and hDD are the coefficients of load pairs for traction, traction-displacement and displacement loads respectively. When the normalized energy release rate is plotted against the normalized crack speed, there is a residual energy left when the crack speed approaches the surface wave speed. In the vacuum crack case considered here, the surface wave speed does not become barrier to the crack propagation which is different in the case of elastodynamics or the case when the crack is in an electrode (Li and Mataga, 1996b). The case of a fixed grip applicable to elastic materials is violated in the case of piezoelectricity with a vacuum crack. There is a continuous energy exchange between a piezoelectric body and its surrounding vacuum. For the electrode crack problem, the fixed grip is forced mathematically and thus the energy exchange between a piezoelectric body and the environment is stopped (Li and Mataga, 1 996a). The energy release rate will be negative for the impermeable case which is in agreement with the result of static piezoelectricity. The stress intensity factor expression for an impermeable moving crack was shown to be (Li and Lee, 2004) (6.89)

where 1 and D are bounded continuous functions associated with the stress and electric displacement intensity factors. The strain energy release rate is (6.90)

42 Y. Eugene Pak, Dhaneshwar Mishra and Seung-Hyun Yoo where

represent the inverse laplace transform of

given by (6.91)

(6.92) (6.93)

The maximum of dynamic stress intensity factors and energy release rate are dependent on the piezoelectric material constant and the angle of incident wave. The dynamic interaction of two cracks decreases the energy release rate more rapidly in the small region due to the wave generated at the crack tip that weakens the energy release rate (Li and Lee, 2004). A study on the asymptotic analysis of a semi-infinite mode III crack growth showed the effect of crack propagation and material property on the dynamic stress intensity factor and the energy release rate (Zhou and Nishioka, 2005). For interacting cracks, dynamic stress intensity factors are not significantly affected when an incident wave frequency is low (Meguid and Wang, 1998). When the wave frequency increases, the phenomenon of dynamic overshoot is observed and this can be observed for a single crack as well as two parallel cracks. The electromechanical coupling has a significant effect on the dynamic stress intensity factors when there are dynamic interactions between the cracks (Wang and Meguid, 2000). A study on two coplanar cracks in a piezoelectric ceramic material under an antiplane mechanical and inplane electrical time dependent load showed that the dynamic electric field will impede or enhance crack propagation at different stages of the load (Chen and Worsowick, 2000). The electromechanical response is greatly affected by the ratio of the crack length to the ligament length between the cracks. It was shown that the stress and electric displacement intensity factors can be expressed together in terms of the energy density factor (EDF).

Figure 6.10.3 Normalized energy release rate verses normalized crack speed (Li and Mataga, 1996b).

43 Fracture Problems in Antiplane Piezoelectricity For dynamic interfacial crack propagation in an elastic-piezoelectric bi-material under uniformly distributed antiplane loadings on the crack face, the dynamic stress intensity factor decreases with the increasing crack propagation velocity (Chen et al., 2008). An investigation of the energy release rate and the bifurcation angles of an antiplane moving crack was carried out by Hou et al. (2001). The study showed singularity at the crack tip and that the coupled elastic and electric fields inside the piezoelectric media depend upon the speed of the crack propagation. The stress intensity factor was shown to be independent of the speed of the crack propagation, which is identical to the conclusion for a purely elastic material. However, when there is no electrical load, the energy release rate is associated with the crack speed, and with an increase in the crack speed the dynamic energy release rate increased. It was also shown that the crack can be propagated into either a curve or a bifurcation if the crack speed is higher than the critical speed. The study on a Griffith moving crack, moving along the interface of two dissimilar piezoelectric materials suggested that the field intensity factors are dependent upon the crack speed and the material coefficients (Chen et al., 1998). When both upper and lower materials connecting the interface are the same, these factors turn out to be the same as the homogeneous piezoelectric materials. The analysis of a moving mode III permeable crack in a functionally graded piezoelectric material was done by Jin and Zhong (2002). They have discussed about the effect of material gradation on dynamic stress intensity factors along with the crack velocity. The Yoffe-type moving crack, in which crack moves with a constant velocity without a change in length, in a functionally graded piezoelectric material for a mode III crack was studied by Li and Weng (2002), and the effect of crack speed on dynamic field intensity factors was presented. The study concluded that with the increased crack speed, both the stress and electric displacement intensity factors decreased, while the electric field intensity factor increased. The same Yoffe-type moving interface crack between two dissimilar functionally graded piezoelectric layers under an electromechanical loading has been studied (Shin and Lee, 2010) by considering the upper and lower layer materials to vary differently in their properties. The dynamic energy release rate was evaluated and the effect of material gradation, crack velocity and the thickness of layers bonded on it have been discussed. Dynamically loaded antiplane cracks in a finite functionally graded piezoelectric material have been studied (Dineva et al., 2010) by developing a numerical technique based on the boundary integral equation method (BIEM). This technique enables the evaluation of the dynamic field intensity factors that can be compared with the experimental and theoretical results. The transient analysis of a semi-infinite propagating crack under a dynamic antiplane concentrated load was considered by Chen et al. (2007) and Ing and Wang (2004). They developed a solution for the transient response of a propagating crack in the Laplace transform domain. The time domain incident wave for dynamic problems can be represented by an exponential functional form in the Laplace transform domain. The transient field can then be constructed by a superposition method. The mixed boundary conditions in the Laplace transform domain for exponentially distributed loading at the crack face (Chen et al., 2007) can be written as


Figure 6.10.4 Normalized dynamic stress intensity factors verses normalized time for different value of crack speed in PZT4 for θ = 1800 (Chen et al., 2007).

(6.94)

where s is the Laplace transform parameter and η is a constant. The coordinate is fixed with respect to the moving crack tip. The overbar symbol is denoted for the transformation on time t. The dynamic crack growth analyses were evaluated in terms of the dynamic stress and electric displacement intensity factors along with the dynamic energy release rate. This can be obtained by solving the governing equations (6.81, 6.82) in the Laplace transform domain utilizing the shear stress and displacement expressions shown in equation (6.94). Values of the dynamic stress intensity and electric displacement intensity factors as well as the energy release rate of a stationary crack approaches the corresponding static value after the surface wave passes the crack tip. Both the dynamic stress intensity factor and the dynamic energy release rate go to zero when the crack propagation speed approaches the Bleustein-Gulyaev piezoelectric surface wave speed (6.85), however, the dynamic electric displacement intensity factor does not (Ing and Wang, 2004; Chen et al., 2007). The dynamic field intensity factors along with the energy release rates can be represented by the product of a universal function, which depends upon the crack speed, the piezoelectric material property, and the corresponding solution for a stationary crack. The effect of a loading angle on dynamic stress and electric displacement intensities factors were discussed by Chen et al. (2007). As the loading angle increased, the crack propagating speed (v) decreased to attain the corresponding static value of the stress and electric displacement intensity factors. The dynamic field intensity factors increased with increase in the loading angle θ. These factors increased when a propagating crack is close to the loading and decreased smoothly after the crack propagates away from the loading. A transient response of the stationary crack increased very rapidly towards the negative value after the electromagnetic waves arrives at the crack tip and then jumps to a positive value and eventually approaches the static value when the piezoelectric shear wave passes through the

45 Fracture Problems in Antiplane Piezoelectricity tip. The transient response of the dynamic intensity factors approached the corresponding static value after the Bleustein-Gulyaev wave has passed the crack tip (Fig. 6.10.4). It was shown that the contribution of electromagnetic waves on dynamic intensity factors is stronger than the electromechanical coupling coefficient.

6.10.3. Transient Analysis Piezoelectric materials have a very wide range of applications in various engineering systems and structures. These systems and structures undergo various types of loading such as static, dynamic, cyclic, impact and so on. The impact loads are applied on the system or structure for a limited period of time with a very high magnitude. These loads can be mechanical impacts, electrical impacts or thermal shocks. These loads can cause transient responses which adversely affect the system causing an abrupt failure. Therefore, studying fracture of piezoelectric materials under various kinds of impact loads that give rise to transient responses can help to understand the performance of these type of materials. Many researchers have focussed their attention on this problem in the context of antiplane fracture of piezoelectric materials. Transient analysis of a piezoelectric strip with a permeable crack under an antiplane impact load was studied by Li and Fan (2002) for the following two cases. First was the strip boundaries free of stresses and the second was the strip boundaries with clamped rigid electrodes. The governing equations (6.81, 6.82) with the following boundary conditions were applied and the dynamic stress intensity factor and dynamic energy release rate were evaluated,

(6.95) where (x) and f(t) are prescribed functions and the superscript c represents the electric quantities in the void inside the crack. They discussed the effect of crack geometry, elapsed time and electromechanical coupling on dynamic stress intensity factor. Transient nature of the stress intensity factor was similar to the pure elastic case where it rises quite rapidly in a short time reaching the peak value, then drops slowly and finally approaching the corresponding static value. The effect of crack geometry, boundary condition, and electromechanical coupling coefficient on dynamic stress intensity factor was also discussed. When the transient response of a cracked piezoelectric strip under an arbitrary electromechanical impact was investigated numerically (Chen and Yu, 1998), it was found that the crack-tip stress and electric field maintain square root singularity. The dynamic stress intensity factor was found to be strongly dependent upon the electrical load and the geometry of the strip, while response of the electric field was solely dependent upon the electrical load. It was also shown that the electric field can promote or retard crack growth depending on the elapsed time. Transient response of a rectangular piezoelectric medium with a center crack has been investigated by Kwon and Lee (2001). Their study showed that for a permeable crack, the

46 Y. Eugene Pak, Dhaneshwar Mishra and Seung-Hyun Yoo dynamic energy release rate and the dynamic stress intensity factors are independent of the electrical loading under an impact stress loading, while they are dependent on the electrical loading under an impact strain loading. These intensity factors also depend upon a resultant stress distribution which is generated by the mechanical stress distribution and the electromechanical interaction. Under a far-field step loading with a time duration 0  t  t0, Figure 6.10.5 shows a larger inertial effects for a longer time duration, where Kst and Gst are, respectively, the static stress intensity factor and the energy release rate. The dynamic stress intensity factor and dynamic energy release rate go to zero when time goes to infinity. For an impermeable crack, the results were in agreement with those presented by Chen and Yu (1998). Similar results were presented by Chen and Karihaloo (1999) in their work on dynamic response of a cracked piezoelectric ceramic with an arbitrary electromechanical impact loading. An eccentric crack in a piezoelectric strip under an antiplane shear impact loading was investigated by Shin et al. (2001) by considering the eccentric crack off the center line. They found that the normalized dynamic stress intensity factor and the energy release rate increased when the crack length and the eccentricity of crack location increased. The larger the crack length and eccentricity of crack location, the faster the time in arriving at the peak values. When a purely electrical load is considered for a mode III impermeable crack, the negative energy release rate implies that a purely electrical impact loading always retards the crack propagation regardless of the load direction (Chen, 2006). This phenomenon is inconsistent with some of the experimental result of the electrical load-induced fracture which prompted Park and Sun (1995) to use the mechanical strain energy release rate as a fracture criterion. The effect of a crack position on transient response of a piezoelectric ceramic strip has been discussed by Li and Tang (2003) by investigating an eccentric crack in a piezoelectric strip under an electromechanical impact. The transient response of a finite piezoelectric strip with coplanar insulating cracks under an electromechanical impact has been investigated by Meguid and Chen (2001) and Su et al. (2003). They discussed the effects of crack spacing, electrical impact loading and electromechanical coupling on the dynamic stress intensity factor. A transient response of a piezoelectric material with a semi-infinite impermeable crack under an impact load has been investigated by Li (2001). The dynamic stress intensity factor and the dynamic mechanical strain energy release rate are presented which can be considered as fundamental solutions for various special cases. A similar analysis for a permeable surface crack has been carried out by Li and Lee (2006) who discussed the effects of the material properties and geometric parameters. A transient response of a sandwiched piezoelectric strip with the crack normal to the interface under an impact load was analyzed by Ueda (2003), and the variations in dynamic energy density factor verses time has been discussed for various geometric parameters. Multiple interfacial collinear cracks in bonded piezoelectric and elastic half spaces were considered and transient response was studied by Zhao et al. (2002). They investigated the effect of the geometry of interacting collinear cracks, the applied electric field and the electrical boundary conditions on dynamic energy release rate and stress intensity factors. For the periodic cracks in a functionally graded piezoelectric material under a shear impact load, the dynamic stress intensity factor and the energy density intensity factor (EDIF) are dependent upon material gradation parameter, crack length and crack span (Chen and Liu,

47 Fracture Problems in Antiplane Piezoelectricity 2005). The dynamic stress intensity factor increased with increasing crack length, but decreased with decreased crack span. The material gradation has different effect on the dynamic stress intensity factor for negative and positive gradation parameters. When the gradation parameter is negative, it first increases then decreases after attaining the maximum while for positive material gradation, it increases with increasing crack length. The increase or decrease in dynamic stress intensity factor cannot be attributed to the crack growth because of different material property at different locations. For the energy density intensity factor (EDIF) criterion of failure, an electrical impact load always enhances the crack growth. A decrease of EDIF with an increase in material gradation implies that a crack growth is impeded with the increased EDIF. The electromechanical field ahead of the crack tip in a multilayered piezoelectric material under a dynamic antiplane shear load was studied by Wang et al. (2000). It was shown that the stress field and the electric displacement fields are decoupled for both homogeneous and non-homogeneous piezoelectric materials. Under an electrical loading, the transient stress and electric displacement are coupled while steady state stress and electric displacement fields are decoupled for a homogeneous piezoelectric material. However, it is coupled for nonhomogeneous piezoelectric materials. It was also shown that the inertia effect has no role on the electric displacement intensity factor. Numerical simulation of a piezoelectric bi-material with a crack under an impact loading was considered by Hu et al. (2007) and presented the finite element formulations and the dynamic field intensity factors for a homogeneous piezoelectric material and a piezoelectric material with bi-material interfaces. Transient dynamic analysis of a cracked piezoelectric solid by a time domain Boundary Element method (Kogl, 2000; Gross et al., 2007; Sanchez et al., 2008) showed the effect of electrical impact loading on the dynamic intensity factors. They concluded that the electrical loading has a very significant effect on the dynamic stress intensity factors which decreases with increasing intensity of the impact electrical loading. They also discussed the presence of electrically neutral points or time ranges where an electrical impact loading has no influence on the dynamic stress intensity factors.

Figure 6.10.5. The normalized dynamic energy release rate and dynamic stress intensity factor for a PZT-5H ceramic under time duration step loading (Kwon and Lee, 2001).


6.11. CONCLUDING REMARKS Fracture problems in piezoelectric materials are reviewed in the context of antiplane mechanical and inplane electrical loading in the presence of cracks and defects. Much argued electrical boundary conditions in the theoretical study of piezoelectric fracture can be broadly classified in three categories. The first category is the impermeable boundary condition. This was first proposed by Deeg (1980) in his Ph.D. thesis which was further explained by Pak (1990a). This boundary condition along with the use of the total energy release rate showed the crack arresting effect in the presence of an electrical load which contributes to the negative energy release rate. This boundary condition also was shown to induce artificial singularity in the crack-tip electrical fields as the limiting process of collapsing an electrically correct (permeable) ellipse to a sharp crack showed (Pak, 2010). Another criterion initially proposed by Parton (1976) is the permeable boundary condition at the crack faces. The third category considered by Zhang and Tong (1995) explained that the permeable and impermeable boundary conditions at the crack face are two extreme cases, and the electric field inside the cavity is uniform and varies with the shape of the ellipse. Landis (2004) also suggested the energetically consistent boundary conditions for electromechanical fracture. In terms of the fracture criteria, Park and Sun (1995), in order to better predict some of the experimental results, suggested the mechanical strain energy release rate (MSERR) since the fracture is a mechanical process. There is another thought on failure criterion given by McMeecking who suggested to use the crack-tip energy release rate in place of the global energy release rate as a failure criterion. The energy release rate is dependent upon the electric charge and thus the parameter introduced later by Kwon (1999) called electric crack conditioning parameter (ECCP) should be linked to the energy release rate. In additions to the electrical boundary condition at the crack face, the orientation and spacing between cracks, the material gradation parameter in functionally graded piezoelectric materials, the material mismatch constant for bi-material interfaces were shown to play an important role in the fracture failure of piezoelectric materials. These parameters can enhance or impede crack growth depending upon the factors mentioned above. The crack-defect interactions such as dislocation-crack interaction or interface-crack interaction or cracks emanating from a hole also have far reaching implications on failure of these materials. These interactions also can enhance or shield crack growth. For dynamic crack case, the crack speed plays an important role along with the elastic waves generated at the crack tip. A simple crack, multi-crack, crack-defect interactions, elliptical defects and dynamic wave scattering problems herein reviewed in the context of linear antiplane piezoelectricity reveal most of the relevant physics and mechanics without much involved mathematics. The implications from these problems can find their counterparts in more realistic inplane loading cases.

6.12. ACKNOWLEDGMENT This work was supported by a grant (2011-P2-21) from the Seoul National University’s Advanced Institutes of Convergence Technology. Authors DM and SHY acknowledge the support provided by the Government of Korea under its BK21 program.


REFERENCE [1]

[2]

[3]

[4] [5]

[6] [7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

Athanasius, L., Ang, W. T., Sridhar, I., 2010. Electro-elastostatic analysis of multiple cracks in an infinitely long piezoelectric strip: A hyper singular integral approach. European Journal of Mechanics A/Solids 29, 4 10-419. Berlincourt, D. A., Curran, D. R., and Jaffe, H., 1964. Piezoelectric and Piezomagnetic Materials and Their Function in Transducers. Physical Acoustics, Mason, W. P., ed. 1A. Academic Press, New York, p. 177. Beom, H. G., Atluri, S. N., 1996. Near-tip fields and intensity factors for interfacial cracks in dissimilar anisotropic piezoelectric media. International Journal of Fracture 75-2, 163-183. Bleustein, J. L., 1968. A new surface wave in piezoelectric materials. Applied Physics Letters 13, 412-413. Chen, B. G., Xiao, Z. M., Liew, K. M., 2004. A screw dislocation interacting with a finite crack in a piezoelectric medium. International Journal of Engineering Science 42, 1325-1345. Chen, B. J., Shu, D., W., Xiao, Z., 2007a. Dislocation interacting with collinear rigid lines in piezoelectric media. Journal of Mechanics of Materials and Structures 2, 23- 42. Chen, B. J., Liew, K. M., Xiao, Z. M., 2005. Unified electrical boundary conditions for a crack interacting with a dislocation in a piezoelectric media. International Journal of Solids and Structures 42, 5118-5128. Chen, X. H., Ing, Y. S., Ma, C. C., 2007b. Transient analysis of dynamic crack propagation in piezoelectric materials. Journal of Chinese Institute of Engineers 3 0-3, 491502. Chen, J., Liu, Z., 2005. On the dynamic behavior of a functionally graded piezoelectric strip with periodic cracks vertical to the boundary. International Journal of Solid and Structures 42, 3133-3146. Chen, Y. H., Hasebe, N., 2005. Current understanding on fracture behaviors of ferroelectric/piezoelectric materials. Journal of Intelligent Material Systems and Structures 16, 673-687. Chen, Z. T., Karihaloo, B. L., 1999. Dynamic response of cracked piezoelectric ceramic under arbitrary electro mechanical impact. International Journal of Solids and Structures 36, 5125-5133. Chen, X. H., Ma, C. C., Ing, Y. S., Tsai, C. H., 2008. Dynamic interfacial crack propagation in elastic-piezoelectric bi-materials subjected to uniformly distributed loading. International Journal of Solids and Structures 45, 959-997. Chen, Z. T., 2006. Dynamic fracture mechanics study of an electrically impermeable mode III crack in a transversely isotropic piezoelectric material under pure electric load. International Journal of Fracture 141, 395-402. Chen, Z. T., Karihaloo, B. L., Yu, S. W., 1999. A Griffith crack moving along the interface of two dissimilar piezoelectric materials. International Journal of Fracture 91, 197-203. Chen, Z. T., Worsowick, M. J., 2000. Antiplane mechanical and inplane electric timedependent load applied to two coplanar cracks in piezoelectric caramic materials. Theoretical and Applied Fracture Mechanics 33, 173-184.

50 Y. Eugene Pak, Dhaneshwar Mishra and Seung-Hyun Yoo [16] Chen, Z. T., Yu, S. W., 1998. Anti plane vibration of cracked piezoelectric materials. Mechanics Research Communications 25-3, 321-327. [17] Chen, Z. T., Yu, S. W., 1998. Transient response of a cracked piezoelectric strip under arbitrary electromechanical impact. Acta Mechanica Sinica (English Series) 14-3, 248256. [18] Chue, C. H., Yeh, C. N., 2010. Angle crack in two bonded functionally graded piezoelectric material under anti-plane shear. Theoretical and applied Fracture Mechanics 53, 233-250. [19] Chue, C. H., Ou, Y. L., 2005. Mode III crack problems for two bonded functionally graded piezoelectric materials. International Journal of Solid and Structures 42, 332 13337. [20] Chung, M. Y., Ting, T. C. T., 1996. Piezoelectric solid with an elliptical inclusion or hole. Internatronal Journal of Solids and Structures 33, 3343-3361. [21] Cherepanov, G. P., 1977. Invariant G-integrals and their application in mechanics. Prikl Mat Mech 41,339-4 12 [in Russian]. [22] Dai, L., Guo, W., Wang, X., 2006. Stress concentration at an elliptic hole in transversely isotropic piezoelectric solids. International Journal of Solids and Structures 43, 18181831. [23] Dai, H. L., Hong, L., Fu, Y. M., Xiao, X., 2010. Analytical solution for electromagnetothermoelastic behavior of a functionally graded piezoelectric hollow cylinder. Applied Mathematical Modeling 34, 343-357. [24] Deeg, W. F., 1980. The Analysis of Dislocation, Crack and Inclusion Problems in Piezoelectric Solids. Ph.D. Thesis, Stanford University Stanford, Ca. [25] Delameter, W. R., Herrmann, G., Barnett, D. M., 1975. Weakening of Elastic Solids by arrays or cracks. Journal of Applied Mechanics 42, 74-80. [26] Denda, M., Mansukh, M., 2005. Upper and lower bounds analysis of electric induction intensity factors for multiple piezoelectric cracks by BEM. Engineering Analysis with Boundary Elements 29, 533-550. [27] Ding, S. H., Li, X., 2008a. Periodic cracks in functionally graded piezoelectric layer bonded to piezoelectric half plane. Theoretical and Applied Fracture Mechanics 49, 313-320. [28] Ding, S. H., Li, X, 2008b. Anti-plane shear crack in bonded functionally graded piezoelectric material under electromechanical loading. Computational Material Science 43, 337-344. [29] Dineva, P., Gross, D., Muller, R., Rangelov, T., 2010. BIEM analysis of dynamically loaded antiplane cracks in graded piezoelectric finite solids. International Journal of Solid and Structures 47, 3150-3165. [30] Du, J., Wang, J., 2006. Dynamic interaction of antiplane shear waves with arc-shaped interfacial crack in piezoelectric media. Ultrasonics 44, e963-e968. [31] Dunn, M., Taya, M, 1993. Micro mechanics predictions of the effective electroelastic modili of piezoelectric composites. International Journal of Solids and Structures 30, 161- 175. [32] Dunn, M., 1994. The effects of crack face boundary conditions on the fracture mechanics of piezoelectric solids. Engineering Fracture Mechanics 48, 25-39.

51 Fracture Problems in Antiplane Piezoelectricity [33] Eshelby, J. D., 1970. Energy relations and the energy-momentum tensor in continuum mechanics. In: Inelastic Behavior of Solids, Eds. Kanninen, M. F. et al. McGraw-Hill, New York, 77-113. [34] Fang, D., Qi, H., Yao, Z., 1998. Numerical analysis of crack propagation in piezoelectric ceramics. Fatigue and Fracture of Engineering Material and Structures 21, 137 1-1380. [35] Fang, Q. H., Feng, H., Liu, Y. W., Jiang, C. P., 2011. Energy density and release rate study of an elliptic-arc interface crack in piezoelectric solid under anti-plane shear. Theoretical and Applied Fracture Mechanics (in Press). [36] Freund, L. B., 1978. Stress-intensity factor calculations based on a conservational integral. International Journal of Solids and Structures 14, 24 1-249. [37] Fulton, C. c., Gao, H., 1998. Nonlinear fracture mechanics of piezoelectric ceramics. Mathematics and Control in Smart structures, Preceedings of SPIE 3323, 119-127. [38] Fulton, C. c., Gao, H., 1999. Electromechanical fracture in piezoelectric ceramics. International Journal of fracture 98, 15-22. [39] Gao, H., Zhang, T.Y., Tong, P., 1997. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. Journal of the Mechanics and Physics of Solids 45, 491-5 10. [40] Gross, D., Dineva, P., Rangelov, T., 2007. BIEM solution of piezoelectric crack finite solid under time harmonic loading. Engineering Analysis with Boundary elements 31, 152- 162. [41] Govorukha, V. B., Loboda, V. V., Kamalah, M., 2006, On the influence of the electric permeability on an interface crack in a piezoelectric bimaterial compound, International Journal of Solid and Structures 43, 1979-1990. [42] Guo, J H., Lu, Z. X., Han, H. T., Yang, Z., 2009. Exact solutions for antiplane problem of two asymmetrical edge cracks emanating from an elliptical hole in a piezoelectric material. International Journal of Solids and Structures 46, 3799-3 809. [43] Guo, J. H., Lu, Z. X., Feng, X., 2010. The fracture behaviors of multiple cracks emanating from circular hole in piezoelectric materials. Acta Mechanica 215, 119-134. [44] Gu, B., Yu, S. W., Feng, X. Q., 2006. Elastic Sv-waves scattering by an interface crack between a piezoelectric layer and elastic substrate. Symposium on Mechanics and Reliability of Actuating Materials, 112-120. [45] Hall, D. A., 2001. Nonlinearity in piezoelectric ceramics. Journal of Material Science 36, 4575-4601. [46] Han, J. C., Wang, B. L., 2005. Electromechanical model of periodic cracks in piezoelectric materials. Mechanics of Materials 37, 1180-1197. [47] Hao, T.-H., Shen, Z.-Y., 1994. A new boundary condition of electric fracture mechanics and its application. Engieering Fracture Mechanics 47, 793-802. [48] Hu, K. Q., Zheng, Y., Yang, J., 2007. A mode III crack in a piezoelectric semiconductor of crystals with 6 mm symmetry. International Journal of Solids and Structures 44, 39283938. [49] Hu, K. Q., Zhong, Z., Jin, B., 2005. Antiplane shear crack in a functionally gradient piezoelectric layer bonded to dissimilar half spaces. International Journal of Mechanical Sciences 47, 8293.

52 Y. Eugene Pak, Dhaneshwar Mishra and Seung-Hyun Yoo [50] Hu, S., Shen, S., Nishioka, T., 2007. Numerical analysis for a crack in piezoelectric material under impact. International Journal of Solids and Structures 44, 84578492. [51] Hwu, C., Ikeda, T., 2008. Electromechanical fracture analysis for corners and cracks in piezoelectric materials. International Journal of Solids and Structures 45, 5744-5764. [52] Hou, M. S., Qian, X. Q., Bian, W. F., 2001. Energy release rate and bifurcation angles of piezoelectric materials with antiplane moving crack. International Journal of Fracture 107, 297-306. [53] Huang, Z., Kuang, Z. B., 2003. A mixed electric boundary value problem for a two dimensional piezoelectric crack. International Journal of Solids and Structures 40, 1533- 1453. [54] Hu, K. Q., Zhong, Z., Jin, B., 2005. Anti-plane shear crack in a functionally graded piezoelectric layer bonded to dissimilar half spaces. International Journal of Mechanical Sciences 47, 82-93. [55] Ikeda, T., 1990. Fundamentals of piezoelectricity. Oxford University Press, Oxford. [56] Ing, Y. S., Wang, M. J., 2004. Transient analysis of mode III crack propagating in piezoelectric medium. International Journal of Solid and Structures 41, 6197-6214. [57] Jin, B., Zhong, Z., 2002. A moving mode-III crack in functionally graded piezoelectric material: permeable problem. Mechanics Research Communications 29, 217224. [58] Jun, L., 2007. Scattering of harmonic antiplane shear waves by a crack in functionally graded piezoelectric/piezomagnetic materials. Acta Mechanica Solida Sinica 20, 75- 85. [59] Kogl, M., Gaul, L., 2000. A boundary element method for transient piezoelectric analysis. Engineering Analysis with Boundary Elements 24, 59 1-598. [60] Kumar S., Singh, R. N., 1996a. Crack propagation in piezoelectric materials under combined under combined mechanical and electrical loadings. Actamateriala 44, 173200. [61] Kumar, S., Singh, R. N., 1996b. Comments on fracture criteria for piezoelectric ceramics. Journal of American Society 74, 1133-1135. [62] Kumar S., Singh, R. N., 1997a. Energy release rate and crack propagation in piezoelectric materials part II: combined electrical and mechanical loads. Actamateriala 45, 859-868. [63] Kumar, S., Singh, R. N., 1997b. Influence of applied electric field and mechanical boundary condition on the stress distribution at the crack tip in piezoelectric materials. Material Science and Engineering A 231, 1-9. [64] Kuna, M., 1999. Finite Element Analyses of crack problems in piezoelectric structures. Computational Material Science 13, 67-80. [65] Kuna, M., 2010. Fracture Mechanics of piezoelectric materials - Where are we right now? Engineering Fracture Mechanics 77, 309-3 26. [66] Kwon, S. M., 2003. Electrical non-linear anti-plane shear crack in a functionally graded piezoelectric strip. International Journal of Solid and Structures 40, 5649-5667. [67] Kwon, S.M., Lee, K. Y., 2000. Analysis of stress and electric fields in a rectangular piezoelectric body with a center crack under antiplane shear loading. International Journal of Solids and Structures 37, 4859-4869. [68] Kwon, S.M., Shin, J. H., 2006. An electrically saturated anti-plane shear crack in a piezoelectric layer constrained between two elastic layers. International Journal of Mechanical Sciences 48, 707-7 16.

53 Fracture Problems in Antiplane Piezoelectricity [69] Kwon, J. H., Lee, K. Y., 2001. Crack problem at interface of piezoelectric strip bonded to elastic layer under antiplane shear. KSME International Journal 15-1, 61-65. [70] Kown, S. M., Lee, K. Y., 2001. Transient response of a rectangular piezoelectric medium with a center crack. European Journal of Mechanics A/Solids 20, 457-468. [71] Landis, C. M., 2004. Energetically consistent boundary conditions for electromechanical fracture. International Journal of Solids and Stuctures 41, 6291-6315. [72] Lee, K. Y., Lee, W. G., Pak, Y. E., 2000. Interaction between a semi-infinite crack and a screw dislocation in a piezoelectric material. Journal of Applied Mechanics 67, 165170. [73] Li, C., Weng, G. J., 2002a. Anti-plane crack problem in functionally graded piezoelectric materials. Journal of Applied Mechanics 69, 48 1-488. [74] Li, C., Weng, G. J., 2002b. Yoffe type moving crack in a functionally graded piezoelectric material. Royal Society of London 458, 381-399. [75] Li, S., Mataga, P. A., 1996. Dynamic crack propagation in piezoelectric materials Part I. Electrode solution. Journal of Mechanics, Physics and Solids 44-11, 1831- 1866. [76] Li, S., Mataga, P. A., 1996. Dynamic crack propagation in piezoelectric materials Part II. Vacuum solution. Journal of Mechanics, Physics and Solids 44-11, 1799-1830. [77] Li, S., To, A. C., Glaser, S. D., 2005. On scattering in a piezoelectric medium by conducting cracks. Journal of Applied Mechanics 72, 943-954. [78] LI, X. F., 2001. Transient response of a piezoelectric material with a semi-infinite mode III crack under impact loads. International Journal of Fracture 111, 119-130. [79] Li, X. F., 2005. Electro-elastic analysis of a cracked piezoelectric ceramic strip sandwiched between two elastic dielectrics. European Journal of Mechanics A/Solids 24, 69-80. [80] Li, X. F., Lee, K. Y., 2004. Fracture Mechanics of cracked piezoelectric materials. International Journal of Solid and Structures 41, 4137-4161. [81] Li, X. F., Lee, K. Y., 2004. Electroelastic behavior of a rectangular piezoelectric ceramic with an anti plane shear crack at arbitrary position. European Journal of Mechanics A/ Solids 23, 645-658. [82] Li, X. F., Lee, K. Y., 2004. Dynamic behavior of piezoelectric ceramic layer with two surface cracks. International Journal of Solids and Structures 41, 3193-3209. [83] Li, X. F., Lee, K. Y., 2005. A piezoelectric material with a periodic distribution of slant mode III cracks. Mechanics of Materials 37, 189-200. [84] LI, X. F., Fan, T. Y., 2000. Semi-Infinite AntiPlane Crack in a Piezoelectric Material. International Journal of Fracture 102-3, 53-60. [85] Li, X. F., Fan, T. Y., 2002. Transient analysis of a piezoelectric strip with a permeable crack under antiplane impact loads. International Journal of Engineering Science 40, 13 1-143. [86] Li, X., Ding, S. H., 2011. Periodically distributed parallel cracks in a functionally graded piezoelectric (FGP) strip bonded to a Functionally graded substrate under static electromechanical load. Computational Material Science 50-4, 1477-1484. [87] Li, Y. D., Lee K. Y., 2009. The shielding effect of the imperfect interface on a mode III permeable crack in layered piezoelectric sensors. Engineering Fracture mechanics 76, 876-883.

54 Y. Eugene Pak, Dhaneshwar Mishra and Seung-Hyun Yoo [88] Li, Y. D., Lee, K., Y., 2009. Fracture analysis on the arc shaped interface in a layered cylindrical piezoelectric sensor polarized along its axis. Engineering Fracture Mechanics 76, 2065-2073. [89] Li, X. F., Wang, B. L., 2007. Antiplane shear crack normal to and terminating at the interface of two bonded piezoelectric solids. International Journal of Solids and Structures 44, 3796-3810. [90] Li, X. F., Tang, G. J., 2003. Transient response of a piezoelectric ceramic strip with an eccentric crack under electromechanical impacts. International Journals of Solids and Structures 40, 3571-3588. [91] Li, Y. D., Lee, K. Y., 2009. Interaction between an electrically permeable crack and the imperfect interface in a functionally graded piezoelectric sensor. International Journal of Engineering Science 47, 363-371. [92] Li, X. F., Lee, K. Y., 2006. Transient response of a semi-infinite piezoelectric layer with a surface permeable crack. Z. angew. Mathematics and Physics 57, 636-651. [93] Liu, R. F., Nam, B. G., Watanabe, K., 2008. Discontinuous crack model for piezoelectric materials and its application to CED evaluation. Engineering Fracture Mechanics 75, 1981-2001. [94] Loboda, V., Lapusta, Y., Sheveleva, A., 2010. Limited permeable crack in an interlayer between piezoelectric materials with different zones of electrical saturation and mechanical yielding. International Journal of Solid and Structures 47, 1795-1806. [95] Loeber, J., Sih, G., 1968. Diffraction of antiplane shear waves by a finite crack. Journal of Acoustical Society of America, 44, 90-9 8. [96] Ma, L., Wu, L. Z., Zhou, Z. G., Guo, L. C., 2005. Fracture analysis of functionally graded piezoelectric strips. Composite Structures 69, 294-300. [97] Ma, H., Wang, B.,2005. The scattering of electro elastic waves by an ellipsoidal inclusion in piezoelectric medium. International Journal of Solids and Structures 42, 4541-4554. [98] Ma, L., Zhi, L., Zhou, Z. G., Guo, L. C., Shi, L., P., 2004. Scattering of the harmonic anti-plane shear waves by a crack in functionally graded piezoelectric materials. European Journal of Mechanics A/Solids 23, 633-643. [99] McMeeking, R. M., 1989. Electrostrictive stresses near crack-like flaws. Journal of Applied Mathematics and Physics (ZAMP) 40, 615-627. [100] McMeeking, R. M., 1999. Crack tip energy release rate for a piezoelectric compact tension specimen, engineering Fracture Mechanics 64, 2 17-244. [101] McMeeking, R. M., 2004. The energy release rate for a Griffith crack in a piezoelectric material, Engineering Fracture Mechanics 71, 1149-1163. [102] McMeeking, R. M., 2001. Towards a fracture mechanics for brittle piezoelectric and dielectric materials. International Journal of Fracture 108, 25-41. [103] Meguid, S. A., Wang, X. D., 1998. Dynamic antiplane behavior of interacting cracks in a piezoelectric medium. International Journal of Fracture 91, 391-403. [104] Meguid, S. A., Chen, Z. T., 2001. Transient response of a finite piezoelectric strip containing coplanar insulating cracks under electromechanical impact. Mechanics of Materials 33, 85-96. [105] Meguid, S. A., Zhong, Z., 1997. Electroelastic analysis of a piezoelectric elliptical inhomogeneity. International Journal of Solid and Structures 26, 3401-34 14.

55 Fracture Problems in Antiplane Piezoelectricity [106] Nam, B. G., Watanabe, K., 2008. Effect of electric boundary conditions on crack energy density and its derivatives for piezoelectric materials. Engineering Fracture Mechanics 75, 207-222. [107] Narita, F., Shindo, Y., 1998. Layered piezoelectric medium with interface crack under antiplane shear. Theoretical and Applied Fracture Mechanics 30, 119-126. [108] Narita, F., Shindo, Y., 1998. Dynamic anti plane shear of cracked piezoelectric ceramic. Theoretical and Applied Fracture Mechanics 29, 169-180. [109] Narita, F., Shindo, Y., 1999. The interface crack problem for bonded piezoelectric and orthotropic layers under antiplane shear loading. International Journal of Fracture 98, 87-101. [110] Narita, F., Shindo, Y., 1999. Scattering of anti plane shear waves by a finite crack in piezoelectric laminates. Acta Mechanica 134,27-43 [111] Ou, Z. C., Chen, Y. H., 2004. Near-tip stress fields and intensity factors for an interface crack in metal/piezoelectric bimaterial. International Journal of Engineering Science 42, 1407-1438. [112] Ou, Y. L., Chue, C. H., 2006. Mode III eccentric crack in a functionally graded piezoelectric strip. International Journal of Solid and Structures 43, 6148-6164. [113] Pak, Y. E., 1990a. Crack extension force in a piezoelectric material. ASME Journal of Applied Mechanics 57, 647-653. [114] Pak, Y. E., 1990b. Force on a piezoelectric screw dislocation. ASME Journal of Applied Mechanics 57, 863-869. [115] Pak, Y. E., 1994. Piezoelectric elliptical inclusion under longitudinal shear. Mechanics and Materials for Electronic Packaging: Coupled Field Behavior in Materials, ASMEAMD 193, 263-268. [116] Pak, Y. E. and Goloubeva, E., 1996. Electroelastsic properties of cracked piezoelectric materials under longitudinal shear. Mechanics of Materials 24, 287-303. [117] Pak, Y. E., 2010. Elliptical inclusion problem in antiplane piezoelectricity: Implications for fracture mechanics. International Journal of Engineering Science 48, 209- 222. [118] Pak, Y. E., Kim, S., 2010. On the use of path-independent integrals in calucalting mixed-mode stress intensity factors for elastic and thermoelastic cases. Journal of Thermal Stresses 33, 661-673. [119] Park, S. B., 1994. Fracture Behavior of piezoelectric materials. PhD thesis, Purdue university. [120] Park, S. B., Sun, C. T., 1995a. Fracture criteria for piezoelectric ceramics. Journal of American Ceramic Society 78, 1475-1480. [121] Park, S. B., Sun, C. T., 1995b. Effect of electric field on fracture of piezoelectric ceramics. International Journal of Fracture 70, 203-2 16. [122] Parton, V. Z., 1976. Fracture Mechanics of Piezoelectric Materials. Acta Astronautica 3, 67 1-683. [123] Parton, V. Z., Kudryavtsev, B. A., 1988. Electromagnetoelasticity. New York: Gordon Breach. [124] Qin, Q. H., 2001. Fracture mechanics of piezoelectric materials. Southampton: WIT Press. [125] Qin, Q. H., Wang, J. S., Li, X. L.,2006. Effect of elastic coating on fracture behavior of piezoelectric fiber with penny-shaped crack. Composite Structures 75, 465-471.

56 Y. Eugene Pak, Dhaneshwar Mishra and Seung-Hyun Yoo [126] Qun, Li, Yiheung, C., 2007. Analysis of crack tip singularities for an interfacial permeable crack in metal/piezoelectric bimaterials. Acta Mechanica Solida Sinica 20- 3, 247-257. [127] Ru, C. Q., 1999. Electric field induced crack closure in linear piezoelectric media. Acta Materialia 47, 4683-4693. [128] Saez, A., Sanchez, F. G., Dominguez, J., 2006. Hyper singular BEM for dynamic fracture in 2-D piezoelectric solids. Computer Methods in Appl. Mech. Engg. 196, 23 5-246. [129] Sanchez, F. G., Zhang, C., Saez, A., 2008. 2-D transient dynamic analysis of cracked piezoelectric solids by a time domain Boundary Element Method. Comput. Methods Appl. Mech. Engrg. 197, 3108-3121. [130] Shenghu, D, Xing, Li, 2007. The periodic crack problem in bonded piezoelectric materials. Acta Mechanica Solida Sinica 20, 172-179. [131] Shindo, Y, Narita, F., Tanka, K., 1996a. Electroelastic intensification near anti-plane shear crack in orthotropic piezoelectric ceramic strip. Theoretical and Applied Fracture Mechanics 25, 65-71. [132] Shindo, Y, Tanka, K., Narita, F., 1996b. Singular stress and electric fields of a piezoelectric ceramic strip with a finite crack under longitudnal shear. Acta Mechanica 120, 31-45. [133] Shindo, Y., Tanka, K., Narita, F., 1997. Singular stress and electric fields of a piezoelectric ceramic strip with a finite crack. Acta Mechanica 120, 31-45. [134] Shindo, Y., Moribayashi, H.; Narita, F., 2002a. Scattering of Antiplane Shear Waves by a Circular Piezoelectric Inclusion Embedded in a Piezoelectric Medium Subjected to a Steady-State Electrical Load. ZAMM. Z. Angew. Mathematics and Mechanics 82-1, 4349. [135] Shindo, Y., Minamida, K., Narita, F., 2002b. Antiplane shear wave scattering from two curved interface cracks between a piezoelectric fiber and an elastic matrix. Smart Material Structure 11, 534-540. [136] Shin, J. W., Lee, Y. S., 2010. A moving interface crack between two dissimilar functionally graded piezoelectric layers under electromechanical loading. International Journal of Solid and Structures 47, 2706-2713. [137] Shin, J.W., Kwon, S.M., Lee, K. Y., 2001. An eccentric crack in a piezoelectric strip under antiplane shear impact loading. International Journal of Solid and Structures 38, 183- 1494. [138] Sih, G. C., Zuo, J. Z., 2000. Multiscale behavior of crack initiation and growth in piezoelectric ceramics. Theoretical and applied Fracture Mechanics 34, 123-141. [139] Singh, B. M., Rokne, J., Dhaliwal, R. S., Vrbik, J., 2008. Anti-plane crack at the interface of two bonded dissimilar graded piezoelectric material. European Journal of Mechanics A/Solids 27, 346-364. [140] Soh, A., K., Fang, D. N., Lee, K. L., 2000. Analysis of bi-piezoelectric ceramic layer with an interfacial crack subjected to anti-plane shear and in-plane electric loading. European Journal of Mechanics A/Solids 19, 96 1-977. [141] Sosa, H. A., 1991. Plane problems in piezoelctric media with defects. International Journal of Solids Structures 28, 49 1-505.

57 Fracture Problems in Antiplane Piezoelectricity [142] Sosa, H. A., Khutoryansky, N., 1996. New development concerning piezoelectric materials with defects. International Journal of Solids and Structures 33, 3399-34 14. [143] Sosa, H. A., Pak, Y. E., 1990. Three-dimensional eigenfunction analysis of a crack in a piezoelectric material. International Journal of Solids and Structures 26-1, 1-15. [144] Sudak, L. J., 2003. Effect of an interphase layer on the electroelastic stresses within a three-phase elliptic inclusion. International Journal of Engineering Science 41, 10191039. [145] Sun, J. L., Zhou, Z. G., Wang, B., 2004. Dynamic behavior of unequal parallel permeable interface multi-cracks in a piezoelectric layer bonded to two piezoelectric material half planes. European Journal of Mechanics A/Solids 23, 993-1005. [146] Su, R. K. L., Feng, W. J., Liu, J. X., Zou, Z. Z., 2003. Transient response of coplanar interfacial cracks between two dissimilar piezoelectric strips under antiplane mechanical and inplane electrical impacts. Acta Mechanica Solida Sinica 16-4, 300-312. [147] Suo, Z., Kuo, C.-M., Barnett, D. M., Willis, J. R., 1992. Fracture mechanics for piezoelectric ceramics. Journal of the Mechanics and Physics of Solids 40, 739-765. [148] Tiersten, H. F., 1969. Linear Piezoelectric Plate Vibrations. Plenum Press, New York. [149] Tong, Z. H., Jiang, C. P., Lo S. H., Cheung, Y. K., 2006. A closed form solution to the anti-plane problem of doubly periodic cracks of unequal size in piezoelectric materials. Mechanics of Materials 38, 269-286. [150] To, A. C., Li, S., Glaser, S. D., 2006. Propagation of a mode III interfacial conductive crack along a conductive interface between piezoelectric materials. Wave Motion 43, 368-386. [151] To, A.,C., Li, S., Glaser, S. D.,2005. On scattering in dissimilar piezoelectric materials by a semi-infinite interfacial crack. Journal of Applied Mechanics and Mathematics 582, 1-23. [152] Tobin, A. G., Pak, Y. E., 1993. Effect of electric fields on fracture behavior of PZT ceramics. SPIE Proceedings 1916, 78-86. [153] Torshizian, M. R., Kargarnovin, M. H., 2010. Anti-plane shear of an arbitrary oriented crack in a functionally graded strip bonded with two dissimilar half-planes. Theoretical and Applied Fracture Mechanics 54, 180-188. [154] Ueda, S., 2003. Crack in functionally graded piezoelectric strip bonded to elastic surface layers under electromechanical loading. Theoretical and Applied Fracture Mechanics 40, 225-236. [155] Ueda, S., 2006. A finite crack in a semi infinite strip of a graded piezoelectric material under electric loading. European Journal of Mechanics A/Solids 25, 250-259. [156] Ueda, S., 2003. Transient response of sandwiched piezoelectric strip with crack normal to interface under impact. Theoretical and Applied Fracture Mechanics 40, 211- 223. [157] Wang, B. L., 2003. A mode III crack in functionally graded piezoelectric materials. Mechanics Research Communications 30, 151159. [158] Wang, B. L., Sun, Y. G., 2004. Out-of plane interface cracks in dissimilar piezoelectric materials. Archieve of Applied Mechanics 74, 2-15. [159] Wang, B. L., Han, J. C., Du, S. Y., 2000. Electroelastic fracture dynamics for multilayered piezoelectric material under dynamic anti plane shearing. International Journal of Solids and Structures 37, 5219-523 1.

58 Y. Eugene Pak, Dhaneshwar Mishra and Seung-Hyun Yoo [160] Wang, X. D., Jiang, L. Y., 2002. Fracture behavior of cracks in piezoelectric media with electromechanically coupled boundary conditions. Proceedings of Royal Society of London 458, 2545-25 60. [161] Wang, H., Singh, R. N., 1994. Piezoelectric ceramics and ceramic composites for intelligent mechanical behaviors. Ceramic Transections 43, 277. [162] Wang, X., Pan, E., Chung, P. W., 2009. On a semi-infinite crack penetrating a piezoelectric circular inhomogeneity with viscous interface. International Journal of Solids and Structures 46, 203-216. [163] Wang, Y. J., Gao, C. F., 2008. The Mode III cracks originating from the edge of a circular hole in piezoelectric solid. International Journals of Solids and Structures 45, 4590-4599. [164] Wang, B. L., Mai, Y. W., 2003. On the electrical boundary conditions on the crack surfaces in piezoelectric ceramics. International Journal of Engineering Science 41, 633-652. [165] Wang, X. D., Meguid, S. A., 2000. Effect of electromechanical coupling on the dynamic interaction of cracks in piezoelectric materials. Acta mechanica 143, 1-15. [166] Wu, X. F., Cohn, S., Dzenis, Y. A., 2003. Screw dislocation interacting with interfacial and interface cracks in piezoelectric bimaterials. International Journal of Engineering Science 41-7, 667-682. [167] Xiao, Z. M., Zhang, H. X., Chen, B. J., 2007. A piezoelectric screw dislocation interacts with interfacial collinear rigid lines in piezoelectric bimaterials. International Journal of Solid and Structures 44, 255-271. [168] Xiao, Z., Zhang, H., Chen, B., 2008. Interaction of dislocation with collinear rigid lines at the interface of piezoelectric media. Journal of Mechanics of Materials and Structures 3, 335-364. [169] Yang, F., Kao, I., 1999. Crack problem in piezoelectric materials: general anti-plane mechanical loading. Mechanics of Materials 31, 395-406. [170] Yang P. S., Liou, J. Y., Sung, J. C., 2008. Infinite sequence of parallel cracks in an anisotropic piezoelectric solid. Computer Methods Appl. Mech. Engrg. 197, 38503861. [171] Yang, P. S., Liou, J., Y., Sung, J. C., 2007. Analysis of a crack in a half-plane piezoelectric solid with traction-induction free boundary. International Journal of Solids and Structures 44, 85568578. [172] Yang, P. S., Liou, J. Y., Sung, J. C.,2008. Subinterface crack in an anisotropic piezoelectric bimaterial. International Journal of Solid and Structures 45, 4990-5014. [173] Yong, H. D., Zhou, Y. H., 2007. A mode III crack in functionally graded piezoelectric strip bonded to two dissimilar piezoelectric half planes. Composite Structures 79,404410. [174] Yu, M., Liu, Y. W., Song, H. P., Xie, C., 2010. Electroelastic interaction between a piezoelectric screw dislocation and a confocal elliptic blunt crack with elliptical inhomogeneity. Theoretical and Applied Fracture Mechanics 54, 156-165. [175] Zhang, T.Y., Hack, J.E., 1992. Mode-III cracks in piezoelectric materials. Journal of Applied Physics 71, 5865-5870. [176] Zhang, T. Y., Tong, P. 1996. Fracture mechanics for a mode III crack in a piezoelectric material. International Journal of Solid and structure, 33-3, 343-359.

59 Fracture Problems in Antiplane Piezoelectricity [177] Zhang, T. Y., Gao, C. F., 2002. Fracture of piezoelectric ceramics. Advanced Applied Mechanics 38, 148-289. [178] Zhang, T. Y., Gao, C. F., 2004. Fracture behaviors of piezoelectric materials. Theoretical and Applied Fracture Mechanics 41, 339-379. [179] Zhengong, Z., Shanyi, D. Biao, W., 2003. On anti-plane shear behavior of a Griffith permeable crack in piezoelectric material by use of the non local theory. Acta MechanicaSinica 19-2, 181-188. [180] Zhao, M. H., Fang, P. Z., Shen, Y. P., 2004. Boundary integral -differential equations and boundary element method for interfacial cracks in three-dimensional piezoelectric media. Engineering Analysis with Boundary Elements 28, 753-762. [181] Zhao, X., Meguid, S. A., Leiw, K. M., 2002. The transient response of bonded piezoelectric and elastic half space with multiple interfacial collinear cracks. Acta Mechanica 159, 11-27. [182] Zhong, X. C., Li, X. F., 2005. Closed form solution for two collinear cracks in a piezoelectric strip. Mechanics Research Communications 32, 401-410. [183] Zhou, Z. H., Xu, X. S., Leung, A. Y. T., 2009a. The mode III stress/electric intensity factors and singularities analysis for edge-cracked circular piezoelectric shafts. International Journal of Solids and Structures 46, 3577-3586. [184] Zhou, Z. G., Guo, Y., Wu, L. Z., 2009b. The behavior of 3 parallel non-symmetric permeable mode III cracks in piezoelectric material plane. Mechanics Research Communications 36, 690-698. [185] Zhou, Z. G.,Zhang, P. W., Li, G., 2009c. Interactions of multiple parallel symmetric permeable mode III cracks in a piezoelectric material plane. European Journal of Mechanics A/Solids 28, 728-737. [186] Zhou, Z. G., Wu, L. Z., Wang, B., 2005. The behavior of crack in functionally graded piezoelectric/piezomagnetic materials under anti-plane shear loading. Achieves of Applied Mechanics 74, 526-535. [187] Zhou, Z. G., Wang, B., 2004. Two parallel symmetric permeable cracks in functionally graded piezoelectric/ piezomagnetic materials under anti-plane shear loading. International Journal of Solid and Structures 41, 4407-4422. [188] Zhou, Z. G., Du, S. H., Wu, L. Z., 2007. Investigation of anti-plane shear behavior of a Griffith crack in functionally graded piezoelectric materials by use of Non local Theory. Composite Structures 78, 575-583. [189] Zhou, Z. G., Wang, B., 2002. The scattering of harmonic elastic anti-plane shear waves by a Griffith crack in a piezoelectric material plane by using the non-local theory. International Journal of engineering Science 40, 303-317. [190] Zhou, Z. G., Li, H. C., Wang, B.,2001. Investigation of the scattering of anti-plane shear waves by two collinear cracks in a piezoelectric material using a new method. Acta Mechanica 147, 87-97. [191] Zhou, Z. D., Nishioka, T., 2005. Asymptotic analysis for steady mode III crack growth in piezoelectric sandwich composite. International Journal of applied Electromagnatics and Mechanics 22 (3-4), 121-132. [192] Zuo, J. Z., Sih, G. C., 2000. Energy density theory formulation and interpretation of cracking behavior for piezoelectric ceramics. Theory of Applied Fracture mechanics 34, 17-33.