The interaction between a screw dislocation and a wedge-shaped crack in one-dimensional hexagonal piezoelectric quasicrystals
College of Mathematics Science, Inner Mongolia Normal University, Huhhot 010022, China
† Corresponding author. E-mail:
1530284866@qq.com guantingliu@imnu.edu.cn
Project supported by the National Natural Science Foundation of China (Grant Nos. 11262017, 11262012, and 11462020), the Natural Science Foundation of Inner Mongolia Autonomous Region, China (Grant No. 2015MS0129), the Programme of Higher-level Talents of Inner Mongolia Normal University (Grant No. RCPY-2-2012-K-035), and the Key Project of Inner Mongolia Normal University (Grant No. 2014ZD03).
1. IntroductionThe interaction of dislocations with defects (cracks, interfaces, inhomogeneities, etc.) is of considerable importance for understanding the physical behavior of materials. In the open literature,[1–5] a lot of work on dislocations and cracks in pure elastic materials, piezoelectric solids, and quasicrystals have been studied in order to build a fracture theory that describes plastic deformation of dislocations near the crack tip. Many methods and techniques, such as the Green function method,[6] integral transformation,[7] and the complex variable method[8–14] can be used to solve these problems. Head[15] first investigated the interaction between a screw dislocation and a bi-material interface. Ohr et al.[16] derived the elastic field of a semi-infinite wedge crack and its interaction with a screw dislocation. The above-mentioned studies are based on the pure elastic materials. For the piezoelectric materials, various types of defects can adversely influence the performance of such piezoelectric devices. Pak[17] obtained closed form solutions for a screw dislocation in a piezoelectric solid and derived the generalized Peach–Koehler forces acting on the screw dislocation subjected to external loads. Zeng et al.[18] studied the interaction between screw dislocations and two asymmetrical interfacial cracks emanating from an elliptic hole under loads at infinity. Lee et al.[19] performed the interaction between a semi-infinite crack and a screw dislocation in a piezoelectric material. Chen et al.[20] investigated the electro–elastic stress on the interaction problem of a screw dislocation near the tip of a semi-infinite wedge-shaped crack in piezoelectric material. Liu et al.[21] derived closed-form solutions of the elastic and electrical fields induced by the screw dislocation using the conformal mapping method in conjunction with image principle. Recently, one-dimensional hexagonal quasicrystals have been paid a great deal of attention. Li et al.[22] performed an elastic field induced by a straight dislocation in a one-dimensional hexagonal quasicrystals with its line parallel to the quasiperiodic axis by superposition of the elastic fields of a pure edge part and a pure screw part. Liu et al.[23] investigated the interaction of defects in one-dimensional (1D) hexagonal quasicrystals using the complex variable function method. Li et al.[24] studied the interaction between the screw dislocation and wedge-shaped crack in 1D hexagonal quasicrystals. These explicit and exact solutions can provide a theoretical analysis for fracture problems.
However, for piezoelectricity of quasicrystals, this problem becomes more complicated due to the introduction of the electric field. Up to now, there has not been any research work on the interaction between a screw dislocation and a wedge-shaped crack in piezoelectricity of one-dimensional hexagonal quasicrystals. In this paper, explicit analytical solutions are obtained for the stress and electric displacement intensity factors of the crack, as well as the force on dislocation.
2. Statement of the problemThe physical problem considered in this paper is shown in Fig. 1. The wedge-shape crack is assumed to infinitely extend in the negative x axis, and the wedge angle is denoted as α0. A charged screw dislocation located at the point
, with the Burgers vectors
is assumed to be straight and infinitely long in the z direction.
As shown in Fig. 1, we have the following boundary conditions:
where
τ1,
τ2, and
D0 are the stress of the phonon field, the stress of the phason field, and the electric displacement, respectively, at the infinite region.
The stress–strain relations for 1D hexagonal piezoelectric quasicrystals, referred to the Cartesian coordinate
with xoy coincident with the periodic plane and the z axis identical to the quasi-periodic direction, read as follows:[25]
The generalized strain-displacement relations are
In the absence of body forces and free charges, the equilibrium equations can be expressed as
where
σij,
εij,
ui are the stress, strain, and displacement of the phonon field, respectively;
Hij,
ωij, and
wi are the stress, strain, and displacement of the phason field, respectively;
Di,
Ei, and
ϕ are the electric displacement, the electric field, and the electric potential, respectively;
Cij,
Ki, and
Ri stand for the phonon elastic, phason elastic, and phonon–phason coupling modulus, respectively;
eij,
dij, and
λij stand for piezoelastic constants and the dielectric permittivity, respectively. All field variables are independent of the longitudinal coordinate
z, say
With the help of Eq. (
7), equations (
2)–(
6) can be decomposed as two uncoupled problems. The first problem is
which is just the classical plane elasticity of conventional hexagonal crystals. The other problem is
which is a phonon–phason–electric coupling anti-plane piezoelasticity problem of 1D hexagonal quasicrystals. Since the first problem described by Eqs. (
8)–(
10) has been studied extensively, we only need to consider the second problem.
Substituting Eq. (12) into Eq. (11) and then into Eq. (13), we have
where
is the two-dimensional Laplace operator. The general solution for the displacement vector
obtained from Eq. (
13) can be written as
where Re is the real part of the complex function, and
,
, and
are arbitrary analytic functions of
z. By using Eqs.(
1), the complex function vector
can be written as
where
is associated with the unperturbed field without the wedge crack and can be chosen as
in which
where
The function
corresponds to the perturbed field due to the existence of the wedge crack. By Eqs. (11), (12), and (15), the resultant force along any arbitrary arc AB is
In which Im stands for the imaginary part and
From Eqs. (
1), we have
in the boundary
, so the boundary condition at the wedge surface
is
It is difficult to directly solve the problem in the
z plane. Therefore, we introduce the conformal mapping function
with
mapping the boundary
in the
z plane into the boundary conditions
in the
ζ plane and the dislocation located at the point
in the
z plane onto
in the
ζ plane. In the
ζ plane of the current problem, the state of stress at
is free from the stress of the phonon field, the stress of the phason field, and the electric displacement, so the boundary condition becomes
where
is along the imaginary axis. From Eq. (
20), we know equation (
24) can be rewritten as
Utilizing Eqs. (
16) and (
25), noting that
holds along the imaginary axis, and using standard analytic continuation arguments,
[26] we obtain
From Eqs. (
16) and (
26), the complex function vector
can be obtained in the
ζ plane as
Therefore, in the physical plane, we have
3. Stress intensity factorsThe entire stress and displacement fields can be calculated by using Eqs. (2)–(5) and (28) (here, we only list the expressions for the stresses in Appendix A). Similar to the classical elasticity theory, we define a complex stress intensity factor vector included by the dislocation as
Since the field variables show
type of singularity near the wedge crack tip, equation (
29) can be rewritten as
Inserting Eq. (
29) into Eq. (
30), we have
We can draw the conclusion from the above results. The stress intensity factor of the wedge-shaped crack is mainly influenced by the screw dislocation along the x axis. It is noted that when
, we gain the stress intensity factor of the semi-infinite crack induced by the dislocation. If there is no applied electric loadings at infinity, the above results equations (31) reduce to the corresponding results[24] of one-dimensional hexagonal quasicrystals. If the phason field is neglected, the present solutions (31) can give the corresponding solutions for a wedge-shaped crack interacting with a screw dislocation in piezoelectric materials, which is new and different from the ones in the open literature.[20] The authors discuss the case of
,
,
, and the line force and the line charge at the dislocation core in the literature.[20]
4. Force on the dislocationAccording to the generalized Peach–Koehler force formula for quasicrystals derived by Li et al.,[22] the image forces acting on the dislocation due to the wedge crack in the piezoelectricity of quasicrystals are
where
,
,
,
,
, and
are the stress field produced by the crack in the absence of the dislocation, for which we will give the detailed expressions in Appendix
B. For the current problem, equation (
32) can be expressed as follows:
When the dislocation is in the positive x-axis direction (
with
), the above results can be simplified as
The results obtained show that the image forces on the dislocation are along the x axis.
5. Numerical examplesClearly, the image forces on the screw dislocation are given explicitly in Eq. (33). They are functions of
,
,
, τ1, τ2, D0,
, and material constants. In order to have a deep investigation on how these parameters affect the image force, a particular one-dimensional hexagonal piezoelectric quasicrystals is used as an example to numerically demonstrate the force on the dislocation. The material constants are selected as follows:[27]
Other parameters are taken to have the values listed below
The forces on the dislocation are normalized with
Figures 2 and 3 show the normalized force on the dislocation versus the angular position of the dislocation when the wedge angle is
. It is found from Figs. 2 and 3 that the magnitudes of the image force on the dislocation increase with decreasing
.
In order to further reveal the variable tendency, figures 4 and 5 depict the variations of the normalized image force versus
when the wedge angle is
. It is seen from Figs. 4 and 5 that the crack always attracts the dislocation in the radial direction when it is far away from the dislocation. We further obtain that the forces along the x axis increase with decreasing
while the forces along the y axis increase with increasing
.
6. ConclusionIn this paper, a screw dislocation near a wedge crack in one-dimensional hexagonal piezoelectric quasicrystals is analyzed. By using the conformal transformation method and the perturbation technique, the stress and electric displacement intensity factors and the image forces on the dislocation are derived. At the same time the explicit analytical expressions for the stress and displacement are also obtained. The influences of the wedge angle and dislocation location on the image force are also discussed. The results are reduced to the new or previous ones when the wedge angle is taken to be the limiting case or the materials are assumed to be one-dimensional hexagonal quasicrystals.
