1 Introduction

The unique structure and physical-mechanical characteristics of quasicrystals open the way to technological applications in many fields^[1]. The icosahedral quasicrystal is one of the most widely used quasicrystal materials, with a good development future^[2-3].

Mechanical properties and fracture behavior of quasicrystals have recently attracted considerable interest^[4-5]. The generalized Dugdale model of quasicrystals has been considered with the help of conformal mapping method^[6]. Li et al.^[7] investigated the buckling and the free vibration of properties of a two-dimensional hexagonal cylindrical shell under axial compression. Collinear periodic crack problems were considered in Ref. [8]. The influence of phason displacement on the related physical quantities was discussed. Fourier images of Green's functions for dynamic plane elasticity problems of decagonal quasicrystals were obtained by using the Fourier transform and matrix transformation method^[9]. The plate bending analysis in one-dimensional orthorhombic quasicrystals under static and transient dynamic loads was studied by the meshless Petrov-Galerkin method^[10]. Using an extended technique of continuous dislocation layers, a moving antiplane shear strip crack problem in one-dimensional hexagonal was investigated^[11].

The coupling behavior between elastic and electric fields of piezoelectric materials brings good prospects in electro-mechanical and electronic devices. Piezoelectricity is an important physical property of quasicrystals^[12]. However, the knowledge and understanding of the piezoelectric behavior of quasicrystals are still limited, especially for two- and three-dimensional quasicrystals. The piezoelectricity tensor associated with the phonon fields was obtained by Rama et al.^[13]. The piezoelasticity of quasicrystals induced by both phonon and phason fields was discussed in Ref. [14]. Three-dimensional fundamental solutions of one-dimensional quasicrystals with the piezoelectric effect were obtained in Refs. [15] and [16]. The general solutions to the piezoelasticity problem for one-dimensional quasicrystals with all point groups were derived based on the operator method and the complex variable function method by Yu et al.^[17]. Then, an antiplane elliptical cavity of one-dimensional hexagonal piezoelectric quasicrystals was investigated^[18]. A circular hole with a straight crack in one-dimensional hexagonal quasicrystals with piezoelectric effects was considered by Yang and Li^[19].

The electrical and elastic fields induced by the dislocation in an icosahedral quasicrystal with piezoelectric effects are derived using the extended Stroh formalism in this paper. Numerical analysis is then conducted to discuss the influence of piezoelectric constants on the displacements and the electric potential.

2 Stroh formalism for two-dimensional deformation of icosahedral quasicrystals with piezoelectric effects

According to the elastic theory of piezoelectric quasicrystals^[14], we have the gradient equations

(1)

the equilibrium equations (if the body force is neglected)

(2)

and the linear electric-elastic coupled constitutive equations

(3)

in which C_ijkl and K_ijkl are elastic constants of the phonon and phason fields, respectively, R_ijkl are the phonon-phason coupling elastic constants, D_i, ϕ, and E_i are the electric displacements, electric potential, and electric fields, respectively, χ_ij is the dielectric permittivity, and e_ijk and d_ijk are piezoelectric constants of the phonon and phason fields, respectively.

Assume that the defect is parallel to the fivefold direction of icosahedral quasicrystals^[20]. Making use of Eqs. (1)-(3), we obtain

(4)

Let

(5)

in which

f(z) is an arbitrary function of z, and p and a_i are the unknown complex constants to be determined. Substituting Eq. (5) into Eq. (4), we have

(6)

where

and the 7× 7 matrices Q, R, and T are given by

(7)

A nontrivial solution for a exists if the determinant of the matrix in Eq. (6) is zero, i.e.,

(8)

The associated eigenvector a can be obtained from Eq. (6). Then, the general solutions can be expressed as

(9)

where

Without loss of generality, let Im p_α >0.

3 Coupled fields caused by line dislocation

The fundamental properties of dislocations in quasicrystals were discussed in Refs. [21] and [22]. The results of the discussion show that the characteristic vector b_*, i.e., the Burgers vector of a dislocation, is a direct sum

where b^|| and b^⊥ describe rigid translations in the physical and complementary subspaces, respectively. The elastic displacement fields induced by a dislocation consist of u and w, i.e., the phonon and phason displacement fields, due to the presence of incommensurate length scales in quasicrystals.

The dislocation is assumed to be infinitely long in the fivefold direction, the core at the origin. Without loss of generality, we take the plane x₂ = 0 and x₁ < 0 to be the slip plane. Introducing a branch cut along the negative x₁-axis, we have r > 0 and -π < θ < π (where x₁ = r cosθ and x₂ = r sinθ).

The boundary conditions of this problem can be expressed as

(10)

in which . b_i^|| and b_i^⊥(i=1, 2, 3) are Burgers components in b^|| and b^⊥, respectively. Δ b_e corresponds to an electric dipole layer along the slip plane.

Let

(11)

where q_α are arbitrary complex constants. Equation (9) can then be written as

(12)

in which 〈ln z_α〉 is the diagonal matrix,

(13)

Substituting Eq. (12) into Eq. (10) yields

(14)

The dielectric constants and piezoelectric constants of phonon and phason fields have not yet been measured for the icosahedral Al-Pd-Mn. The numerical results are presented to show the effects of the phonon piezoelectric constant e₁₄ and the phason piezoelectric constant d₁₂₃ on the displacements and the electric potential. The material properties^[23] are given by

In the computation, we assume that the phason piezoelectric constant d₁₃₂=13 C· m^-2 and the dielectric constant χ₁₁=60×10^-10 C· (V·m)^-1. Other parameters are taken to have the values below

The normalized displacements u_x^* and w_x^* (i.e., u_x^*=u_x/10^-10 m and w_x^*=w_x/10^-10 m) and the electric potential ϕ versus the angular θ under different piezoelectric constants of phonon and phason fields are depicted in Figs. 1-6, where r is fixed at r=15× 10^-9 m. The coupling constant R of the quasicrystal has not yet been measured. In the computation, we assume that R=951 MPa.

The figures show that the phonon displacements, phason displacements, and the electric potential are significantly affected by the piezoelectric constants. With the variation of the phonon piezoelectric constant (see Figs. 1 and 2), a different changing tendency of the phonon displacement u_x and phason displacement w_x is observed. From Figs. 3 and 4, we draw the conclusion that the displacements u_x and w_x have the same tendency with the variation of the phason piezoelectric constants, but different in the extent. Figures 5 and 6 show that the phason piezoelectric constant has more influence on the electric potential than the phonon piezoelectric constant.

The effects of the phonon-phason coupling elastic constant R on the normalized displacements u_x^* and w_x^* are discussed in Figs. 7 and 8. As the value of R increases, the absolute values of the normalized displacements u_x^* and w_x^* decrease.

4 Conclusions and discussion

A theoretical and numerical treatment of a line dislocation in icosahedral quasicrystals with piezoelectric effects is presented by using the extended Stroh formalism. The effects of the piezoelectric constants on the displacements and the electric potential are discussed. The results are reduced to the classical ones when the electric field is neglected. These results are valuable for understanding the effects of dislocations on icosahedral quasicrystals with piezoelectric effects.