On the Constitutive Modelling of Piezoelectric Quasicrystals

Abstract

Quasicrystals endowed with piezoelectric properties belong nowadays to novel piezoelectric materials. In this work, the basic framework of generalized piezoelectricity theory of quasicrystals is investigated by providing an improvement of the existing constitutive modelling. It is shown, for the first time, that the tensor of phason piezoelectric moduli is fully asymmetric without any major or minor symmetry, which has important consequences on the constitutive relations as well as on its classification with respect to the crystal systems and Laue classes. The exploration of the tensor of phason piezoelectric moduli has a significant impact on the understanding of the piezoelectric properties of quasicrystals. Using the group representation theory, the classification of the tensor of phason piezoelectric moduli with respect to the crystal systems and Laue classes is given for one-dimensional quasicrystals. The number of independent components of the phason piezoelectric moduli is determined for all 31 point groups of one-dimensional quasicrystals. It is proven that the 10 centrosymmetric crystallographic point groups have no piezoelectric effects and that the remaining 21 non-centrosymmetric crystallographic point groups exhibit piezoelectric effects due to both phonon and phason fields. Moreover, the constitutive relations for one-dimensional hexagonal piezoelectric quasicrystals of Laue class 9 with point group 6 and Laue class 10 with point group $6mm$ are explicitly derived, showing that the constitutive relations for piezoelectric quasicrystals depend on the considered Laue class as well as on the point group. Comparisons with existing results in the literature and discussion are also given.

1. Introduction

It is well-known that an important class of materials with numerous applications in technology are the piezoelectric crystals (e.g., [1,2,3]). For a piezoelectric material, the elastic and electric response is coupled and anisotropic. Piezoelectricity exists only in crystals with non-centrosymmetric point groups [2]. Nowadays, industrial, manufacturing, and the automotive industry are the largest application market for piezoelectric devices.

Nowadays, an interesting class of novel materials are the quasicrystals. Quasicrystals were discovered by Shechtman in 1982 and later reported by Shechtman et al. [4] in 1984. In 2011, Shechtman received the Nobel Prize in Chemistry for the discovery of quasicrystals. Quasicrystals belong to aperiodic crystals and possess long-range orientational order but no translational symmetry in the aperiodic directions. The basis of the continuum theory of solid quasicrystals is set up by two elementary excitations: the phonons and the phasons [5,6]. Quasicrystals can have crystallographic and non-crystallographic point groups. There exist one-, two- and three-dimensional quasicrystals. In particular, for one-dimensional quasicrystals, there exist 31 point groups, which are crystallographic point groups consisting of triclinic, monoclinic, orthorhombic, tetragonal, trigonal and hexagonal systems and 10 Laue classes [7]. The two-dimensional quasicrystals have 57 point groups, consisting of 31 crystallographic point groups and 26 non-crystallographic point groups (pentagonal, decagonal, octagonal, dodecagonal systems) [8]. The three-dimensional quasicrystals have 60 point groups, consisting of 32 crystallographic point groups (i.e., cubic system) and 28 non-crystallographic point groups (i.e., icosahedral system) [9]. To study the elastic properties of quasicrystals, the framework of linear elasticity theory of quasicrystals has been investigated. The elastic energy density was derived for two-dimensional pentagonal and three-dimensional icosahedral quasicrystals by Levine et al. [6] and for octagonal and dodecagonal quasicrystals by Socolar [10]. The generalized linear elasticity theory of quasicrystals was first proposed by Ding et al. [11]. Yang et al. [12] investigated the linear elasticity theory of cubic quasicrystals. The generalized linear elasticity theory of one-dimensional quasicrystals was developed by Wang et al. [7].

The desirable properties of the quasicrystalline materials belong to their piezoelectric properties, as pointed out by Yang et al. [13,14], Hu et al. [15,16], and Gong et al. [17]. In the theory of piezoelectricity of quasicrystals, the phonon fields as well as the phason fields are coupled with the electric field. Due to the piezoelectric effects, quasicrystals may have applications in electronic systems such as sensor and actuator devices, by which an electric voltage can induce an elastic deformation and vice versa. Wang and Pan [18] derived analytical solutions for the fields involved in the problem of uniformly moving screw dislocations in a one-dimensional hexagonal piezoelectric quasicrystal. By using the generalized Stroh formalism, analytical solutions of the electric and elastic fields induced by a straight dislocation in a one-dimensional piezoelectric quasicrystal have been derived by Yang et al. [19]. Quasicrystals with piezoelectric properties represent novel piezoelectric materials. The search for a clean, reusable source of energy is driving the interest in the exploration of piezoelectricity of quasicrystals. Quasicrystals may be used to develop new piezoelectric materials with promising applications in technology that could include energy harvesting practices in the future [20].

Using group representation theory, Hu et al. [15] explored the piezoelectric effects in quasicrystals, including the contribution from both phonon and phason fields, and investigated piezoelectric effects in all two-dimensional quasicrystals with crystallographic and non-crystallographic symmetries and three-dimensional icosahedral and cubic quasicrystals. Hu et al. [15] found that the piezoelectric behavior of quasicrystals is more complicated than that of ordinary crystals due to the presence of the phason field and gave the non-vanishing phonon and phason piezoelectric constants for two-dimensional quasicrystals and three-dimensional icosahedral and cubic quasicrystals. In particular, Hu et al. [15] showed that two-dimensional quasicrystals with non-centrosymmetric crystallographic point groups have piezoelectric effects induced by both phonon and phason fields. Two-dimensional quasicrystals with non-crystallographic point groups can be divided into four classes. The first class consists of quasicrystals with central symmetry ($\overline{5},\overline{5}m,N/m,N/mmm$ for $N=10,8,12$) having no piezoelectric effects. The second class consists of quasicrystals with five-fold symmetry ($5,52,5m$) having piezoelectric effects induced by both phonon and phason fields. The third class consists of quasicrystals with even-fold rotoinversion symmetry ($\overline{N},\overline{N}m2$ for $N=10,8,12$) having a piezoelectric effect induced only by the phason field. The fourth class consists of quasicrystals with even-fold proper rotation symmetry ($N,N22,Nmm$ for $N=10,8,12$) having a piezoelectric effect induced only by the phonon field. On the other hand, three-dimensional quasicrystals can be divided into three classes. The first class ($m\overline{3},m\overline{3}m,235,m3\overline{5}$) has no piezoelectric effects. The second class ($23,\overline{4}3m$) has piezoelectric effects induced by both phonon and phason fields. The third class (432) has a piezoelectric effect induced only by the phason field.

Concerning one-dimensional quasicrystals, Wang et al. [7] derived the independent components of the elastic constitutive tensors for the 31 point groups of one-dimensional quasicrystals based on the fact that “all the point groups belonging to the same Laue class possess the same elastic properties” given by Yang et al. [21]. However, this does not hold for the piezoelectric quasicrystals, since the piezoelectric properties are not the same for point groups belonging to the same Laue class. This happens because the tensor of phonon piezoelectric moduli (the corresponding “usual” tensor of piezoelectric moduli for crystals) depends on the point group (see Nye [2]). An important aspect that is highlighted in this work is that, for the derivation of the explicit form of the constitutive relations for piezoelectric quasicrystals, Laue classes as well as the corresponding point groups have to be taken into account. For the classification of the elastic constitutive tensors $C_{ijkl},E_{\alpha j\gamma l},D_{ij\gamma l}$, the Laue class is enough. On the other hand, in general, for the classification of the phonon and phason piezoelectric constitutive tensors $e_{jkl}$ and $f_{j\gamma l}$, the Laue class is not enough and the classification has to be given with respect to the point groups. Only for one-dimensional quasicrystals, the classification of the tensor of phason piezoelectric moduli $f_{j3l}$ is enough with respect to the Laue class as we will show in this work. The above aspects are often ignored in the literature of elasticity (see, e.g., [9,22,23]) and piezoelectricity (see, e.g., [24,25,26,27]) of one-dimensional hexagonal quasicrystals. The significance of the classification of the constitutive tensors and its influence on the constitutive relations is revealed in Section 3.

As far as one-dimensional piezoelectric quasicrystals are concerned, the independent components of the piezoelectric constitutive tensors have been given by Li and Liu [28] in matrix form for all 31 point groups of one-dimensional quasicrystals. However, the tensor of phason piezoelectric moduli has been erroneously given in [28] as a $3\times 6$ matrix instead of a $3\times 3$ matrix, as it will be shown in this work, leading to an incorrect classification of the tensor of phason piezoelectric moduli and consequently to an incorrect interpretation of the piezoelectric properties of one-dimensional quasicrystals.

On the other hand, concerning the state of the art of constitutive modelling in the literature of piezoelectric quasicrystals, Altay and Dökmeci [29] developed the three-dimensional basic equations in differential and variational invariant forms for piezoelectric quasicrystals. However, they provided the constitutive relations for general quasicrystals with the constitutive tensor of phason piezoelectric moduli possessing an invalid minor symmetry with respect to the last two indices, and all these basic equations have been adopted by many researchers in the field of piezoelectric quasicrystals.

In this work, developing the constitutive modelling of piezoelectric quasicrystals, it is shown first of all that the tensor of phason piezoelectric moduli is an asymmetric, two-point tensor of rank 3 without possessing any minor symmetry in contrast to the statement given by Altay and Dökmeci [29]. The fully asymmetric character of the tensor of phason piezoelectric moduli has a significant impact on its classification and this in turn has physical consequences for the piezoelectric properties of quasicrystals and mathematical consequences for the number of independent components of the phason piezoelectric moduli and consequently for the explicit expressions of the corresponding constitutive relations.

This paper is organized as follows. In Section 2, the basic framework of generalized piezoelectricity theory of quasicrystals is given by improving the constitutive modelling concerning the symmetries of the tensor of phason piezoelectric moduli. In Section 3, the piezoelectric behavior of one-dimensional quasicrystals is investigated. By using group representation theory, the piezoelectric constitutive tensors as well as the tensors with “phasonic character” are studied with a focus on the classification of the tensor of phason piezoelectric moduli for one-dimensional piezoelectric quasicrystals, giving rise to the examination of the piezoelectric behavior of one-dimensional quasicrystals. Based on this classification, the constitutive relations for one-dimensional hexagonal piezoelectric quasicrystals are explicitly derived for Laue class 9 with point group 6 and Laue class 10 with point group $6mm$. Conclusions are given in Section 4. The classification of the constitutive tensors with “phasonic character” for one-dimensional piezoelectric quasicrystals is given in the Appendix A, Appendix B and Appendix C.

2. Generalized Piezoelectricity Theory of Quasicrystals

In this section, we give the basic framework of generalized linear piezoelectricity theory of quasicrystals, which is the generalization of linear (compatible) elasticity of quasicrystals toward the piezoelectricity of quasicrystals. The focus of this section lies on the constitutive modelling and its improvement and the matrix representation form of the phason piezoelectric moduli tensor.

2.1. Basic Framework

An $(n-3)$-dimensional quasicrystal can be generated by the projection of an n-dimensional periodic structure to the three-dimensional physical space ($n=4,5,6$). The n-dimensional hyperspace $E^n$ can be decomposed into the direct sum of two orthogonal subspaces,

$$\begin{array}{c}\hfill E^n=E_{\parallel }^3\oplus E_{\perp }^{(n-3)},\end{array}$$

(1)

where $E_{\parallel }^3$ is the three-dimensional physical or parallel space of the phonon fields and $E_{\perp }^{(n-3)}$ is the $(n-3)$-dimensional perpendicular space of the phason fields. For $n=4,5,6$, we speak of one-dimensional, two-dimensional, and three-dimensional quasicrystals and the dimension of the hyperspace is 4D, 5D, and 6D, respectively. It is considerably helpful to use two different coordinate systems with different indices [30,31]. Indices in the parallel space are denoted by small Latin letters $i,j,k$ with $i,j,k=1,2,3$. Indices in the perpendicular space are denoted by small Greek letters $\alpha ,\gamma$ with $\alpha ,\gamma =3$ for one-dimensional quasicrystals (with quasiperiodicity in z-direction), $\alpha ,\gamma =1,2$ for two-dimensional quasicrystals, and $\alpha ,\gamma =1,2,3$ for three-dimensional quasicrystals. The reason for this distinction is based on the fact that the phonon and phason fields transform, in general, under different representations, as it has been shown by Yang et al. [14,30] using group representation theory. In particular, the phonon fields having indices $i,j,k$ transform under ${\Gamma }_A$, where ${\Gamma }_A$ is a three-dimensional representation of the symmetry group in the parallel space, and the phason fields having indices $\alpha ,\gamma$ transform under ${\Gamma }_B$, where ${\Gamma }_B$ is a representation in the perpendicular space (three-dimensional representation for three-dimensional quasicrystals, two-dimensional representation for two-dimensional quasicrystals, one-dimensional representation for one-dimensional quasicrystals). Irreducible representations of phonon tensor fields and of phason tensor fields “live” in the parallel tensor space and in the perpendicular tensor space, respectively. It is important to highlight here that indices with small Latin letters cannot be interchanged with indices with small Greek letters, since they transform under different representations and “live” in different spaces. Throughout the article, phonon fields will be denoted by ${(·)}^{\parallel }$ and phason fields by ${(·)}^{\perp }$. All field quantities (phonon and phason fields) depend on the so-called material space coordinates (or spatial coordinates) $\mathit{x}\in {\mathbb{R}}^3$. Notice that, in the linear theory of quasicrystals, the material space coincides with the parallel space.

The (elastic) phonon and phason distortion tensors, ${\beta }_{kl}^{\parallel }$ and ${\beta }_{\gamma l}^{\perp }$, are defined as the gradient of the phonon displacement vector $u_k^{\parallel }$ and of the phason displacement vector$u_{\gamma }^{\perp }$, respectively,

$$\begin{array}{c}\hfill {\beta }_{kl}^{\parallel }:=u_{k,l}^{\parallel },{\beta }_{\gamma l}^{\perp }:=u_{\gamma ,l}^{\perp },\end{array}$$

(2)

and they fulfill the following compatibility conditions:

$$\begin{array}{c}\hfill {ϵ}_{ijl}{\beta }_{kl,j}^{\parallel }=0,{ϵ}_{ijl}{\beta }_{\gamma l,j}^{\perp }=0,\end{array}$$

(3)

where ${ϵ}_{ijl}$ denotes the three-dimensional Levi-Civita tensor. The subscript comma denotes the partial differentiation ${\partial }_j=\partial /\partial x_j$ with respect to the spatial coordinates $x_j$. In the geometrically linearized theory of elasticity, the symmetric part of ${\beta }_{kl}^{\parallel }=u_{k,l}^{\parallel }$ is the phonon strain tensor

$$\begin{array}{c}\hfill e_{kl}^{\parallel }=\frac{1}{2}(u_{k,l}^{\parallel }+u_{l,k}^{\parallel }).\end{array}$$

(4)

With the use of the two different coordinate systems and indices, it becomes evident that the phason distortion tensor ${\beta }_{\gamma l}^{\perp }$ is an asymmetric two-point tensor, that is,

$$\begin{array}{c}\hfill {\beta }_{\gamma l}^{\perp }\ne {\beta }_{l\gamma }^{\perp }\end{array}$$

(5)

and consequently a phason strain tensor, a quantity that is often misleadingly used in the literature, cannot be defined. A simple example is the case of one-dimensional quasicrystals, where $\gamma =3$ and $l=1,2,3$ and an interchange of the indices is not possible (e.g., ${\beta }_{31}^{\perp }$ exists but ${\beta }_{13}^{\perp }$ does not exist). The lack of distinction between phonon and phason indices has caused a great confusion in the literature of quasicrystals.

The electric field strength vector $E_l$ is defined as the negative gradient of the electrostatic potential $\phi$ as follows:

$$\begin{array}{c}\hfill E_l:=-{\phi }_{,l}\end{array}$$

(6)

and satisfies the Bianchi identity

$$\begin{array}{c}\hfill {ϵ}_{ijl}{E}_{l,j}=0,\end{array}$$

(7)

which is the compatibility condition for the electric field. Physically, Equation (7) is the electrostatic version of the Faraday law [32].

2.2. Constitutive Modelling

We consider a variational problem with the Lagrangian density $\mathcal{L}$ depending on the variables ${\mathit{u}}^{\parallel }$, ${\mathit{u}}^{\perp }$ and $\phi$ for given body forces and body charges

$$\begin{array}{c}\hfill \mathcal{L}=\mathcal{L}(u_i^{\parallel },u_{i,j}^{\parallel },u_{\alpha }^{\perp },u_{\alpha ,j}^{\perp },\phi ,{\phi }_{,j})=-\left[\Psi (e_{ij}^{\parallel },{\beta }_{\alpha j}^{\perp },E_j)+V(u_i^{\parallel },u_{\alpha }^{\perp },\phi )\right],\end{array}$$

(8)

where $\Psi$ is the so-called electric enthalpy density for piezoelectric quasicrystals and V is the electro-elastic potential of (phonon and phason) body forces and body charges. For vanishing phason fields, the above electric enthalpy density and the electro-elastic potential reduce to the corresponding forms of classical piezoelectricity [33,34,35].

For a linear piezoelectric quasicrystal, the electric enthalpy density $\Psi$ is given by the following quadratic form:

$$\begin{array}{c}\hfill \Psi =\frac{1}{2}{C}_{ijkl}{e}_{ij}^{\parallel }{e}_{kl}^{\parallel }+D_{ij\gamma l}{e}_{ij}^{\parallel }{\beta }_{\gamma l}^{\perp }+\frac{1}{2}{E}_{\alpha j\gamma l}{\beta }_{\alpha j}^{\perp }{\beta }_{\gamma l}^{\perp }-e_{jkl}{E}_j{e}_{kl}^{\parallel }-f_{j\gamma l}{E}_j{\beta }_{\gamma l}^{\perp }-\frac{1}{2}{\epsilon }_{jl}{E}_j{E}_l,\end{array}$$

(9)

where the constitutive tensors, which are the physical property tensors, are defined by

$$\begin{array}{c}\hfill C_{ijkl}=\frac{{\partial }^2\Psi }{\partial e_{ij}^{\parallel }\partial e_{kl}^{\parallel }},E_{\alpha j\gamma l}=\frac{{\partial }^2\Psi }{\partial {\beta }_{\alpha j}^{\perp }\partial {\beta }_{\gamma l}^{\perp }},D_{ij\gamma l}=\frac{{\partial }^2\Psi }{\partial e_{ij}^{\parallel }\partial {\beta }_{\gamma l}^{\perp }},\end{array}$$

(10)

$$\begin{array}{c}\hfill e_{jkl}=-\frac{{\partial }^2\Psi }{\partial E_j\partial e_{kl}^{\parallel }},f_{j\gamma l}=-\frac{{\partial }^2\Psi }{\partial E_j\partial {\beta }_{\gamma l}^{\perp }},{\epsilon }_{jl}=-\frac{{\partial }^2\Psi }{\partial E_j\partial E_l}.\end{array}$$

(11)

$C_{ijkl}$ is the tensor of the elastic moduli of phonons, $E_{\alpha j\gamma l}$ is the tensor of the elastic moduli of phasons, $D_{ij\gamma l}$ is the tensor of the elastic moduli of the phonon–phason coupling, $e_{jkl}$ is the tensor of the phonon piezoelectric moduli, $f_{j\gamma l}$ is the tensor of the phason piezoelectric moduli, and ${\epsilon }_{jl}$ is the tensor of the dielectric moduli. For these constitutive tensors, it holds:

• The tensor of the elastic moduli of phonons possesses the major symmetry and the minor symmetries (see, e.g., [36])

  $$\begin{array}{c}\hfill C_{ijkl}=C_{klij},C_{ijkl}=C_{ijlk}=C_{jikl},\end{array}$$

  (12)

  since the electric enthalpy density $\Psi$ is quadratic in terms of the phonon strain tensor $e_{ij}^{\parallel }$ and the tensor $e_{ij}^{\parallel }$ is symmetric, respectively. The tensor $C_{ijkl}$ has 21 independent components.
• The tensor of the elastic moduli of phasons possesses only a major symmetry (see also [11])

  $$\begin{array}{c}\hfill E_{\alpha j\gamma l}=E_{\gamma l\alpha j},\end{array}$$

  (13)

  since the electric enthalpy density is quadratic in terms of the phason distortion tensor ${\beta }_{\alpha j}^{\perp }$, but ${\beta }_{\alpha j}^{\perp }$ is an asymmetric tensor.
• The tensor of the elastic moduli of the phonon–phason coupling possesses only a minor symmetry with respect to the first two indices (see also [11])

  $$\begin{array}{c}\hfill D_{ij\gamma l}=D_{ji\gamma l},\end{array}$$

  (14)

  due to the symmetric character of $e_{ij}^{\parallel }$.
• The tensor of the phonon piezoelectric moduli possesses a minor symmetry with respect to the last two indices (see, e.g., [2])

  $$\begin{array}{c}\hfill e_{kij}=e_{kji},\end{array}$$

  (15)

  since $e_{ij}^{\parallel }$ is a symmetric tensor. The tensor $e_{kji}$ has $3\times 6=18$ independent components.
• The tensor of the phason piezoelectric moduli $f_{j\gamma l}$ is an asymmetric tensor without a minor symmetry with respect to the last two indices

  $$\begin{array}{c}\hfill f_{j\gamma l}\ne f_{jl\gamma },\end{array}$$

  (16)

  since the phason distortion tensor ${\beta }_{\gamma l}^{\perp }$ is an asymmetric two-point tensor (see Equation (5) and also Remark 2).
• The tensor of the dielectric moduli exhibits the major symmetry (see, e.g., [2])

  $$\begin{array}{c}\hfill {\epsilon }_{ij}={\epsilon }_{ji},\end{array}$$

  (17)

  since the electric enthalpy density is quadratic with respect to the electric field strength $E_j$ and it has six independent components.

Evidently, the tensors $D_{ij\gamma l},e_{kij},f_{j\gamma l}$ do not possess a major symmetry, since they are coupling tensors (see also Remark 3).

To sum up, it holds for the symmetries of the constitutive or physical property tensors of piezoelectric quasicrystals:

$$\begin{array}{cc}\hfill & C_{ijkl}=C_{klij}=C_{ijlk}=C_{jikl},E_{\alpha j\gamma l}=E_{\gamma l\alpha j},D_{ij\gamma l}=D_{ji\gamma l},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & e_{kij}=e_{kji},f_{j\gamma l}\ne f_{jl\gamma },{\epsilon }_{ij}={\epsilon }_{ji}.\hfill \end{array}$$

(19)

Remark 1.

The number of components of the constitutive tensors with phasonic character depends, on the first step, on the dimension of the considered quasicrystal, that is if it is one-dimensional, two-dimensional, or three-dimensional. Accordingly, on the second step, the number of the independent components depends on the considered symmetry and Laue class (see, e.g., the classification of the constitutive tensors $D_{ij3l}$ and $f_{j3l}$ for one-dimensional piezoelectric quasicrystals in Appendix B and Appendix C, respectively).

Remark 2.

As it has been shown above, the tensor of phason piezoelectric moduli $f_{j\gamma l}$ is not symmetric with respect to the last two indices, as erroneously postulated (see Equation (2.30)) by Altay and Dökmeci [29] and adopted by other researchers (see, e.g., [25,37]), leading to erroneous constitutive relations. The impact of the asymmetric character of phason piezoelectric moduli will become clear in Section 3.

The three response quantities are defined through the following linear constitutive relations:

$$\begin{array}{cc}\hfill & {\sigma }_{ij}^{\parallel }=\frac{\partial \Psi }{\partial e_{ij}^{\parallel }}=C_{ijkl}{e}_{kl}^{\parallel }+D_{ij\gamma l}{\beta }_{\gamma l}^{\perp }-e_{lij}{E}_l,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & {\sigma }_{\alpha j}^{\perp }=\frac{\partial \Psi }{\partial {\beta }_{\alpha j}^{\perp }}=D_{kl\alpha j}{e}_{kl}^{\parallel }+E_{\alpha j\gamma l}{\beta }_{\gamma l}^{\perp }-f_{l\alpha j}{E}_l,\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & D_j=-\frac{\partial \Psi }{\partial E_j}=e_{jkl}{e}_{kl}^{\parallel }+f_{j\gamma l}{\beta }_{\gamma l}^{\perp }+{\epsilon }_{jl}{E}_l,\hfill \end{array}$$

(22)

where ${\sigma }_{ij}^{\parallel }$ and ${\sigma }_{\alpha j}^{\perp }$ are the phonon and phason stress tensors, respectively, and $D_j$ is the electric displacement vector or electric excitation. From Equations (20) and (21), it is obvious that the phonon stress tensor is symmetric, ${\sigma }_{ij}^{\parallel }={\sigma }_{ji}^{\parallel }$, whereas the phason stress tensor ${\sigma }_{\alpha j}^{\perp }$ is an asymmetric two-point tensor, ${\sigma }_{\alpha j}^{\perp }\ne {\sigma }_{j\alpha }^{\perp }$ (see also [31,38]). Moreover, one can see in Equations (20)–(22) that the electric field strength gives a contribution to the phonon and phason stress fields and the elastic phonon strain tensor and phason distortion tensor contribute to the electric displacement vector, as it should be for a piezoelectric quasicrystal by its definition.

Using the constitutive relations (20)–(22), the electric enthalpy density (9) can be rewritten in the following “compact” form:

$$\begin{array}{c}\hfill \Psi (e_{ij}^{\parallel },{\beta }_{\alpha j}^{\perp },E_j)=\frac{1}{2}{\sigma }_{ij}^{\parallel }{e}_{ij}^{\parallel }+\frac{1}{2}{\sigma }_{\alpha j}^{\perp }{\beta }_{\alpha j}^{\perp }-\frac{1}{2}{D}_j{E}_j.\end{array}$$

(23)

The electro-elastic potential V is defined by

$$\begin{array}{c}\hfill V(u_i^{\parallel },u_{\alpha }^{\perp },\phi )=-f_i^{\parallel }{u}_i^{\parallel }-f_{\alpha }^{\perp }{u}_{\alpha }^{\perp }+q\phi ,\end{array}$$

(24)

where $f_i^{\parallel }$ is the conventional phonon body force density, $f_{\alpha }^{\perp }$ is the phason body force density (e.g., [38,39]), and q denotes the body charge density. The phonon and phason body force densities and the body charge density are given as the excitations due to the phonon displacement, phason displacement, and electrostatic potential, respectively, as follows:

$$\begin{array}{c}\hfill f_i^{\parallel }=-\frac{\partial V}{\partial u_i^{\parallel }},f_{\alpha }^{\perp }=-\frac{\partial V}{\partial u_{\alpha }^{\perp }},q=\frac{\partial V}{\partial \phi }.\end{array}$$

(25)

The associated Euler–Lagrange equations are as follows:

$$\begin{array}{cc}\hfill E_i^{u^{\parallel }}\left(\mathcal{L}\right)& =\frac{\partial \mathcal{L}}{\partial u_i^{\parallel }}-{\partial }_j\left(\frac{\partial \mathcal{L}}{\partial u_{i,j}^{\parallel }}\right)=0,\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill E_{\alpha }^{u^{\perp }}\left(\mathcal{L}\right)& =\frac{\partial \mathcal{L}}{\partial u_{\alpha }^{\perp }}-{\partial }_j\left(\frac{\partial \mathcal{L}}{\partial u_{\alpha ,j}^{\perp }}\right)=0,\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill E^{\phi }\left(\mathcal{L}\right)& =\frac{\partial \mathcal{L}}{\partial \phi }-{\partial }_j\left(\frac{\partial \mathcal{L}}{\partial {\phi }_{,j}}\right)=0.\hfill \end{array}$$

(28)

Using Equations (8), (9), (20)–(22), (24), and (25) as well as the relations (2), (4), and (6), Equations (26)–(28) give the (force) equilibrium conditions

$$\begin{array}{cc}\hfill {\sigma }_{ij,j}^{\parallel }+f_i^{\parallel }& =0,\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill {\sigma }_{\alpha j,j}^{\perp }+f_{\alpha }^{\perp }& =0,\hfill \end{array}$$

(30)

and the Gauss law of electrostatics (e.g., [32])

$$\begin{array}{cc}\hfill & D_{j,j}=q.\hfill \end{array}$$

(31)

Substituting Equations (20)–(22) into Equations (29)–(31) and using Equations (2), (4), and (6), we obtain the coupled equations of equilibrium for the phonon and phason displacement fields and the electrostatic potential:

$$\begin{array}{cc}\hfill C_{ijkl}{u}_{k,lj}^{\parallel }+D_{ij\gamma l}{u}_{\gamma ,lj}^{\perp }+e_{lij}{\phi }_{,lj}& =-f_i^{\parallel },\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill D_{kl\alpha j}{u}_{k,lj}^{\parallel }+E_{\alpha j\gamma l}{u}_{\gamma ,lj}^{\perp }+f_{l\alpha j}{\phi }_{,lj}& =-f_{\alpha }^{\perp },\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill e_{jkl}{u}_{k,lj}^{\parallel }+f_{j\gamma l}{u}_{\gamma ,lj}^{\perp }-{\epsilon }_{jl}{\phi }_{,lj}& =q,\hfill \end{array}$$

(34)

which are non-homogeneous partial differential equations with sources being the body forces and body charges.

In absence of body forces and body charges, the corresponding coupled equations of equilibrium are given by the following homogeneous partial differential equations:

$$\begin{array}{cc}\hfill C_{ijkl}{u}_{k,lj}^{\parallel }+D_{ij\gamma l}{u}_{\gamma ,lj}^{\perp }+e_{lij}{\phi }_{,lj}& =0,\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill D_{kl\alpha j}{u}_{k,lj}^{\parallel }+E_{\alpha j\gamma l}{u}_{\gamma ,lj}^{\perp }+f_{l\alpha j}{\phi }_{,lj}& =0,\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill e_{jkl}{u}_{k,lj}^{\parallel }+f_{j\gamma l}{u}_{\gamma ,lj}^{\perp }-{\epsilon }_{jl}{\phi }_{,lj}& =0.\hfill \end{array}$$

(37)

Remark 3.

It should be noticed that the electric enthalpy density $\Psi$ given by Equation (9) is in accordance with the corresponding one given by Altay and Dökmeci [29] (see Equation (2.26) in [29]). However, attention should be given to the last terms of the constitutive Equations (20) and (21) in comparison with the corresponding terms in Equations (2.27) and (2.28) in [29], since there is a difference in the order of indices. In particular, in Equations (2.27) and (2.28) in [29], a major symmetry has been used for two coupling tensors, the tensors of phonon and phason piezoelectric moduli. However, these major symmetries do not exist.

2.3. Matrix Representation Form of the Tensor of Phason Piezoelectric Moduli $f_{i\alpha j}$

Special attention is given here to the tensor of phason piezoelectric moduli $f_{i\alpha j}$, since its asymmetric character has important physical consequences. The tensor of phason piezoelectric moduli $f_{i\alpha j}$ is a physical property tensor with the indices i and j to transform under ${\Gamma }_A$ and the index $\alpha$ to transform under ${\Gamma }_B$. Hence, $f_{i\alpha j}$ transforms under the representation ${\Gamma }_A\times {\Gamma }_B\times {\Gamma }_A$. Let us write the tensor $f_{i\alpha j}$ in a matrix representation form.

• For three-dimensional triclinic piezoelectric quasicrystals, the tensor of phason piezoelectric moduli $f_{i\alpha j}(i=1,2,3,\alpha =1,2,3,j=1,2,3)$ has $3\times 3\times 3=27$ independent components. Following [16] for the arrangement of the last two indices in the order

  $$\begin{array}{c}\hfill \left(\alpha j\right)=11,22,33,23,31,12,32,13,21,\end{array}$$

  the tensor $f_{i\alpha j}$ can be written in a $3\times 9$ matrix representation as follows:

  $$\begin{array}{c}\hfill f_{i\alpha j}={\left(\begin{array}{ccccccccc}{f}_{111}& f_{122}& f_{133}& f_{123}& f_{131}& f_{112}& f_{132}& f_{113}& f_{121}\\ f_{211}& f_{222}& f_{233}& f_{223}& f_{231}& f_{212}& f_{232}& f_{213}& f_{221}\\ f_{311}& f_{322}& f_{333}& f_{323}& f_{331}& f_{312}& f_{332}& f_{313}& f_{321}\end{array}\right)}_{\left(27\right)}.\end{array}$$

  (38)
  The subscript $(·)$ in the matrices stands for the number of independent components. The classification of the tensor of phason piezoelectric moduli $f_{i\alpha j}$ has been given by Hu et al. [15] for three-dimensional icosahedral and cubic piezoelectric quasicrystals.
• For two-dimensional triclinic piezoelectric quasicrystals, the tensor $f_{i\alpha j}(i=1,2,3,\alpha =1,2,j=1,2,3)$ has $3\times 2\times 3=18$ independent components. Following [16] for the arrangement of the last two indices in the order

  $$\begin{array}{c}\hfill \left(\alpha j\right)=11,22,23,12,13,21,\end{array}$$

  the tensor $f_{i\alpha j}$ can be written in a $3\times 6$ matrix representation as follows:

  $$\begin{array}{c}\hfill f_{i\alpha j}={\left(\begin{array}{cccccc}{f}_{111}& f_{122}& f_{123}& f_{112}& f_{113}& f_{121}\\ f_{211}& f_{222}& f_{223}& f_{212}& f_{213}& f_{221}\\ f_{311}& f_{322}& f_{323}& f_{312}& f_{313}& f_{321}\end{array}\right)}_{\left(18\right)},\end{array}$$

  (39)

  which is in full agreement with the corresponding representation ($d_{ijk}^{\left(2\right)}$) given by Hu et al. [15]. The classification of the tensor of phason piezoelectric moduli $f_{i\alpha j}$ has been given by Hu et al. [15] for all two-dimensional piezoelectric quasicrystals with crystallographic and non-crystallographic symmetries.
• For one-dimensional triclinic piezoelectric quasicrystals, the tensor $f_{i\alpha j}(i=1,2,3,\alpha =3,j=1,2,3)$ has $3\times 1\times 3=9$ independent components. Since $\alpha =3$ is a fixed index, the tensor $f_{i3j}$ is a tensor of rank 2 and can be written as usual in a $3\times 3$ matrix representation

  $$\begin{array}{c}\hfill f_{i3j}={\left(\begin{array}{ccc}{f}_{131}& f_{132}& f_{133}\\ f_{231}& f_{232}& f_{233}\\ f_{331}& f_{332}& f_{333}\end{array}\right)}_{\left(9\right)}.\end{array}$$

  (40)
  Therefore, it is clear that the phason piezoelectric moduli for one-dimensional piezoelectric quasicrystals is represented by $3\times 3$ matrices and not by $3\times 6$ matrices, as it has been erroneously given by Li and Liu [28]. The classification of the tensor of phason piezoelectric moduli $f_{i3j}$ for one-dimensional piezoelectric quasicrystals is given in Section 3, which is devoted exclusively to one-dimensional piezoelectric quasicrystals.

3. Piezoelectricity of One-Dimensional Quasicrystals

Based on the improved constitutive modelling that has been developed in the previous section, we give here the basic equations governing the one-dimensional piezoelectric quasicrystals and explore their piezoelectric behavior through the classification of the tensors with “phasonic character”. This gives rise to show, considering one-dimensional hexagonal piezoelectric quasicrystals as a representative example, that the constitutive relations depend on the considered Laue class as well as on the point group.

3.1. Basic Equations

In the class of one-dimensional quasicrystals, the atom arrangement is quasiperiodic in one direction, for example, in the $x_3$-direction, and periodic in the plane perpendicular to this direction ($x_1{x}_2$-plane). One-dimensional quasicrystals are quasicrystals with crystallographic symmetries. For one-dimensional quasicrystals, the phason displacement field is a scalar field, $u_{\alpha }^{\perp }\equiv u_3^{\perp }$, the phason distortion tensor becomes a vector ${\beta }_{3l}^{\perp }=u_{3,l}^{\perp }$, and the indices $\alpha ,\gamma =3$.

From the viewpoint of group representation theory, $u_i^{\parallel },i=1,2,3$ transforms under a three-dimensional representation ${\Gamma }_A$, whereas $u_3^{\perp }$ transforms under a one-dimensional representation ${\Gamma }_B$. The latter may be an identity representation or sometimes a one-dimensional non-identity representation with:

(i)

A character 1 for the first-type operations, which are proper rotations n ($n=1,2,3,4,6$) about the vertical axis and reflections m across vertical mirror planes;

(ii)

A character $-1$ for the second-type operations, which are compound operations being a product of inversion $\overline{1}$ and one of the first-type operations [7].

For one-dimensional piezoelectric quasicrystals, the set of the equilibrium Equations (29)–(31) becomes

$$\begin{array}{cc}\hfill {\sigma }_{ij,j}^{\parallel }+f_i^{\parallel }& =0,\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill {\sigma }_{3j,j}^{\perp }+f_3^{\perp }& =0,\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill D_{j,j}& =q\hfill \end{array}$$

(43)

and the corresponding constitutive relations (20)–(22) take the form

$$\begin{array}{cc}\hfill & {\sigma }_{ij}^{\parallel }=C_{ijkl}{e}_{kl}^{\parallel }+D_{ij3l}{\beta }_{3l}^{\perp }-e_{lij}{E}_l,\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill & {\sigma }_{3j}^{\perp }=D_{kl3j}{e}_{kl}^{\parallel }+E_{3j3l}{\beta }_{3l}^{\perp }-f_{l3j}{E}_l,\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill & D_j=e_{jkl}{e}_{kl}^{\parallel }+f_{j3l}{\beta }_{3l}^{\perp }+{\epsilon }_{jl}{E}_l.\hfill \end{array}$$

(46)

Accordingly, the field Equations (32)–(34) read

$$\begin{array}{cc}\hfill C_{ijkl}{u}_{k,lj}^{\parallel }+D_{ij3l}{u}_{3,lj}^{\perp }+e_{lij}{\phi }_{,lj}& =-f_i^{\parallel },\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill D_{kl3j}{u}_{k,lj}^{\parallel }+E_{3j3l}{u}_{3,lj}^{\perp }+f_{l3j}{\phi }_{,lj}& =-f_3^{\perp },\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill e_{jkl}{u}_{k,lj}^{\parallel }+f_{j3l}{u}_{3,lj}^{\perp }-{\epsilon }_{jl}{\phi }_{,lj}& =q.\hfill \end{array}$$

(49)

In the constitutive relations (44)–(46), the constitutive tensors $C_{ijkl}$, $D_{ij3l}$, $E_{3j3l}$, $e_{lij}$, $f_{l3j}$, and ${\epsilon }_{jl}$ appear. It is remarkable to observe that all these tensors transform as tensors in the three-dimensional physical or parallel space, since only the indices in the parallel space are varying. The classification of the tensors of the elastic moduli $C_{ijkl},D_{ij3l}$, and $E_{3j3l}$ for one-dimensional quasicrystals has been given by Wang et al. [7]. On the other hand, the tensors of the phonon piezoelectric moduli $e_{lij}$ and the dielectric moduli ${\epsilon }_{jl}$ are actually the tensors of classical piezoelectricity in the absence of phason fields and follow the classification of the corresponding tensors of classical piezoelectricity of crystals, as it can be found in Nye [2]. Therefore, what remains to be given is the classification of the tensor of phason piezoelectric moduli $f_{l3j}$, which follows in the next section.

3.2. Piezoelectric Behavior of One-Dimensional Quasicrystals

The tensors of elastic moduli of phonons $C_{ijkl}$, phonon piezoelectric moduli $e_{ijk}$, and dielectric moduli ${\epsilon }_{ij}$ are actually the tensors of classical piezoelectricity in absence of phason fields and follow the classification of the corresponding tensors of classical piezoelectricity of crystals, as it is given by Nye [2]. Our primary interest here for the examination of the piezoelectric behavior of one-dimensional quasicrystals is the classification of the tensor of phason piezoelectric moduli. In order to illustrate the method clearly, we give the classification of all three constitutive tensors with, let us say, "phasonic character", that is, $D_{ij3l}$, $E_{3j3l}$, and $f_{l3j}$, following Nye [2].

Let $A_{ij}$ be a coordinate transformation matrix in the parallel space, $B=1$ for first-type operations in the perpendicular space and $B=-1$ for second-type operations in the perpendicular space of one-dimensional quasicrystals.

• The tensor $D_{ij3l}$ has 18 independent components and transforms under the representation $\left\{{\Gamma }_A\times {\Gamma }_A\right\}\times {\Gamma }_B\times {\Gamma }_A$, where the symbol $\left\{·\right\}$ denotes the symmetric part. Therefore, $D_{ij3l}$ transforms as a tensor of rank 3 in the parallel space times a scalar in the perpendicular space according to the following law:

  $$\begin{array}{c}\hfill D_{ij3l}^{\prime }=A_{ik}{A}_{jm}B{A}_{ln}{D}_{km3n}.\end{array}$$

  (50)
• The tensor $E_{3j3l}$ has six independent components and transforms under the representation $\left\{\left({\Gamma }_B\times {\Gamma }_A\right)\times \left({\Gamma }_B\times {\Gamma }_A\right)\right\}$. Accordingly, $E_{3j3l}$ transforms as tensor of rank 2 in the parallel space times a scalar in the perpendicular space two times according to the following law:

  $$\begin{array}{c}\hfill E_{3j3l}^{\prime }=B{A}_{jm}B{A}_{ln}{E}_{3m3n}.\end{array}$$

  (51)
• The tensor of phason piezoelectric moduli $f_{i3j}$ has nine independent components and transforms under the representation ${\Gamma }_A\times {\Gamma }_B\times {\Gamma }_A$. Therefore, $f_{i3j}$ transforms as tensor of rank 2 in the parallel space times a scalar in the perpendicular space according to the following law:

  $$\begin{array}{c}\hfill f_{i3j}^{\prime }=A_{ik}B{A}_{jl}{f}_{k3l}.\end{array}$$

  (52)

Let us now consider a one-dimensional quasicrystal possessing a center of symmetry (inversion). The corresponding transformation reads

$$\begin{array}{c}\hfill A_{ik}=-{\delta }_{ik},B=-1,A_{jl}=-{\delta }_{jl},\end{array}$$

(53)

where ${\delta }_{ik}$ is the Kronecker delta. We examine the behavior of the tensors $D_{ij3l}$, $E_{3j3l}$, and $f_{i3j}$ under this transformation. Using the transformation (53), the transformed tensors (50) and (51) read

$$\begin{array}{cc}\hfill D_{ij3l}^{\prime }& =D_{ij3l},\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill E_{3j3l}^{\prime }& =E_{3j3l}.\hfill \end{array}$$

(55)

It can be seen that both tensors $D_{ij3l}$ and $E_{3j3l}$ remain unchanged under the action of the symmetry operation “inversion”. Therefore, “elastic properties possess an intrinsic (or inherent) centrosymmetry, and hence all point groups belonging to the same Laue class possess the same elastic properties”, as mentioned by Hu et al. [8]. Therefore, $D_{ij3l}$ and $E_{3j3l}$ are non-zero for one-dimensional quasicrystals with centrosymmetric point groups. The independent elastic constants and invariants for each Laue class of one-dimensional quasicrystals have been given by Wang et al. [7]. The corresponding classification of the tensors $D_{ij3l}$ and $E_{3j3l}$ in matrix form is given in Appendix B and Appendix A, respectively.

Next, we explore how the phonon and phason fields influence the piezoelectric behavior of one-dimensional quasicrystals through the tensors of phonon and phason piezoelectric moduli, $e_{lij}$ and $f_{i3j}$. Using the transformation (53), the transformed tensor of phason piezoelectric moduli (52) reads

$$\begin{array}{c}\hfill f_{i3j}^{\prime }=-f_{i3j}.\end{array}$$

(56)

Since the considered one-dimensional quasicrystal possesses a center of symmetry, the transformation does not change the components: $f_{i3j}$ must transform to itself, that is,

$$\begin{array}{c}\hfill f_{i3j}^{\prime }=f_{i3j}.\end{array}$$

(57)

Therefore, from Equations (56) and (57), we obtain that

$$\begin{array}{c}\hfill f_{i3j}=0\end{array}$$

(58)

for one-dimensional quasicrystals with a center of symmetry. This means that one-dimensional quasicrystals with centrosymmetric point groups cannot exhibit piezoelectric effects induced by the phason fields. The same argument holds for the phonon piezoelectric moduli which obeys the transformation law

$$\begin{array}{c}\hfill e_{lij}^{\prime }=A_{lk}{A}_{im}{A}_{jn}{e}_{kmn},\end{array}$$

(59)

which, under the transformation of inversion (53), yields $e_{lij}^{\prime }=-e_{lij}$. Therefore, $e_{lij}=0$ for one-dimensional quasicrystals with a center of symmetry. Hence, under the transformation of inversion, both piezoelectric moduli become zero and this prevents the appearance of piezoelectricity in one-dimensional quasicrystals with centrosymmetric crystallographic point groups. The outcome of the aforementioned analysis is: for one-dimensional quasicrystals with centrosymmetric point groups both phonon and phason piezoelectric moduli are zero and consequently they do not exhibit piezoelectricity.

One-dimensional quasicrystals have 31 point groups; 21 of them are non-centrosymmetric. The classification of the tensor of phonon piezoelectric moduli $e_{lij}$ is the same as in classical piezoelectricity and can be found in Nye [2]. Note that the classification of the phonon piezoelectric moduli depends not only on the Laue class but also on the point group. Using group representation theory [2,40], we give the classification of the tensor of phason piezoelectric moduli $f_{i3j}$ with respect to the crystal system and Laue classes of one-dimensional quasicrystals with non-centrosymmetric crystallographic point groups in Appendix C. The results of this classification are collected in

Table 1. Matrix form of the tensor of phason piezoelectric moduli $f_{i3j}$ for one-dimensional piezoelectric quasicrystals with respect to the crystal system and Laue classes.

System	Laue Class	Matrix Form of ${\mathit{f}}_{\mathit{i}3\mathit{j}}$	Orientation of Axes
$\begin{array}{c}\mathrm{Triclinic}\end{array}$	1	${\left(\begin{array}{ccc}{f}_{131}& f_{132}& f_{133}\\ f_{231}& f_{232}& f_{233}\\ f_{331}& f_{332}& f_{333}\end{array}\right)}_{\left(9\right)}$	arbitrary
$\begin{array}{c}\mathrm{Monoclinic}\end{array}$	$\begin{array}{c}2\end{array}$	${\left(\begin{array}{ccc}{f}_{131}& f_{132}& 0\\ f_{231}& f_{232}& 0\\ 0& 0& f_{333}\end{array}\right)}_{\left(5\right)}$	$2\parallel x_3$
$\begin{array}{c}\mathrm{Monoclinic}\end{array}$	$\begin{array}{c}3\end{array}$	${\left(\begin{array}{ccc}{f}_{131}& 0& f_{133}\\ 0& f_{232}& 0\\ f_{331}& 0& f_{333}\end{array}\right)}_{\left(5\right)}$	$2_h\parallel x_2$
$\begin{array}{c}\mathrm{Orthorhombic}\end{array}$	4	${\left(\begin{array}{ccc}{f}_{131}& 0& 0\\ 0& f_{232}& 0\\ 0& 0& f_{333}\end{array}\right)}_{\left(3\right)}$	$2\parallel x_i$
$\begin{array}{c}\mathrm{Tetragonal}\\ \mathrm{Trigonal}\\ \mathrm{Hexagonal}\end{array}$	$\begin{array}{c}5\\ 7\\ 9\end{array}$	${\left(\begin{array}{ccc}{f}_{131}& f_{132}& 0\\ -f_{132}& f_{131}& 0\\ 0& 0& f_{333}\end{array}\right)}_{\left(3\right)}$	$\begin{array}{c}4\parallel x_3\hfill \\ 3\parallel x_3\hfill \\ 6\parallel x_3\hfill \end{array}$
$\begin{array}{c}\mathrm{Tetragonal}\\ \mathrm{Trigonal}\\ \mathrm{Hexagonal}\end{array}$	$\begin{array}{c}6\\ 8\\ 10\end{array}$	${\left(\begin{array}{ccc}{f}_{131}& 0& 0\\ 0& f_{131}& 0\\ 0& 0& f_{333}\end{array}\right)}_{\left(2\right)}$	$\begin{array}{c}4\parallel x_3\hfill \\ 3\parallel x_3\hfill \\ 6\parallel x_3\hfill \end{array}$

.

In addition, all 31 point groups of one-dimensional quasicrystals and the corresponding numbers of the independent phonon and phason piezoelectric moduli show exactly which point groups exhibit piezoelectric effects and which not, which is detailed in

Table 2. The 31 point groups of one-dimensional quasicrystals and piezoelectricity. ($n_e$: number of independent phonon piezoelectric moduli $e_{ijk}$. $n_f$: number of independent phason piezoelectric moduli $f_{i3j}$).

System	Laue Class	Point Group	Inversion	${\mathit{n}}_{\mathit{e}}$	${\mathit{n}}_{\mathit{f}}$
Triclinic	1	1	–	18	9
		$\overline{1}$	$\overline{1}$	0	0
Monoclinic	2	2 ($2\parallel x_3$)	–	8	5
		$m_h$	–	10	5
		$2/m_h$	$\overline{1}$	0	0
	3	$2_h$ ($2\parallel x_2$)	–	8	5
		m	–	10	5
		$2_h/m$	$\overline{1}$	0	0
Orthorhombic	4	$2_h{2}_{h}2$	–	3	3
		$mm2$ ($2\parallel x_3$)	–	5	3
		$2_{h}m{m}_h$ ($2\parallel x_1$)	–	5	3
		$mm{m}_h$	$\overline{1}$	0	0
Tetragonal	5	4	–	4	3
		$\overline{4}$	–	4	3
		$4/m_h$	$\overline{1}$	0	0
	6	${42}_h{2}_h$	–	1	2
		$4mm$	–	3	2
		$\overline{4}{2}_{h}m$ ($2\parallel x_1$)	–	2	2
		$4/m_{h}mm$	$\overline{1}$	0	0
Trigonal	7	3	–	6	3
		$\overline{3}$	$\overline{1}$	0	0
	8	${32}_h$ ($2\parallel x_1$)	–	2	2
		$3m$	–	4	2
		$\overline{3}m$	$\overline{1}$	0	0
Hexagonal	9	6	–	4	3
		$\overline{6}$	–	2	3
		$6/m_h$	$\overline{1}$	0	0
	10	${62}_h{2}_h$	–	1	2
		$6mm$	–	3	2
		$\overline{6}m{2}_h$	–	1	2
		$6/m_{h}mm$	$\overline{1}$	0	0

. It can be seen in Table 2 that the 21 non-centrosymmetric point groups of one-dimensional quasicrystals produce piezoelectricity due to both phonon and phason fields. Note that for the usual crystals, only 20 from 21 non-centrosymmetric point groups possess piezoelectricity (see, e.g., Nye [2]).

Remark 4.

The latter result is in contradiction with the result of Li and Liu [28], where there exist two point groups, namely hexagonal Laue class 9 with point group $\overline{6}$ and hexagonal Laue class 10 with point group $\overline{6}m{2}_h$, possessing piezoelectric effects only due to phonon fields, since the phonon piezoelectric moduli is non-zero whereas the phason piezoelectric moduli is zero for these two point groups; a case that cannot be true as it has been shown above.

Remark 5.

Comparing Table 2 with the corresponding Table 1 in [28], one can see that there are differences in the number of the independent components of phason piezoelectric moduli. Notice that the tensor of phason piezoelectric moduli, as a tensor of rank 2, has the same properties, that is, the same number of independent components for point groups belonging to the same Laue class. However, this is not the case in [28] (see also Remark 6).

To sum up: from the 31 crystallographic point groups of one-dimensional quasicrystals, the 10 centrosymmetric crystallographic point groups have no piezoelectric effects and the remaining 21 non-centrosymmetric crystallographic point groups exhibit piezoelectric effects due to both phonon and phason fields.

Furthermore, it is worth mentioning that the 21 non-centrosymmetric point groups which produce piezoelectricity due to phonon and phason fields in one-dimensional quasicrystals (see Table 2) are the same as the 21 non-centrosymmetric point groups which produce piezoelectricity due to phonon and phason fields in two-dimensional quasicrystals with crystallographic symmetries, as shown by Hu et al. [15].

Remark 6.

Notice that the phason piezoelectric moduli for one-dimensional hexagonal piezoelectric quasicrystals of Laue class 9 with point group $\overline{6}$ and Laue class 10 with point group $\overline{6}m{2}_h$ are zero according to Li and Liu [28], producing no piezoelectric effect due to the phasons. However, as it can be seen in Table 1, there exist three and two independent components of phason piezoelectric moduli for one-dimensional hexagonal piezoelectric quasicrystals of Laue class 9 (independent of the point group) and Laue class 10 (independent of the point group), respectively, and therefore there is a piezoelectric contribution due to phasons. The misleading result of Li and Liu [28] can also be explained by the fact that point groups belonging to the same Laue class, such as hexagonal Laue class 9 and Laue class 10, have different number of independent components of phason piezoelectric moduli. However, the independent components of the tensor of phason piezoelectric moduli, as an asymmetric tensor of rank 2, are the same for the point groups belonging to the same Laue class in the case of one-dimensional quasicrystals (see also Table 2).

3.3. One-Dimensional Hexagonal Piezoelectric Quasicrystals

In order to gain more insight into the significance of the piezoelectric constitutive tensors, we proceed to explicitly give the constitutive relations for one-dimensional hexagonal piezoelectric quasicrystals of Laue classes 9 and 10. The forthcoming comparisons between the two sets of the constitutive relations reveal important differences between the two Laue classes, which are often ignored in the literature, not only in the framework of piezoelectricity (e.g., [24,25,26,27]), but also in the framework of elasticity (see e.g., [9,22,23]) of one-dimensional hexagonal quasicrystals.

For one-dimensional piezoelectric quasicrystals, Laue classes are not enough for the explicit form of the constitutive relations, since the tensor of phonon piezoelectric moduli $e_{lij}$ depends on the considered point group (see [2]). In what follows, we examine two Laue classes of one-dimensional piezoelectric quasicrystals with a representative point group.

• For Laue class 9 with point group 6, using Equations (A5), (A11), and (A19) and the expressions for the tensors of the elastic moduli and phonon piezoelectric moduli given by Nye [2], the constitutive relations (44)–(46) take the following explicit form:

  $$\begin{array}{cc}\hfill & {\sigma }_{11}^{\parallel }=C_{1111}{e}_{11}^{\parallel }+C_{1122}{e}_{22}^{\parallel }+C_{1133}{e}_{33}^{\parallel }+D_{1133}{\beta }_{33}^{\perp }-e_{311}{E}_3,\hfill \end{array}$$

  (60)

  $$\begin{array}{cc}\hfill & {\sigma }_{22}^{\parallel }=C_{1122}{e}_{11}^{\parallel }+C_{1111}{e}_{22}^{\parallel }+C_{1133}{e}_{33}^{\parallel }+D_{1133}{\beta }_{33}^{\perp }-e_{311}{E}_3,\hfill \end{array}$$

  (61)

  $$\begin{array}{cc}\hfill & {\sigma }_{33}^{\parallel }=C_{3311}(e_{11}^{\parallel }+e_{22}^{\parallel })+C_{3333}{e}_{33}^{\parallel }+D_{3333}{\beta }_{33}^{\perp }-e_{333}{E}_3,\hfill \end{array}$$

  (62)

  $$\begin{array}{cc}\hfill & {\sigma }_{23}^{\parallel }=2C_{2323}{e}_{23}^{\parallel }+D_{2331}{\beta }_{31}^{\perp }+D_{2332}{\beta }_{32}^{\perp }-e_{123}{E}_1-e_{113}{E}_2,\hfill \end{array}$$

  (63)

  $$\begin{array}{cc}\hfill & {\sigma }_{13}^{\parallel }=2C_{2323}{e}_{13}^{\parallel }+D_{2332}{\beta }_{31}^{\perp }-D_{2331}{\beta }_{32}^{\perp }-e_{113}{E}_1+e_{123}{E}_2,\hfill \end{array}$$

  (64)

  $$\begin{array}{cc}\hfill & {\sigma }_{12}^{\parallel }=2C_{1212}{e}_{12}^{\parallel },\hfill \end{array}$$

  (65)

  $$\begin{array}{cc}\hfill & {\sigma }_{31}^{\perp }=2D_{2331}{e}_{23}^{\parallel }+2D_{2332}{e}_{13}^{\parallel }+E_{3131}{\beta }_{31}^{\perp }-f_{131}{E}_1+f_{132}{E}_2,\hfill \end{array}$$

  (66)

  $$\begin{array}{cc}\hfill & {\sigma }_{32}^{\perp }=2D_{2332}{e}_{23}^{\parallel }-2D_{2331}{e}_{13}^{\parallel }+E_{3131}{\beta }_{32}^{\perp }-f_{132}{E}_1-f_{131}{E}_2,\hfill \end{array}$$

  (67)

  $$\begin{array}{cc}\hfill & {\sigma }_{33}^{\perp }=D_{1133}\left(e_{11}^{\parallel }+e_{22}^{\parallel }\right)+D_{3333}{e}_{33}^{\parallel }+E_{3333}{\beta }_{33}^{\perp }-f_{333}{E}_3,\hfill \end{array}$$

  (68)

  $$\begin{array}{cc}\hfill & D_1=2e_{123}{e}_{23}^{\parallel }+2e_{113}{e}_{13}^{\parallel }+f_{131}{\beta }_{31}^{\perp }+f_{132}{\beta }_{32}^{\perp }+{\epsilon }_{11}{E}_1,\hfill \end{array}$$

  (69)

  $$\begin{array}{cc}\hfill & D_2=2e_{113}{e}_{23}^{\parallel }-2e_{123}{e}_{13}^{\parallel }-f_{132}{\beta }_{31}^{\perp }+f_{131}{\beta }_{32}^{\perp }+{\epsilon }_{11}{E}_2,\hfill \end{array}$$

  (70)

  $$\begin{array}{cc}\hfill & D_3=e_{311}\left(e_{11}^{\parallel }+e_{22}^{\parallel }\right)+e_{333}{e}_{33}^{\parallel }+f_{333}{\beta }_{33}^{\perp }+{\epsilon }_{33}{E}_3\hfill \end{array}$$

  (71)

  with five elastic moduli of phonons, $C_{1111}$, $C_{1122}$, $C_{1133}$, $C_{3333}$, $C_{2323}$, $C_{1212}=(C_{1111}-C_{1122})/2$; two elastic moduli of phasons, $E_{3131}$ and $E_{3333}$; four elastic moduli of phonon–phason coupling, $D_{2331}$, $D_{2332}$, $D_{1133}$, $D_{3333}$; four phonon piezoelectric moduli, $e_{311}$, $e_{333}$, $e_{113}$, $e_{123}$; three phason piezoelectric moduli, $f_{131}$, $f_{132}$, $f_{333}$; and two dielectric moduli, ${\epsilon }_{11}$ and ${\epsilon }_{33}$.
  Thus, a one-dimensional hexagonal piezoelectric quasicrystal of Laue class 9 with point group 6 has 20 material moduli.
• For Laue class 10 with point group $6mm$, using Equations (A5), (A12), and (A20) and the expressions for the tensors of the elastic moduli and phonon piezoelectric moduli given by Nye [2], the constitutive relations (44)–(46) take the following explicit form:

  $$\begin{array}{cc}\hfill & {\sigma }_{11}^{\parallel }=C_{1111}{e}_{11}^{\parallel }+C_{1122}{e}_{22}^{\parallel }+C_{1133}{e}_{33}^{\parallel }+D_{1133}{\beta }_{33}^{\perp }-e_{311}{E}_3,\hfill \end{array}$$

  (72)

  $$\begin{array}{cc}\hfill & {\sigma }_{22}^{\parallel }=C_{1122}{e}_{11}^{\parallel }+C_{1111}{e}_{22}^{\parallel }+C_{1133}{e}_{33}^{\parallel }+D_{1133}{\beta }_{33}^{\perp }-e_{311}{E}_3,\hfill \end{array}$$

  (73)

  $$\begin{array}{cc}\hfill & {\sigma }_{33}^{\parallel }=C_{3311}(e_{11}^{\parallel }+e_{22}^{\parallel })+C_{3333}{e}_{33}^{\parallel }+D_{3333}{\beta }_{33}^{\perp }-e_{333}{E}_3,\hfill \end{array}$$

  (74)

  $$\begin{array}{cc}\hfill & {\sigma }_{23}^{\parallel }=2C_{2323}{e}_{23}^{\parallel }+D_{2332}{\beta }_{32}^{\perp }-e_{113}{E}_2,\hfill \end{array}$$

  (75)

  $$\begin{array}{cc}\hfill & {\sigma }_{13}^{\parallel }=2C_{2323}{e}_{13}^{\parallel }+D_{2332}{\beta }_{31}^{\perp }-e_{113}{E}_1,\hfill \end{array}$$

  (76)

  $$\begin{array}{cc}\hfill & {\sigma }_{12}^{\parallel }=2C_{1212}{e}_{12}^{\parallel },\hfill \end{array}$$

  (77)

  $$\begin{array}{cc}\hfill & {\sigma }_{31}^{\perp }=2D_{2332}{e}_{13}^{\parallel }+E_{3131}{\beta }_{31}^{\perp }-f_{131}{E}_1,\hfill \end{array}$$

  (78)

  $$\begin{array}{cc}\hfill & {\sigma }_{32}^{\perp }=2D_{2332}{e}_{23}^{\parallel }+E_{3131}{\beta }_{32}^{\perp }-f_{131}{E}_2,\hfill \end{array}$$

  (79)

  $$\begin{array}{cc}\hfill & {\sigma }_{33}^{\perp }=D_{1133}\left(e_{11}^{\parallel }+e_{22}^{\parallel }\right)+D_{3333}{e}_{33}^{\parallel }+E_{3333}{\beta }_{33}^{\perp }-f_{333}{E}_3,\hfill \end{array}$$

  (80)

  $$\begin{array}{cc}\hfill & D_1=2e_{113}{e}_{13}^{\parallel }+f_{131}{\beta }_{31}^{\perp }+{\epsilon }_{11}{E}_1,\hfill \end{array}$$

  (81)

  $$\begin{array}{cc}\hfill & D_2=2e_{113}{e}_{23}^{\parallel }+f_{131}{\beta }_{32}^{\perp }+{\epsilon }_{11}{E}_2,\hfill \end{array}$$

  (82)

  $$\begin{array}{cc}\hfill & D_3=e_{311}\left(e_{11}^{\parallel }+e_{22}^{\parallel }\right)+e_{333}{e}_{33}^{\parallel }+f_{333}{\beta }_{33}^{\perp }+{\epsilon }_{33}{E}_3\hfill \end{array}$$

  (83)

  with five elastic moduli of phonons, $C_{1111}$, $C_{1122}$, $C_{1133}$, $C_{3333}$, $C_{2323}$, $C_{1212}=(C_{1111}-C_{1122})/2$; two elastic moduli of phasons, $E_{3131}$ and $E_{3333}$; three elastic moduli of phonon–phason coupling, $D_{2332}$, $D_{1133}$, $D_{3333}$; three phonon piezoelectric moduli, $e_{311}$, $e_{333}$, $e_{113}$; two phason piezoelectric moduli $f_{131}$ and $f_{333}$; and two dielectric moduli, ${\epsilon }_{11}$ and ${\epsilon }_{33}$.
  Thus, a one-dimensional hexagonal piezoelectric quasicrystal of Laue class 10 with point group $6mm$ has 17 material moduli.

In the framework of piezoelectricity, the constitutive relations for one-dimensional hexagonal piezoelectric quasicrystals of Laue class 10 with point group $6mm$, Equations (72)–(83), are in accordance with the corresponding ones given by Wang and Pan [18], where $D_{1133}=R_1$, $D_{3333}=R_2$, $D_{2332}=R_3$, $E_{3131}=K_2$, $E_{3333}=K_1$, $e_{311}=e_{31}^{\left(1\right)}$, $e_{333}=e_{33}^{\left(1\right)}$, $e_{113}=e_{15}^{\left(1\right)}$, $f_{131}=e_{15}^{\left(2\right)}$, $f_{333}=e_{33}^{\left(2\right)}$. The constitutive relations for one-dimensional hexagonal piezoelectric quasicrystals of Laue class 9 with point group 6, Equations (60)–(71), are given here for the first time in the literature.

It can be seen that the constitutive relations (60)–(71) for one-dimensional hexagonal piezoelectric quasicrystals of Laue class 9 with point group 6 are different than the constitutive relations (72)–(83) for the hexagonal piezoelectric quasicrystals of Laue class 10 with point group $6mm$. In particular, comparing the above two sets of the constitutive relations, we observe that two additional terms appear in each of the Equations (63), (64), (66), (67), (69), and (70) due to the different matrix representation of three tensors, namely the tensor of elastic moduli of phonon–phason coupling $D_{ij3l}$ (see Equations (A11) and (A12)), the tensor of phason piezoelectric moduli $f_{i3j}$ (see Equations (A19) and (A20)), and finally the tensor of phonon piezoelectric moduli $e_{lij}$. This difference in the constitutive relations is often ignored and neglected in the literature of one-dimensional piezoelectric quasicrystals (e.g., [24,25,26,27]). In general, for one-dimensional hexagonal piezoelectric quasicrystals, there are five non-centrosymmetric point groups (see Table 2), leading to different constitutive relations. For the explicit form of the constitutive relations of piezoelectric quasicrystals, the corresponding point group is relevant and must be taken into account.

Furthermore, as it has been mentioned above, a reason for the difference in the two sets of the constitutive relations lies on the different matrix representation of the tensor of the elastic moduli of phonon–phason coupling $D_{ij3l}$. Therefore, a difference in the constitutive relations of one-dimensional hexagonal quasicrystals between the two Laue classes 9 and 10 is expected also in the framework of elasticity. This is a fact that is often ignored in the literature of elasticity theory of one-dimensional hexagonal quasicrystals, since no attention is given to the distinction of the two Laue classes (see, e.g., [9,22,23]). The constitutive relations of one-dimensional hexagonal quasicrystals for the two Laue classes 9 and 10 are derived from Equations (60)–(71) and Equations (72)–(83), respectively, for vanishing electric fields, and they are given explicitly below.

• The constitutive relations for one-dimensional hexagonal quasicrystals of Laue class 9 are as follows:

  $$\begin{array}{cc}\hfill & {\sigma }_{11}^{\parallel }=C_{1111}{e}_{11}^{\parallel }+C_{1122}{e}_{22}^{\parallel }+C_{1133}{e}_{33}^{\parallel }+D_{1133}{\beta }_{33}^{\perp },\hfill \end{array}$$

  (84)

  $$\begin{array}{cc}\hfill & {\sigma }_{22}^{\parallel }=C_{1122}{e}_{11}^{\parallel }+C_{1111}{e}_{22}^{\parallel }+C_{1133}{e}_{33}^{\parallel }+D_{1133}{\beta }_{33}^{\perp },\hfill \end{array}$$

  (85)

  $$\begin{array}{cc}\hfill & {\sigma }_{33}^{\parallel }=C_{3311}(e_{11}^{\parallel }+e_{22}^{\parallel })+C_{3333}{e}_{33}^{\parallel }+D_{3333}{\beta }_{33}^{\perp },\hfill \end{array}$$

  (86)

  $$\begin{array}{cc}\hfill & {\sigma }_{23}^{\parallel }=2C_{2323}{e}_{23}^{\parallel }+D_{2331}{\beta }_{31}^{\perp }+D_{2332}{\beta }_{32}^{\perp },\hfill \end{array}$$

  (87)

  $$\begin{array}{cc}\hfill & {\sigma }_{13}^{\parallel }=2C_{2323}{e}_{13}^{\parallel }+D_{2332}{\beta }_{31}^{\perp }-D_{2331}{\beta }_{32}^{\perp },\hfill \end{array}$$

  (88)

  $$\begin{array}{cc}\hfill & {\sigma }_{12}^{\parallel }=2C_{1212}{e}_{12}^{\parallel },\hfill \end{array}$$

  (89)

  $$\begin{array}{cc}\hfill & {\sigma }_{31}^{\perp }=2D_{2331}{e}_{23}^{\parallel }+2D_{2332}{e}_{13}^{\parallel }+E_{3131}{\beta }_{31}^{\perp },\hfill \end{array}$$

  (90)

  $$\begin{array}{cc}\hfill & {\sigma }_{32}^{\perp }=2D_{2332}{e}_{23}^{\parallel }-2D_{2331}{e}_{13}^{\parallel }+E_{3131}{\beta }_{32}^{\perp },\hfill \end{array}$$

  (91)

  $$\begin{array}{cc}\hfill & {\sigma }_{33}^{\perp }=D_{1133}\left(e_{11}^{\parallel }+e_{22}^{\parallel }\right)+D_{3333}{e}_{33}^{\parallel }+E_{3333}{\beta }_{33}^{\perp }\hfill \end{array}$$

  (92)

  with five elastic moduli of phonons, $C_{1111}$, $C_{1122}$, $C_{1133}$, $C_{3333}$, $C_{2323}$, $C_{1212}=(C_{1111}-C_{1122})/2$; two elastic moduli of phasons, $E_{3131}$ and $E_{3333}$; and four elastic moduli of phonon–phason coupling, $D_{2331}$, $D_{2332}$, $D_{1133}$, $D_{3333}$.
  Thus, a one-dimensional hexagonal quasicrystal of Laue class 9 has 11 material moduli.
• The constitutive relations of one-dimensional hexagonal quasicrystals of Laue class 10 read:

  $$\begin{array}{cc}\hfill & {\sigma }_{11}^{\parallel }=C_{1111}{e}_{11}^{\parallel }+C_{1122}{e}_{22}^{\parallel }+C_{1133}{e}_{33}^{\parallel }+D_{1133}{\beta }_{33}^{\perp },\hfill \end{array}$$

  (93)

  $$\begin{array}{cc}\hfill & {\sigma }_{22}^{\parallel }=C_{1122}{e}_{11}^{\parallel }+C_{1111}{e}_{22}^{\parallel }+C_{1133}{e}_{33}^{\parallel }+D_{1133}{\beta }_{33}^{\perp },\hfill \end{array}$$

  (94)

  $$\begin{array}{cc}\hfill & {\sigma }_{33}^{\parallel }=C_{3311}(e_{11}^{\parallel }+e_{22}^{\parallel })+C_{3333}{e}_{33}^{\parallel }+D_{3333}{\beta }_{33}^{\perp },\hfill \end{array}$$

  (95)

  $$\begin{array}{cc}\hfill & {\sigma }_{23}^{\parallel }=2C_{2323}{e}_{23}^{\parallel }+D_{2332}{\beta }_{32}^{\perp },\hfill \end{array}$$

  (96)

  $$\begin{array}{cc}\hfill & {\sigma }_{13}^{\parallel }=2C_{2323}{e}_{13}^{\parallel }+D_{2332}{\beta }_{31}^{\perp },\hfill \end{array}$$

  (97)

  $$\begin{array}{cc}\hfill & {\sigma }_{12}^{\parallel }=2C_{1212}{e}_{12}^{\parallel },\hfill \end{array}$$

  (98)

  $$\begin{array}{cc}\hfill & {\sigma }_{31}^{\perp }=2D_{2332}{e}_{13}^{\parallel }+E_{3131}{\beta }_{31}^{\perp },\hfill \end{array}$$

  (99)

  $$\begin{array}{cc}\hfill & {\sigma }_{32}^{\perp }=2D_{2332}{e}_{23}^{\parallel }+E_{3131}{\beta }_{32}^{\perp },\hfill \end{array}$$

  (100)

  $$\begin{array}{cc}\hfill & {\sigma }_{33}^{\perp }=D_{1133}\left(e_{11}^{\parallel }+e_{22}^{\parallel }\right)+D_{3333}{e}_{33}^{\parallel }+E_{3333}{\beta }_{33}^{\perp }\hfill \end{array}$$

  (101)

  with five elastic moduli of phonons, $C_{1111}$, $C_{1122}$, $C_{1133}$, $C_{3333}$, $C_{2323}$, $C_{1212}=(C_{1111}-C_{1122})/2$; two elastic moduli of phasons, $E_{3131}$, and $E_{3333}$; and three elastic moduli of phonon–phason coupling, $D_{2332}$, $D_{1133}$, $D_{3333}$.
  Thus, a one-dimensional hexagonal quasicrystal of Laue class 10 has 10 material moduli.

It has to be mentioned that: (i) the non-zero elastic constants given above for both Laue classes 9 and 10 are in accordance with the corresponding non-zero elastic constants given by Wang et al. [7]; (ii) the constitutive relations (93)–(101) are in accordance with the constitutive relations of one-dimensional hexagonal quasicrystals of Laue class 10 given by Wang et al. [7].

4. Conclusions

The main favorable conclusions of this work are summarized as follows:

• It is proven that the tensor of phason piezoelectric moduli $f_{i\alpha j}$ is fully asymmetric without any major or minor symmetry in contrast to the work of Altay and Dökmeci [29], where it is claimed that the tensor $f_{i\alpha j}$ is symmetric with respect to the last two indices. Therefore, the tensor of phason piezoelectric moduli $f_{i\alpha j}$ is a two-point tensor of rank 3 in the case of three-dimensional piezoelectric quasicrystals and is represented by a $3\times 9$ matrix with 27 independent components. It is not represented by a $3\times 6$ matrix with 18 independent components, which would be the case if possessing a minor symmetry with respect to the last two indices, as claimed in [29], which is however not the case.
• The classification of the tensor of phason piezoelectric moduli $f_{i3j}$ for one-dimensional piezoelectric quasicrystals has been derived in a $3\times 3$ matrix form for all crystal systems and Laue classes. Hence, $f_{i3j}$ cannot be represented in a $3\times 6$ matrix form as claimed in [28].
• It has been shown that all non-centrosymmetric crystallographic point groups of one-dimensional quasicrystals exhibit piezoelectric effects induced by both phonon and phason fields in contrast to the result of Li and Liu [28], where there are two cases of one-dimensional hexagonal piezoelectric quasicrystals, namely of Laue class 9 with point group $\overline{6}$ and Laue class 10 with point group $\overline{6}m{2}_h$, in which there is a piezoelectric effect induced only by the phonon and not by the phason fields.
• It has been shown that one-dimensional quasicrystals with centrosymmetric crystallographic point groups have no piezoelectric effects.
• The piezoelectric behavior of one-dimensional quasicrystals has been explored here. It is shown that one-dimensional quasicrystals can be divided into two classes: the first class, consisting of one-dimensional quasicrystals with centrosymmetry, has no piezoelectric effects and the second class, consisting of one-dimensional quasicrystals without centrosymmetry, has piezoelectric effects induced by both phonon and phason fields.
• For the derivation of the constitutive relations of elasticity of quasicrystals, the Laue class has to be taken into account. As a representative example, the explicit constitutive relations for Laue class 9 and Laue class 10 have been derived. A one-dimensional hexagonal quasicrystal of Laue class 9 has 11 material moduli, whereas that of Laue class 10 has 10 material moduli.
• For the derivation of the constitutive relations of piezoelectricity of quasicrystals, the point group has to be taken into account in addition to the Laue class. As a representative example, the explicit constitutive relations for Laue class 9 with point group 6 and Laue class 10 with point group $6mm$ have been derived. A one-dimensional hexagonal piezoelectric quasicrystal of Laue class 9 with point group 6 has 20 material moduli, whereas that of Laue class 10 with point group $6mm$ has 17 material moduli.