# Research on Multiple Griffith Cracks at the Interface of Fine-Grained Piezoelectric Coating/Substrate

Authors: Shuaishuai Hu; Jiansheng Liu; Junlin Li
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The behavior of a fine-grained piezoelectric coating/substrate with multiple Griffith interface cracks under electromechanical loads is investigated. In this work, double coupled singular integral equations are proposed to solve the fracture problems. Both the singular integral equation and single-valued conditions are simplified into an algebraic equation and solved by numerical calculation. Thereby, the intensity factors of electric displacement and stress obtained are used to obtain the expression of the energy release rate. Furthermore, numerical results of the energy release rate with material parameters are demonstrated. Based on the obtained results, it could be concluded that the energy release rate is closely related to the size of the interface cracks and the mechanical-electrical loading. For a bimaterial structure, the fine-grained piezoelectric structure exhibited better material performance compared to the large one.


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## Body

## 1. Introduction

Piezoelectric bimaterials have been widely applied in many fields, such as sensors, actuators, filters, and storage devices. Owing to the reliability of composite materials, double-layer bonded bimaterial composite structures have played an important role in smart structures. However, piezoelectric composites tend to crack, fracture, and debond at the interface during the process of fabricating defects and load conditions [1]. Therefore, the issue of defects of piezoelectric materials under electromechanical load has attracted wide attention [2–7]. Viun et al. [8] handled the interface problems of bi-piezoelectric materials under tensile mechanical loads and electric displacements. The influence of material parameters on finite-sized piezoelectric plates is analyzed by a finite-element method. Li et al. [9] analyzed the mechanical properties of static and dynamic interface cracks of functionally graded coatings/substrates. In this work, the Fourier transform and Laplace transform were used as the fundamental solutions to solve the problem under static and dynamic loads. Soh et al. [10] studied the behavior of bi-piezoelectric ceramic strips with mode III central interface cracks, and the influences of thickness and material coefficients of piezoelectric strips on intensity factors and stress were obtained. Shavlakadze [11] demonstrated the electroelastic contact problem of piezoelectric plates with elastic coatings. By using the method of analytic function theory, the problem was reduced to a singular integrodifferential equation with fixed singularity. Shin and Kim [12] studied the transient response of mode III interface cracks between the piezoelectric layer and the functional orthotropic material layer. The integral equations were solved by the integral method, and it was concluded that the material parameters could prevent the transient fracture of the interface crack. Gherrous and Ferdjani [13] investigated a Griffith interface crack of bi-piezoelectric materials, and the singular integral equations were reduced into algebraic equations by using Chebyshev polynomials. Nan and Wang [14] studied the effect of residual surface stress on the stress and electric field strength of conductive cracks in piezoelectric nanomaterials. The numerical results showed that the residual stress had a significant effect on the piezoelectric nanomaterials, and the piezoelectric nanomaterials had stronger electromechanical coupling compared to the classical fracture mechanics. Nourazar and Ayatollahi [15] examined the effects of multiple interface cracks in orthotropic-strip-bonded bi-piezoelectric strips, and the results reflected the crack size had a great influence on the intensity factor and the hoop stress of the cavities. Tian and Rajapakse [16] studied a penny-shaped interface crack of double piezoelectric materials, and the Hankel integral transformation method was used to establish a mixed boundary-value problem corresponding to an interface crack. Gao et al. [17] investigated a permeable crack problem in infinite magnetoelectroelastic solids with linear coupling between the elastic and electromagnetic fields, and the results showed in the most general cases that the singularity of the electric-magnetic field was always dependent on that of stress. Kozinov et al. [18] considered a periodic set of limited electrically permeable cracks at the interface of two different piezoelectric semi-infinite spaces and analyzed the influence of electric permeability of cracks on electromechanical fields and the fracture mechanical parameters.However, the piezoelectric materials with defects studied above are large-grained materials composed of multidomain grains from several microns to tens of microns, and the size could not meet the needs. Reducing the particle size to the submicron level can improve the machinability of the material and the mechanical strength of the device. Kim et al. studied the piezoelectric characteristics of fine-grained piezoelectric materials compared with conventionally prepared large-grained piezoelectric materials under certain conditions [19]. In this latter work, the influence of grain size on density, dielectric constant, phase structure, and piezoelectric properties was investigated [20–23]. However, the mechanical properties of fine-grained piezoelectric materials with defects have not been studied.Although many researchers have studied the interface fracture problems of piezoelectric bi-materials [1], few studies have been done on the interface fracture of fine-grained piezoelectric coating/substrate smart structures. As an important fracture parameter with which to measure structural safety, the energy release rate plays a crucial role in safety checking of structures. In this paper, a fine-grained piezoelectric coating/substrate mechanical model is established to describe the interaction of multiple Griffith interface cracks. The Fourier integral transformation method is used to transform the problem into a singular integral equation to obtain the energy release rate conveniently. Furthermore, the interaction between the energy release rate and material parameters is investigated. The results of the studies indicate that the appropriate stress and electric displacement loads could prevent the growth of cracks and that a thinner coating thickness is more conducive to the safety of the structure. In addition, fine-grained piezoelectric materials have more safety than large-grained ones.


## 2. Problem Formulation

As shown in Figure1, the fine-grained ceramic powder was sprayed uniformly to the surface of the piezoelectric substrate by plasma spraying technology to form a coating. A fine-grain piezoelectric coating/substrate structure was obtained by polarization treatment of coating. The thickness of the coating is h1 and of the substrate is h2. The rectangular coordinate system was built with the Χ-axis along the structure interface and the Y-axis along the structure thickness direction. The multiple Griffith interface cracks are 2-dimensional and all within the same orientation and occupying the intervals x∈aj,bj j=1,2,…,n, respectively. The fine-grained piezoelectric coating and piezoelectric substrate are polarized along the Z-axis, and they are transversely isotropic.Figure 1
Multiple Griffith interface cracks between the fine-grained piezoelectric coating and the substrate.Assume that the problem of the fine-grained piezoelectric coating/substrate interface cracks is considered as only the out-of-plane displacementW and the in-plane electric fields Φ such that(1)U=V=0,W=Wx,y,Φ=Φx,y.In this case, the constitutive equations become [4](2)σxz=c44k∂W∂x+e15k∂Φ∂x,(3)Dx=e15k∂W∂x−ε11k∂Φ∂x,(4)σyz=c44k∂W∂y+e15k∂Φ∂y,(5)Dy=e15k∂W∂y−ε11k∂Φ∂y,where σxz and σyz are the stress components, Dx and Dy are the electric displacement components, c44k is the elastic modulus, e15k and ε11k are the piezoelectric and dielectric constants, and the superscript kk=1,2 stands for the fine-grained piezoelectric coating and piezoelectric substrate, respectively.According to Park, the governing equations can be written as follows:(6)c44k∇2W+e15k∇2Φ=0,(7)e15k∇2W−ε11k∇2Φ=0,k=1,2,where ∇2 is the two-dimensional Laplacian operator.Assume that there is no loading on the surface of the coating/substrate and the antiplane shear stressσ0 and in-plane electric displacement D0 are loaded on the surface of the Griffith interface cracks. Therefore, the mixed boundary conditions of the problems can be written as follows:(8)σyzx,h1=σyzx,−h2=0,(9)Dyx,h1=Dyx,−h2=0,(10)σyzx,0+=σyzx,0−,x∉aj,bj,(11)Dyx,0+=Dyx,0−,x∉aj,bj,(12)Wx,0+=Wx,0−,x∉aj,bj,(13)Φx,0+=Φx,0−,x∉aj,bj,(14)σyzx,0+=σyzx,0−=−σ0,x∈aj,bj,(15)Dyx,0+=Dyx,0−=−D0,x∈aj,bj.


## 3. Solution to the Problem

By Fourier integral transform of equations (6) and (7), the loading displacement and electric field are obtained, respectively:(16)Wx,y=12π∫−∞+∞A1keαy+B1ke−αye−iαxdα,(17)Φx,y=12π∫−∞+∞A2keαy+B2ke−αye−iαxdα,where A1kα, B1kα, A2kα, and B2kα are the unknown functions k=1,2.Substitution formulas (16) and (17) into equations (4) and (5), the stress and the electric displacement are obtained, respectively:(18)σyz=12π∫−∞+∞c44kA1kαeαy−B1kαe−αy+e15kA2kαeαy−B2kαe−αye−iαxdα,(19)Dy=12π∫−∞+∞e15kA1kαeαy−B1kαe−αy+ε11kA2kαeαy−B2kαe−αye−iαxdα.In order to solve the problem easily, we utilize continuously distributed dislocations to simulate collinear cracks. For this, we introduce the density function as follows:(20)gWjx=ddxWjx,0+−Wjx,0−,(21)gΦjx=ddxΦjx,0+−Φjx,0−,j=1,2,…,n.According to equations (12) and (13), the dislocation density function should make the following single-valued conditions valid:(22)∫ajbjgWjtdt=0,gWjx=0,forx∉aj,bj,j=1,2,…,n,∫ajbjgΦjtdt=0,gΦjx=0,forx∉aj,bj,j=1,2,…,n.Using the boundary conditions (8) and (9), we obtain(23)A11α=B11αe−2αh1,A21α=B21αe−2αh1,B12α=A12αe−2αh2,B22α=A22αe−2αh2.Inserting equations (16) and (17) into equations (20) and (21) and inserting equations (18) and (19) into the boundary conditions (8)–(11), (14), and (15), we obtain(24)c441A11αeαh1−B11αe−αh1+e151A21αeαh1−B21αe−αh1=0,(25)c442A12αe−αh2−B12αeαh2+e152A22αe−αh2−B22αeαh2=0,(26)e151A11αeαh1−B11αe−αh1+ε111A21αeαh1−B21αe−αh1=0,(27)e152A12αe−αh2−B12αeαh2−ε112A22αe−αh2−B22αeαh2=0,(28)c441A11α−B11α+e151A21α−B21α=c442A12α−B12α+e152A22α−B22α,(29)e151A11α−B11α−ε111A21α−B21α=e152A12α−B12α−ε112A22α−B22α,(30)A11α+B11α−A12α−B12α=iα∫ajbjgWjteiαtdt=F1,(31)A21α+B21α−A22α−B22α=iα∫ajbjgΦjteiαtdt=F2.Equations (25)–(32) can be written in the form of matrix as follows:(32)A⋅B=C,where(33)A=c441eαh1−c441eαh100e151eαh1−e151eαh10000c442e−αh2−c442e−αh200e152e−αh2−e152e−αh2e151eαh1−e151eαh100−ε111eαh1ε111eαh10000e152e−αh2−e152e−αh200−ε112e−αh2ε112e−αh2c441−c441−c442c442e151−e151−e152e152e151−e151−e152e152−ε111ε111ε112−ε11211−1−10000000011−1−1,B=A11,B11,A12,B12,A21,B21,A22,B22T,C=0,0,0,0,0,0,F1,F2T.Solving the linear equation (32), we obtain(34)A11α=H1F1+H2F2H,A12α=H3F1+H4F2H,B11α=H5F1+H6F2H,B12α=H7F1+H8F2H,A21α=H9F1+H10F2H,A22α=H11F1+H12F2H,B21α=H13F1+H14F2H,B22α=H15F1+H16F2H.The expressions ofH and Hq,q=1,2,…,16, are given in Appendix.Obviously, the unknown functionsAm1α, Am2α and Bm1α, Bm2α m=1,2 depend on F1 and F2 and, consequently, on the density functions gWjx and gΦjx. Once F1 and F2 are determined, the stress expressions and electric displacement expressions could be obtained.For this, by substituting equations (16) and (17) into equations (4) and (5), we obtain(35)σyzx,y=12π∫−∞+∞c442αeαyHH3−H7e−2αyF1+H4−H8e−2αyF2+e152αeαyHH11−H15e−2αyF1+H12−H16e−2αyF2e−iαxdα,(36)Dyx,y=12π∫−∞+∞e152αeαyHH3−H7e−2αyF1+H4−H8e−2αyF2+ε112αeαyHH11−H15e−2αyF1+H12−H16e−2αyF2e−iαxdα.Next, we need to transform equations (35) and (36) into singular integral equations of the first kind with Cauchy kernel. In another form, the integral equations (35) and (36) will be(37)σyzx,0−=limy⟶0−12π∫−∞+∞αeαyHc442H3−c442H7+e152H11−e152H15F1+c442H4−c442H8+e152H12−e152H16F2e−iαxdα=−σ0,(38)Dyx,0−=limy⟶0−12π∫−∞+∞αeαyHe152H3−e152H7+ε112H11−ε112H15F1+e152H4−e152H8+ε112H12−ε112H16F2e−iαxdα=−D0,where(39)Q1α=c442H3−c442H7+e152H11−e152H15,Q2α=c442H4−c442H8+e152H12−e152H16,Q3α=e152H3−e152H7+ε112H11−ε112H15,Q4α=e152H4−e152H8+ε112H12−ε112H16.Equations (37) and (38) can be rewritten in the following forms:(40)12π∫−∞+∞αeαyHQ1F1e−iαxdα+12π∫−∞+∞αeαyHQ2F2e−iαxdα=−σ0,(41)12π∫−∞+∞αeαyHQ3F1e−iαxdα+12π∫−∞+∞αeαyHQ4F2e−iαxdα=−D0.By substituting the expressions ofF1 and F2 into equations (40) and (41), we obtain(42)12π∫−∞+∞iαQ1eαyαH∫ajbjgWjteiαtdte−iαxdα+12π∫−∞+∞αQ2eαyαH∫ajbjgΦjteiαtdte−iαxdα=−σ0,12π∫−∞+∞iαQ3eαyαH∫ajbjgWjteiαtdte−iαxdα+12π∫−∞+∞αQ4eαyαH∫ajbjgΦjteiαtdte−iαxdα=−D0.Through changing the order of the integration, we obtain(43)12π∫ajbj∫−∞+∞iαQ1eαyαHeiαt−xdαgWjtdt+12π∫ajbj∫−∞+∞αQ2eαyαHeiαt−xdαgΦjtdt=−σ0,12π∫ajbj∫−∞+∞iαQ3eαyαHeiαt−xdαgWjtdt+12π∫ajbj∫−∞+∞αQ4eαyαHeiαt−xdαgΦjtdt=−D0.On the cracks’ surface,(45)∫ajbjQ1∗H∗1t−x+k1x,tgWjtdt+∫ajbjQ2∗H∗1t−x+k2x,tgΦjtdt=−πσ0,(46)∫ajbjQ3∗H∗1t−x+k3x,tgWjtdt+∫ajbjQ4∗H∗1t−x+k4x,tgΦjtdt=−πD0,where(47)Q1∗=c441c442ε111−ε112−e151c442−e151e152c442−c441−c441e152e152−e151,Q2∗=e151c442ε111−ε112+ε111e152c442−c441−ε111c442+e151e152e152−e151,Q3∗=c441e152ε111−ε112−e151e152−e151e152c442−c441−c441ε112e152−e151,Q4∗=e151e152ε111−ε112+ε111ε112c442−c441−ε111e152+e151ε112e152−e151,H∗=e152−e1512−c442−c441ε111−ε112,k1x,t=∫0∞Q1αH−Q1∗H∗sinαt−xdα,k2x,t=∫0∞Q2αH−Q2∗H∗sinαt−xdα,k3x,t=∫0∞Q3αH−Q3∗H∗sinαt−xdα,k4x,t=∫0∞Q4αH−Q4∗H∗sinαt−xdα.Ultimately, equations (4) and (5) are reduced to coupling the first-kind Cauchy singular integral equations.In equations (45) and (46), gWjt and gΦjt are unknown functions with the following conditions:(48)∫ajbjgWjtdt=0,∫ajbjgΦjtdt=0,j=1,2,…,n.By the principle of superposition and mixed boundary conditions, equations (45) and (46) can be formulated as(49)∑j=1n∫ajbjQ1∗H∗1tj−xk+k1xk,tjgWjtjdtj+∫ajbjQ2∗H∗1tj−xk+k2xk,tjgΦjtjdtj=−πσ0,xk∈ak,bk,j,k=1,2,…,n,(50)∑j=1n∫ajbjQ3∗H∗1tj−xk+k3xk,tjgWjtjdtj+∫ajbjQ4∗H∗1tj−xk+k4xk,tjgΦjtjdtj=−πD0,xk∈ak,bk,j,k=1,2,…,n.


## 4. Resolution of Singular Integral Equations

### 4.1. Normalization of Singular Integral Equations

For the convenience of numerical calculations, we define the normalized quantities as follows:(51)rj=xj−c0ja0j,sj=tj−c0ja0j,(52)gWjx=f1jx,gΦjx=f2jx,kix,t=Lir,s,i=1,2,3,4,(53)a0j=bj−aj2,c0j=bj+aj2,j=1,2,…,n.Using equations (51) and (52), the integral equations (49) and (50) can be expressed as(54)∑j=1n∫−11Q1∗H∗a0ja0jsj+c0j−a0krk−c0k+a0jL1rk,sjf1jsjdsj+∫−11Q2∗H∗a0ja0jsj+c0j−a0krk−c0k+a0jL2rk,sjf2jsjdsj=−πσ0,(55)∑j=1n∫−11Q3∗H∗a0ja0jsj+c0j−a0krk−c0k+a0jL3rk,sjf1jsjdsj+∫−11Q4∗H∗a0ja0jsj+c0j−a0krk−c0k+a0jL4rk,sjf2jsjdsj=−πD0.The dislocation density functionsf1jsj and f2jsj j=1,2,...,n in equations (54) and (55) have the squareroot-type singularity. So we can express f1jsj=1/1−sj2⋅g1jsjπσ0, where g1jsj is a continuous function defined in the interval [−1, 1]. In the same way, f2jsj may be expressed as(56)f1jsj=g1jsj1−sj2πσ0,(57)f2jsj=g2jsj1−sj2πD0,j=1,2,…,n.Using the Chebyshev collocation method, one can transform equations (54), (55), and (48) into the system of algebraic equations as follows:(58)1N∑R=0NχR∑j=1nQ1∗H∗a0ja0jsjR+c0j−a0krkq−c0k+a0jL1rkq,sjRg1jsjR+Q2∗H∗a0ja0jsjR+c0j−a0krkq−c0k+a0jL2rkq,sjRg2jsjR=−1,(59)1N∑R=0NχR∑j=1nQ3∗H∗a0ja0jsjR+c0j−a0krkq−c0k+a0jL3rkq,sjRg1jsjR+Q4∗H∗a0ja0jsjR+c0j−a0krkq−c0k+a0jL4rkq,sjRg2jsjR=−1,(60)∑R=0NχRg1kskR=0,∑R=0NχRg2kskR=0,k=1,2,…,n,q=1,2,…,N,where(61)χ0=χN=12,χ1=⋯=χN−1=1,rkq=cos2q−1π2N,sjR=cosRπN.N is the node number of the quadrature formula and sjR and rkq are the zero points of the first and second kinds of Chebyshev polynomials. Equations (58)–(60) are solved numerically to get the solutions of g1jsj and g2jsj, which can permit to get the stress and electric displacement.


### 4.2. Fracture Parameters

The intensity factors are defined by [4](62)Kσaj=limx⟶aj−2πaj−xσyzx,0,(63)Kσbj=limx⟶bj+2πx−bjσyzx,0,(64)KDaj=limx⟶aj−2πaj−xDyx,0,(65)KDbj=limx⟶bj+2πx−bjDyx,0,j=1,2,...,n.According to equations (45) and (46), the singular parts of σyzx,0 and Dyx,0 are(66)limx⟶aj−σyzx,0=Q1∗H∗limsj⟶−1−∫−11f1jsjsj−xjdsj+Q2∗H∗limsj⟶−1−∫−11f2jsjsj−xjdsj,(67)limx⟶aj−Dyx,0=Q3∗H∗limsj⟶−1−∫−11f1jsjsj−xjdsj+Q4∗H∗limsj⟶−1−∫−11f2jsjsj−xjdsj,(68)limx⟶bj+σyzx,0=Q1∗H∗limsj⟶1+∫−11f1jsjsj−xjdsj+Q2∗H∗limsj⟶1+∫−11f2jsjsj−xjdsj,(69)limx⟶bj+Dyx,0=Q3∗H∗limsj⟶1+∫−11f1jsjsj−xjdsj+Q4∗H∗limsj⟶1+∫−11f2jsjsj−xjdsj.Substituting equations (56) and (57) into equations (66)–(69), we obtain(70)limx⟶aj−σyzx,0=σ0Q1∗H∗limsj⟶−1−g1j−1sj2−1+σ0Q2∗H∗limsj⟶−1−g2j−1sj2−1,(71)limx⟶bj+σyzx,0=σ0Q1∗H∗limsj⟶1+−g1j1sj2−1+σ0Q2∗H∗limsj⟶−1−−g2j1sj2−1,(72)limx⟶aj−Dyx,0=D0Q3∗H∗limsj⟶−1−g1j−1sj2−1+D0Q4∗H∗limsj⟶−1−g2j−1sj2−1,(73)limx⟶bj+Dyx,0=D0Q3∗H∗limsj⟶1+−g1j1sj2−1+D0Q4∗H∗limsj⟶−1−−g2j1sj2−1.Substituting equations (70)–(73) into equations (62)–(65), we obtain(74)Kσaj=σ0πa0jQ1∗H∗g1j−1+Q2∗H∗g2j−1,Kσbj=−σ0πa0jQ1∗H∗g1j1+Q2∗H∗g2j1,KDaj=D0πa0jQ3∗H∗g1j−1+Q4∗H∗g2j−1,KDbj=−D0πa0jQ3∗H∗g1j1+Q4∗H∗g2j1,j=1,2,…,n.The energy release rate of piezoelectric materials has the following form [24]:(75)G=14ω1Kσ2+ω2KσKD+ω3KD2,where(76)K=Kσ,KD,ω1=Q4∗H∗Λ,ω2=Q3∗−Q1∗H∗Λ,ω3=−Q1∗H∗Λ,Λ=Q1∗H∗Q2∗H∗Q3∗H∗Q4∗H∗.


## 4.1. Normalization of Singular Integral Equations

For the convenience of numerical calculations, we define the normalized quantities as follows:(51)rj=xj−c0ja0j,sj=tj−c0ja0j,(52)gWjx=f1jx,gΦjx=f2jx,kix,t=Lir,s,i=1,2,3,4,(53)a0j=bj−aj2,c0j=bj+aj2,j=1,2,…,n.Using equations (51) and (52), the integral equations (49) and (50) can be expressed as(54)∑j=1n∫−11Q1∗H∗a0ja0jsj+c0j−a0krk−c0k+a0jL1rk,sjf1jsjdsj+∫−11Q2∗H∗a0ja0jsj+c0j−a0krk−c0k+a0jL2rk,sjf2jsjdsj=−πσ0,(55)∑j=1n∫−11Q3∗H∗a0ja0jsj+c0j−a0krk−c0k+a0jL3rk,sjf1jsjdsj+∫−11Q4∗H∗a0ja0jsj+c0j−a0krk−c0k+a0jL4rk,sjf2jsjdsj=−πD0.The dislocation density functionsf1jsj and f2jsj j=1,2,...,n in equations (54) and (55) have the squareroot-type singularity. So we can express f1jsj=1/1−sj2⋅g1jsjπσ0, where g1jsj is a continuous function defined in the interval [−1, 1]. In the same way, f2jsj may be expressed as(56)f1jsj=g1jsj1−sj2πσ0,(57)f2jsj=g2jsj1−sj2πD0,j=1,2,…,n.Using the Chebyshev collocation method, one can transform equations (54), (55), and (48) into the system of algebraic equations as follows:(58)1N∑R=0NχR∑j=1nQ1∗H∗a0ja0jsjR+c0j−a0krkq−c0k+a0jL1rkq,sjRg1jsjR+Q2∗H∗a0ja0jsjR+c0j−a0krkq−c0k+a0jL2rkq,sjRg2jsjR=−1,(59)1N∑R=0NχR∑j=1nQ3∗H∗a0ja0jsjR+c0j−a0krkq−c0k+a0jL3rkq,sjRg1jsjR+Q4∗H∗a0ja0jsjR+c0j−a0krkq−c0k+a0jL4rkq,sjRg2jsjR=−1,(60)∑R=0NχRg1kskR=0,∑R=0NχRg2kskR=0,k=1,2,…,n,q=1,2,…,N,where(61)χ0=χN=12,χ1=⋯=χN−1=1,rkq=cos2q−1π2N,sjR=cosRπN.N is the node number of the quadrature formula and sjR and rkq are the zero points of the first and second kinds of Chebyshev polynomials. Equations (58)–(60) are solved numerically to get the solutions of g1jsj and g2jsj, which can permit to get the stress and electric displacement.


## 4.2. Fracture Parameters

The intensity factors are defined by [4](62)Kσaj=limx⟶aj−2πaj−xσyzx,0,(63)Kσbj=limx⟶bj+2πx−bjσyzx,0,(64)KDaj=limx⟶aj−2πaj−xDyx,0,(65)KDbj=limx⟶bj+2πx−bjDyx,0,j=1,2,...,n.According to equations (45) and (46), the singular parts of σyzx,0 and Dyx,0 are(66)limx⟶aj−σyzx,0=Q1∗H∗limsj⟶−1−∫−11f1jsjsj−xjdsj+Q2∗H∗limsj⟶−1−∫−11f2jsjsj−xjdsj,(67)limx⟶aj−Dyx,0=Q3∗H∗limsj⟶−1−∫−11f1jsjsj−xjdsj+Q4∗H∗limsj⟶−1−∫−11f2jsjsj−xjdsj,(68)limx⟶bj+σyzx,0=Q1∗H∗limsj⟶1+∫−11f1jsjsj−xjdsj+Q2∗H∗limsj⟶1+∫−11f2jsjsj−xjdsj,(69)limx⟶bj+Dyx,0=Q3∗H∗limsj⟶1+∫−11f1jsjsj−xjdsj+Q4∗H∗limsj⟶1+∫−11f2jsjsj−xjdsj.Substituting equations (56) and (57) into equations (66)–(69), we obtain(70)limx⟶aj−σyzx,0=σ0Q1∗H∗limsj⟶−1−g1j−1sj2−1+σ0Q2∗H∗limsj⟶−1−g2j−1sj2−1,(71)limx⟶bj+σyzx,0=σ0Q1∗H∗limsj⟶1+−g1j1sj2−1+σ0Q2∗H∗limsj⟶−1−−g2j1sj2−1,(72)limx⟶aj−Dyx,0=D0Q3∗H∗limsj⟶−1−g1j−1sj2−1+D0Q4∗H∗limsj⟶−1−g2j−1sj2−1,(73)limx⟶bj+Dyx,0=D0Q3∗H∗limsj⟶1+−g1j1sj2−1+D0Q4∗H∗limsj⟶−1−−g2j1sj2−1.Substituting equations (70)–(73) into equations (62)–(65), we obtain(74)Kσaj=σ0πa0jQ1∗H∗g1j−1+Q2∗H∗g2j−1,Kσbj=−σ0πa0jQ1∗H∗g1j1+Q2∗H∗g2j1,KDaj=D0πa0jQ3∗H∗g1j−1+Q4∗H∗g2j−1,KDbj=−D0πa0jQ3∗H∗g1j1+Q4∗H∗g2j1,j=1,2,…,n.The energy release rate of piezoelectric materials has the following form [24]:(75)G=14ω1Kσ2+ω2KσKD+ω3KD2,where(76)K=Kσ,KD,ω1=Q4∗H∗Λ,ω2=Q3∗−Q1∗H∗Λ,ω3=−Q1∗H∗Λ,Λ=Q1∗H∗Q2∗H∗Q3∗H∗Q4∗H∗.


## 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 have a single interface crack. Then, the influence of material constants on the energy release rate is studied when we have multiple interface cracks.Initially, we assume that the piezoelectric substrate is the PZT-5H ceramic. Its parameters are given as [2](77)c442=3.53×1010N/m2,e152=17.0C/m2,ε112=1.51×10−8C/Vm,Gcr=5N/m,where superscript 2 represents the substrate material.


### 5.1. A Single Interface Crack

To validate the solution, we chose the case that material 1 is a fine-grained coating made of PZT-5H ceramics by plasma spraying technology and material 2 is normal PZT-5H ceramics (where 1 and 2 represent the fine-grained coating and substrate material, respectively), which means that material 1 and material 2 have approximately the same piezoelectric and dielectric constants under suitable conditions, and the elastic modulus of material 1 is larger than that of material 2. Soc441/c442=2.5, h1/h2=1/100, and the single interface crack of l0=5mm loaded by an antiplane stress of σ0=4.2MPa and a variable electric displacement D0 are considered. The curves of the energy release rate G/Gcr as a function of D0 under different values of l0/h1 are plotted (Figure 2). Then, the same interface crack is loaded by an electric displacement D0of10−3c/m2 and a variable stress σ0. For different values of l0/h1, the curves G/Gcr as a function of σ0 are plotted (Figure 3).Figure 2
Effects of electric displacementD0 on the energy release rate for different l0/h1.Figure 3
Effects of stressσ0 on the energy release rate for different l0/h1.


#### 5.1.1. Studies of the Effect of Stressσ0 and Electric Displacement D0 on Energy Release Rate

Effects of electric displacementD0 and stress σ0 on the energy release rate for different l0/h1 are shown in Figures 2 and 3, respectively.


### 5.2. Multiple Interface Cracks

The interaction between multiple interface cracks is investigated. The distance between double interface crack centers is chosen asa/l0=a/l1=6/5 (where l0=l1=5mm, l2=3mm, and h1=5mm and other material parameters are the same as in the previous selection). The influences of electric displacement D0 on the normalized energy release rate G/Gcr are shown in Figures 4 and 5 and the change rules of the normalized energy release rate G/G0 and the value of l/h1 are shown in Figures 6–8 for the interaction between three identical and different Griffith cracks. The rules of the normalized energy release rate G/G0 versus the value of l/h1 are shown in Figures 9 and 10 (where a/l0=a/l1=6/5, l0=l1=5mm, l2=3mm, l3=2.5mm, h1=5mm, and G0=liml/h1⟶0G and other material parameters are the same as in the previous selection).Figure 4
Effects of stressσ0 on the energy release rate for double identical interface cracks.Figure 5
Effects of stressσ0 on the energy release rate for double different interface cracks.Figure 6
Effects ofl0/h1 on the energy release rate for double identical interface cracks.Figure 7
Effects ofl1/h1 on the energy release rate for double different interface cracks.Figure 8
Effects ofl1/h1 on the energy release rate for double different interface cracks.Figure 9
Effects ofl1/h1 on the energy release rate for three identical interface cracks.Figure 10
Effects ofl1/h1 on the energy release rate for three different interface cracks.


#### 5.2.1. Studies of the Effect of Stressσ0 and Electric Displacement D0 on Energy Release Rate for Double Griffith Interface Cracks

Effects of electric displacementD0 and stress σ0 on the energy release rate for double Griffith interface cracks are shown in Figures 4 and 5.


#### 5.2.2. Studies of the Influence ofl/h1 on Energy Release Rate for Double Interface Cracks

Effects ofl/h1 on the energy release rate for double Griffith interface cracks are shown in Figures 6–8.


#### 5.2.3. Studies of the Influence ofl/h1 on Energy Release Rate for Three Interface Cracks

Effects ofl/h1 on the energy release rate for three interface cracks are shown in Figures 9 and 10.


## 5.1. A Single Interface Crack

To validate the solution, we chose the case that material 1 is a fine-grained coating made of PZT-5H ceramics by plasma spraying technology and material 2 is normal PZT-5H ceramics (where 1 and 2 represent the fine-grained coating and substrate material, respectively), which means that material 1 and material 2 have approximately the same piezoelectric and dielectric constants under suitable conditions, and the elastic modulus of material 1 is larger than that of material 2. Soc441/c442=2.5, h1/h2=1/100, and the single interface crack of l0=5mm loaded by an antiplane stress of σ0=4.2MPa and a variable electric displacement D0 are considered. The curves of the energy release rate G/Gcr as a function of D0 under different values of l0/h1 are plotted (Figure 2). Then, the same interface crack is loaded by an electric displacement D0of10−3c/m2 and a variable stress σ0. For different values of l0/h1, the curves G/Gcr as a function of σ0 are plotted (Figure 3).Figure 2
Effects of electric displacementD0 on the energy release rate for different l0/h1.Figure 3
Effects of stressσ0 on the energy release rate for different l0/h1.


### 5.1.1. Studies of the Effect of Stressσ0 and Electric Displacement D0 on Energy Release Rate

Effects of electric displacementD0 and stress σ0 on the energy release rate for different l0/h1 are shown in Figures 2 and 3, respectively.


## 5.1.1. Studies of the Effect of Stressσ0 and Electric Displacement D0 on Energy Release Rate

Effects of electric displacementD0 and stress σ0 on the energy release rate for different l0/h1 are shown in Figures 2 and 3, respectively.


## 5.2. Multiple Interface Cracks

The interaction between multiple interface cracks is investigated. The distance between double interface crack centers is chosen asa/l0=a/l1=6/5 (where l0=l1=5mm, l2=3mm, and h1=5mm and other material parameters are the same as in the previous selection). The influences of electric displacement D0 on the normalized energy release rate G/Gcr are shown in Figures 4 and 5 and the change rules of the normalized energy release rate G/G0 and the value of l/h1 are shown in Figures 6–8 for the interaction between three identical and different Griffith cracks. The rules of the normalized energy release rate G/G0 versus the value of l/h1 are shown in Figures 9 and 10 (where a/l0=a/l1=6/5, l0=l1=5mm, l2=3mm, l3=2.5mm, h1=5mm, and G0=liml/h1⟶0G and other material parameters are the same as in the previous selection).Figure 4
Effects of stressσ0 on the energy release rate for double identical interface cracks.Figure 5
Effects of stressσ0 on the energy release rate for double different interface cracks.Figure 6
Effects ofl0/h1 on the energy release rate for double identical interface cracks.Figure 7
Effects ofl1/h1 on the energy release rate for double different interface cracks.Figure 8
Effects ofl1/h1 on the energy release rate for double different interface cracks.Figure 9
Effects ofl1/h1 on the energy release rate for three identical interface cracks.Figure 10
Effects ofl1/h1 on the energy release rate for three different interface cracks.


### 5.2.1. Studies of the Effect of Stressσ0 and Electric Displacement D0 on Energy Release Rate for Double Griffith Interface Cracks

Effects of electric displacementD0 and stress σ0 on the energy release rate for double Griffith interface cracks are shown in Figures 4 and 5.


### 5.2.2. Studies of the Influence ofl/h1 on Energy Release Rate for Double Interface Cracks

Effects ofl/h1 on the energy release rate for double Griffith interface cracks are shown in Figures 6–8.


### 5.2.3. Studies of the Influence ofl/h1 on Energy Release Rate for Three Interface Cracks

Effects ofl/h1 on the energy release rate for three interface cracks are shown in Figures 9 and 10.


## 5.2.1. Studies of the Effect of Stressσ0 and Electric Displacement D0 on Energy Release Rate for Double Griffith Interface Cracks

Effects of electric displacementD0 and stress σ0 on the energy release rate for double Griffith interface cracks are shown in Figures 4 and 5.


## 5.2.2. Studies of the Influence ofl/h1 on Energy Release Rate for Double Interface Cracks

Effects ofl/h1 on the energy release rate for double Griffith interface cracks are shown in Figures 6–8.


## 5.2.3. Studies of the Influence ofl/h1 on Energy Release Rate for Three Interface Cracks

Effects ofl/h1 on the energy release rate for three interface cracks are shown in Figures 9 and 10.


## 6. Results and Discussion

Figure2 displays the variation of the normalized energy release rate G/Gcr against the applied electric displacement D0. It can be clearly seen that by varying l0/h1, the curves G/Gcr are parabolas, and as D0 decreases from zero, G/Gcr always decreases. However, when D0 increases from zero, G/Gcr firstly increases and then begins to decrease. The results show that negative electric displacement loading always prevents crack propagation and positive electric displacement loading may prompt or prevent crack propagation, which are consistent with the conclusions of Pak et al. [4, 10, 13]. However, the peak energy release rate of the fine-grained piezoelectric coating/substrate structure is higher than that of the large-grained piezoelectric coating/substrate structure.Figure3 indicates the effects of applied stress σ0 on the normalized energy release rate G/Gcr. It can be easily seen that as stress σ0 increases from zero, G/Gcr always increases. When the stress σ0 decreases from zero, the G/Gcr first decreases and then increases, and it indicates the initial closure of a crack followed by a type of crack propagation due to compressive loading. The results indicate that stress loading may prompt or retard crack propagation.The diversification of the normalized energy release rateG/Gcr versus the applied electric displacement D0 for double identical and different interface cracks is shown in Figures 4 and 5. It is found that the variation of the energy release rate against the applied electric displacement is similar to that shown in Figure 2. However, the normalized energy release rate of the double interface crack has a larger peak compared to that shown in Figure 2. Moreover, the peak values of the energy release rate near the crack tips b1, a2 are larger than those near the crack tips a1, b2. Because of symmetry when the length of double interface cracks is equal, the variations of the energy release rate near the crack tips b1, a2 and a1, b2 are the same.Figures6–8 present the effects of the values of l/h1 (where l stands for the crack length) on the normalized energy release rate G/G0 for double identical and different interface cracks. We found that the energy release rate increased with the increase of l/h1. The change rate of the inner crack tips is higher with the variation of l/h1 than that of the outer crack tips. The lines in Figure 8 intersect with the variation of l/h1 in this case; if the double cracks have the same length, they are consistent with previous conclusions.Figures9 and 10 illustrate the change rule of the normalized energy release rate G/G0 and l/h1 for three identical and different interface cracks. The results reflect that the values of the energy release rate susceptible to l/h1 near the inner crack tips are larger than those that approach the outer crack tips, regardless of the number of cracks. At the same time, we noticed that when the value of thickness of coating h1 was small, the value of the energy release rate was large, which indicated that the structure was more safe when the coating was thin. This reflects the superiority of the fine-grained piezoelectric material structure because it can make the structure thinner.


## 7. Conclusions

In this paper, multiple Griffith cracks at the interface of a fine-grained piezoelectric coating/substrate are studied. The variation of the energy release rate with applied stress, electric displacement, crack length, and coating thickness is demonstrated. Based on the change rule of the energy release rate and various parameters, the following conclusions are drawn:(1)
The interaction between cracks, applied electromechanical loads, and coating thickness has a great influence on the energy release rate for Griffith interface cracks. Moreover, the negative electric displacement loading always prevents crack propagation, and positive electric displacement loading may prompt or delay crack propagation. Furthermore, the appropriate stress loading can restrain the crack growth. Therefore, in the practical application of engineering, we can choose the appropriate load to ensure the safety of the structure.(2)
For a bimaterial structure, compared with previous works [10, 13], we found that the peak of the energy release rate at the crack tips of a fine-grained piezoelectric coating/substrate interface is larger than that at the crack tips of a large-grained piezoelectric material/substrate structure interface, and thus, the fine-grained piezoelectric coating/substrate structure has safer material properties. Therefore, in order to better resist fractures, fine-grained piezoelectric materials are a better choice in engineering.



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*Source: 2547891-2019-08-19.xml*