# Interaction of Screw Dislocation with Edge Interfacial Crack on Fine-Grained Piezoelectric Coating/Substrate

Authors: Shuaishuai Hu; Jiansheng Liu; Junlin Li; Xiufeng Xie
Journal: Advances in Materials Science and Engineering
 (2019)

---

## Abstract

The interaction between micro- and macrocracks in a fine-grained piezoelectric coating/substrate under remote antiplane mechanical and in-plane electrical loadings was studied. The principle of superposition and a mapping function method was used to transform the fine-grained coating/substrate structure containing the screw dislocation and the edge interfacial crack into the right semi-infinite plane piezoelectric bimaterial with screw dislocation to simplify the problem. Furthermore, the electric field, displacement field, intensity factors, and image force of these two problems were established. In addition, numerical calculations were then given graphically to study the effects of the elastic modulus of the material, the size of the crack, the thickness of the coating, and the screw dislocation angle on the edge interface crack and dislocation.


---

## Body

## 1. Introduction

Piezoelectric materials have been widely applied to modern devices, such as sensors, actuators, and transducers, due to their intrinsic electromechanical coupling ability. At the same time, it is easy to affect the performance and strength of the devices and structures on a number of scales (micro and macro, e.g., dislocations and cracks) based on the characteristics of the piezoelectric material and the defects of the processing technology. Therefore, the defects of piezoelectric materials under mechanical and electrical loads have attracted wide attention [1–4]. Pak [5] derived the interaction energy for two different internal stress-field systems and utilized the standard method and a generalized path-independent integral to calculate the force on a piezoelectric screw dislocation subjected to external mechanical and electrical loads. Liu et al. [6] revealed the degree of influence of electromechanical coupling behavior on the image force on a screw dislocation. Jin and Fang [7] investigated the electroelastic coupling interaction between multiple screw dislocations that may be located either outside or inside the inclusion and a circular inclusion with an imperfect interface in a piezoelectric solid. Wang and Xu [8] analytically discovered the contribution of surface piezoelectricity to the interaction between a piezoelectric screw dislocation and a finite crack in a hexagonal piezoelectric solid. Bagheri et al. [9] explored the problem of a screw dislocation located in a substrate imperfectly bonded to a coating, and the dislocation density was used to determine the field intensity factors. Zhang et al. [10] investigated the interaction between a screw dislocation and a circular nano-inhomogeneity with a semi-infinite wedge crack penetrating the interface, and the influence of elastic mismatch of materials, inhomogeneity size, interface stress, and wedge-crack opening angle was studied. Shen and Hung [11] examined the interaction between a screw dislocation and an oblique edge crack in a semi-infinite piezoelectric medium under remote antiplane mechanical and in-plane electrical loadings. Liu et al. [12] dealt with the influence of a Kelvin-type viscoelastic interface on the generation of screw dislocations near the interfacial blunt crack tip in light of a pair of concentrated loads via the integral transform and conformal mapping. Fan et al. [13, 14] investigated a curved interface crack between a circular inclusion and an infinite matrix and discussed the influence of the material properties and other geometric parameters on the stress intensity factors, plastic zone size, and crack tip opening displacement. In another article considered, the elastic-plastic stress analysis on a matrix crack interacting with nearby circular inclusions in fiber-reinforced composites was considered, and with the aid of the distributed dislocation method, the effective stress intensity factor, PZS, and crack tip opening displacement were evaluated.However, all of polycrystalline materials studied above composed of multidomain grains of size from several microns to tens of micrometers, and the sizes cannot meet the requirements of the present work. Reducing the particle size to the submicron level can improve the machinability of the material and the mechanical strength of the resulting device. In recent years, fine-grained piezoelectric materials have been used for cutting and grinding, and some high-frequency transducers, microbrakes, and thin buzzers have been fabricated in which the advantages of fine-grained piezoelectric materials have been proved. In these works, the effects of grain size on the phase structure, density, and the dielectric and piezoelectric properties were investigated [15–17]. However, the mechanical properties of fine-grained piezoelectric materials with defects were not studied, and fine-grained piezoelectric materials are required to be in the form of coatings, especially in some special applications.In the work described in this paper, the fine-grained piezoelectric coating/substrate mechanical model was established, in which the interaction between the screw dislocation and the edge interfacial crack was presented. The superposition principle and the mapping function method were utilized to transform the coating/substrate structure into the right semi-infinite plane piezoelectric bimaterial structure, and the analytical expressions of the image force and intensity factor for the mechanical model were obtained. Furthermore, the influence of various material parameters on image force and intensity factor was analyzed by numerical calculation.


## 2. Problem Formulation

As shown in Figure1, the coating and the substrate are bonded together by plasma spraying, while the edge crack with a length of a is along the interface between the coating and substrate. The coating/substrate structure is considered as a strip with limited thickness along Y axis and infinite length along X axis. The thicknesses of the coating and the substrate are h1 and h2, respectively. Assume that the poling direction Z normal to the X−Y plane and the screw dislocation b=bz,bϕ with dislocation line parallel to the poling direction are situated at z0 in fine-grained piezoelectric coating, in which bz and bϕ are the classical elastic displacement jump and the electric potential jump, respectively. The coating/substrate structure are linearly elastic, homogeneous, isotropic, and piezoelectric. Their physical properties are defined by the elastic constants (c44), the piezoelectric constants (e15), and the dielectric constants (ε11).Figure 1
The fine-grained piezoelectric coating/substrate structure.The piezoelectric boundary value problem is only considered in the case of out of plane displacementw and in-plane electric potential ϕ field such that(1)U=V=0,W=Wx,y,Φ=Φx,y.In this case, the constitutive equations become(2)σxz=c44n∂W∂x+e15n∂Φ∂x,(3)Dx=e15n∂W∂x−ε11n∂Φ∂x,(4)σyz=c44n∂W∂y+e15n∂Φ∂y,(5)Dy=e15n∂W∂y−ε11n∂Φ∂y.In equations (2)–(5), σmz is the stress tensor, Dmm=x,y is the electric displacement vector, c44n is the elastic modulus measured at constant electric field, e15n and ε11n are the piezoelectric and dielectric constants, and the superscript nn=1,2 stands for the upper and lower piezoelectric materials, respectively.The electric field can be given by the following form:(6)Ex=−∂Φ∂x,Ey=−∂Φ∂y.According to [1], the governing equations are simplified to(7)c44n∇2W+e15n∇2Φ=0,(8)e15n∇2W−ε11n∇2Φ=0,n=1,2.The governing equations (7) and (8) can be reduced to the more compact form:(9)∇2W=0,∇2Φ=0,where ∇2 is the two-dimensional Laplacian operator.For simplicity, we define the displacement vectoru=W,ΦΤ and the stress vector tm=σmz,DmΤ, where the Τ represents the transposing of the matrix.Consequently equations (2)–(5) and (9) can be made in the forms(10)∇2u=0,tm=Ln∂u∂m,m=x,y,where(11)Ln=c44ne15ne15n−ε11n,n=1,2.According to the theory of complex functions, the general solution to equation (10) takes the following form:(12)u=Imfz,in which fz=f1z,f2zΤ is an analytical function vector of the complex variable z=x+iy, where Im stands for the imaginary part and i=−1. Substituting equations (11) into (10), the expressions of stress and electric displacement are obtained:(13)ty+itx=Lnf′z.By using the principle of superposition, the problem can be regarded as the sum of a homogeneous problem without crack and a nonhomogeneous problem of edge interface crack and screw dislocation.The boundary conditions can be written as(14)σyzx,h1=σyzx,−h2=0,0≤x<+∞,Dyx,h1=Dyx,−h2=0,0≤x<+∞,σyzx,0+=σyzx,0−,x>a,Dyx,0+=Dyx,0−,x>a,Wx,0+=Wx,0−,x>a,Φx,0+=Φx,0−,x>a,σyzx,0+=σyzx,0−=0,x>a,Dyx,0+=Dyx,0−=0,x>a,∮Ωdu=b,∮Ωtydx−txdy=0,where Ω stands for the any closed curve around the dislocation z0.


## 3. Solution to the Problem

### 3.1. Transformation of Problem

According to the above discussion, we only need to determine the analytic function offz. By using the principle of superposition, the problem stated in the previous section can be expressed in Figure 2.Figure 2
Superposition scheme: (a) a screw dislocation near interface of the fine-grained piezoelectric coating/substrate; (b) an edge interfacial crack of the half semi-infinite plane piezoelectric bimaterials.
(a)
(b)In order to obtain the solution of the problem shown in Figure2, two mapping functions were used to simplify the problem:(15)ζ=−i1+ieπ/h1+h22−h1−h2/2i1−ieπ/h1+h22−h1−h2/2i2,(16)ξ=ζ2−a2,which transforms the regions occupied by piezoelectric coating and dislocation in the z-plane into the right semi-infinite of the ξ plane and dislocation point changed from z0 to ξ0, as follows, in Figures 3(c) and 3(d), where η and ς are the real and imaginary parts of the corresponding complex variables, respectively. Equations (15) and (16) are explained in Figures 4 and 5 in Appendix.Figure 3
A screw dislocation interaction with an edge interfacial crack in the right semi-infinite plane piezoelectric bimaterials.Figure 4Figure 5


### 3.2. Screw Dislocation of the Right Semi-Infinite Piezoelectric Composite Medium

From the above statement, it can be noted that what we only need is to get the screw dislocation in the right semi-infinite piezoelectric composite medium.According to the boundary conditions, assume that the unknown functionsf1z and f2z have the following forms in the mapped plane [6]:(17)f1z=f1sz+f1pz,(18)f2z=f2pz,(19)f1pz=Alnz−z0−lnz+z0¯,(20)f1sz=G⋅f1p¯z,(21)f2pz=I−G⋅f1pz,A=b2π,where f1sz represents the function associated with the unperturbed fields which are related to the solution of the corresponding homogeneous media subject to a piezoelectric screw dislocation and holomorphic in the entire domain except at z0, f1pz, and f2pz are the holomorphic function corresponding to the perturbed fields in the domain where the coating and substrate are located, the overbar denotes the complex conjugate, and I is a 2×2 unit matrix. The matrix G is(22)G=L1+L2−1L2−L1.Substituting equation (19)–(20) into equations (17) and (18), the functions f1z and f2z in the z-plane can be written as(23)f1z=b2πlnz−z0−lnz+z0¯+Glnz−z0¯−lnz+z0,(24)f2z=b2πI−Glnz−z0−lnz+z0¯.By equation (13), the stresses and electric displacements can be obtained.


## 3.1. Transformation of Problem

According to the above discussion, we only need to determine the analytic function offz. By using the principle of superposition, the problem stated in the previous section can be expressed in Figure 2.Figure 2
Superposition scheme: (a) a screw dislocation near interface of the fine-grained piezoelectric coating/substrate; (b) an edge interfacial crack of the half semi-infinite plane piezoelectric bimaterials.
(a)
(b)In order to obtain the solution of the problem shown in Figure2, two mapping functions were used to simplify the problem:(15)ζ=−i1+ieπ/h1+h22−h1−h2/2i1−ieπ/h1+h22−h1−h2/2i2,(16)ξ=ζ2−a2,which transforms the regions occupied by piezoelectric coating and dislocation in the z-plane into the right semi-infinite of the ξ plane and dislocation point changed from z0 to ξ0, as follows, in Figures 3(c) and 3(d), where η and ς are the real and imaginary parts of the corresponding complex variables, respectively. Equations (15) and (16) are explained in Figures 4 and 5 in Appendix.Figure 3
A screw dislocation interaction with an edge interfacial crack in the right semi-infinite plane piezoelectric bimaterials.Figure 4Figure 5


## 3.2. Screw Dislocation of the Right Semi-Infinite Piezoelectric Composite Medium

From the above statement, it can be noted that what we only need is to get the screw dislocation in the right semi-infinite piezoelectric composite medium.According to the boundary conditions, assume that the unknown functionsf1z and f2z have the following forms in the mapped plane [6]:(17)f1z=f1sz+f1pz,(18)f2z=f2pz,(19)f1pz=Alnz−z0−lnz+z0¯,(20)f1sz=G⋅f1p¯z,(21)f2pz=I−G⋅f1pz,A=b2π,where f1sz represents the function associated with the unperturbed fields which are related to the solution of the corresponding homogeneous media subject to a piezoelectric screw dislocation and holomorphic in the entire domain except at z0, f1pz, and f2pz are the holomorphic function corresponding to the perturbed fields in the domain where the coating and substrate are located, the overbar denotes the complex conjugate, and I is a 2×2 unit matrix. The matrix G is(22)G=L1+L2−1L2−L1.Substituting equation (19)–(20) into equations (17) and (18), the functions f1z and f2z in the z-plane can be written as(23)f1z=b2πlnz−z0−lnz+z0¯+Glnz−z0¯−lnz+z0,(24)f2z=b2πI−Glnz−z0−lnz+z0¯.By equation (13), the stresses and electric displacements can be obtained.


## 4. Image Force and Intensity Factor

### 4.1. Image Force on a Screw Dislocation

As shown in Figures3(c) and 3(d), in order to get the image force of the problem, in equations (23) and (24), use ξ instead of z. We got(25)f1ξ=b2πlnξ−ξ0−lnξ+ξ0¯+Glnξ−ξ0¯−lnξ+ξ0,(26)f2ξ=b2πI−Glnξ−ξ0−lnξ+ξ0¯.Substituting (16) into equations (25) and (26), we got(27)f1ζ=b2πlnζ2−a2−ζ02−a2−lnζ2−a2+ζ0¯2−a2+Glnζ2−a2−ζ0¯2−a2−lnζ2−a2+ζ02−a2,(28)f2ζ=b2πI−Glnζ2−a2−ζ02−a2−lnζ2−a2+ζ0¯2−a2.Take the derivative of equations (27) and (28):(29)f1′ζ=b2πζζ2−a2X1−GY1,f2′ζ=b2πI−Gζζ2−a2X1,where(30)X1=1ζ2−a2−ζ02−a2−1ζ2−a2+ζ0¯2−a2,Y1=1ζ2−a2−ζ0¯2−a2−1ζ2−a2+ζ02−a2.According to the generalized Peach–Koehler formula derived in [5] for piezoelectric materials, we have(31)Fx=bΤtyp1ζ0,Fy=−bΤtxp1ζ0,where tmp1ζ0m=x,y is the perturbed stresses and electric displacements at z0.By equation (13), the following equation can be obtained:(32)Fx−iFy=bΤtyp1ζ0+itxp1ζ0=bΤL1f1′−f1∞′,with(33)f1∞=b2πlnζ2−a2−ζ02−a2.From equations (32), the image force of the right semi-infinite plane piezoelectric bimaterial can be obtained:(34)Fx−iFy=bbΤ2πL1−1ζ02−a2+ζ0¯2−a2−GY1⋅ζ0ζ02−a2,(35)Fx−iFy=bbΤ2π−1−c121+c441υ2c12+c441υ1+c12Y1−1−c12ζ02−a2+ζ0¯2−a2ζ02−ζ0¯2⋅ζ0ζ02−a2,(36)c12=c441c442,υ=ε111e1512.Here, we only need to get the real and imaginary parts of equation (34), and then the image force of the screw dislocation as shown in Figure 2(b) can be obtained.Substituting equation (15) into equations (27) and (28), we got(37)f1z=b2πln1+ieπ/h1+h2z−h1−h2/2i1−ieπ/h1+h2z−h1−h2/2i4−a2−1+ieπ/h1+h2z0−h1−h2/2i1−ieπ/h1+h2z0−h1−h2/2i4−a2−ln1+ieπ/h1+h2z−h1−h2/2i1−ieπ/h1+h2z−h1−h2/2i4−a2+1+ieπ/h1+h2z0¯−h1−h2/2i1−ieπ/h1+h2z0¯−h1−h2/2i4−a2+Gln1+ieπ/h1+h2z−h1−h2/2i1−ieπ/h1+h2z−h1−h2/2i4−a2−1+ieπ/h1+h2z0¯−h1−h2/2i1−ieπ/h1+h2z0¯−h1−h2/2i4−a2−ln1+ieπ/h1+h2z−h1−h2/2i1−ieπ/h1+h2z−h1−h2/2i4−a2+1+ieπ/h1+h2z0−h1−h2/2i1−ieπ/h1+h2z0−h1−h2/2i4−a2,(38)f2z=b2πI−Gln1+ieπ/h1+h2z−h1−h2/2i1−ieπ/h1+h2z−h1−h2/2i4−a2−1+ieπ/h1+h2z0−h1−h2/2i1−ieπ/h1+h2z0−h1−h2/2i4−a2−ln1+ieπ/h1+h2z−h1−h2/2i1−ieπ/h1+h2z−h1−h2/2i4−a2+1+ieπ/h1+h2z0¯−h1−h2/2i1−ieπ/h1+h2z0¯−h1−h2/2i4−a2.Take the derivative of equations (37) and (38):(39)f1′z=−2beπ/h1+h2z−h1−h2/2ih1+h21−ieπ/h1+h2z−h1−h2/2i2M11−N11+GM12−N12⋅P1,f2′z=−2beπ/h1+h2z−h1−h2/2ih1+h21−ieπ/h1+h2z−h1−h2/2i2I−GM11−N11⋅P1,where(40)M11=11+ieπ/h1+h2z−h1−h2/2i/1−ieπ/h1+h2z−h1−h2/2i4−a2−1+ieπ/h1+h2z−h1−h2/2i/1−ieπ/h1+h2z0−h1−h2/2i4−a2,N11=11+ieπ/h1+h2z−h1−h2/2i/1−ieπ/h1+h2z−h1−h2/2i4−a2+1+ieπ/h1+h2z0¯−h1−h2/2i/1−ieπ/h1+h2z0¯−h1−h2/2i4−a2,M12=11+ieπ/h1+h2z−h1−h2/2i/1−ieπ/h1+h2z−h1−h2/2i4−a2−1+ieπ/h1+h2z0¯−h1−h2/2i/1−ieπ/h1+h2z0¯−h1−h2/2i4−a2,N12=11+ieπ/h1+h2z−h1−h2/2i/1−ieπ/h1+h2z−h1−h2/2i4−a2+1+ieπ/h1+h2z0−h1−h2/2i/1−ieπ/h1+h2z0−h1−h2/2i4−a2,P1=11+ieπ/h1+h2z−h1−h2/2i/1−ieπ/h1+h2z−h1−h2/2i4−a2⋅1+ieπ/h1+h2z−h1−h2/2i1−ieπ/h1+h2z−h1−h2/2i3.From the previous discussion, the image force of the fine-grained piezoelectric coating/substrate can be expressed as(41)Fx−iFy=−2L1G⋅eπ/h1+h2z0−h1−h2/2ibbΤh1+h21−ieπ/h1+h2z0−h1−h2/2i2−N21+GM22−N22⋅P2,Fx−iFy=−2eπ/h1+h2z0−h1−h2/2ibbΤh1+h21−ieπ/h1+h2z0−h1−h2/2i21−c121+c441υ2c12+c441υ1+c12⋅M22−N22−1−c12N21⋅P2,where(42)c12=c441c442,υ=ε111e1512,N21=11+ieπ/h1+h2z0−h1−h2/2i/1−ieπ/h1+h2z0−h1−h2/2i4−a2+1+ieπ/h1+h2z0¯−h1−h2/2i/1−ieπ/h1+h2z0¯−h1−h2/2i4−a2,M22=11+ieπ/h1+h2z0−h1−h2/2i/1−ieπ/h1+h2z0−h1−h2/2i4−a2−1+ieπ/h1+h2z0¯−h1−h2/2i/1−ieπ/h1+h2z0¯−h1−h2/2i4−a2,N22=11+ieπ/h1+h2z0−h1−h2/2i/1−ieπ/h1+h2z0−h1−h2/2i4−a2+1+ieπ/h1+h2z0−h1−h2/2i/1−ieπ/h1+h2z0−h1−h2/2i4−a2,P2=11+ieπ/h1+h2z0−h1−h2/2i/1−ieπ/h1+h2z0−h1−h2/2i4−a2⋅1+ieπ/h1+h2z0−h1−h2/2i1−ieπ/h1+h2z0−h1−h2/2i3.


### 4.2. Intensity Factors

In studying the interaction between dislocation and crack, the intensity factor can be used to analyze the shielding or antishielding effect of dislocation on cracks. In this section, our attention will focus on the elector-elastic fields near the tip of the edge crack where local elector-elastic field intensification usually occurs. For the antiplane problem of the piezoelectric materials, the intensity factors are defined by(43)K=kσ,kDΤ=limr⟶02πrtyr,0,where kσ is the classical SIF, kD is the EDIF, and ty=σyz,Dy. According to the boundary continuity conditions, tyr,0 in equation (43) can be replaced by t1yr,0 or t2yr,0 (here 1 and 2 represent upper and lower materials, respectively):(44)K=limr⟶02πrt1yr,0=limr⟶02πrt2yr,0.Equation (43) is expressed in the Cartesian coordinate systems, as follows:(45)K=limz⟶a2πz−at1yz.


#### 4.2.1. Intensity Factor of the Right Semi-Infinite Plane Piezoelectric Bimaterial with Screw Dislocation

Using equation (13), equation (45) can be expressed as(46)K1=limz⟶a2πz−at1yz=a4πc441b2ε111+c441υ+1+c12ε1112c12υ+c441υ1+c12⋅Re1z02−a2−z0¯2−a2,where(47)c12=c441c442,υ=ε111e1512.


#### 4.2.2. Intensity Factor of the Fine-Grained Piezoelectric Coating/Substrate with Edge Interfacial Crack and Screw Dislocation

According to the above discussion, substituting equation (15) into equation (46), the intensity factor can be expressed as(48)K2=limz⟶a2πz−at1yz=−c441baaπh1+h22ε111+c441υ+1+c12ε1112c12υ+c441υ1+c12Re11+ieπ/h1+h2z−h1−h2/2i/1−ieπ/h1+h2z−h1−h2/2i4−a2−1+ieπ/h1+h2z0¯−h1−h2/2i/1−ieπ/h1+h2z0¯−h1−h2/2i4−a2,where(49)c12=c441c442,υ=ε111e1512.


## 4.1. Image Force on a Screw Dislocation

As shown in Figures3(c) and 3(d), in order to get the image force of the problem, in equations (23) and (24), use ξ instead of z. We got(25)f1ξ=b2πlnξ−ξ0−lnξ+ξ0¯+Glnξ−ξ0¯−lnξ+ξ0,(26)f2ξ=b2πI−Glnξ−ξ0−lnξ+ξ0¯.Substituting (16) into equations (25) and (26), we got(27)f1ζ=b2πlnζ2−a2−ζ02−a2−lnζ2−a2+ζ0¯2−a2+Glnζ2−a2−ζ0¯2−a2−lnζ2−a2+ζ02−a2,(28)f2ζ=b2πI−Glnζ2−a2−ζ02−a2−lnζ2−a2+ζ0¯2−a2.Take the derivative of equations (27) and (28):(29)f1′ζ=b2πζζ2−a2X1−GY1,f2′ζ=b2πI−Gζζ2−a2X1,where(30)X1=1ζ2−a2−ζ02−a2−1ζ2−a2+ζ0¯2−a2,Y1=1ζ2−a2−ζ0¯2−a2−1ζ2−a2+ζ02−a2.According to the generalized Peach–Koehler formula derived in [5] for piezoelectric materials, we have(31)Fx=bΤtyp1ζ0,Fy=−bΤtxp1ζ0,where tmp1ζ0m=x,y is the perturbed stresses and electric displacements at z0.By equation (13), the following equation can be obtained:(32)Fx−iFy=bΤtyp1ζ0+itxp1ζ0=bΤL1f1′−f1∞′,with(33)f1∞=b2πlnζ2−a2−ζ02−a2.From equations (32), the image force of the right semi-infinite plane piezoelectric bimaterial can be obtained:(34)Fx−iFy=bbΤ2πL1−1ζ02−a2+ζ0¯2−a2−GY1⋅ζ0ζ02−a2,(35)Fx−iFy=bbΤ2π−1−c121+c441υ2c12+c441υ1+c12Y1−1−c12ζ02−a2+ζ0¯2−a2ζ02−ζ0¯2⋅ζ0ζ02−a2,(36)c12=c441c442,υ=ε111e1512.Here, we only need to get the real and imaginary parts of equation (34), and then the image force of the screw dislocation as shown in Figure 2(b) can be obtained.Substituting equation (15) into equations (27) and (28), we got(37)f1z=b2πln1+ieπ/h1+h2z−h1−h2/2i1−ieπ/h1+h2z−h1−h2/2i4−a2−1+ieπ/h1+h2z0−h1−h2/2i1−ieπ/h1+h2z0−h1−h2/2i4−a2−ln1+ieπ/h1+h2z−h1−h2/2i1−ieπ/h1+h2z−h1−h2/2i4−a2+1+ieπ/h1+h2z0¯−h1−h2/2i1−ieπ/h1+h2z0¯−h1−h2/2i4−a2+Gln1+ieπ/h1+h2z−h1−h2/2i1−ieπ/h1+h2z−h1−h2/2i4−a2−1+ieπ/h1+h2z0¯−h1−h2/2i1−ieπ/h1+h2z0¯−h1−h2/2i4−a2−ln1+ieπ/h1+h2z−h1−h2/2i1−ieπ/h1+h2z−h1−h2/2i4−a2+1+ieπ/h1+h2z0−h1−h2/2i1−ieπ/h1+h2z0−h1−h2/2i4−a2,(38)f2z=b2πI−Gln1+ieπ/h1+h2z−h1−h2/2i1−ieπ/h1+h2z−h1−h2/2i4−a2−1+ieπ/h1+h2z0−h1−h2/2i1−ieπ/h1+h2z0−h1−h2/2i4−a2−ln1+ieπ/h1+h2z−h1−h2/2i1−ieπ/h1+h2z−h1−h2/2i4−a2+1+ieπ/h1+h2z0¯−h1−h2/2i1−ieπ/h1+h2z0¯−h1−h2/2i4−a2.Take the derivative of equations (37) and (38):(39)f1′z=−2beπ/h1+h2z−h1−h2/2ih1+h21−ieπ/h1+h2z−h1−h2/2i2M11−N11+GM12−N12⋅P1,f2′z=−2beπ/h1+h2z−h1−h2/2ih1+h21−ieπ/h1+h2z−h1−h2/2i2I−GM11−N11⋅P1,where(40)M11=11+ieπ/h1+h2z−h1−h2/2i/1−ieπ/h1+h2z−h1−h2/2i4−a2−1+ieπ/h1+h2z−h1−h2/2i/1−ieπ/h1+h2z0−h1−h2/2i4−a2,N11=11+ieπ/h1+h2z−h1−h2/2i/1−ieπ/h1+h2z−h1−h2/2i4−a2+1+ieπ/h1+h2z0¯−h1−h2/2i/1−ieπ/h1+h2z0¯−h1−h2/2i4−a2,M12=11+ieπ/h1+h2z−h1−h2/2i/1−ieπ/h1+h2z−h1−h2/2i4−a2−1+ieπ/h1+h2z0¯−h1−h2/2i/1−ieπ/h1+h2z0¯−h1−h2/2i4−a2,N12=11+ieπ/h1+h2z−h1−h2/2i/1−ieπ/h1+h2z−h1−h2/2i4−a2+1+ieπ/h1+h2z0−h1−h2/2i/1−ieπ/h1+h2z0−h1−h2/2i4−a2,P1=11+ieπ/h1+h2z−h1−h2/2i/1−ieπ/h1+h2z−h1−h2/2i4−a2⋅1+ieπ/h1+h2z−h1−h2/2i1−ieπ/h1+h2z−h1−h2/2i3.From the previous discussion, the image force of the fine-grained piezoelectric coating/substrate can be expressed as(41)Fx−iFy=−2L1G⋅eπ/h1+h2z0−h1−h2/2ibbΤh1+h21−ieπ/h1+h2z0−h1−h2/2i2−N21+GM22−N22⋅P2,Fx−iFy=−2eπ/h1+h2z0−h1−h2/2ibbΤh1+h21−ieπ/h1+h2z0−h1−h2/2i21−c121+c441υ2c12+c441υ1+c12⋅M22−N22−1−c12N21⋅P2,where(42)c12=c441c442,υ=ε111e1512,N21=11+ieπ/h1+h2z0−h1−h2/2i/1−ieπ/h1+h2z0−h1−h2/2i4−a2+1+ieπ/h1+h2z0¯−h1−h2/2i/1−ieπ/h1+h2z0¯−h1−h2/2i4−a2,M22=11+ieπ/h1+h2z0−h1−h2/2i/1−ieπ/h1+h2z0−h1−h2/2i4−a2−1+ieπ/h1+h2z0¯−h1−h2/2i/1−ieπ/h1+h2z0¯−h1−h2/2i4−a2,N22=11+ieπ/h1+h2z0−h1−h2/2i/1−ieπ/h1+h2z0−h1−h2/2i4−a2+1+ieπ/h1+h2z0−h1−h2/2i/1−ieπ/h1+h2z0−h1−h2/2i4−a2,P2=11+ieπ/h1+h2z0−h1−h2/2i/1−ieπ/h1+h2z0−h1−h2/2i4−a2⋅1+ieπ/h1+h2z0−h1−h2/2i1−ieπ/h1+h2z0−h1−h2/2i3.


## 4.2. Intensity Factors

In studying the interaction between dislocation and crack, the intensity factor can be used to analyze the shielding or antishielding effect of dislocation on cracks. In this section, our attention will focus on the elector-elastic fields near the tip of the edge crack where local elector-elastic field intensification usually occurs. For the antiplane problem of the piezoelectric materials, the intensity factors are defined by(43)K=kσ,kDΤ=limr⟶02πrtyr,0,where kσ is the classical SIF, kD is the EDIF, and ty=σyz,Dy. According to the boundary continuity conditions, tyr,0 in equation (43) can be replaced by t1yr,0 or t2yr,0 (here 1 and 2 represent upper and lower materials, respectively):(44)K=limr⟶02πrt1yr,0=limr⟶02πrt2yr,0.Equation (43) is expressed in the Cartesian coordinate systems, as follows:(45)K=limz⟶a2πz−at1yz.


### 4.2.1. Intensity Factor of the Right Semi-Infinite Plane Piezoelectric Bimaterial with Screw Dislocation

Using equation (13), equation (45) can be expressed as(46)K1=limz⟶a2πz−at1yz=a4πc441b2ε111+c441υ+1+c12ε1112c12υ+c441υ1+c12⋅Re1z02−a2−z0¯2−a2,where(47)c12=c441c442,υ=ε111e1512.


### 4.2.2. Intensity Factor of the Fine-Grained Piezoelectric Coating/Substrate with Edge Interfacial Crack and Screw Dislocation

According to the above discussion, substituting equation (15) into equation (46), the intensity factor can be expressed as(48)K2=limz⟶a2πz−at1yz=−c441baaπh1+h22ε111+c441υ+1+c12ε1112c12υ+c441υ1+c12Re11+ieπ/h1+h2z−h1−h2/2i/1−ieπ/h1+h2z−h1−h2/2i4−a2−1+ieπ/h1+h2z0¯−h1−h2/2i/1−ieπ/h1+h2z0¯−h1−h2/2i4−a2,where(49)c12=c441c442,υ=ε111e1512.


## 4.2.1. Intensity Factor of the Right Semi-Infinite Plane Piezoelectric Bimaterial with Screw Dislocation

Using equation (13), equation (45) can be expressed as(46)K1=limz⟶a2πz−at1yz=a4πc441b2ε111+c441υ+1+c12ε1112c12υ+c441υ1+c12⋅Re1z02−a2−z0¯2−a2,where(47)c12=c441c442,υ=ε111e1512.


## 4.2.2. Intensity Factor of the Fine-Grained Piezoelectric Coating/Substrate with Edge Interfacial Crack and Screw Dislocation

According to the above discussion, substituting equation (15) into equation (46), the intensity factor can be expressed as(48)K2=limz⟶a2πz−at1yz=−c441baaπh1+h22ε111+c441υ+1+c12ε1112c12υ+c441υ1+c12Re11+ieπ/h1+h2z−h1−h2/2i/1−ieπ/h1+h2z−h1−h2/2i4−a2−1+ieπ/h1+h2z0¯−h1−h2/2i/1−ieπ/h1+h2z0¯−h1−h2/2i4−a2,where(49)c12=c441c442,υ=ε111e1512.


## 5. Numerical Process and Results

This section is divided into two parts. Firstly, some numerical applications to verify the obtained solutions are given, in the case when we reduce the current cases to simpler one (as shown in Figure3(b)). Then, the influence of material constants, crack length, coating thickness on the image force, and intensity factor is studied.


### 5.1. Study of a Screw Dislocation in Two Bonded Piezoelectric Right Semi-Infinite Planes as Shown in Figure3(b)

As shown in Figure3(b), by equations (23) and (24), based on the previous discussion, we got the image force acting on the dislocation near a traction-free and a charge-free boundary as(50)Fx−iFy=bbΤ2πL1−1ξ0+ξ0¯+G1ξ0−ξ0¯−1ξ0+ξ0.When the material parameters of upper and lower materials are same, i.e.,L1=L2, we obtained(51)Fx=ReFx−iFy=RebbΤ2πL1−1ξ0+ξ0¯+G1ξ0−ξ0¯−1ξ0+ξ0=−bbΤL12π1ξ0+ξ0¯,Fy=ImFx−iFy=ImbbΤ2πL1−1ξ0+ξ0¯+G1ξ0−ξ0¯−1ξ0+ξ0=0.The above results agree with the ones given by Pak [5], Liu et al. [6], and Chen et al. [18].The strength of piezoelectricity is characterized by the electric-mechanical coupling factork=e152/c44ε11. As shown in Figure 3(d), we choose the PZT-6B ceramic and the PZT-5H ceramic as material 1, respectively. The ceramic’s properties are given by Chen et al. [18] as follows: c441=2.71×1010N/m2, e151=4.6C/m2, ε111=3.6×10−9C/Vm, k=0.271, c441=3.53×1010N/m2, e151=17.0C/m2, ε111=1.51×10−8C/Vm, and k=0.542.The force on the dislocation is normalized by(52)F∗1=bz24πr0,F∗2=bφ24πr0.The normalized force on the dislocation alongx-axis versus angular position θ (in degree) is shown in Figures 6 and 7.Figure 6
Force on the dislocationbz along the x-axis versus angular position θ.Figure 7
Force on the dislocationbφ along the x-axis versus angular position θ.It is observed that the force on the dislocation along thex-axis decreases with the increasing angle θ. It means the electric-free surface always attracts the dislocation. This conclusion agree with the ones given by Chen et al. [18].


### 5.2. Study of the Interactions of the Dislocation with the Fine-Grained Piezoelectric Coating/Substrate Structure as Shown in Figure1

It is known from the characteristics of fine-grained piezoelectric materials that the piezoelectric and dielectric constants of the material reach the level of the large-grained piezoelectric material, and that the elastic modulus is much larger than that of the large-grained piezoelectric material. Therefore, in the following discussion, we assume that the piezoelectric coating/substrate has the same piezoelectric and dielectric constants.In this part, some numerical applications were firstly considered to be presented. When we have a screw dislocation loading in a semi-infinite plane with an edge interface crack, the lower half of the plane, we choose the PZT-5H ceramic, and in the lower half plane, suitable piezoelectric materials can be selected. The PZT-5H ceramic’s properties are given by Pak [1]: c442=3.53×1010N/m2, e152=17.0C/m2, and ε112=1.51×10−8C/Vm.Assume thath2/h1=100, r0/h1=0.1, and b=bz,0.


#### 5.2.1. Study of the Influence of the Parameterc441/c442

Relationship between different image forces and angleθ for different values of the parameter c441/c442 is discussed in this section.


#### 5.2.2. Study of the Influence of the Crack Length

Relationship between different image forces and angleθ for different values of the crack length a is discussed in this section.


#### 5.2.3. Study of the Influence of Parameterh1

Relationship between different image forces and angleθ for different values of the parameter h1 is discussed in this section.


### 5.3. Discussion of Results

Figures8 and 9 show the changes of the image force Fx and Fy, respectively, with angle θ when the elastic modulus changes. It can be seen from the figures that the elastic modulus has a significant effect on the image force, and the image force near the interface is the largest (0.2<θ<0.6). However, when the θ increases to a certain value (θ>0.6), the influence of the elastic modulus on the image force is very small, which is caused by the shielding effect of the crack on the dislocation (where a = 1 mm, h1=1mm, and F0=bz2/h1+h2).Figure 8
Relationship between image forceFx and angle θ for different values of the parameter c441/c442.Figure 9
Relationship between image forceFy and angle θ for different values of the parameter c441/c442.It can be seen in Figure10 that the intensity factor K2 decreases with increase in elastic modulus and the intensity factor is negative, indicating that the dislocation plays a shielding effect on the crack. In addition, when the angle θ is greater than 0.25, the change of the elastic modulus has little effect on the intensity factor (where a = 1 mm, h1=1mm, and F0=bz2/h1+h2).Figure 10
Relationship between IFsK2 and angle θ for different values of the parameter c441/c442.In Figures11 and 12, the law of variation of Fx and Fy with angle θ is given when the crack length has different values. When a = 1 mm, the influence of θ on the image force is very small. With the change of crack length, the image force is greatly influenced by θ. However, the image force gradually tends to be balanced with the increase in angle θ, as in previous studies (where c441/c442=1.2, h1=1mm, and F0=bz2/h1+h2).Figure 11
Relationship between image forceFx and angle θ for different values of the crack length a.Figure 12
Relationship between image forceFy and angle θ for different values of the crack length a.In Figure13, it can be seen that the crack length has a great influence on the intensity factor K2. With the increasing crack length, the strength factor first increases and then decreases, and the dislocation near the crack is a maximum. This indicates that the dislocation can not only delay the crack growth but it can also promote it. However, the intensity factor tends to be balanced with the increasing angle θ, when the dislocation is far away from the crack, at which point it has little effect on the intensity factor (where c441/c442=1.2, h1=1mm, and F0=bz2/h1+h2).Figure 13
Relationship between IFsK2 and angle θ for different values of the crack length a.Figures14 and 15 show that the coating thickness has a great influence on the image force Fx and Fy, respectively, and decreases with the increasing coating thickness. The maximum value of the image force is near the crack, indicating that the dislocation can affect the crack. However, with the increasing angle θ, the influence of coating thickness on image force gradually decreases. When the angle θ is greater than 0.8, the coating thickness has almost no effect on image force (where c441/c442=1.2, a = 1 mm, and F0=bz2/h1+h2).Figure 14
Relationship between image forceFy and angle θ for different values of parameter h1.Figure 15
Relationship between image forceFx and angle θ for different values of parameter h1.Figure16 shows that the intensity factor K2 decreases with the increasing coating thickness and the maximum of the intensity factor at the dislocations near the interface and the strength factor gradually tends to be stable with the increasing angle θ also indicating that dislocation near a crack can prevent crack growth (where c441/c442=1.2, a = 1 mm, and K0=aaπ/2h1+h2c441bz).Figure 16
Relationship between IFsK2 and angle θ for different values of parameter h1.Finally, it can be concluded that the elastic modulus of the material and the coating thickness plays an important role in fracture mechanics for the fine-grained piezoelectric coating/substrate structure.


## 5.1. Study of a Screw Dislocation in Two Bonded Piezoelectric Right Semi-Infinite Planes as Shown in Figure3(b)

As shown in Figure3(b), by equations (23) and (24), based on the previous discussion, we got the image force acting on the dislocation near a traction-free and a charge-free boundary as(50)Fx−iFy=bbΤ2πL1−1ξ0+ξ0¯+G1ξ0−ξ0¯−1ξ0+ξ0.When the material parameters of upper and lower materials are same, i.e.,L1=L2, we obtained(51)Fx=ReFx−iFy=RebbΤ2πL1−1ξ0+ξ0¯+G1ξ0−ξ0¯−1ξ0+ξ0=−bbΤL12π1ξ0+ξ0¯,Fy=ImFx−iFy=ImbbΤ2πL1−1ξ0+ξ0¯+G1ξ0−ξ0¯−1ξ0+ξ0=0.The above results agree with the ones given by Pak [5], Liu et al. [6], and Chen et al. [18].The strength of piezoelectricity is characterized by the electric-mechanical coupling factork=e152/c44ε11. As shown in Figure 3(d), we choose the PZT-6B ceramic and the PZT-5H ceramic as material 1, respectively. The ceramic’s properties are given by Chen et al. [18] as follows: c441=2.71×1010N/m2, e151=4.6C/m2, ε111=3.6×10−9C/Vm, k=0.271, c441=3.53×1010N/m2, e151=17.0C/m2, ε111=1.51×10−8C/Vm, and k=0.542.The force on the dislocation is normalized by(52)F∗1=bz24πr0,F∗2=bφ24πr0.The normalized force on the dislocation alongx-axis versus angular position θ (in degree) is shown in Figures 6 and 7.Figure 6
Force on the dislocationbz along the x-axis versus angular position θ.Figure 7
Force on the dislocationbφ along the x-axis versus angular position θ.It is observed that the force on the dislocation along thex-axis decreases with the increasing angle θ. It means the electric-free surface always attracts the dislocation. This conclusion agree with the ones given by Chen et al. [18].


## 5.2. Study of the Interactions of the Dislocation with the Fine-Grained Piezoelectric Coating/Substrate Structure as Shown in Figure1

It is known from the characteristics of fine-grained piezoelectric materials that the piezoelectric and dielectric constants of the material reach the level of the large-grained piezoelectric material, and that the elastic modulus is much larger than that of the large-grained piezoelectric material. Therefore, in the following discussion, we assume that the piezoelectric coating/substrate has the same piezoelectric and dielectric constants.In this part, some numerical applications were firstly considered to be presented. When we have a screw dislocation loading in a semi-infinite plane with an edge interface crack, the lower half of the plane, we choose the PZT-5H ceramic, and in the lower half plane, suitable piezoelectric materials can be selected. The PZT-5H ceramic’s properties are given by Pak [1]: c442=3.53×1010N/m2, e152=17.0C/m2, and ε112=1.51×10−8C/Vm.Assume thath2/h1=100, r0/h1=0.1, and b=bz,0.


### 5.2.1. Study of the Influence of the Parameterc441/c442

Relationship between different image forces and angleθ for different values of the parameter c441/c442 is discussed in this section.


### 5.2.2. Study of the Influence of the Crack Length

Relationship between different image forces and angleθ for different values of the crack length a is discussed in this section.


### 5.2.3. Study of the Influence of Parameterh1

Relationship between different image forces and angleθ for different values of the parameter h1 is discussed in this section.


## 5.2.1. Study of the Influence of the Parameterc441/c442

Relationship between different image forces and angleθ for different values of the parameter c441/c442 is discussed in this section.


## 5.2.2. Study of the Influence of the Crack Length

Relationship between different image forces and angleθ for different values of the crack length a is discussed in this section.


## 5.2.3. Study of the Influence of Parameterh1

Relationship between different image forces and angleθ for different values of the parameter h1 is discussed in this section.


## 5.3. Discussion of Results

Figures8 and 9 show the changes of the image force Fx and Fy, respectively, with angle θ when the elastic modulus changes. It can be seen from the figures that the elastic modulus has a significant effect on the image force, and the image force near the interface is the largest (0.2<θ<0.6). However, when the θ increases to a certain value (θ>0.6), the influence of the elastic modulus on the image force is very small, which is caused by the shielding effect of the crack on the dislocation (where a = 1 mm, h1=1mm, and F0=bz2/h1+h2).Figure 8
Relationship between image forceFx and angle θ for different values of the parameter c441/c442.Figure 9
Relationship between image forceFy and angle θ for different values of the parameter c441/c442.It can be seen in Figure10 that the intensity factor K2 decreases with increase in elastic modulus and the intensity factor is negative, indicating that the dislocation plays a shielding effect on the crack. In addition, when the angle θ is greater than 0.25, the change of the elastic modulus has little effect on the intensity factor (where a = 1 mm, h1=1mm, and F0=bz2/h1+h2).Figure 10
Relationship between IFsK2 and angle θ for different values of the parameter c441/c442.In Figures11 and 12, the law of variation of Fx and Fy with angle θ is given when the crack length has different values. When a = 1 mm, the influence of θ on the image force is very small. With the change of crack length, the image force is greatly influenced by θ. However, the image force gradually tends to be balanced with the increase in angle θ, as in previous studies (where c441/c442=1.2, h1=1mm, and F0=bz2/h1+h2).Figure 11
Relationship between image forceFx and angle θ for different values of the crack length a.Figure 12
Relationship between image forceFy and angle θ for different values of the crack length a.In Figure13, it can be seen that the crack length has a great influence on the intensity factor K2. With the increasing crack length, the strength factor first increases and then decreases, and the dislocation near the crack is a maximum. This indicates that the dislocation can not only delay the crack growth but it can also promote it. However, the intensity factor tends to be balanced with the increasing angle θ, when the dislocation is far away from the crack, at which point it has little effect on the intensity factor (where c441/c442=1.2, h1=1mm, and F0=bz2/h1+h2).Figure 13
Relationship between IFsK2 and angle θ for different values of the crack length a.Figures14 and 15 show that the coating thickness has a great influence on the image force Fx and Fy, respectively, and decreases with the increasing coating thickness. The maximum value of the image force is near the crack, indicating that the dislocation can affect the crack. However, with the increasing angle θ, the influence of coating thickness on image force gradually decreases. When the angle θ is greater than 0.8, the coating thickness has almost no effect on image force (where c441/c442=1.2, a = 1 mm, and F0=bz2/h1+h2).Figure 14
Relationship between image forceFy and angle θ for different values of parameter h1.Figure 15
Relationship between image forceFx and angle θ for different values of parameter h1.Figure16 shows that the intensity factor K2 decreases with the increasing coating thickness and the maximum of the intensity factor at the dislocations near the interface and the strength factor gradually tends to be stable with the increasing angle θ also indicating that dislocation near a crack can prevent crack growth (where c441/c442=1.2, a = 1 mm, and K0=aaπ/2h1+h2c441bz).Figure 16
Relationship between IFsK2 and angle θ for different values of parameter h1.Finally, it can be concluded that the elastic modulus of the material and the coating thickness plays an important role in fracture mechanics for the fine-grained piezoelectric coating/substrate structure.


## 6. Conclusions

The interaction between micro- and macroscopic crack (i.e., Griffith-type interface cracks and screw dislocation) in fine-grained piezoelectric coating/substrate was considered in this paper. The mapping function method is utilized to transform the complex geometric model into a relatively simple model, and the image force, intensity factor of the right semi-infinite piezoelectric bimaterials with the edge interfacial crack, and the finite thickness coating/substrate structure with the screw dislocation are obtained. Based on the solutions for these two problems, the relationship between microscopic defects and macroscopic cracks in the fine-grained piezoelectric coating/substrate is provided.A numerical application was used to analyze the case of the screw dislocation and the edge interfacial crack in a finite thickness coating/substrate, and the results show that the coating thickness, elastic modulus, and the length of the crack have a significant influence on the dislocation and interface crack. The numerical results also show that, to better resist fracture, it is necessary to choose a coating material with a larger elastic modulus and smaller thickness to realize better resistance to fracturing of the coating/substrate structure. These characteristics reflect the superior properties of fine-grained piezoelectric coating materials.



---

*Source: 3848240-2019-11-19.xml*