# Exact Solutions for Interaction of Parallel Screw Dislocations with a Wedge Crack in One-Dimensional Hexagonal Quasicrystal with Piezoelectric Effects

Authors: Xin Lv; Guan-Ting Liu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The purpose of this paper is to consider the interaction between many parallel dislocations and a wedge-shaped crack and their collective response to the external applied generalized stress in one-dimensional hexagonal piezoelectric quasicrystal, employing the complex variable function theory and the conformal transformation method; the problem for the crack is reduced to the solution of singular integral equations, which can be further reduced to solving Riemann–Hilbert boundary value problems. The analytical solutions of the generalized stress field are obtained. The dislocations are subjected to the phonon field line force, phason field line force, and line charge at the core. The positions of the dislocations are arbitrary, but the dislocation distribution is additive. The dislocation is not only subjected to the external stress and the internal stress generated by the crack, but also to the force exerted on it by other dislocations. The closed-form solutions are obtained for field intensity factors and the image force on a screw dislocation in the presence of a wedge-shaped crack and a collection of other dislocations. Numerical examples are provided to show the effects of wedge angle, dislocation position, dislocation distribution containing symmetric configurations and dislocation quantities on the field intensity factors, energy release rate, and image force acting on the dislocation. The principal new physical results obtained here are (1) the phonon stress, phason stress, and electric displacement singularity occur at the crack tip and dislocations cores, (2) the increasing number of dislocations always accelerates the crack propagation, (3) the effect of wedge angle on crack propagation is related to the distribution of dislocations, and (4) the results of the image force on the dislocation indicate that the dislocations can either be attracted or rejected and reach stable positions eventually.


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## Body

## 1. Introduction

As a new structure of solid, quasicrystal (QC) was discovered by Shechtman et al. [1] in 1984. QC is a new solid structure with long-range orientation order and without translational symmetry which differs from crystals. The elastic problem of QC is related not only to the phonon field which describes lattice vibration, but also to the phason field which is about the quasiperiodic arrangement of atoms. As soon as the quasicrystal was discovered, it attracted wide attention because of its special symmetry and good performance. In the early discovery of the quasicrystal, the defects in the quasicrystal were observed [2]. Due to the unique properties of quasicrystal materials, many achievements have been made in the study of quasicrystal elasticity and fracture mechanics [3–9]. To name a few recent researches, Lou et al. [10] proposed a model of a thin elastic inclusion embedded in an infinite 1D hexagonal quasicrystal. Wang and Schiavone [11] investigated the elastic field near the tip of an anticrack in a homogeneous decagonal quasicrystal material subjected to plane strain deformations. Zhao et al. [12] studied the frictional contact problem and the adhesive contact problem in three-dimensional (3D) icosahedral quasicrystal. Li et al. [13] have given analytical solutions for arbitrarily shaped planar cracks in a two-dimensional (2D) hexagonal QC media.Quasicrystal is sensitive to force, electricity, magnetism, heat, and light, and these physical properties have been extensively studied. It is well known that QCs with piezoelectric effects have better performance in smart composite structures. Therefore, the study of quasicrystal materials considering the piezoelectric effect is of great significance. A lot of work on QC with piezoelectric effects has been published in the open literature. Li et al. [14] obtained three-dimensional general solutions to static problems of a 1D hexagonal piezoelectric QC. Yu et al. [15] considered an antiplane crack problem by the semi-inverse method in a 1D hexagonal piezoelectric QC. Yu et al. [16] also discussed the complex variable method for an antiplane elliptical cavity. Yang et al. [17] presented the fracture behavior of two asymmetrical limited permeable cracks emanating from an elliptical hole in a one-dimensional hexagonal quasicrystal with piezoelectric effects. Yang et al. [18] derived the closed-form solution for a multilayered two-dimensional decagonal QC plate. Guo et al. [19] analyzed an elliptical inclusion embedded in an infinite 1D hexagonal piezoelectric QC matrix. Li et al. [20] have given the exact solution for a layered piezoelectric quasicrystal nanoplate with nonlocal effect in cylindrical bending. Zhang et al. [21] addressed the static bending deformation of a functionally graded multilayered 1D hexagonal piezoelectric QC simply supported nanoplate. Zhou and Li [22] dealt with the problem of two collinear cracks in a one-dimensional hexagonal piezoelectric QC strip under mode III far-field loading, while they analyzed the electroelastic problem related to two collinear cracks of equal length and normal to the boundaries of a 1D hexagonal piezoelectric quasicrystal layer with the aid of a singular integral equation in ref. [23]. Besides, Zhou and Li [24] examined the electroelastic behavior induced by a penny-shaped dielectric crack of a 1D hexagonal piezoelectric quasicrystal plate of finite thickness, and they considered a Yoffe-type moving crack in a 1D hexagonal piezoelectric quasicrystal by using the Fourier transform technique in ref. [25]. Li and Liu [26] derived the electroelastic field of a straight dislocation in an icosahedral piezoelectric QC. Li et al. [27] displayed the interaction between a screw dislocation and an elliptical hole with two asymmetrical cracks in a 1D hexagonal quasicrystal with piezoelectric effects. Hu et al. [28] studied an antiplane interface crack between dissimilar 1D hexagonal piezoelectric QC.Engineering materials always contain some defects, such as dislocations, inclusions, and cracks. The existence of these defects often seriously affects the safety and reliability of materials. There is not only one defect in the material, but usually different types of defects also exist at the same time, and the interaction between multiple defects has a very important impact on the mechanical properties and physical properties of the material. Dislocation activity is the cause of plastic deformation at the crack tip, which determines the crack propagation behavior, so the interaction between crack and dislocation has attracted more and more attention. Concerning the wedge-shaped crack which is an important geometric form of structures interacting with a single dislocation, Ohr et al. [29] performed the elastic analysis of the interaction between a wedge crack and a screw dislocation. Zhang et al. [30] obtained an analytical solution of the interaction of edge dislocation with the wedge crack by using the elastic technique of Muskhelishvili. Chen et al. [31] derived the closed solution of the electroelastic field for a screw dislocation situated near the tip two-bonded wedge-shaped dissimilar piezoelectric materials. Furthermore, they deduced the electroelastic stress field in the interaction of a piezoelectric screw dislocation with a wedge-shaped bimaterial interface in ref. [32]. Li and Zhao [33] coped with the problem of the interaction of a screw dislocation with the interface and wedge-shaped cracks in a one-dimensional hexagonal piezoelectric QC biomaterial using the conformal mapping method in conjunction with the image principle. Jiang and Liu [34] investigated the interaction between a screw dislocation and a wedge-shaped crack in the 1D hexagonal piezoelectric QC based on the conformal mapping method and the perturbation technique.The research studies mentioned above were mainly focused on the single screw dislocation interacting with the various types of cracks. In engineering practice, structures with pairs of dislocations are very common. Dislocation emission and dislocation pile-up at the crack tip will result in crack blunting, inhibiting the growth of the crack and then improving the fracture toughness of the material. Concerning multiple dislocations interacting with cracks, the dislocation is not only subjected to the external stress and the internal stress generated by the crack, but also to the force exerted on it by other dislocations. Lin and Thomson [35] earlier investigated the elastic interaction between pairs of dislocations and a cleavage crack under external loads. Later, Juang and Lee [36] examined the interaction of the two-dislocation model and a surface crack by using the dislocation-modelling technique and the complex variable method. More recently, Liu and Yang [37] made a detailed analysis of the stress field of infinitely many parallel screw dislocations near a semi-infinite crack tip without applied loads, and dislocations considered here are located at particular points to make the total stress finite. Lv and Liu [38] addressed the analytical solution of the generalized stress field for parallel screw dislocations near a semi-infinite crack in magnetic-electric-mechanical coupling fields. However, for the piezoelectric QC, these problems become more complicated than the conventional material due to the introduction of the phason field and the electric field. To the best of the authors’ knowledge, the study on the interaction between the wedge-shaped crack and multiple dislocations in a 1D hexagonal QC with piezoelectric effect has not been reported yet. In this paper, through the singular integral technique, we consider the antiplane problem of the interaction between many parallel screw dislocations and a wedge-shaped crack under remote uniform loadings, and the dislocations are subjected to the phonon field line force, phason field line force, and line charge at the core. The positions of the dislocations are arbitrary. The advantage of the present model is demonstrated when dealing with many dislocations because the dislocation distribution is additive. The expressions for the generalized stress field, field intensity factors, energy release rate, and image force are derived in a closed form. The exact solutions are presented and compared with those already obtained by other authors. Numerical examples are presented for various parameters of the problem.


## 2. Basic Equations

The quasicrystal piezoelectric material considered here is transversely isotropic with a hexagonal symmetry, which has an isotropic basal plane ofx1−x2 plane and an electric field poling direction and quasiperiodic direction of the x3 axis. When the defects penetrate along the x3 axis, the quasicrystal piezoelectric boundary value problem is treated as in the case of out-of-plane phonon field displacement u3x1,x2, out-of- plane phason field displacement w3x1,x2, and in-plane electric field φx1,x2. The basic equations can be written as follows [39].In this case, the constitutive relations become(1)σ3k=C44u3,k+R3w3,k−e15φ,k,H3k=R3u3,k+K2w3,k−d15φ,k,Dk=e15u3,k+d15w3,k+λ11φ,k,k=1,2.Generalized strain-displacement relations are given by(2)εij=ui,j+uj,i2,ωij=wi,j,Ei=−ϕ,i,i=1,2,3.Equilibrium equations are(3)σij,i=0,Hij,i=0,Di,i=0,where a comma represents differentiation with respect toxii=1,2,3; the repeated indices denote that summation σ3k and ε3k are the phonon field stress and strain; H3k and ω3k are the phason field stress and strain; Dk and Ek are the electric displacement and the electric field; C44, K2, and R3 are the phonon field elastic, phason field elastic, and phonon-phason field coupling coefficients; e15 and d15 are the piezoelectric constants; λ11 is the dielectric constant.Substituting equations (1) and (2) to (3), the governing equation becomes(4)C∇2u=0,in which(5)u=u3w3φT,∇2=∂2∂x12+∂2∂x22,C=C44R3e15R3K2d15e15d15−λ11.In general,(6)detC=−C44d152−K2e152+2R3d15e15−C44K2λ11+R32λ11≠0.Equation (4) can be written as(7)∇2u=0.


## 3. Description of the Problem and Its Solutions

### 3.1. The Electroelastic field of Parallel Screw Dislocations

The physical problem is shown in Figure1. There are n parallel screw dislocations located at ziri,θii=1,2,...,n with the Burgers vectors 0,0,b3i,b⊥i,bφi in the vicinity of a wedge-shaped crack in the 1D hexagonal QC piezoelectric material which is under the far-field uniform loading σ32∞,H32∞,E2∞. The dislocations are subjected to the phonon field line force p, phason field line force q, and line charge v at the core. The wedge-shaped crack is infinitely long along the negative real axis, and α denotes the wedge angle.Figure 1
The interaction between a wedge-shaped crack and many parallel screw dislocations.As shown in Figure1, we have the following crack boundary conditions:(8)x12+x22⟶∞:σ32=σ32∞,H32=H32∞,E2=E2∞,x,y∈θ=±π−α2:σ32=0,H32=0,E2=0.Proposing a conformal transformation function(9)z=mζ=ζ1/λ,λ=2π2π−α,which maps the wedge-shaped crack in the z plane to a semi-infinite crack along the negative real axis in the ζ plane (see Figure 2). The problem turns into the interaction of parallel screw dislocations with a semi-infinite crack.Figure 2
Conformal transformation.In theζ plane, we first consider the analytical solution of the elastic field of a screw dislocation. Assuming that one of the dislocations is situated at the point ζd with the Burgers vector 0,0,0,b3,b⊥,bφ, u can be expressed by the imaginary part of an arbitrary analytic function fζ following the theory of a complex variable function, and ζ=ξ1+ξ2 and(10)u=Imfζ.It is reasonable to assume that the potential function is composed of two parts; that is,(11)fζ=f0ζ+f1ζ,where f0ζ is the complex potential for the dislocation in an infinite medium and f1ζ is the additional term which assumes that the stress boundary conditions on the cleavage surfaces are satisfied. Based on the force conditions of the dislocation, f0ζ can be chosen as(12)f0ζ=Alnζ−ζd,with(13)A=A1+iA2,B1+iB2,C1+iC2Τ.Then,(14)Im∮ζdf0′ζdζ=Im∮ζdAζ−ζddζ=Im2πiA=b3,b⊥,bφΤ,Im∮ζd−iCf0′ζdζ=Im∮ζd−iCAζ−ζddζ=Im2πCA=−p,−q,vΤ.From equations (10)–(14), we have(15)A1=b32π,A2=K2λ11+d152p−R3λ11+e15d15q−K2e15−R3d15v2πdetC,B1=b⊥2π,B2=−λ11R3+e15d15p+λ11C44+e152q−C44d15−R3e15v2πdetC,C1=bφ2π,C2=e15K2−R3d15p+C44d15−R3e15q+C44K2−R32v2πdetC.Introducing the generalized strainsZ2=ε32,ω32,−E2 and Z1=ε32,ω32,−E2and the generalized stresses Σ2=σ32,H32,D2Τ and Σ1=σ31,H31,D1Τ, the generalized strain field and the stress field can be written as(16)Ζ2+iΖ1=f0′ζ=Aζ−ζd,(17)Σ2+iΣ1=Cf0′ζ=CAζ−ζd.According to equation (17), the generalized stress stimulated by dislocation ζi which is working on ζ1 is(18)Fi1=CAζ1−ζi.Now, we study the electroelastic solution of parallel screw dislocations. Taking the form of the corresponding matrix asAi, the generalized stress generated by dislocation ζi exerted on the arbitrary point ζ is(19)F0ζ=∑i=1nCAiζ−ζi.


### 3.2. The Interaction ofn Parallel Screw Dislocations with Crack

It can be seen from the above discussion that the distribution force induced by dislocations on a semi-infinite crack surface is(20)F0ζ=∑i=1nCAiζ−ζi+CA¯iζ−ζ¯i,−∞<ξ1<0,ξ2=0.Here, the condition of the free crack surface is considered; that is,(21)Ref′ζΤ=0,−∞<ξ1<0,ξ2=0,where Re denotes the real part of an analytical function vector.The physical meaning of the boundary condition equation (21) is that the additional distribution force F1 represents the cancelling force exerted on the boundary equal to the negative of F0. With the aid of equation (20), the boundary condition equation (21) leads to a system of the Riemann–Hilbert problem as(22)F1+ξ1+F1−ξ1=−F0ξ1.Using the standard methods given by Muskhelishvili [40], the general solution of equation (21) arrives at(23)F1ζ=−Xζ2πi∫−∞0F0ξ1X+ξ1ξ1−ζdξ1+PζXζ,where(24)Xζ=ζ−1/2,and we have considered the single-valued condition of generalized displacements, i.e., Pζ=0.The solution of equation (20) together with equations (23) and (24) gives(25)F1ζ=−∑i=1nC2ζAiζ+ζi+A¯iζ+ζ¯i.Considering the stress field of the crack due to the far-field uniform loadingsσ32∞,H32∞,E2∞, equation (20) combined with equation (25) yields(26)Ftotalζ=∑i=1nC2ζAiζ−ζi+A¯iζ−ζ¯i+CBζ,in which B=A3,B3,C3Τ and(27)A3=σ32∞−R3ω32∞+e15E2∞C44,B3=ω32∞,C3=−E2∞.Therefore, the analytic solution of electroelastic interaction of many parallel screw dislocations with the crack is(28)Ftotalz=Ftotalζm′ζ=∑i=1nλC2z1−λ/2Aizλ/2−ziλ/2+A¯izλ/2−z¯iλ/2+λCBz1−λ/2.Ifλ=1 and i=1, equation (28) reduces to the result in ref. [41] under some restrictions.


## 3.1. The Electroelastic field of Parallel Screw Dislocations

The physical problem is shown in Figure1. There are n parallel screw dislocations located at ziri,θii=1,2,...,n with the Burgers vectors 0,0,b3i,b⊥i,bφi in the vicinity of a wedge-shaped crack in the 1D hexagonal QC piezoelectric material which is under the far-field uniform loading σ32∞,H32∞,E2∞. The dislocations are subjected to the phonon field line force p, phason field line force q, and line charge v at the core. The wedge-shaped crack is infinitely long along the negative real axis, and α denotes the wedge angle.Figure 1
The interaction between a wedge-shaped crack and many parallel screw dislocations.As shown in Figure1, we have the following crack boundary conditions:(8)x12+x22⟶∞:σ32=σ32∞,H32=H32∞,E2=E2∞,x,y∈θ=±π−α2:σ32=0,H32=0,E2=0.Proposing a conformal transformation function(9)z=mζ=ζ1/λ,λ=2π2π−α,which maps the wedge-shaped crack in the z plane to a semi-infinite crack along the negative real axis in the ζ plane (see Figure 2). The problem turns into the interaction of parallel screw dislocations with a semi-infinite crack.Figure 2
Conformal transformation.In theζ plane, we first consider the analytical solution of the elastic field of a screw dislocation. Assuming that one of the dislocations is situated at the point ζd with the Burgers vector 0,0,0,b3,b⊥,bφ, u can be expressed by the imaginary part of an arbitrary analytic function fζ following the theory of a complex variable function, and ζ=ξ1+ξ2 and(10)u=Imfζ.It is reasonable to assume that the potential function is composed of two parts; that is,(11)fζ=f0ζ+f1ζ,where f0ζ is the complex potential for the dislocation in an infinite medium and f1ζ is the additional term which assumes that the stress boundary conditions on the cleavage surfaces are satisfied. Based on the force conditions of the dislocation, f0ζ can be chosen as(12)f0ζ=Alnζ−ζd,with(13)A=A1+iA2,B1+iB2,C1+iC2Τ.Then,(14)Im∮ζdf0′ζdζ=Im∮ζdAζ−ζddζ=Im2πiA=b3,b⊥,bφΤ,Im∮ζd−iCf0′ζdζ=Im∮ζd−iCAζ−ζddζ=Im2πCA=−p,−q,vΤ.From equations (10)–(14), we have(15)A1=b32π,A2=K2λ11+d152p−R3λ11+e15d15q−K2e15−R3d15v2πdetC,B1=b⊥2π,B2=−λ11R3+e15d15p+λ11C44+e152q−C44d15−R3e15v2πdetC,C1=bφ2π,C2=e15K2−R3d15p+C44d15−R3e15q+C44K2−R32v2πdetC.Introducing the generalized strainsZ2=ε32,ω32,−E2 and Z1=ε32,ω32,−E2and the generalized stresses Σ2=σ32,H32,D2Τ and Σ1=σ31,H31,D1Τ, the generalized strain field and the stress field can be written as(16)Ζ2+iΖ1=f0′ζ=Aζ−ζd,(17)Σ2+iΣ1=Cf0′ζ=CAζ−ζd.According to equation (17), the generalized stress stimulated by dislocation ζi which is working on ζ1 is(18)Fi1=CAζ1−ζi.Now, we study the electroelastic solution of parallel screw dislocations. Taking the form of the corresponding matrix asAi, the generalized stress generated by dislocation ζi exerted on the arbitrary point ζ is(19)F0ζ=∑i=1nCAiζ−ζi.


## 3.2. The Interaction ofn Parallel Screw Dislocations with Crack

It can be seen from the above discussion that the distribution force induced by dislocations on a semi-infinite crack surface is(20)F0ζ=∑i=1nCAiζ−ζi+CA¯iζ−ζ¯i,−∞<ξ1<0,ξ2=0.Here, the condition of the free crack surface is considered; that is,(21)Ref′ζΤ=0,−∞<ξ1<0,ξ2=0,where Re denotes the real part of an analytical function vector.The physical meaning of the boundary condition equation (21) is that the additional distribution force F1 represents the cancelling force exerted on the boundary equal to the negative of F0. With the aid of equation (20), the boundary condition equation (21) leads to a system of the Riemann–Hilbert problem as(22)F1+ξ1+F1−ξ1=−F0ξ1.Using the standard methods given by Muskhelishvili [40], the general solution of equation (21) arrives at(23)F1ζ=−Xζ2πi∫−∞0F0ξ1X+ξ1ξ1−ζdξ1+PζXζ,where(24)Xζ=ζ−1/2,and we have considered the single-valued condition of generalized displacements, i.e., Pζ=0.The solution of equation (20) together with equations (23) and (24) gives(25)F1ζ=−∑i=1nC2ζAiζ+ζi+A¯iζ+ζ¯i.Considering the stress field of the crack due to the far-field uniform loadingsσ32∞,H32∞,E2∞, equation (20) combined with equation (25) yields(26)Ftotalζ=∑i=1nC2ζAiζ−ζi+A¯iζ−ζ¯i+CBζ,in which B=A3,B3,C3Τ and(27)A3=σ32∞−R3ω32∞+e15E2∞C44,B3=ω32∞,C3=−E2∞.Therefore, the analytic solution of electroelastic interaction of many parallel screw dislocations with the crack is(28)Ftotalz=Ftotalζm′ζ=∑i=1nλC2z1−λ/2Aizλ/2−ziλ/2+A¯izλ/2−z¯iλ/2+λCBz1−λ/2.Ifλ=1 and i=1, equation (28) reduces to the result in ref. [41] under some restrictions.


## 4. Field Intensity Factors and Energy Release Rate

### 4.1. Field Intensity Factors

In thez plane, according to ref. [31], the phonon field stress, the phason field stress, and the electric displacement intensity factor at the wedge-shaped crack tip are defined as(29)K=Kσ32+iKσ31KH32+iKH31KD2+iKD1=limz⟶02πz1−λ/2Ftotal.Actually, onlyε32, ω32, E2, σ32, H32, and D2 are singular at the crack tip. Substituting equation (28) to (29) and setting zi=rieiθi, we obtain(30)Kσ32=λ2πriλ/2∑i=1npsinλ2θi−C44b3+R3b⊥+e15bφcosλ2θi+λ2π1−λ/2σ32∞,(31)KH32=λ2πriλ/2∑i=1nqsinλ2θi−R3b3+K2b⊥+d15bφcosλ2θi−λ2π1−λ/2d15E2∞−K2ω32∞−R3e15E2∞−R3ω32∞+σ32∞C44,(32)KD2=−λ2πriλ/2∑i=1nvsinλ2θi+e15b3+d15b⊥−λ11bφcosλ2θi+λ2π1−λ/2λ11E2∞+d15ω32∞+e15e15E2∞−R3ω32∞+σ32∞C44.Equations (30)–(32) agree with Chen’s results [31] if there is only one dislocation and the phason field and the applied generalized stress are not considered. If the applied loading is σ32∞,H32∞,E2∞ and the matrix only contains one dislocation whose core are not under the line force, the above equations can reduce to Li’s results [3]. On the basis of the above restrictions, if the phason field is neglected, the solutions agree with those in ref. [34]. Moreover, equations (30)–(32) suggest that Kσ32 is independent of q and v, KH32 is independent of p and v, and KD2 is independent of p and q, if all b3, b⊥, bφ, p, q, and v are supposed to be positive.


### 4.2. Energy Release Rate

The energy release rate can be calculated as [42](33)G=14KΤHK,where H=2C−1.Thus, the energy release rate for the interaction of the wedge-shaped crack with screw dislocations is derived as(34)G=12detC−d152−K2λ11Kσ322+−e152−C44λ11KH322+C44K2−R32KD22+2d15e15+R3λ11Kσ32KH32+2−C44d15+e15R3KH32KD2+2−e15K2+d15R3Kσ32KD2.


## 4.1. Field Intensity Factors

In thez plane, according to ref. [31], the phonon field stress, the phason field stress, and the electric displacement intensity factor at the wedge-shaped crack tip are defined as(29)K=Kσ32+iKσ31KH32+iKH31KD2+iKD1=limz⟶02πz1−λ/2Ftotal.Actually, onlyε32, ω32, E2, σ32, H32, and D2 are singular at the crack tip. Substituting equation (28) to (29) and setting zi=rieiθi, we obtain(30)Kσ32=λ2πriλ/2∑i=1npsinλ2θi−C44b3+R3b⊥+e15bφcosλ2θi+λ2π1−λ/2σ32∞,(31)KH32=λ2πriλ/2∑i=1nqsinλ2θi−R3b3+K2b⊥+d15bφcosλ2θi−λ2π1−λ/2d15E2∞−K2ω32∞−R3e15E2∞−R3ω32∞+σ32∞C44,(32)KD2=−λ2πriλ/2∑i=1nvsinλ2θi+e15b3+d15b⊥−λ11bφcosλ2θi+λ2π1−λ/2λ11E2∞+d15ω32∞+e15e15E2∞−R3ω32∞+σ32∞C44.Equations (30)–(32) agree with Chen’s results [31] if there is only one dislocation and the phason field and the applied generalized stress are not considered. If the applied loading is σ32∞,H32∞,E2∞ and the matrix only contains one dislocation whose core are not under the line force, the above equations can reduce to Li’s results [3]. On the basis of the above restrictions, if the phason field is neglected, the solutions agree with those in ref. [34]. Moreover, equations (30)–(32) suggest that Kσ32 is independent of q and v, KH32 is independent of p and v, and KD2 is independent of p and q, if all b3, b⊥, bφ, p, q, and v are supposed to be positive.


## 4.2. Energy Release Rate

The energy release rate can be calculated as [42](33)G=14KΤHK,where H=2C−1.Thus, the energy release rate for the interaction of the wedge-shaped crack with screw dislocations is derived as(34)G=12detC−d152−K2λ11Kσ322+−e152−C44λ11KH322+C44K2−R32KD22+2d15e15+R3λ11Kσ32KH32+2−C44d15+e15R3KH32KD2+2−e15K2+d15R3Kσ32KD2.


## 5. Image Force

The image force is expressed as [43](35)F1=b3σ32Τ+b⊥H32Τ+bφD2Τ+pε31Τ+qω31Τ+vE1Τ,F2=−b3σ31Τ−b⊥H31Τ−bφD1Τ+pε32Τ+qω32Τ+vE2Τ,where stresses and strains with superscript “Τ” are the perturbation stresses and strains at the dislocation point.Therefore, the image force exerted on the dislocation atzd (zd is excluded from zi) in the presence of a wedge-shaped crack and a collection of other dislocations at zi can be expressed as(36)f¯d=F1−iF2=bCm′ζd−A4ζd+A¯2ζdζd−ζ¯d+ib′m′ζd−A4ζd+A¯2ζdζd−ζ¯d+∑ibCm′ζdAi2ζdζd−ζi+A¯i2ζdζd−ζ¯i+ib′m′ζdAi2ζdζd−ζi+A¯i2ζdζd−ζ¯i+bCB+ib′Bm′ζdζd,where(37)b=b3,b⊥,bφ,b′=p,q,v.Noting that the above solution can be reduced to Lin and Thomson [35] in the case of λ=1 and the material is pure elastic. Furthermore, equation (36) is comprised of three terms. The first term containing the quadratic of b,b′ coming from the image dislocation. The second term containing b,b′ and bi,b′i which are the interaction forces from the other dislocations acting on the dislocation zd. The last one containing b,b′ which is induced by the applied generalized stress.When all the dislocations are located at the real axis, the solution can be simplified as(38)f¯d=−bCA4m′ξdξd−ib′A4m′ξdξd+∑ibCAim′ξdξdξd−ξi+ib′Aim′ξdξdξd−ξi+bCB+ib′Bm′ξdξd,i.e.,(39)f¯d=−bCA4xd−ib′A4xd+∑ibCAixd1−λ/2xdλ/2−xiλ/2+ib′Aixd1−λ/2xdλ/2−xiλ/2+bCB+ib′Bxd1−λ/2.


## 6. Numerical Example

Although the field intensity factors, energy release rate, and the image force acting on the dislocation are expressed explicitly in the previous sections, it is not straightforward that how the dislocation location, dislocation distribution, the wedge angle, and the number of dislocations affect them. The numerical examples are therefore performed in this section. The material constants we choose are listed in Table1 [14].Table 1
Material constants in the 1D hexagonal piezoelectric quasicrystal.
C44Nm−2R3Nm−2K2Nm−2e15Cm−2d15Cm−2λ11C2N−1m−25×10101.20×1090.30×109−0.138−0.1682.6×10−12The Burgers vectors, line forces, and line charge at the dislocation core are(40)b3=b⊥=1.0×10−9m,bφ=1.0×10−9V,p=q=10N/m,v=1.0×10−3C/m.The critical energy release rate is taken asGr=5N/m and the wedge angle in the following numerical discussions is π/6. For the sake of discussion, the magnitude of all Burgers vectors is regarded as 10−9 and all the dislocation have the same value of p,q,v.Figures3–5, respectively, show the phonon field stress, phason field stress, and electric displacement component contours due to the dislocations. When the dislocations are located at the points 0,1 and 0,−1, one may note that as the lines of σ32, H32, and D2 get closer to the crack surface, they become parallel to the surface and the generalized stress becomes smaller. The phonon field stress σ32 vanishes when it reaches the crack surface, as well as the phason field stress H32 and electric displacement D2, so that the boundary condition is satisfied. The lines of σ31, H31, and D1 are antisymmetric about the straight line x2=0, but the lines of σ32, H32, and D2 are symmetric about the straight line x2=0. These results also reveal that the phonon field stress, phason field stress, and the electric displacement field show square root singularity at the crack tip while having one order singularity at the location of dislocation. The phonon field stress contours have the same trend as the phason field stress contours but is opposite to the electric field. In addition, the magnitude of the electric displacement is less than the magnitude of the phason field stress, but the magnitude of the phason field stress is less than the magnitude of the phonon field stress.Figure 3
Contours of phonon field stress components are induced by dislocations situated at0,1 and 0,−1, (a) σ32 and (b) σ31, and at 1,0 and 2,0, (c) σ32 and (d) σ31.
(a)(b)(c)(d)Figure 4
Contours of phason field stress components are induced by dislocations situated at0,1 and 0,−1, (a) H32/10−2 and (b) H31/10−2, and at 1,0 and 2,0, (c) H32/10−2 and (d) H31/10−2.
(a)(b)(c)(d)Figure 5
Contours of electric displacement components are induced by dislocations situated at0,1 and 0,−1, (a) D2/10−11 and (b) D1/10−11, and at 1,0 and 2,0, (c) D2/10−12 and (d) D1/10−12.
(a)(b)(c)(d)Supposing there are already two dislocations located atx1=1000×b3 and x2=2000×b3, respectively, the relationship between the field intensity factors and the positions of the other dislocation is exhibited in Figures 6–8 without remote uniform loading. Let us define the normalized phonon field intensity factor, phason field intensity factor, and electric displacement intensity factor as K1∗=Kσ32/C44b3, K2∗=KH32/C44b3, and K3∗=KD2/C44b3, respectively. It can be seen that the value of K1∗ and K3∗ is always negative, but the value of K2∗ is always positive. We find that the effects of the dislocation position on the three intensity factors are different. Figures 6(a) and Figure 6(b) show that the shielding effect in the phonon field quickly decreases with the value of the normalized distance rd/b3 increasing, while decreases with the value of θd increasing. From Figures 7(a) and Figure 7(b), it is interesting to note that K2∗ increases with the increasing θd, while decreases with the increasing rd/b3. When the dislocation is located at the real axis, the stress intensity factor is positive. Figures 8(a) and Figure 8(b) indicate that the shielding effect on the electric field decreases as rd/b3 increases, while it decreases as θd decreases. Taking the stress intensity factor of the phonon field as an example, Figure 9(a) illustrates that K1∗ increases with α, and the larger the wedge angle is, the stronger the shielding effect will become. The variations in the three field intensity factors with the wedge angle are plotted in Figure 9(b). When the two dislocations are located at x1=1 and x2=2, it can be found that the phonon field stress intensity factor increases as the wedge angle increases, but λ has little effect on the phason field stress intensity factor and the electric displacement intensity factor since the material constants related to the phonon field are far greater than those related to the phason field and the electric field.Figure 6
(a)K1∗ varies with rd/b3. (b) K1∗ varies with θd.
(a)(b)Figure 7
(a)K2∗ varies with rd/b3. (b) K2∗ varies with θd.
(a)(b)Figure 8
(a)K3∗/10−5 varies with rd/b3. (b) K3∗/10−6 varies with θd.
(a)(b)Figure 9
(a)K1∗ varies with rd/b3 of different α. (b) Field intensity factors vary with λ.
(a)(b)Similarity, there exist two dislocations located atx1=1000×b3 and x2=2000×b3, respectively. In this case, Figures 10(a) and 10(b) more clearly show the retarding effects, respectively, of rd/b3 and θd on the energy release rate without far-field loading. The normalized energy release rate G/Gr increases as d θ decreases. In contrast, G/Gr increases as rd/b3 increases. Figure 10(a) implies that rd/b3 contributes none to the energy release rate when its value increases to a certain extent. Figure 10(b) illustrates that rd/b3 does not affect the energy rate when all the dislocations are situated at the real axis. Figures 11(a) and Figure 11(b) depict the effect of the wedge angle on the energy release rate when the dislocations are in different positions. The energy release rate is positive whether the dislocations are located at the real axis or symmetrically distributed at the imaginary axis, which means the wedge angle accelerates the crack propagation. G increases as the number of dislocations increases. Also, G increases with the increasing λ when all the dislocations are located at the real axis, while it increases first and then decreases when the dislocations are situated at the imaginary axis symmetrically.Figure 10
(a)G/Gr varies with rd/b3. (b) G/Gr/10−5 varies with θd.
(a)(b)Figure 11
(a) Energy release rate varies withλ when all the dislocations are located at the real axis. (b) Energy release rate varies with λ when all the dislocations are located at the imaginary axis symmetrically.
(a)(b)Making all dislocations located at the real axis, andx1=1 and x2=2, in the absence of the remote applied generalized stress, Figure 12(a) shows that f¯d is always negative when the dislocation xd is between the crack tip and dislocation x1, which means dislocation xd is always attracted. When the dislocation xd from x2 moves to x1, it is first rejected and then attracted, and the attractive force is gradually increased and the repulsive force is gradually reduced. When the dislocation xd is farther away from x2, it is always rejected and the repulsive force is reduced. Figure 12(b) implies that the force on the dislocation is always positive and the magnitude decreases when increasing the wedge angle.Figure 12
(a) The force acting on the dislocationxdvaries with the distance from the dislocation xd to the crack tip when the other dislocations are located at 1,0 and 2,0. (b) The force acting on the dislocation xdvaries with λ when the other dislocations are located at1,0 and2,0.
(a)(b)


## 7. Conclusions

In this paper, the fracture mechanic model of the two-dimensional crack and parallel dislocations is established. The interaction between parallel screw dislocations and a wedge-shaped crack in a one-dimensional hexagonal quasicrystal with piezoelectric effects is studied, and the closed-form solutions of the electroelastic field, field intensity factors, energy release rate, and generalized Peach-Koehler force are obtained. The analytical solutions can reduce to the results from the existing literature. The disadvantage of the present solution is that the dislocations are considered to distribute uniformly in the numerical examples which do not reflect the microstructure of the material. The solutions are limited in numbers of dislocations, and only the same sign dislocations are considered here. From the numerical illustrations, some important results are summarized in the following outline:(1)
The trend of contours of the phonon field stress is similar to the phason field stress but is different from the electric displacement.(2)
The stress concentration occurs at the crack tip and dislocations cores. The effects of the location of dislocations on the three field intensity factors are different, and the wedge angle always increases the shielding effect.(3)
The increasing number of dislocations always accelerates the crack propagation. The wedge angle accelerates the crack growth when dislocations are located at the real axis, but it first promotes crack growth and then demotes crack propagation when dislocations are located at the imaginary axis symmetrically.(4)
The image force acting on the dislocation in the presence of a crack and other dislocations decreases when increasing the wedge angle. The dislocation is either attracted or rejected in the process of moving from left to right along the real axis.



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