# Dynamic Propagation Characteristics of a Mode-III Interfacial Crack in Piezoelectric Bimaterials

Authors: Yani Zhang; Junlin Li; Xiufeng Xie
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

This article presents the dynamic behavior of a semi-infinite interfacial crack in piezoelectric bimaterials under impact loading. With the help of the transform methods (the Laplace transforms and Fourier transforms), the problem is studied with the Wiener–Hopf technique. This strict proof guarantees the feasibility of this approach. The dynamic stress intensity factor and dynamic electric displacement intensity factor of the interfacial crack propagation characteristics are expressed. Finally, several classic numerical examples are mentioned and discussed to demonstrate that the theoretical deduction is highly accurate for interfacial crack analysis of piezoelectric bimaterials. The results show that the crack propagation is affected by the electromechanical coupling coefficient. In addition, if the velocity of the dynamic crack propagation reaches the generalized Raleigh wave speed, the dynamic stress intensity factor will disappear. Furthermore, for a given time, the ratio of the dynamic stress intensity factor to load increases with the electromechanical coupling coefficient decreasing. Numerical examples are presented to highlight the result.


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## Body

## 1. Introduction

### 1.1. Background

Curie brothers have presented the piezoelectric effect. Due to their coupling behaviors, piezoelectric materials have been widely used in modern smart devices and structures such as actuators, sensors, and ultrasonic generators. At present, piezoelectric materials widely used in engineering practice are mainly piezoelectric ceramics and piezoelectric composites made of piezoelectric ceramics and polymers. Piezoelectric ceramics are brittle materials and have the disadvantage of low fracture toughness. However, defects often cause the structural failure of piezoelectric materials. Cracks as their main disadvantages in piezoelectric media have been widely investigated [1–9]. One of the challenging problems is the fracture behavior of interfacial crack in piezoelectric bimaterials. A few fundamental problems of the crack in homogeneous and inhomogeneous piezoelectric bimaterials have been discussed [10–13]. The fracture analyses of these piezoelectric materials have provided much knowledge for improving the performance of piezoelectric devices. However, most results have focused on static propagation.Dynamic fracture can be divided into three types according to the time dependence of load and crack state: the mechanical-electric load is time-independent and the crack propagates rapidly [14]; not only is the electromechanical load time-dependent but also the crack propagates rapidly [15]; the crack is stable and the electromechanical load varies with time [16].A great deal of research has been done on the first and the second kind of fracture conditions [17, 18]. For the third type of dynamic fracture, the dynamic propagation of electroacoustic wave (elastic wave) [19] and impact is included. The crack propagation becomes dynamic behavior under the action of explosion or impact load. The propagation behavior of the crack was analyzed under a concentrated point load [20, 21]. The solutions are derived in an explicit way. When the “electrode” Bleustein–Gulyaev wave velocity is(1)cbg=c¯44ρ1−ke2,above the crack propagation speed, the stress displacement and electric displacement are exhibited r−1/2, which is similar to that of static fracture behavior. Narita [22] researched the crack in a piezoelectric layer. The interface was normal to the interface. The results show it is feasible and practical for the design of piezoelectric devices.The interfacial crack propagation in piezoelectric bimaterials is the most complicated problem in the dynamic fracture of piezoelectric media. Ray investigated the dynamic crack propagation behaviors in piezoelectric layers [23]. The transient response of a Griffith crack between two piezoelectric layers under finite widths was investigated [24]. The problem was formulated using integral transforms, and the path-independent integral G was evaluated at the crack tip to obtain the dynamic energy release rate. The dynamic fracture toughness of a mode-III interfacial crack in two piezoelectric half-spaces was analyzed [25, 26]. Shen et al. provided a unified method to analyze a interfacial crack propagating in piezoelectric bimaterials [27]. The problem of an antiplane Griffith crack moving along with the interface of dissimilar piezoelectric materials was performed using the integral transform technique [28]. A semi-infinite crack in piezoelectric and piezomagnetic bimaterials was researched through the Wiener–Hopf techniques together with the integral transform method [29]. Nourazar and Ayatollahi analyzed the multiple moving cracks along with the interface of two piezoelectric layers under electrically impermeable or permeable loading, respectively [30].At the same time, numerical simulations were used for studying the crack. Fracture in piezoelectric materials was analyzed by using a cell-based smoothed X-FEM [31]. Numerical methods were required to deal with the dynamic fracture behavior under loading conditions and general geometries. The dynamic problems of piezoelectric materials were researched by using the X-FEM [32], and it is effective in solving the dynamic fracture problems of isotropic materials. The SGBEM was raised about the dynamic crack in the piezoelectric composite materials [33].However, piezoelectric bimaterials are widely used in modern life. The crack propagation caused by impact load is often hard to observe, and the crack runs through the interface of bimaterials, which is a complex transient process. The fracture mechanism and process of the material cannot be observed. Therefore, the macroscopic process of crack propagation is carried out to research questions about the materials.The interfacial fracture of piezoelectric bimaterials is caused by its mismatching characteristics under working loading conditions. There are few kinds of research on the mechanical properties of piezoelectric bimaterials and a variation of the crack as time changes. Therefore, the analysis of dynamic crack along with the interface in piezoelectric bimaterials is essential to the practical engineering design.


### 1.2. Outline

The presentation of the study is structured as follows. In Section2, the mixed initial boundary conditions and constitutive equations are presented in detail. In Section 3, the solution method of dynamic crack propagation in the transformed domain is presented. The dynamic stress intensity factor and electric displacement intensity factor of the crack propagation behavior in the interface are given in Section 4. Section 5 presents the numerical results. In Section 6, we give some immediate conclusions.


## 1.1. Background

Curie brothers have presented the piezoelectric effect. Due to their coupling behaviors, piezoelectric materials have been widely used in modern smart devices and structures such as actuators, sensors, and ultrasonic generators. At present, piezoelectric materials widely used in engineering practice are mainly piezoelectric ceramics and piezoelectric composites made of piezoelectric ceramics and polymers. Piezoelectric ceramics are brittle materials and have the disadvantage of low fracture toughness. However, defects often cause the structural failure of piezoelectric materials. Cracks as their main disadvantages in piezoelectric media have been widely investigated [1–9]. One of the challenging problems is the fracture behavior of interfacial crack in piezoelectric bimaterials. A few fundamental problems of the crack in homogeneous and inhomogeneous piezoelectric bimaterials have been discussed [10–13]. The fracture analyses of these piezoelectric materials have provided much knowledge for improving the performance of piezoelectric devices. However, most results have focused on static propagation.Dynamic fracture can be divided into three types according to the time dependence of load and crack state: the mechanical-electric load is time-independent and the crack propagates rapidly [14]; not only is the electromechanical load time-dependent but also the crack propagates rapidly [15]; the crack is stable and the electromechanical load varies with time [16].A great deal of research has been done on the first and the second kind of fracture conditions [17, 18]. For the third type of dynamic fracture, the dynamic propagation of electroacoustic wave (elastic wave) [19] and impact is included. The crack propagation becomes dynamic behavior under the action of explosion or impact load. The propagation behavior of the crack was analyzed under a concentrated point load [20, 21]. The solutions are derived in an explicit way. When the “electrode” Bleustein–Gulyaev wave velocity is(1)cbg=c¯44ρ1−ke2,above the crack propagation speed, the stress displacement and electric displacement are exhibited r−1/2, which is similar to that of static fracture behavior. Narita [22] researched the crack in a piezoelectric layer. The interface was normal to the interface. The results show it is feasible and practical for the design of piezoelectric devices.The interfacial crack propagation in piezoelectric bimaterials is the most complicated problem in the dynamic fracture of piezoelectric media. Ray investigated the dynamic crack propagation behaviors in piezoelectric layers [23]. The transient response of a Griffith crack between two piezoelectric layers under finite widths was investigated [24]. The problem was formulated using integral transforms, and the path-independent integral G was evaluated at the crack tip to obtain the dynamic energy release rate. The dynamic fracture toughness of a mode-III interfacial crack in two piezoelectric half-spaces was analyzed [25, 26]. Shen et al. provided a unified method to analyze a interfacial crack propagating in piezoelectric bimaterials [27]. The problem of an antiplane Griffith crack moving along with the interface of dissimilar piezoelectric materials was performed using the integral transform technique [28]. A semi-infinite crack in piezoelectric and piezomagnetic bimaterials was researched through the Wiener–Hopf techniques together with the integral transform method [29]. Nourazar and Ayatollahi analyzed the multiple moving cracks along with the interface of two piezoelectric layers under electrically impermeable or permeable loading, respectively [30].At the same time, numerical simulations were used for studying the crack. Fracture in piezoelectric materials was analyzed by using a cell-based smoothed X-FEM [31]. Numerical methods were required to deal with the dynamic fracture behavior under loading conditions and general geometries. The dynamic problems of piezoelectric materials were researched by using the X-FEM [32], and it is effective in solving the dynamic fracture problems of isotropic materials. The SGBEM was raised about the dynamic crack in the piezoelectric composite materials [33].However, piezoelectric bimaterials are widely used in modern life. The crack propagation caused by impact load is often hard to observe, and the crack runs through the interface of bimaterials, which is a complex transient process. The fracture mechanism and process of the material cannot be observed. Therefore, the macroscopic process of crack propagation is carried out to research questions about the materials.The interfacial fracture of piezoelectric bimaterials is caused by its mismatching characteristics under working loading conditions. There are few kinds of research on the mechanical properties of piezoelectric bimaterials and a variation of the crack as time changes. Therefore, the analysis of dynamic crack along with the interface in piezoelectric bimaterials is essential to the practical engineering design.


## 1.2. Outline

The presentation of the study is structured as follows. In Section2, the mixed initial boundary conditions and constitutive equations are presented in detail. In Section 3, the solution method of dynamic crack propagation in the transformed domain is presented. The dynamic stress intensity factor and electric displacement intensity factor of the crack propagation behavior in the interface are given in Section 4. Section 5 presents the numerical results. In Section 6, we give some immediate conclusions.


## 2. Governing Equations

According to [34] and the variational principle, the energy conservation laws can be written as(2)ddtξ+Ξ=∫SτjinjvidS+∫VEiDi.dV,where ξ=1/2ρ∫VvividS, Ξ=∫VUdS, S, τ, ρ, E, D, and v are, respectively, the kinetic energy, internal energy (including electrical and mechanical energies), a surface in the volume domain V, outward normal vector, the stress tensor, density, electric field, electric displacement, and velocity vector.The basic equations for linear piezoelectric medium can be written as follows.Kinematic equations:(3)τij,j=ρwi..,where w is the mechanical displacement.Electrostatic charge conservation:(4)Dj,j=0.Gradient equations:(5)∈ij=wi,j+wj,i2,(6)Ei=−φ,i,where w is the mechanical displacement, ∈ij is the strain tensor, and φ is the electric potential.From equations (2) and (3), we will obtain the local form of the energy conservation laws:(7)dU=τijd∈ij+EidDi.We introduce electric enthalpy densityH∈ij,Ei,(8)H=U−EiDi,(9)dH=∂H∂∈ijd∈ij+∂H∂EidEi.The electric enthalpy densityH∈ij,Ei is expanded in Taylor series near the natural state (τij=0, Ei=0):(10)H=12cijkl∈ij∈kl−ekij∈ijEk+12εijEiEj,where cijkl are the elastic stiffness constants, ekij are the piezoelectric constants, and εij are the dielectric constants.By substituting equation (7) to (8), the change of H can be written as follows:(11)dH=τijd∈ij−DidEi.It is noted that(12)τij=∂H∂τij,Di=−∂H∂Ei.According to equations (10) and (12), the nontrivial constitutive equations can be obtained:(13)τij=cijkl∈kl−ekijEk,(14)Di=eikl∈kl+εikEk.Using the symmetry of strain tensor and stress tensor, the symmetry relation of material constants can be obtained:(15)cijkl=cjikl=cjilk=cklij,ekij=ekji,εij=εji.From equations (3)-(4) and (13)-(15), we have the following basic governing equations of piezoelectric theory:(16)cijklwk,lj+ekijφ,kj=ρwi..,eiklwk,li−εikφ,ki=0.For the transversely isotropic piezoelectric medium, the constitutive equations [35] are(17)τxx=c11∈xx+c12∈yy+c13∈zz−e31Ez,τyy=c12∈xx+c11∈yy+c13∈zz−e31Ez,τzz=c13∈xx+c13∈yy+c33∈zz−e33Ez,τyz=2c44∈yz−e15Ey,τzx=2c44∈zx−e15Ex,τxy=c11−c12∈xy,Dx=2e15∈zx+ε11Ex,Dy=2e15∈yz+ε11Ey,Dz=e31∈xx+e31∈yy+e33∈zz+ε33Ez.If only the in-plane electric fields and the out-of-plane displacement are considered, the dynamic antiplane governing equations for the transversely isotropic piezoelectric material can be described by(18)c44∂2w∂x2+∂2w∂y2+e15∂2φ∂x2+∂2φ∂y2=ρ∂2w∂t2,e15∂2w∂x2+∂2w∂y2−ε11∂2φ∂x2+∂2φ∂y2=0.The constitutive equations become(19)τzx=2c44∈zx−e15Ex=c44∂w∂x+e15∂φ∂x,Dx=2e15∈zx+ε11Ex=e15∂w∂x−ε11∂φ∂x,τyz=2c44∈yz−e15Ey=c44∂w∂y+e15∂φ∂y,Dy=2e15∈yz+ε11Ey=e15∂w∂y−ε11∂φ∂y.In Figure1, the piezoelectric material one and piezoelectric material two occupy the domains Ω1 and Ω2, respectively. A semi-infinite crack along with the interface in the transversely isotropic piezoelectric bimaterials is considered. It is located at Y=0,X<0. The crack at any time t<0 is in a static equilibrium state in Figure 1. The crack tip begins to move when t=0. The position is x=vt when t>0.Figure 1
A semi-infinite mode-III interfacial propagating crack in piezoelectric bimaterials.If only the in-plane electric fields and the out-of-plane displacement are considered, the dynamic antiplane governing equations for the transversely isotropic piezoelectric bimaterials can be described by(20)c44k∂2w∂X2+∂2w∂Y2+e15k∂2φ∂X2+∂2φ∂Y2=ρ∂2w∂t2,(21)e15k∂2w∂X2+∂2w∂Y2−ε11k∂2φ∂X2+∂2φ∂Y2=0.For the abovementioned equations,w=wx,y,t is the mechanical displacements, and φx,y,t is the electric potential. ρ is the mass density, c44k are the elastic stiffness constants, e15k are the piezoelectric constants, and ε11k are the dielectric constants, where k=1,2 signify piezoelectric material one and piezoelectric material two, respectively.The shear stress componentsτ,xzτyz and electric displacements Dx,Dy are related to the mechanical displacements and electric potential by the nontrivial constitutive equations:(22)τxz=c44k∂w∂x+e15k∂φ∂x,(23)Dx=e15k∂w∂x−ε11k∂φ∂x,(24)τyz=c44k∂w∂y+e15k∂φ∂y,(25)Dy=e15k∂w∂y−ε11k∂φ∂y.For uniform crack propagation, a moving coordinate system is commonly used:(26)x=X−vt,y=Y,z=Z.By using transformations, equation (21) may be converted to(27)1−vck¯2∂2w∂x2+∂2w∂y2+e15kc44k∂2φ∂x2+∂2φ∂y2=1ck¯2∂2w∂t2−2v∂2w∂x∂t,(28)e15kc44k∂2w∂x2+∂2w∂y2−ε11kc44k∂2φ∂x2+∂2φ∂y2=0,where ck¯=c44k/ρ1/2.The boundary conditions of mixed mechanical are(29)τyzx,0+,t=τyzx,0−,t=−τ0Ht,−∞<x<0,(30)Dyx,0+,t=Dyx,0−,t,−∞<x<0,(31)wx,0+,t=wx,0−,t=0,0<x<+∞,(32)φx,0+,t=φx,0−,t=0,0<x<+∞,(33)τyzx,0+,t=τyzx,0−,t,0<x<+∞,(34)Dyx,0+,t=Dyx,0−,t,0<x<+∞,where Ht is the Heaviside step function of the time t.The mechanical displacements and electric potential will disappear wheny⟶∞:(35)τyzx,y,t=0,wx,y,t=0,Dyx,y,t=0,φx,y,t=0,y⟶∞.The static initial conditions are(36)wx,y,0=0,∂wx,y,0∂t=0,−∞<x,y<+∞,(37)φx,y,0=0,∂φx,y,0∂t=0,−∞<x,y<+∞.


## 3. Solution to the Problem

Following [20, 21], a function is introduced as(38)ϕ=φ−e15kε11kw.The equations are converted into the canonical form by substituting equations (38) into (27)-(28).(39)1−vck¯2+e15kε11ke15kc44k∂2w∂x2+1+e15kε11ke15kc44k∂2w∂y2−1ck¯2∂2w∂t2+2vck¯2∂2w∂x∂t=0,(40)∂2ϕ∂x2+∂2ϕ∂y2=0.To obtain the solution of equations (22)–(25), the following are the Fourier transform concerning x and Laplace transform concerning the time t:(41)f∗x,y,p=∫0+∞fx,y,te−ptdt,fx,y,t=12πi∫Brf∗x,y,peptdp,(42)Fζ,y,p=∫−∞+∞f∗x,y,peiζxdx,f∗x,y,p=12π∫iζ2−∞iζ2+∞Fζ,y,pe−iξxdζ,where ξ=ξ1+iξ2 is complex, and Br denotes the Bromwich path of integration.From equations (41) and (42), the governing equations (27), (28), and (38) in the transformed domain can be obtained as follows:(43)w∗ζ,y,p=∫−∞+∞∫0+∞wx,y,te−pteiζxdtdx,(44)φ∗ζ,y,p=∫−∞+∞∫0+∞φx,y,te−pteiζxdtdx,(45)ϕ∗ζ,y,p=∫−∞+∞∫0+∞φx,y,te−pteiζxdtdx−e15kε11k∫−∞+∞∫0+∞wx,y,te−pteiζxdtdx.By substituting equations (43) and (45) to equations (39) and (40), respectively, using zero initial conditions in equations (36) and (37), the transformed governing equations become(46)1001∂2w∗∂y2∂2ϕ∗∂y2−ζ21−v/ck¯2+e15ke15k/c44kε11k+p2/ck¯2ζ2+2vip/ck¯ζ1+e15ke15k/c44kε11k001w∗ϕ∗=0,where (47)w∗ζ,y,p=∫−∞+∞∫0+∞wx,y,te−pteiζxdtdx,(48)φ∗ζ,y,p=∫−∞+∞∫0+∞φx,y,te−pteiζxdtdx,(49)ϕ∗ζ,y,p=∫−∞+∞∫0+∞φx,y,te−pteiζxdtdx−e15kε11k∫−∞+∞∫0+∞wx,y,te−pteiζxdtdx.Applying the conditions given in equations (35), (36), and (37) at infinity, the solutions of equation (46) are(50)w∗ζ,y,p=A1ke3−2kakζy,ϕ∗ζ,y,p=B1ke3−2kbkζy,where(51)akζ=fk−v/ck¯2ζ2+p2/ck¯2+2vipζ/ck¯2fk=skζ−ip/ck+vζ+ip/ck−v=ak+ζak−ζ,sk=1−v/ck2,fk=1+e15ke15kc44kε11k,bkζ=limλ⟶0λ2−ζ2=λ+ζλ−ζ=bk+ζbk−ζ,where ck=fkck¯2 denotes the shear bulk wave velocity.We have the following form with the inverse Fourier transform to equation (50):(52a)w∗x,y,p=12π∫iζαk−∞iζαk+∞A1ke3−2kakye−iζxdζ,(52b)ϕ∗x,y,p=12π∫iςβk−∞iςβk+∞B1ke3−2kbkye−iζxdζ.From equations (52a) and (52b), we have the following form:(53)φ∗x,y,p=e15kε11kwk∗x,y,p+ϕk∗x,y,p=e15kε11k12π∫iζαk−∞iζαk+∞A1ke3−2kakye−iζxdζ+12π∫iζβk−∞iζβk+∞B1ke3−2kbkye−iζxdζin which(54)−pck−v<ζαk<pck+v−ζ<ζβk<ζ.By using equations (52a), (52b) and (53), the transformed results of the stress and electric displacement are(55)τyz∗x,y,p=c44k+e15ke15kε11k3−2k12π∫iζαk−∞iζαk+∞A1kake3−2kakye−iζxdζ+e15k3−2k12π∫iζβk−∞iζβk+∞B1kbke3−2kbkye−iζxdζ(56)Dy∗x,y,p=−ε11k3−2k12π∫iζβk−∞iζβk+∞B1kbke3−2kbkye−iζxdζ.To further improve the speed for solving problems, the boundary conditions equations (29)–(34) expressed in the Laplace transform domain are(57)τyz∗x,0+,p=τyz∗x,0−,p=−τ0p,x<0,(58)Dy∗x,0+,p=Dy∗x,0−,p,x<0,(59)w∗x,0+,p=w∗x,0−,p=0,x>0,(60)φ∗x,0+,p=φ∗x,0−,p=0,x>0,(61)τyz∗x,0+,p=τyz∗x,0−,p,x>0,(62)Dy∗x,0+,p=Dy∗x,0−,p,x>0.By substituting equations (59) and (60) into equations (52a) and (53), respectively, we have(63)12π∫iζαk−∞iζαk+∞A1ke−iζxdζ=0,x>0,12π∫iζβk−∞iζβk+∞B1ke−iζxdζ=0,x>0.Substitution equations (57) and (58) into equations (55) and (56), we will have(64)3−2k2πc44k+e15ke15kε11k∫iζαk−∞iζαk+∞akA1ke−iζxdζ+e15k∫iζβk−∞iζβk+∞B1kbke−iζxdζ=−τ0p,x<0.From equation (64), we have(65)12π∫iζαk−∞iζαk+∞A1kake−iζxdζ=−ε11kc44kε11k+e15ke15k13−2kτ0p,x<0.Then, rewritingak, we have(66)ak=p−iζgkp−iζhk,in which(67)gk=v+fkck¯2,hk=−v+fkck¯2.Whenv=0, gk and hk become(68)gk=fkck¯2=hk=ck.It can be verified that this result is consistent with reference [36] in the case v=0. Now, let us calculate the case v≠0. Given the four boundary conditions equations (59)–(62), the displacement and stress boundary conditions must extend to the entire range of the x-axis. Then, we introduce two unknown functions:(69)w∗x,0,p=0w−∗x,0,px≥0x<0,(70)τyz∗x,0,p=τx+∗x,0,p−τ0px≥0x<0,(71)φ∗x,0,p=0φ−∗x,0,px≥0x<0.Equations (69), (70), and (71) can be written as(72)w∗x,0,p=w−∗x,0,p,(73)τyz∗x,0,p=τx+∗x,0,p−τ0pH−x,(74)φ∗x,0,p=φ−∗x,0,p.Substituting equations (52b) and (56) into equation (74), respectively, we can obtain(75)12π∫iζβk−∞iζβk+∞B1ke−iζxdζ=φ−∗x,0,p.Substituting equations (52a), (52b), (55), and (64) into equation (73), we can obtain(76)12π∫iζαk−∞iζαk+∞A1ke−iζxdζ=w−∗x,0,p,(77)12π∫iζαk−∞iζαk+∞A1kake−iζxdζ=τx+∗x,0,p−τ0/pH−xc44k+e15ke15k/ε11k3−2k.Equations (76) and (77) are a set of dual integral equations. The inverse Fourier transform is applied to equations (76)-(77):(78)Wk−∗ζ,0,p=A1k=∫−∞0w−∗x,0,peiζxdx,(79)c44kε11k+e15ke15kε11k3−2kA1kak=∫0+∞τx+∗x,0,peiζxdx−τ0ipζ+e15kε11k∫0+∞Dy+∗x,0,peiζxdxEquation (79) can be rewritten as(80)Wk−∗Mk=Rk+∗−τ0ipζ+qkDk+∗,where(81)Mk=pkskζ−ip/dk+vζ+ip/dk−vζ−ip/ck+vζ+ip/ck−vζ−ip/dk+vζ+ip/dk−v,pk=c44k+e15ke15kε11k3−2k,qk=e15kε11k,Rk+∗=∫0+∞τx+∗x,0,peiζxdx,Dk+∗=∫0+∞Dy+∗x,0,peiξxdx,dk=c44kc44kε11kε11kc44k+2c44kε11ke15ke15kc44kε11k+e15ke15k2ρ.dk are the speeds of the Bleustein–Gulyaev wave.Ifa1ζ−ke12b1ζ+a2ζ−ke22b2ζ=0 exists a real root v, then the real root v is called the Maerfeld–Tournois (MT) wave speed. If the Maerfeld wave exists, we define c=ce; otherwise, c=c1. The functions Hζ (Appendix) can be defined as follows:(82)Hζ=zc+v−ipzc−v+ipzc1−v+ipzc1+v−ipcc1akv−kek2bkv=H+∗ζH−∗ζrkwhere(83)kek=e15ke15kc44kε11k+e15ke15k,S±∗=exp−1π∫1/cmin∓v1/cmax∓varctanImHipsReHipsdss−iζ/p,O∗=d1d2ζd1+v−ipζd2+v−ip1ζd1−v+ipζd2−v+ip=O+∗O−∗,where cmin and cmax are the minimum and maximum of the wave speeds ck and dk (k=1,2), respectively. Since Mkζ (Maerfeld–Tournois wave function) depends on the MT wave, Mkζ are defined and coefficiented as follows:(84)Mk=dk−vck+vp−iζdk+vp+iζck−vH+∗H−∗O+∗O−∗dk+vck−vp+iζdk−vp−iζck+vC1+∗C2+∗S+∗C1−∗C2−∗S−∗rk=Fk+∗Fk−∗Lk∗L−∗rkThe product decomposition of equation (80) is given by [20, 21, 37](85)pkrkWk−∗Fk+∗Fk−∗L+∗L−∗=Rk+∗−τ0ipζ+qkDk+∗,where(86)Fk+∗=pd+kv−iζζ1+v/c−ip/cζ1+v/c1−ip/c1pc+kv−iζζ1+v/d1−ip/d1ζ1+v/d2−ip/d2,Fk−∗=pc−kv+iζζc−v+ipζc1−v+ippd−kv+iζζd1−v+ipζd2−v+ip,L+∗=C1+∗C2+∗Sk+∗,L−∗=C1−∗C2−∗Sk−∗.Moreover, equation (85) is expressed as follows:(87)pkrkPk−∗Wk−∗+Pk+∗0τ0ipζ=Pk+∗Rk+∗−Pk+∗ζ−Pk+∗0τ0ipζ+qkPk+∗Dk+∗.On the basis of equation (41), the Laplace transform of τyzx,y,t is(88)τyz∗x,y,p=∫0+∞τyzx,y,te−ptdt,where p is a positive real variable in the case.From the integrable energy density and continuity of displacement, the displacement functionw−∗x,0,p and stress function τx+∗x,0,p satisfy the following conditions:(89)w−∗x,0,p∼oxθ,x⟶0−,y=0,θ>0,τx+∗x,0,p∼ox−1/2,x⟶0+,y=0.Using the Able theorem [38], we can obtain(90)limx⟶0+τx+∗x,0,px1/2∼limζ⟶∞Rx∗ζ,0,pζ1/2,limx⟶0+w−∗x,0,px−θ∼limζ⟶∞W−∗ζ,0,pζ1+θ.It follows that asζ⟶∞,(91)pkrkPk−∗W−∗ζ+Pk+∗0τ0ipζ∼oζ−1/2−θ,Pk+∗Rk+∗−Pk+∗ζ−Pk+∗0τ0ipζ+qkPk+∗Dk+∗∼oζ−1.According the extended Liouville’s theorem [37], the polynomial must be less than the maximum value (−1/2−θ, −1). It can only be equal to zero in the case that(92)W−∗ζ,0,p=1/d1−v−iζ/p1/d2−v−iζ/ppck−v+iζ1/c1−v−iζ/p1/c−v−iζ/ppdk−v+iζτ0ipζ1−v/d11−v/d21−v/c11/21−v/cippkrkL−∗.The functionsAkζ are obtained by substituting equation (92) into (78):(93)A1kζ=1/d1−v−iζ/p1/d2−v−iζ/ppck−v+iζ1/c1−v−iζ/p1/c−v−iζ/ppdk−v+iζ1−v/d11−v/d21−v/c11/21−v/cippkrkL−∗τ0ipζ.According to the procedure by Li and Mataga [20, 21], the functions Bkζ can be obtained:(94)B1kζ=−1/d1−v−iζ/p1/d2−v−iζ/ppck−v+iζ1/c1−v−iζ/p1/c−v−iζ/ppdk−v+iζ1−v/d11−v/d2qk1−v/c11/21−v/cippkrkL−∗τ0ipζ.


## 4. Intensity Factors

Substituting equations (93) and (94) into equations (55) and (56), respectively, and applying the inverse Fourier transform, Dy∗x,p and τyz∗x,p can be obtained. According to Abel’s theorem [37], we can obtain the stress intensity factor and electric displacement intensity factor as follows:(95)KIIIp=limx⟶0+2πxτyz∗x,p=21−v/d11−v/d21−v/c11−v/cL−∗vτ0p3/2,(96)KD1p=limx⟶0+2πxDy∗x,p=e151ε111c441ε111+e151e15121−v/d11−v/d2s1−ke121−v/c11−v/cL−∗vτ0p3/2,(97)KD2p=limx⟶0+2πxDy∗x,p=e152ε112c442ε112+e152e15221−v/d11−v/d2s2−ke221−v/c21−v/cL−∗vτ0p3/2,Using the inverse Laplace transform to equations (95)–(97) leads to(98)KIIIt,v=221−v/d11−v/d21−v/c11−v/cL−∗vπτ0t,KD1t,v=e151ε111c441ε111+e151e151221−v/d11−v/d2s1−ke12π1−v/c11−v/cL−∗vτ0t,KD2t,v=e152ε112c442ε112+e152e152221−v/d11−v/d2s2−ke22π1−v/c11−v/cL−∗vτ0t.According to the procedure in [38] for the problem of the static semi-infinite crack propagation, we introduce a normalization.The stress intensity factor for the static semi-infinite crack is(99)KIIIt,0=22L−∗0πτ0t.The electric displacement intensity factor for a stationary semi-infinite crack is(100)KD1t,0=e151ε111c441ε111+e151e151221−ke12πL−∗0τ0t,KD2t,0=e152ε112c442ε112+e152e152221−ke22πL−∗0τ0t.As a result, the dynamic stress intensity factor and the dynamic electric displacement intensity factors can be represented as(101)KIIIt,v=fvKIIIt,0,KD1t,v=g1vKD1t,0,KD2t,v=g2vKD2t,0,where(102)fv=1−v/d11−v/d21−v/c11−v/cL−∗v,g1v=1−ke12s1−ke12fv,g2v=1−ke22s2−ke22fv,are the nondimensional dynamic intensity factors.


## 5. Numerical Examples

The material properties and the mass density [20, 36], elastic stiffness constant, piezoelectric constant, dielectric constant, electromechanical coupling coefficient, and shear wave speed are given in Table 1. Incidentally, the values c and d for PZT65/35 and ZnO, respectively, in [20] are incorrect, and the correct values are given in Table 1.Table 1
Material properties and electroacoustic constants.
ConstantMaterialPZT-4PZT-5HZnOBaTiO3PZT65/35PZT-2ρ103kg/m37.57.55.685.77.8257.6ε110−9F/m6.463415.0520.07579.87225.664.4624c441010N/m22.562.34.2474.43.8902.22e15C/m12.7170.4811.48.3879.8k=e152/c44ε11+e1520.70260.67450.25860.47990.49210.7016c=c44+e152/ε11/ρ103m/s2.59632.37212.83073.16682.56112.3985d=c1−k4103m/s2.25792.11232.82443.08172.48492.0878There are three parts in this section. First, when the research is degenerated into the simple case, some numerical applications are provided to verify the conclusions obtained. Second, some analyses and comparisons are made between the results obtained by us and the existing references [25, 26] (the methods used are different from those we used). Third, the influence of material constants, velocity of the crack propagation, and time of impact are discussed. The researches have led to some new conclusions.


### 5.1. Study of Degenerate into the Simple Case

When bimaterials degenerate into a single material, it follows that(103)KIIIt,v=fvKIIIt,0,KDt,v=gvKDt,0,where fv=1−v/d/1−v/cL−∗v, gv=1−ke2/s−ke2fv.It is noted that the conclusions agree with [20, 21]. We choose the single material PZT-4 and PZT-5H, respectively, for research. The corresponding analysis diagram is shown in Figure 2. They show the variations fv and gv with the normalized speed v/d of the dynamic crack propagation. As shown in Figure 2(a), fv decreases with increasing velocity v of the dynamic crack propagation. The variations fv vanish when v reaches Bleustein–Gulyaev d. Furthermore, the nondimensional function of the stress fv for an antiplane crack propagating in piezoelectric solids as considered is plotted for comparison [20, 21]. Their changes are consistent. However, as shown in Figure 2(a), when the speed of the dynamic crack propagation reaches Bleustein–Gulyaev d, the variations gv do not have to go to zero. It has an initial decrease as the speed v of the dynamic crack propagation increases from zero. The variations gv increase again as the speed v of the dynamic crack propagation reaches Bleustein–Gulyaev d. Furthermore, the variations gv for the crack propagating in piezoelectric solids [20, 21] as considered are plotted for comparison. Their changes are consistent.Figure 2
Variations offv and gv versus the nondimensional velocity v/d for a single material PZT-4 or PZT-5H.
(a)(b)


### 5.2. Analysis and Comparison with the Existing References

Some analyses and comparisons are made between the results obtained by us and the existing references [25, 26] (the methods used are different from those we used). We choose PZT-4&PZT65/35 and PZT-2&ZnO, respectively, for research. The corresponding analysis diagram is shown in Figure 3. In Figures 3(a) and 3(c), it represents fv, g1v, and g2v of PZT-4&PZT65/35 versus the normalized speed v/d and v/c of the dynamic crack for two cases. In Figures 3(b) and 3(d), it represents fv, g1v, and g2v of PZT-2&ZnO versus the normalized speed v/d and v/c of the dynamic crack for two cases.Figure 3
Variations offv, g1v, and g2v versus the nondimensional velocity of the dynamic crack propagation for different piezoelectric bimaterials. (a-b) v/d. (c-d) v/c.
(a)(b)(c)(d)It is obvious that the nondimension functionsfv and g1v decrease with the velocity v of the crack propagation increasing, as shown in Figure 3. The nondimension functions fv and g1v reach zero, with the velocity v of the dynamic crack propagation reaching the slower velocity d. It is consistent with [25, 26]. However, the nondimension function g2v does not necessarily decrease to zero and is relatively complex, with the dynamic crack speed reaching the Bleustein–Gulyaev wave speed d. It has an initial decrease as the speed v of the dynamic crack propagation increases from zero. The variations increase again as the speed v of the dynamic crack propagation reaches Bleustein–Gulyaev d.The whole process of changes are consistent with [25, 26].


### 5.3. The Influence of Material Constants, Velocity of the Crack Propagation, and Time of Impact

Figure4(a) shows the change of the nondimension functions fv with the electromechanical coefficient k, with each plotted against the nondimension velocity v/c for five different piezoelectric bimaterials. Suppose the values of k are larger, the shape of the curve flats. Figure 4(b) shows the change of the nondimension functions fv with the electromechanical coefficient k, with each plotted against the nondimensional velocity v/d for five different piezoelectric bimaterials. As shown, the nondimension functions fv are controlled by the electromechanical coefficient k, that is, if the values of k are larger, the nondimension functions fv are lower.Figure 4
Variations offv versus the nondimensional velocity of the dynamic crack propagation for different piezoelectric bimaterials. (a) v/c. (b) v/d.
(a)(b)Figures5(a) and 5(b) show the change of the nondimensional functions gv with the electromechanical coefficient k, with each plotted against the nondimension velocity v/c (v/d) for five different piezoelectric bimaterials. In Figure 5, the change of each function gv remains almost constant as k increases. The change of each gv drops sharply as k approaches unity. The Bleustein–Gulyaev wave speed d and the shear wave speed c both increase as k increases. The change of gv decreases as c increases, and the change of gv increases as c increases. However, the effect is not an obvious effect when k is at low values. The variation of gv is complex, and the nondimension functions gv attains a finite nonzero value with v approaches c or d. The change of gv is affected by the Bleustein–Gulyaev wave term 1−kk2. Since 1−kk2 go to zero with v approaching the velocity d of Bleustein–Gulyaev wave, the functions gv do not have to go to zero at large v=d.Figure 5
Variations ofgv versus the nondimensional velocity of the dynamic crack propagation for different piezoelectric bimaterials. (a) v/c. (b) v/d.
(a)(b)It is obvious that the nondimension functionsfv and g1v decrease with the velocity v of the crack propagation increasing, as shown in Figures 6(a) and 6(b). The nondimension functions fv and g1v reach zero, with the velocity v of the dynamic crack propagation reaching the slower velocity d. It is similar to the nondimension function KIIIv of purely elastic solids. However, the nondimension function g2v does not necessarily decrease to zero and is relatively complex, with the dynamic crack speed reaching the Bleustein–Gulyaev wave speed d. In Figures 6(c) and 6(d), it represents fv, g1v, and g2v versus the normalized speed v/d of the dynamic crack for two cases. It is clear that the function fv decreases as the velocity of the dynamic crack propagation increases. Because the material is consistent, the speed of the Bleustein–Gulyaev wave and the speed of the Maerfeld–Tournois wave are the same. The nondimension functions g1v and g2v are identical in the condition. It has an initial decrease as the speed v of the dynamic crack propagation increases from zero. The variations increase again as the speed v of the dynamic crack propagation reaches the Bleustein–Gulyaev d.Figure 6
Example: the nondimension functions,fv, g1v, and g2v, for the following pairs of half-spaces: (a) PZT-4 & PZT-5H. (b) PZT-5H & ZnO. (c) BaTiO3 & BaTiO3 (g1v  =  g2v  =  gv). (d) PZT-5H & PZT-5H (g1v  =  g2v  =  gv).
(a)(b)(c)(d)To verify the accuracy of the proposed theoretical calculation, a central crack of the piezoelectric bimaterials is under a mechanical impact loading. The impact is caused by mechanical loadingτt=τ0Ht, where τ0 is the loading amplitude. The M is a parameter of v/c, and a is a parameter to measure the strength of the dynamic crack propagation. In this section, we examine the performance of fv through six different piezoelectric bimaterials for which the experimental results are available. Figures 7(a) – 7(c) show that the time t of six different piezoelectric bimaterials take to attain a=1cm in M=0.3, M=0.5, and M=0.8, respectively. The following characteristics are obvious: (a) the dependence of fv on the electromechanical coefficient k is significant for a given M; (b) the values of fv decrease with increasing the electromechanical coefficient k for a given time and a given M; (c) it is found that for a given strength of the crack, the length of time is independent of the electromechanical coefficient k. This characteristic has also not been reported in the piezoelectric bimaterials.Figure 7
The time of six different piezoelectric bimaterials inM when a takes to attain 1 cm: (a) M=0.3. (b) M=0.5. (c) M=0.8.
(a)(b)(c)In order to analyzefv with M=0.3, M=0.5, and M=0.8 of the proposed method, we examined PZT-4&PZT-4, ZnO&ZnO, and PZT65/35&PZT65/35, respectively. The experimental setup is shown in Figures 8(a)–8(c). The following characteristics are obvious: (a) the dependence of fv on M is significant for given piezoelectric bimaterials; (b) the values of fv decrease with increasing M; (c) it is found that for a given strength of the crack, the larger M is, the smaller the time is, under the same loading. This characteristic has also not been reported in the piezoelectric bimaterials.Figure 8
Variation offv in M=0.3, M=0.5, and M=0.8 with time when a takes to attain 1 cm: (a) PZT-4 & PZT-4. (b) ZnO & ZnO. (c) PZT65/35 & PZT65/35.
(a)(b)(c)In Figures9(a)–9(c), the effects of the electromechanical coupling coefficient k on KIIIt,v/τ0 are investigated numerically by calculating KIIIt,v/τ0 of six different piezoelectric bimaterials, respectively. In this study, we specify M=0.5, M=0.6, and M=0.7, respectively, and the corresponding results of KIIIt,v/τ0 are shown in Figures 9(a)–9(c). The calculated results KIIIt,v/τ0 of six different curves are shown. The following characteristics are obvious: (a) the dependence of KIIIt,v/τ0 on the electromechanical coefficient k is significant; (b) it can also be observed that KIIIt,v/τ0 are equal to zero at t=0. The results show that the electromechanical coupling coefficient k of the piezoelectric half-space can be increased from zero to unity. In contrast, the material properties of the elastic half-space are held constant; (c) the values of KIIIt,v/τ0 decrease with increasing electromechanical coefficient k for a given time and a given v. This characteristic has also not been reported in the piezoelectric bimaterials.Figure 9
Variation ofKIIIt,v/τ0 with time t for six different piezoelectric bimaterials: (a) M=0.5. (b) M=0.6. (c) M=0.7.
(a)(b)(c)In Figures10(a)–10(c), the calculated results KIIIt,v/τ0 of four different curves are shown. The following characteristics are obvious: (a) the dependence of KIIIt,v/τ0 and M is significant; (b) the values of the KIIIt,v/τ0 decrease with increasing electromechanical coefficient k; (c) the values of KIIIt,v/τ0 increase with decreasing M; (d) as a consequence, the effects of the electromechanical coefficient k on KIIIt,v/τ0 in piezoelectric bimaterials are significant.Figure 10
Variation ofKIIIt,v/τ0 with time t in M=0.5 and M=0.6: (a) PZT-4 & PZT-4 and PZT-4 & BaTiO3. (b) PZT-5H and PZT-5H & PZT-5H & ZnO. (c) PZT-2 & PZT-2 and PZT-2 & ZnO.
(a)(b)(c)


## 5.1. Study of Degenerate into the Simple Case

When bimaterials degenerate into a single material, it follows that(103)KIIIt,v=fvKIIIt,0,KDt,v=gvKDt,0,where fv=1−v/d/1−v/cL−∗v, gv=1−ke2/s−ke2fv.It is noted that the conclusions agree with [20, 21]. We choose the single material PZT-4 and PZT-5H, respectively, for research. The corresponding analysis diagram is shown in Figure 2. They show the variations fv and gv with the normalized speed v/d of the dynamic crack propagation. As shown in Figure 2(a), fv decreases with increasing velocity v of the dynamic crack propagation. The variations fv vanish when v reaches Bleustein–Gulyaev d. Furthermore, the nondimensional function of the stress fv for an antiplane crack propagating in piezoelectric solids as considered is plotted for comparison [20, 21]. Their changes are consistent. However, as shown in Figure 2(a), when the speed of the dynamic crack propagation reaches Bleustein–Gulyaev d, the variations gv do not have to go to zero. It has an initial decrease as the speed v of the dynamic crack propagation increases from zero. The variations gv increase again as the speed v of the dynamic crack propagation reaches Bleustein–Gulyaev d. Furthermore, the variations gv for the crack propagating in piezoelectric solids [20, 21] as considered are plotted for comparison. Their changes are consistent.Figure 2
Variations offv and gv versus the nondimensional velocity v/d for a single material PZT-4 or PZT-5H.
(a)(b)


## 5.2. Analysis and Comparison with the Existing References

Some analyses and comparisons are made between the results obtained by us and the existing references [25, 26] (the methods used are different from those we used). We choose PZT-4&PZT65/35 and PZT-2&ZnO, respectively, for research. The corresponding analysis diagram is shown in Figure 3. In Figures 3(a) and 3(c), it represents fv, g1v, and g2v of PZT-4&PZT65/35 versus the normalized speed v/d and v/c of the dynamic crack for two cases. In Figures 3(b) and 3(d), it represents fv, g1v, and g2v of PZT-2&ZnO versus the normalized speed v/d and v/c of the dynamic crack for two cases.Figure 3
Variations offv, g1v, and g2v versus the nondimensional velocity of the dynamic crack propagation for different piezoelectric bimaterials. (a-b) v/d. (c-d) v/c.
(a)(b)(c)(d)It is obvious that the nondimension functionsfv and g1v decrease with the velocity v of the crack propagation increasing, as shown in Figure 3. The nondimension functions fv and g1v reach zero, with the velocity v of the dynamic crack propagation reaching the slower velocity d. It is consistent with [25, 26]. However, the nondimension function g2v does not necessarily decrease to zero and is relatively complex, with the dynamic crack speed reaching the Bleustein–Gulyaev wave speed d. It has an initial decrease as the speed v of the dynamic crack propagation increases from zero. The variations increase again as the speed v of the dynamic crack propagation reaches Bleustein–Gulyaev d.The whole process of changes are consistent with [25, 26].


## 5.3. The Influence of Material Constants, Velocity of the Crack Propagation, and Time of Impact

Figure4(a) shows the change of the nondimension functions fv with the electromechanical coefficient k, with each plotted against the nondimension velocity v/c for five different piezoelectric bimaterials. Suppose the values of k are larger, the shape of the curve flats. Figure 4(b) shows the change of the nondimension functions fv with the electromechanical coefficient k, with each plotted against the nondimensional velocity v/d for five different piezoelectric bimaterials. As shown, the nondimension functions fv are controlled by the electromechanical coefficient k, that is, if the values of k are larger, the nondimension functions fv are lower.Figure 4
Variations offv versus the nondimensional velocity of the dynamic crack propagation for different piezoelectric bimaterials. (a) v/c. (b) v/d.
(a)(b)Figures5(a) and 5(b) show the change of the nondimensional functions gv with the electromechanical coefficient k, with each plotted against the nondimension velocity v/c (v/d) for five different piezoelectric bimaterials. In Figure 5, the change of each function gv remains almost constant as k increases. The change of each gv drops sharply as k approaches unity. The Bleustein–Gulyaev wave speed d and the shear wave speed c both increase as k increases. The change of gv decreases as c increases, and the change of gv increases as c increases. However, the effect is not an obvious effect when k is at low values. The variation of gv is complex, and the nondimension functions gv attains a finite nonzero value with v approaches c or d. The change of gv is affected by the Bleustein–Gulyaev wave term 1−kk2. Since 1−kk2 go to zero with v approaching the velocity d of Bleustein–Gulyaev wave, the functions gv do not have to go to zero at large v=d.Figure 5
Variations ofgv versus the nondimensional velocity of the dynamic crack propagation for different piezoelectric bimaterials. (a) v/c. (b) v/d.
(a)(b)It is obvious that the nondimension functionsfv and g1v decrease with the velocity v of the crack propagation increasing, as shown in Figures 6(a) and 6(b). The nondimension functions fv and g1v reach zero, with the velocity v of the dynamic crack propagation reaching the slower velocity d. It is similar to the nondimension function KIIIv of purely elastic solids. However, the nondimension function g2v does not necessarily decrease to zero and is relatively complex, with the dynamic crack speed reaching the Bleustein–Gulyaev wave speed d. In Figures 6(c) and 6(d), it represents fv, g1v, and g2v versus the normalized speed v/d of the dynamic crack for two cases. It is clear that the function fv decreases as the velocity of the dynamic crack propagation increases. Because the material is consistent, the speed of the Bleustein–Gulyaev wave and the speed of the Maerfeld–Tournois wave are the same. The nondimension functions g1v and g2v are identical in the condition. It has an initial decrease as the speed v of the dynamic crack propagation increases from zero. The variations increase again as the speed v of the dynamic crack propagation reaches the Bleustein–Gulyaev d.Figure 6
Example: the nondimension functions,fv, g1v, and g2v, for the following pairs of half-spaces: (a) PZT-4 & PZT-5H. (b) PZT-5H & ZnO. (c) BaTiO3 & BaTiO3 (g1v  =  g2v  =  gv). (d) PZT-5H & PZT-5H (g1v  =  g2v  =  gv).
(a)(b)(c)(d)To verify the accuracy of the proposed theoretical calculation, a central crack of the piezoelectric bimaterials is under a mechanical impact loading. The impact is caused by mechanical loadingτt=τ0Ht, where τ0 is the loading amplitude. The M is a parameter of v/c, and a is a parameter to measure the strength of the dynamic crack propagation. In this section, we examine the performance of fv through six different piezoelectric bimaterials for which the experimental results are available. Figures 7(a) – 7(c) show that the time t of six different piezoelectric bimaterials take to attain a=1cm in M=0.3, M=0.5, and M=0.8, respectively. The following characteristics are obvious: (a) the dependence of fv on the electromechanical coefficient k is significant for a given M; (b) the values of fv decrease with increasing the electromechanical coefficient k for a given time and a given M; (c) it is found that for a given strength of the crack, the length of time is independent of the electromechanical coefficient k. This characteristic has also not been reported in the piezoelectric bimaterials.Figure 7
The time of six different piezoelectric bimaterials inM when a takes to attain 1 cm: (a) M=0.3. (b) M=0.5. (c) M=0.8.
(a)(b)(c)In order to analyzefv with M=0.3, M=0.5, and M=0.8 of the proposed method, we examined PZT-4&PZT-4, ZnO&ZnO, and PZT65/35&PZT65/35, respectively. The experimental setup is shown in Figures 8(a)–8(c). The following characteristics are obvious: (a) the dependence of fv on M is significant for given piezoelectric bimaterials; (b) the values of fv decrease with increasing M; (c) it is found that for a given strength of the crack, the larger M is, the smaller the time is, under the same loading. This characteristic has also not been reported in the piezoelectric bimaterials.Figure 8
Variation offv in M=0.3, M=0.5, and M=0.8 with time when a takes to attain 1 cm: (a) PZT-4 & PZT-4. (b) ZnO & ZnO. (c) PZT65/35 & PZT65/35.
(a)(b)(c)In Figures9(a)–9(c), the effects of the electromechanical coupling coefficient k on KIIIt,v/τ0 are investigated numerically by calculating KIIIt,v/τ0 of six different piezoelectric bimaterials, respectively. In this study, we specify M=0.5, M=0.6, and M=0.7, respectively, and the corresponding results of KIIIt,v/τ0 are shown in Figures 9(a)–9(c). The calculated results KIIIt,v/τ0 of six different curves are shown. The following characteristics are obvious: (a) the dependence of KIIIt,v/τ0 on the electromechanical coefficient k is significant; (b) it can also be observed that KIIIt,v/τ0 are equal to zero at t=0. The results show that the electromechanical coupling coefficient k of the piezoelectric half-space can be increased from zero to unity. In contrast, the material properties of the elastic half-space are held constant; (c) the values of KIIIt,v/τ0 decrease with increasing electromechanical coefficient k for a given time and a given v. This characteristic has also not been reported in the piezoelectric bimaterials.Figure 9
Variation ofKIIIt,v/τ0 with time t for six different piezoelectric bimaterials: (a) M=0.5. (b) M=0.6. (c) M=0.7.
(a)(b)(c)In Figures10(a)–10(c), the calculated results KIIIt,v/τ0 of four different curves are shown. The following characteristics are obvious: (a) the dependence of KIIIt,v/τ0 and M is significant; (b) the values of the KIIIt,v/τ0 decrease with increasing electromechanical coefficient k; (c) the values of KIIIt,v/τ0 increase with decreasing M; (d) as a consequence, the effects of the electromechanical coefficient k on KIIIt,v/τ0 in piezoelectric bimaterials are significant.Figure 10
Variation ofKIIIt,v/τ0 with time t in M=0.5 and M=0.6: (a) PZT-4 & PZT-4 and PZT-4 & BaTiO3. (b) PZT-5H and PZT-5H & PZT-5H & ZnO. (c) PZT-2 & PZT-2 and PZT-2 & ZnO.
(a)(b)(c)


## 6. Conclusions

The dynamic behavior of an interfacial mode-III crack under mechanical impact loading is considered. The dynamic stress intensity factor, the dynamic electric displacement intensity factor, and their nondimension functions are derived with the analytical expressions. According to the solution established in the study, we have some conclusions: (1) the behavior of the dynamic fracture in piezoelectric bimaterials is more to do with the electromechanical coefficientk; (2) the dynamic stress intensity factor always vanishes with the velocity of dynamic crack propagation arriving at the speed of the generalized Raleigh wave; (3) it is verified that the nondimensional stress intensity factors are controlled by the electromechanical coefficient k, that is, the bigger the electromechanical coefficient k, the smaller the nondimensional stress intensity factor; (4) the existence of the Maerfeld–Tournois wave will increase the value of the dynamic intensity factors; (5) the values of KIIIt,v/τ0 increase with M decreasing; (6) consequently, the effects of the electromechanical coefficient k on KIIIt,v/τ0 in piezoelectric bimaterials are significant; (7) it is found that for a given strength of the crack, the larger M is, the smaller the time is, under the same loading. In a word, it analyzes the relationship between the dynamic crack change of piezoelectric bimaterials with time, which makes the theoretical analysis more perfect and provides more detailed theoretical support for engineering applications.The stress and electric field concentration caused by cracks will lead to the loss of its design function. The fracture of piezoelectric materials we studied is mainly a piezoelectric solid with cracks. The propagation of cracks is studied to provide theoretical reference for reliability analysis and prolonging life-span. The cracks of piezoelectric materials include mode-I, mode-II, mode-III, and compound. The research idea is consisted of analyzing fracture, selecting the type of crack, establishing the model, defining the boundary conditions, and confirming fracture criterion. In this study, the existing crack is mode-III crack. Dynamic propagation characteristics of the crack under impact loading are presented. It is considered under the in-plane electric field and out-of-plane displacement. The mode- I, mode-II, and composite cracks will be analyzed and discussed in the future research. At the same time, nanoscale piezoelectric bimaterials are widely used in daily life. Our analysis in this section does not apply to nanomaterials. In the next research, we can consider discussing the interface cracks of nanoscale piezoelectric bimaterials in not only analyzing the semi-infinite III cracks but also thinking about deeply the influence of nanomaterial thickness on material design.



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*Source: 1733011-2022-03-27.xml*