# Interaction of a Screw Dislocation with Interface and Wedge-Shaped Cracks in One-Dimensional Hexagonal Piezoelectric Quasicrystals Bimaterial

Authors: Lian he Li; Yue Zhao
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

Interaction of a screw dislocation with wedge-shaped cracks in one-dimensional hexagonal piezoelectric quasicrystals bimaterial is considered. The general solutions of the elastic and electric fields are derived by complex variable method. Then the analytical expressions for the phonon stresses, phason stresses, and electric displacements are given. The stresses and electric displacement intensity factors of the cracks are also calculated, as well as the force on dislocation. The effects of the coupling constants, the geometrical parameters of cracks, and the dislocation location on stresses intensity factors and image force are shown graphically. The distribution characteristics with regard to the phonon stresses, phason stresses, and electric displacements are discussed in detail. The solutions of several special cases are obtained as the results of the present conclusion.


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## Body

## 1. Introduction

As a new structure of solid in nature, quasicrystals (QCs) differ from ordinary crystals and noncrystals. QCs have many excellent properties, such as high strength, high wear resistance, and low heat-transfer, which have great development trends and been applied extensively in the fields of technology and engineering. Since the discovery of QCs [1], generalized elasticity theory of QCs has been established [2, 3]. Using the mathematical theory of elasticity of QCs, Fan et al. [4, 5] studied the moving screw dislocation and straight dislocation in one-dimensional (1D) hexagonal QCs. Enrico et al. [6] studied the linear crack problem in ten symmetric two-dimensional quasicrystals by using the Stroh method. Gao et al. [7] solved the fracture mechanics problem of cubic quasicrystals with an elliptical hole or a crack. Wang et al. [8] coped with the problem involving the interaction between a screw dislocation and a semi-infinite crack in decagonal QCs. Liu et al. [9] studied the interaction of dislocations with cracks in one-dimensional hexagonal QCs based on the analytic function theory. Li et al. [10] investigated the interaction of a screw dislocation with an elliptical hole in icosahedral quasicrystals.Piezoelectricity is an important physical property of QCs. The piezoelectric QCs [11–14] have been investigated to a certain degree. It is well known that the interaction among defects is of considerable importance in understanding the fracture mechanical behavior of materials. Few results have been reported with regard to the interaction problem of defects in piezoelectric QCs.Wedge-shaped crack is an important geometric form of many composites and structures. The discontinuity of material properties or the abrupt change in geometry makes the stresses field at the top of wedge singular under certain conditions, which means that the wedge tip is prone to cracking. Therefore, it is of great significance for understanding the properties to study a screw dislocation interacting with interface and interfacial wedge-shaped cracks in piezoelectric quasicrystal bimaterial. The general expressions of the complex potentials are obtained and fracture parameters such as intensity factors and image force are given in closed-form. Three special cases, i.e., the screw dislocation interacting, respectively, with semi-infinite, finite, and wedge-shaped cracks in 1D hexagonal piezoelectric QCs bimaterial is also considered.


## 2. Problem Description and Basic Equations

The physical model considered in this section is shown in Figure1. An interfacial crack of length l and a wedge-shaped crack whose angle is 2α lie along a part of the interface between two 1D hexagonal piezoelectric QCs materials. We suppose that the interface is perfectly bonded along x-axis when x≥0. A screw dislocation of Burgers vector b=(b3,b⊥,bϕ)T is located at the point z0(r,θ) in material 1 and the dislocation line is perpendicular to the z-plane. The crack surfaces are considered traction-free and no electric charge.Figure 1
A screw dislocation interaction with wedge-shaped cracks in 1D hexagonal piezoelectric QCs bimaterial.The problem considered is an antiplane elastic deformation problem. The nonzero field quantities(σzβ,εzβ,uz), (Hzβ,ωzβ,wz) and (Dβ,Eβ,ϕ), β=1,2, should satisfy the following basic equations [12]:constitutive equations(1)σzβ=2C44εzβ+R3ωzβ-e15EβHzβ=2R3εzβ+K2ωzβ-d15EβDβ=2e15εzβ+d15ωzβ+λ11Eβgeometry equations (2)εzβ=12uz,β,ωzβ=wz,β,Eβ=-ϕ,βequilibrium equations (if the body force is neglected)(3)σzβ,β=0,Hzβ,β=0,Dβ,β=0where a comma represents partial differentiation and the repeated indices denote summation; σzβ, εzβ, and uz are the stresses, strains, and displacement of the phonon field, respectively, with β=x,y; Hzβ, ωzβ, and wz are the stresses, strains, and displacement of the phason field, respectively; Dβ, Eβ, and ϕ represent the electric displacements, electric fields and electric potential, respectively; C44, K2, and R3 denote the phonon elastic, phason elastic, and phonon-phason coupling constants, respectively; e15, d15 are the piezoelectric constants in the phonon field and phason field, respectively; λ11 denotes the dielectric permittivity.Substituting (1) and (2) into (3), the governing equations are obtained as follows: (4)B∇2u=0where u=uz,wz,ϕT, ∇2=∂2/∂x2+∂2/∂y2 is the two-dimensional Laplace operator and(5)B=C44R3e15R3K2d15e15d15-λ11is the material constants matrix.By using the properties of analytic function, in view of (4), one obtains (6)u=Im⁡fzwhere Im represents the imaginary part of an analytic function.Introducing the generalized strainsZx=[εzx,ωzx,-Ex]T, Zy=[εzy,ωzy,-Ey]T and the generalized stresses Σx=[σzx,Hzx,Dx]T, Σy=[σzy,Hzy,Dy]T, we have the following relationships:(7)Zy+iZx=f′zΣy+iΣx=Bf′zwhere the superscript prime denotes the derivative with respect to the argument.By (7), the resultant force and the normal component of the electric displacement along any arc AB can be represented as(8)T=∫ABΣxdy-Σydx=BRe⁡fzABwhere Re indicates the real part of an analytical function.The boundary conditions of the present problem can be expressed as follows:(9)∫ldu1=b,∫lΣy1dx-Σx1dy=0(10)u1x,0+=u2x,0-,T1x,0+=T2x,0-,0≤x≤∞(11)T1x,0=T2x,0=0,-l≤x≤0(12)Σx1nx1+Σy1ny1=0,y=-x+ltan⁡αΣx2nx2+Σy2ny2=0,y=x+ltan⁡αin which all quantities with superscripts j(j=1,2) represent material 1 and material 2, respectively. The following main task is to determine the unknown functions f(j)(z) satisfying the boundary conditions.


## 3. Solution to the Problem

Introduce a conformal mapping function as [15](13)ζ=z+lδ-lδ,δ=ππ-α,1≤δ≤2,ζ=ξ+iηwhich transforms the infinite plane which consists of material 1 and material 2 in the z-plane into the upper and lower quarter planes in the ζ-plane and the point z0 in the z-plane into the point ζ0=(z0+l)δ-lδ in the ζ-plane.By (10), (11), and (12), we have (14)u1ξ,0+=u2ξ,0-,T1ξ,0+=T2ξ,0-,0≤ξ≤∞(15)T10,η=0,η≥0T20,η=0,η≤0Assume that the unknown functionsf(j) in the ζ-plane can be expressed as(16)fjζ=fbjζ+fcjζwhere fb(j)(ζ) represent the functions of a screw dislocation in an infinite bimaterial, while fc(j)(ζ) are the functions corresponding to the perturbed fields due to the free boundary ζ=0.We first consider a screw dislocation in a infinite 1D hexagonal piezoelectric QCs bimaterial. In this case, the functionsfb(j)(ζ), which satisfy the conditions (9) and (14), are assumed as (17)fb1ζ=fm1ζ+fp1ζ,fb2ζ=fp2ζwhere fm(1)(ζ) is connected to the unperturbed field and it is analytic apart from ζ0 point in the complex plane and fp(j)(ζ) is related to the perturbed fields in the material j.By (9), fm(1)(ζ) is given as(18)fm1ζ=12πbln⁡ζ-ζ0Inserting (6) and (8) into (14), we have (19)Im⁡fb1ζ=Im⁡fb2ζ,B1Re⁡fb1ζ=B2Re⁡fb2ζUsing (17), (18), and (19), analytical extension techniques [16], and noting ζ-=ζ holds along the real axis, we have (20)fb1ζ=12πIln⁡ζ-ζ0+Π12ln⁡ζ-ζ-0bfb2ζ=12πI-Π12bln⁡ζ-ζ0where the overbar is the conjugate of a complex function, I is a 3×3 unit matrix, and matrix Π12 is(21)Π12=B2+B1-1B2-B1By (8), (15), and (16), noting ζ-=-ζ holds along the imaginary axis, one can obtain(22)fc1ζ=-12πIln⁡ζ+ζ-0-1+Π12ln⁡ζ+ζ0bfc2ζ=-12πI-Π12ln⁡ζ+ζ-0bSubstituting (20) and (22) into (16) and by (13), the functions f(j)(z) in the z-plane are obtained as(23)f1z=12πIln⁡z+lδ-z0+lδz+lδ-z-0+lδ+I+Π12ln⁡ζ-ζ-0ζ+ζ0bf2z=12πI-Π12ln⁡ζ-ζ0ζ+ζ-0bWith (23), the elastic and electric fields can be derived in terms of (6)-(7). For the generalized stress fields, one has (24)Σy1+iΣx1=δz+lδ-12πB11z+lδ-z0+lδ-1z+lδ-z-0+lδ+Beff1ζ-ζ-0-1ζ+ζ012ζbΣy2+iΣx2=δz+lδ-12πBeff12ζ1ζ-ζ0-1ζ+ζ-0bwith Beff=2B(2)(B(2)+B(1))-1B(1)=2B(1)(B(2)+B(1))-1B(2).


## 4. Stress Intensity Factors and Image Force on the Dislocation

The phonon field stresses, phason field stresses, and electric displacements intensity factors at the crack tip are defined as(25)K=KσKHKD=limz→02πzΣyj,j=1,2Substituting (24) into (25) will yield(26)K=-12δlδ-12πBeff1ζ0+1ζ-0bAccording to the generalized Peach-Koehler formula [14], the image force is expressed as(27)Fx-iFy=bTΣyp1z0+iΣxp1z0in which Σkp1z0k=x,y are the perturbation stresses of the phonon fields, phason fields, and the perturbed electric displacements at the dislocation point z0. With (7), (24), and (27), we have(28)Fx-iFy=δz+lδ-12πbTB1δ-12δz0+lδ-1z0+lδ-z-0+lδ+Beff1ζ0-ζ-0-12ζ012ζ0bThe analytical solutions of some special cases can be derived from the present results. Discussions in detail are given as follows:(1)
If the length of the interfacial crackl tends to zero, (24) reduces to(29)Σy1+iΣx1=δ4πz1-δ/2B11zδ-z0δ-1zδ+z-0δ+B1Π121zδ-z-0δ-1zδ+z0δbEquation (29) represents the case of the interaction of a screw dislocation with a wedge-shaped crack. If B(1)=B(2), the result is in excellent agreement with the result in [17].(2)
Whenβ=0, (24) degenerates into(30)Σy1+iΣx1=12πB11z-z0-1z-z-0+Beff1z-z-0-1z+z012zbEquation (30) gives the case of a dislocation interacting with interface and a semi-infinite interfacial crack.(3)
Whenβ=π/2, (24) becomes(31)Σy1+iΣx1=12πB11z-z0+1z-z0+2l-1z-z-0-1z-z-0+2l+Beff1zz+2l-z-0z-0+2l-1zz+2l+z0z0+2lz+lzz+2lbEquation (31) represents the result of the interaction of a dislocation with interface and a finite interfacial crack.


## 5. Numerical Results

In this section, the selected numerical results are presented to analyze the fracture mechanic behavior of the 1D hexagonal piezoelectric QCs bimaterial. The material properties of the bimaterial are listed in Table1 [16, 18–20].Table 1
Material properties of two 1D hexagonal piezoelectric QCs.
C 44 ( G P a ) K 2 ( G P a ) e 15 ( C ⋅ m - 2 ) d 15 ( C ⋅ m - 2 ) λ 11 ( 1 0 - 9 C 2 N - 1 m - 2 ) Material 1 50 0.3 -0.318 -0.16 0.0826 Material 2 70.19 24 11.6 1.16 5The coupling constantR has great influence on the mechanical behaviors of QCs, and it has not been measured yet. In the computation, the coupling constant of material 2 is assumed as R3(2)=0.8846GPa [21].Other parameters are selected as follows:(32)b3=1.6×10-9m,b⊥=10.7×10-9m,bϕ=1×10-9V.Figures2 and 3 show the variations of the normalized phonon and phason stresses intensity factors with the angular position θ of the dislocation for different values of R3(1) at r/l=0.1, α=π/6. It is found that the normalized phonon and phason stresses intensity factors are always negative, indicating that screw dislocation can reduce the shielding effect.Figure 2
Normalized phonon stresses intensity factor(2πl)1/2Kσ/(C44(1)b3) versus θ.Figure 3
Normalized phason stresses intensity factor(2πl)1/2KH/(C44(1)b3) versus θ.Figures4 and 5 depict the normalized force on the dislocation versus θ for different values of the wedge angle α at r/l=0.5. It can be seen from Figure 4 that the normalized image force along x-axis is always negative and the absolute value decreases with the increase of the wedge angle α when 0<θ<π/2 and increases with the increase of α when π/2<θ<π. Figure 5 shows that the wedge angle α has no obvious effect on the normalized image force along y-axis.Figure 4
Normalized image force2πlFx/(C44(1)b32) versus θ.Figure 5
Normalized image force2πlFy/(C44(1)b32) versus θ.Figures6, 7, and 8 show the distribution of the phonon stresses, phason stresses, and electric displacements with r=1, θ=π/6, and r/l=0.5. It can be seen that the phonon stresses, phason stresses, and electric displacements are singular at the crack tip z=0 and dislocation point z=z0. σzy, Hzy, and Dy are continuous at the interface y=0, while σzx, Hzx, and Dx are discontinuous at the interface y=0.Figure 6
Contours of the phonon stresses.
(a)
1 0 - 10 ( 2 π σ z y / b 3 ) (b)
1 0 - 10 ( 2 π σ z x / b 3 )Figure 7
Contours of the phason stresses.
(a)
1 0 - 10 ( 2 π H z y / b 3 ) (b)
1 0 - 10 ( 2 π H z x / b 3 )Figure 8
Contours of the electric displacements.
(a)
2 π D y / b 3 (b)
2 π D x / b 3


## 6. Conclusions

The wedge-shaped cracks interacting with a screw dislocation in 1D hexagonal piezoelectric QCs bimaterial are presented in this paper. The analytic expressions of the electrical field, phason, and phonon field have been derived. The solutions obtained in present paper can be generalized to other QCs bimaterial problem. These results provide the theoretical basis for the application of the QCs material.



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