1 Introduction

Quasicrystal (QC) was discovered by Shechtman et al.^[1] in 1982, and Shechtman won the Nobel Prize in 2011. QCs possess not only quasi-periodic long-range translational symmetry but also noncrystallo-graphic rotational symmetry. The one-dimensional (1D) or two-dimensional (2D) QCs are the ones in which the atomic structures of the materials are quasiperiodic in one or two directions. According to the Landau density wave theory^[2], the total displacement field in QCs is the sum of the two displacement vectors u and w. One displacement belongs to the phonon field, which represents the displacement of lattice point deviated from its equilibrium position due to the vibration of the lattice. The other is the phason field, which describes the local rearrangement of atoms in a unit cell based on the Penrose tiling. Since the discovery of QCs, great progress has been made in the elastic theory for many years^[3-7]. For the elastic problems of three-dimensional (3D) QCs, Fan and Guo^[8] derived the final governing equation and the fundamental solution of plane elasticity of icosahedral QCs. QCs are sensitive to magnetic, electrical, mechanical, thermal, and optical effects. Due to their unique structure, QCs are very attractive for their performance properties such as high hardness, low thermal conductivity, and oxidation resistance^[9-11]. Recent experiments^[12-14] showed that QCs could be used as particle reinforced phases embedded in polymer, metal matrix composites, or alloys to enhance the mechanical properties of these materials. Therefore, it is of significance to study the micromechanics of composite materials made up of QCs and other materials so that the experimental phenomenon is well understood.

There is evidence that piezoelectricity is an important physical property of QCs^[15-19]. Due to the excellent coupling among the phonon field, phason field, and electric field, the piezoelectric (PE) effect of QCs has attracted much attention recently so that piezoelectric quasicrystal (PQC) materials can play a key role in new structures and systems. Altay and Dömeci^[20] established the basic equations for 3D PQC materials. Grimmer^[21] addressed the stress-strain relationship of the second-order PE effect of crystal and QC with multiple symmetries. Zhang et al.^[22] studied the plane elasticity of 1D orthorhombic QCs with the PE effect and obtained the closed-form solutions of wedge problems or half-plane problems. Yang and Li^[23] investigated the anti-plane shear problem for a crack originating from a circular hole in 1D hexagonal QCs with the PE effects by means of the complex variable function method and the technique of conformal mapping. Li et al.^[24] obtained the 3D basic solution of 1D hexagonal PQC. Yu et al.^[25] presented the final governing equations for the plane problem of 1D PQC and their general solutions. Furthermore, they solved the fracture behavior of an elliptical hole embedded in 1D hexagonal PQC and derived its exact solutions by the complex variable function method^[26]. More recently, Zhang et al.^[27] derived the Green's functions of 1D QC bimaterial with the PE effect based on the Stroh formalism.

The investigations mentioned above are mainly limited at macro-scale. To explain the experimental phenomenon of composites involving QC and PE materials, it is necessary to study micromechanics of PQC composites. Many micromechanical schemes^[28-31] have been successfully used for obtaining the effective modulus of composites, such as the generalized self-consistent, self-consistent, Mori-Tanaka, differential, and dilute methods. It is reported^[30-31] that the generalized self-consistent method is in better agreement with the experimental data than the others. Recently, Guo et al.^[32] firstly considered an anti-plane elliptic inclusion embedded in 1D hexagonal PQC composites. Then, Guo and Pan^[33] developed a three-phase cylindrical model of 1D hexagonal PQC composites and predicted the effective modulus of PQC composites based on the generalized self-consistent method. However, no work on the three-phase elliptical cylinder model of QC composites has been reported up to now, especially involving the PE effect. Actually, the shape of inclusion or fiber is not circular but arbitrarily embedded in the matrix. Thus, the three-phase elliptical cylinder model^[34-35] is an important one to describe the physical properties of microstructure of composite, which is fully different from the two-phase model by Guo et al.^[32].

In this work, we develop a three-phase confocal elliptical cylinder model with any section orientation for PQC composites and obtain the exact closed-form solutions for this model subject to antiplane mechanical loads of phonon and phason fields, and in-plane electrical loads at infinity by utilizing the technique of conformal mapping and Laurent expansion^[36]. Furthermore, based on the generalized self-consistent method, the effective moduli of some new composites made up of PQC, QC, and PE are predicted to account for the influence of fiber section shape and volume fraction on the effective electroelastic constants of these composites, which may be useful for designing new composites in engineering practice.

2 Problem description and basic equations for PQC

Consider a three-phase confocal elliptical cylinder model made up of PQC in each phase, as shown in Fig. 1(a). The elliptical region (called the inclusion or fiber phase) S^Ⅰ is encircled by L₁, and the elliptical ring region (called the matrix phase) S^M is encircled by L₁ and L₂. L₁ and L₂ have the common foci O₁ and O₂. In the representative unit cell of whole composites, the volume fraction of the inclusions is λ=a₁b₁/(a₂b₂), where (a₁, b₁) and (a₂, b₂) are the semimajor axes and semiminor axes of L₁ and L₂, respectively. The infinite section S^C outside L₂ represents the PQC composite with unknown effective electroelastic constants to be determined. A local Cartesian coordinate system Ox₁x₂x₃ makes an arbitrary angle β with the global Cartesian coordinate system OX₁X₂X₃ in the isotropic plane. The PQC composite is subject to antiplane shear loadings of the phonon and phason fields and in-plane electric loads at infinity.

We assume that the quasi-periodic arrangement of atom and the polarized direction of the electric field are along the x₃-direction and that all the physical quantities are independent of the variable x₃. For the antiplane problem of PQC composites, the basic equations can be simplified to^[26-27]

(1)

(2)

(3)

where j=1, 2, the superscript T denotes the transpose of the vector or matrix, and the material matrix is expressed as

(4)

In Eqs. (1)-(4), the comma denotes differentiation with respect to x_j, and the repeated indices imply summation. σ_ij and u_i are the stress tensors and elastic displacements of the phonon field, respectively. H_ij and ω_ij are the stress and strain tensors of the phason field, respectively. w_i are the elastic displacements of the phason field. D_i, E_i, and ϕ are the electric displacements, the electric fields, and the electric potential, respectively. C₄₄ and K₂ are elastic constants of the phonon and phason fields, respectively, R₃ is the phonon-phason coupling elastic constant, e₁₅ and d₁₅ are the PE coefficients, and λ₁₁ is the dielectric permittivity.

Substituting Eq. (2) into Eq. (3) then into Eq. (1), we derive the governing equations for the extended displacement vector u=[u₃, w₃, ϕ]^T as

(5)

where

is a 2D Laplace operator.

According to the theory of complex variable function, the general solution to Eq. (5) can be expressed by the real or imaginary part of an analytical function vector F(z), where z=x₁+ix₂ is the complex variable, i.e.,

(6)

where Re denotes the real part of a complex function. From Eqs. (2), (3), and (6), the constitutive equations can also be expressed in a complex potential form as

(7)

where the superscript prime denotes differentiation with respect to the argument.

For convenient analysis later on, we also introduce vectors

so that the above equations are rewritten as

(8)

where Im denotes the imaginary part of complex function, and [F(Z)]_A^B represents the change in the function F(z) from point A to point B along any arc AB.

As shown in Fig. 1(a), we assume that the different phases are perfectly bonded at their interfaces so that we have the following continuity conditions:

(9)

(10)

where the superscripts I, M, and C refer to the inclusion, matrix, and composite of the model, respectively.

3 Exact solutions of electroelastic fields of PQC composites

To obtain the electroelastic fields of three-phase confocal elliptical cylinder model, we introduce the following conformal mapping that transforms the elliptical cross section of z-plane into the circular ring in the ζ-plane (see Fig. 1(b)),

(11)

in which R is a constant to be determined.

where ρ and θ are the polar coordinates in the complex ζ-plane. The regions S^Ⅰ, S^M, and S^C divided by the imaginary cut O₁O₂ and the elliptical contours L₁ and L₂ in the z-plane are mapped onto the circular ring regions S^I', S^M', and S^C', with the radii ρ₀=1 and ρ₁ and ρ₂ in the ζ-plane, respectively.

From the transformation (11), we have

(12)

where ρ_k is the radius of the circle L'_k in the ζ-plane.

Denote the aspect ratio of fiber by γ=b₁/a₁. We further obtain

(13)

The complex potential vector

where the superscript i=I, M, or C, F in Eq. (6) can be expanded into the Laurent series (two items are enough for obtaining the exact solutions, which can be seen in Appendix A) as in the ζ-plane,

(14)

where Cⁱ (i represents I, M, or C) and Dⁱ (i represents M or C) are 3×1 complex constant vectors to be determined.

With substitution of Eq. (14) into Eqs. (6) and (8), then into the continuity conditions (9) and (10), the constant vectors can be expressed as

(15)

where

(16)

in which I is the unit matrix, and the superscript "-1" denotes the inverse of a matrix. The overbar denotes the conjugate of a complex number.

If the three-phase confocal elliptical cylinder model is subject to a uniform antiplane shear stress of phonon and phason fields and a uniform in-plane electrical displacement in the global coordinate system, i.e., Σ⁰ =[σ₃₁^∞, H₃₁^∞, D₁^∞]^T at infinity, as shown in Fig. 1, the far-field condition in the local coordinate system can be expressed as Σ₁^∞-iΣ₂^∞=Σ⁰e^iβ. Σ_i^∞ (i=1, 2) denote the far-field boundary conditions along the X_i-direction in the z-plane.

By taking z→∞, from Eqs. (7) and (14), we have

(17)

Substituting Eq. (15) into Eq. (17), we can obtain

(18)

where

(19)

Substituting Eq. (18) into Eq. (15), we find

(20)

Substituting the first two formulae in Eq. (20) into the last formula in Eq. (15), we can easily obtain

(21)

Thus, all the unknown constant vectors in the expressions of the complex potential vector (14) have been determined exactly. Consequently, the stresses of the phonon, phason, and electric fields can be obtained from Eqs. (8) and (14) exactly.

4 Effective electroelastic constants of PQC composites

In this section, the effective electroelastic constants of PQC composites are predicted by the generalized self-consistent method^[31] which can be derived from

(22)

where and refer to the extended averaging stress and strain in the global coordinate system, respectively, and B₀^C denotes the effective constant matrix to be determined.

According to the generalized self-consistent method, the averaged extended stress in a representative unit cell is expressed as

(23)

where λ=a₁b₁/(a₂b₂) is the volume fraction of the inclusions, represents the averaged stress along the X₁-direction in the global coordinate system, and the superscripts I and M denote the inclusion and matrix of the model, respectively.

It is noted that the extended stress in the global coordinate system is related to the orientation angle β, which is based on the assumption of statistical uniformity. Thus, we have

(24)

where S is the area of the region under consideration in the z-plane. From Eqs. (22)-(24), the effective electroelastic constants of PQC composites can be obtained from the following expression:

(25)

From Eqs. (7), (14), (17), and (18) and combining with the far-field condition in the local coordinate system, we can obtain

(26)

where Σ_i (i=1, 2) are the extended stresses in the local coordinate system.

Substituting Eq. (26) into Eq. (24), and noting that the extended stress between the global coordinate system and the local coordinate system has the relationship Σ_1g-iΣ_2g=(Σ₁-iΣ₂)e^-iβ, we find (see Appendix B)

(27)

where

ρ represents the radius of the integral area, and θ is the angle that ρ turns over in the ζ-plane.

Substituting Eq. (26) into Eq. (27) yields

(28)

It is obvious that and in Eq. (28) are zeros when the model is subject to the uniform far-field extended stress Σ⁰ only.

Finally, substituting Eq. (28) into Eq. (25), by the generalized self-consistent method, the effective electroelastic constants of PQC composites are derived as

(29)

It is noted that M₁, M₂, ..., M₆ include the unknown effective electroelastic constant matrix B₀^C. Therefore, Eq. (29) is an implicit matrix equation. We take the Voigt approximation^[31]

(30)

as the initial value of iteration to obtain the effective electroelastic constants of PQC composites.

5 Numerical examples

The three-phase confocal elliptical cylinder model proposed in this paper plays an important role in predicting the effective electroelastic constants of PQC composites. In this section, some numerical examples are presented to illustrate the effects of the volume fraction and the cross-sectional shape of inclusion (or fiber) on the effective electroelastic constants of several composites made up of PQC, QC, and PE, which are useful in designing new composites with better physical properties. The material properties of these materials^[33] are listed in Tables 1 and 2.

First of all, a comparison of the generalized self-consistent method with the other micromechanical methods is made for predicting the effective electroelastic constants of PQC composites, as shown in Fig. 2. It is obvious that both the generalized self-consistent and Mori-Tanaka methods are always consistent to predict the effective electroelastic constants of PQC1/PQC2 composites. As a special case, the classical elasticity results^[37] reduced from the present model with γ=b₁/a₁=0.2 based on the generalized self-consistent method are in reasonable agreement with the experimental results^[38]. Figure 3 shows a comparison of the shear modulus of circular inclusion (γ=0.999 9) in carbon/epoxy composites by four micromechanical methods with the experimental data^[39] and numerical results^[40]. It can be found that the results of the generalized self-consistent and Mori-Tanaka methods agree well with the experimental data and numerical results. Furthermore, it is reported^[30-31] that the difference between the generalized self-consistent method and the Mori-Tanaka method is small for predicting uncoupled effective properties and the former is in even better agreement with the experimental data. Therefore, we display the effects of the volume fraction and the cross-sectional shape of inclusion (or fiber) on the effective electroelastic constants of several composites made up of PQC, QC, and PE, based on the generalized self-consistent method as follows, which need to be justified by future experimental results.

Figure 4 shows the variation of the effective electroelastic constants of phonon, phason, and electric fields for several composites made up of PQC, QC, and PE materials with the fiber volume fraction for a given γ=0.2. Two important features can be observed from Fig. 4(a): (ⅰ) The QC/PE composite displays the highest effective elastic constant of phonon field, but the lowest effective elastic constant in the PE/QC composite is found if the inclusion and matrix are changed. Thus, QCs are suitable for particle reinforced phases embedded in matrix composites or alloys to enhance their mechanical properties, which is in good agreement with the recent experiments^{[9, 12-14]}. (ⅱ) With the increasing fiber volume fraction, the effective elastic constant of QC/PQC2 composites is larger than that of PE/PQC2. It can be found from Fig. 4(b) that the effective phonon-phason coupling elastic constant always decreases with the increasing fiber volume fraction. If a PE is used as inclusion, the effective phonon-phason coupling elastic constant reduces to zero when the fiber volume fraction is large enough as expected. This phenomenon is also observed from Fig. 4(c) for the effective elastic constant of phason field. Furthermore, the effective elastic constant of phason field for PQC1/PQC2 and QC/PQC2 nearly increases with the increasing fiber volume fraction (see Fig. 4(c)). We can see from Figs. 4(d)-4(e) that the effective PE constant is almost similar except for PQC1/PQC2 composites. It can be found from Fig. 4(f) that the effective dielectric constant of PE/PQC2 reaches its maximum at about λ=0.3, and it has the same value as PQC1/PQC2 when the fiber volume fraction is large enough (i.e., λ=0.95). Furthermore, as we expect, there is no effective dielectric constant when QCs are used as inclusion.

Figure 5 shows the variation of the effective electroelastic constants of phonon, phason, and electric fields for several composites made up of PQC, QC, and PE materials with the aspect ratio of the fiber for a given λ=0.1. It can be seen from Fig. 5(a) that the effective elastic constant of phonon field always decreases with the increasing aspect ratio of the fiber and QC/PQC2 shows the highest effective elastic constant. The variation of aspect ratio of the fiber has a slight effect on the effective elastic constant of PQC1/PQC2, QC/PE, and PE/QC composites. An increase of aspect ratio of the fiber results in the decrease of the effective phonon-phason coupling elastic constant of QC/PQC2 composite, but in the increase of the effective phonon-phason coupling elastic constant of PE/PQC2 and PE/QC composites, as shown in Fig. 5(b). It can be found from Fig. 5(c) that the effective elastic constant of phason field for PQC1/PQC2 and QC/PQC2 composites is larger than that of PE/PQC2 and PE/QC composites. It can be further seen from Figs. 5(d)-5(e) that the PQC1/PQC2 composite shows the lowest effective PE constant d₁₅^*/d₁₅^M but the highest effective PE constant e₁₅^*/e₁₅^M. It is interesting to note that the effective dielectric constant is only sensitive to the composite made up of PE inclusion, as shown in Fig. 5(f).

6 Conclusions

A three-phase confocal elliptical cylinder model is proposed to consider micromechanics of PQC composites, and the phonon, phason, and electric fields are derived exactly by using the conformal mapping and the Laurent expansion technique under far-field anti-plane mechanical and in-plane electric loadings. Based on the generalized self-consistent method, the effective electroelastic constants of several new composites made up of PQC, QC, and PE materials are predicted. Some important conclusions can be drawn from numerical examples:

(ⅰ) Compared with other micromechanical methods, the generalized self-consistent method and the Mori-Tanaka method can predict the effective modulus of the PQC composites excellently.

(ⅱ) QCs are suitable for particle reinforced phases embedded in matrix composites or alloys to enhance their mechanical properties, which is in good agreement with the experimental results.

(ⅲ) The fiber volume fraction and the aspect ratio of fiber greatly affect the effective electroelastic constants of composites made up of PQC, QC, and PE, which would be useful for designing new composites in engineering practice.

Appendix A Derivation of Eq. (14)

In the ζ-plane, the analytical function F^Ⅰ(ζ) in the regions S'_I relevant to the fiber can be expanded into the Laurent series^[36-37],

(A1)

where

Since O₁O₂ in the z-plane is only an imaginary cut, ζ=e^iθ and ζ=e^-iθ in the ζ-plane are corresponding to the same point in the z-plane^[37]. Therefore, the function F^Ⅰ must satisfy

(A2)

From Eqs. (A1) and (A2), it can be found that

(A3)

As shown in Fig. 1, the geometry of a unit cell (elliptical inclusion) is symmetric with respect to the x₁- and x₂-axes. Thus, the resultant extended stress components on the unit cell section are only dependent on the far-field loadings

which are determined by

(A4)

To make the extended stress finite when ζ→∞, the constants in Eq. (A4) require

(A5)

Hence, the exact solutions can be obtained by taking the items (k=-1, 1) in the expansion (A1), i.e.,

(A6)

Similarly, the functions F^M(ζ) and F^C(ζ) in the regions S'_M and S'_C in the ζ-plane, respectively, can be expressed as

(A7)

Appendix B Derivation of Eq. (27)

The averaging extended stress in the global coordinate system should be based on the assumption of statistical uniformity^[31],

(B1)

Therefore, we have

(B2)

Note that the extended stress between the global coordinate system and the local coordinate system has the following relationship:

(B3)

The averaging extended stress transformed from the global coordinate system in the z-plane into that in the ζ-plane can be derived as

(B4)

where

(B5)

Thus, we have

(B6)