1 Introduction

Quasicrystals (QCs) in the Al-Mn phase were discovered firstly in 1982 by Shechtman et al.^[1], which exhibit the forbidden rotational symmetry in the conventional crystallography and lack translational symmetry. The ordered but aperiodic atomic arrangement in QCs enables them to display some attractive properties^[2-5], such as high hardness, high wear resistance, low thermal conductivity, low electrical conductivity, low surface energy, and high infra-red absorption. Due to these special properties, QCs have many potential applications^[6-9], including thermal barrier coatings, wear resistant coatings, and reinforcements in composites and thin film. The multilayered plate model offers guidance in understanding the deformation of QC coatings or thin film.

A one-dimensional (1D) QC refers to a three-dimensional (3D) structure with atomic arrangement quasi-periodically in one direction and periodically in the plane perpendicular to that direction. For the plate model in QCs, Gao et al.^[10] applied the reciprocal theorem to consider the plate bending of 1D hexagonal QCs. Furthermore, Gao and Ricoeur^[11] established a refined theory of thick plates for 1D QCs. Sladek et al.^[12] used the meshless Pethov-Galerkin method to analyze the bending problem in 1D orthorhombic QCs under static and transient dynamic loads. Waksmanski et al.^[13] obtained an exact closed-form solution of free vibration of a simply-supported and multilayered 1D QC plate by using the pseudo-Stroh formulation and propagator matrix method. Yang et al.^[14-15] derived an exact closed-form solution for a simply-supported and multilayered two-dimensional (2D) decagonal QC plate and 1D orthorhombic QC plate by utilizing the pseudo-Stroh formalism.

The studies showed that piezoelectricity is an important physical property of QCs^[16-20]. The inherent piezoelectric (PE) coupling effect of QCs has attracted much attention, and exciting advances have been achieved. For example, Altay and Dökmeci^[21] addressed the 3D basic equations of piezoelectric quasicrystals (PQCs). Li et al.^[22] obtained the 3D general solutions to a static problem of 1D hexagonal PQCs by utilizing the operator theory. Yu et al.^[23] derived the governing equations of the plane electroelastic problem of 1D PQCs and their general solutions. Based on these theories, the eletroelastic behaviors of 1D PQC involving dislocations^[24-25], cracks^[23-26], holes^[27-28], and inclusions^[29-30] have been investigated extensively. More recently, by using an extended dislocation layer method, Tupholme^[31] derived the closed-form expressions of the stress and electric fields for a moving non-constantly loaded antiplane, Griffith-type strip crack in 1D PQCs. Yang et al.^[32] used the pseudo-Stroh formalism and propagator matrix method to present an exact elastic analysis of a multilayered 2D decagonal QC plate subject to patch loading with simply-supported boundary conditions.

As mentioned above, although some plate models in QCs were conducted, to the best of the authors' knowledge, the static deformation of 1D PQC plate in a 3D finite space has not been reported in the literature. In fact, PQC plates process the coupling effect among phonon, phason, and electric fields, which is superior to QC plates with phonon-phason coupling in engineering practice. To better understand the unique physical properties of QCs to meet our increasing demands or new materials, it is of significance to study multilayered PQC plates. Therefore, in this work, we derive an exact closed-form solution of static deformation for 3D multilayered plates made of 1D PQCs, QCs, and PE with simply-supported boundary conditions.

2 Problem description and basic equations

We consider a simply-supported, transversely isotropic, and multilayered 1D hexagonal PQC plate with horizontal dimensions L_x and L_y and a total thickness H, as shown in Fig. 1. A Cartesian coordinate system is placed on the horizontal Ox₁x₂-plane of the bottom surface, and the positive x₃-direction is along the thickness of the plate. The horizontal Ox₁x₂-plane is the periodic plane, and both the quasi-periodic direction and the electric poling direction are along the x₃-axis. There are N layers with the thickness h_j=z_j-z_j-1 (j=1, 2, ..., N), and each layer is bonded perfectly on its interfaces. Thus, we assume that the extended displacement and traction are continuous across the interfaces, i.e.,

(1)

Furthermore, the simply-supported boundary conditions can be written as

(2)

where x₁=0 and x₁= L_x denote the left and right edges of the plate, respectively, and x₂=0 and x₂=L_y denote the back and front edges of the plate, respectively. At these edges, the displacements u_i (i=1, 2, 3) of the phonon field, the displacement w₃ of the phason field, and the electric potential ϕ are zeros.

The basic equations for static deformation of PQCs presented by Altay and Dökmeci^[21] include the equilibrium equations

(3)

the gradient equations

(4)

and the constitutive equations

(5)

In (1)-(5), a comma denotes differentiation with respect to x_i (i=1, 2, 3), repeated indices imply summation (from 1 to 3), σ_ij, ε_ij, and u_i are the components of the stress, strain, and displacement of the phonon field, respectively, H_ij, ω_ij, and w_i are the components of the stress, strain, and displacement of the phason field, respectively, D_i, E_i, and ϕ are the electric displacements, electric fields, and electric potential, respectively, C_ijkl and K_ijkl are the elastic constants of the phonon field and phason field, respectively, R_ijkl are the phonon-phason coupling elastic constants, e_ijk and d_ijk are the PE coefficients, and λ_ij is the dielectric permittivity.

3 General solutions for a homogeneous 1D hexagonal PQC plate

To satisfy the simply-supported boundary conditions (2), the general solutions of the extended displacement vector for a homogeneous PQC plate can be assumed in the form of

(6)

where

(7)

in which n and m are two positive integers. Also in (6), s is the eigenvalue, and a₁, a₂, ..., a₅ are unknown constants to be determined. Notice that by taking the summation over n and m values, we then have the general solution in terms of 2D Fourier series expansion.

From the gradient equation (4), the constitutive equation (5), and the general solution (6) of the extended displacement, the extended traction vector can be expressed as

(8)

Two group coefficients of (6) and (8) are written in the forms of vectors a=[a₁, a₂, ..., a₅]^T and b=[b₁, b₂, ..., b₅]^T with the following relationship:

(9)

where the superscript T denotes matrix transpose, and the matrices R, T, and Q are

(10)

(11)

(12)

in which

where the short notations for the subscripts are used from 4 to 2, i.e., 11→1, 22→2, 33→3, 23→4, 31→5, and 12→6. The latter two numbers of e_kij and d_kij are also simplified similarly.

Substituting the extended traction vector in (8) into the equilibrium equation (3), we have

(13)

From (9) and (13), the final governing equation can be derived as

(14)

Similar to the method of Pan^[33], (9) and (14) can be transformed to a linear eigensystem as follows:

(15)

where

(16)

If we solve the eigenvalues and eigenvectors from (15), the general solution for a 1D homogenous PQC plate can be obtained as

(17)

In (17), K is a constant vector to be determined by the external loads on the surfaces of the plates, and the matrices A and B and the diagonal matrix are

(18)

4 Exact closed-form solution of multilayered PQC plates

According to the boundary condition of the bottom surface of plate, we can obtain the constant vector K from (17), i.e.,

(19)

Substituting (19) into (17), the general solutions of the extended displacement and traction become

(20)

where P(z) is the propagator matrix, i.e.,

(21)

Making use of the following characteristic of the propagator matrix^[33]:

(22)

we can propagate the solution from the bottom surface z=0 to the top surface z=H of the multilayered PQC plate by using the propagating relation (20) repeatedly, i.e.,

(23)

where

(24)

To solve the boundary value problem above, the extended stress boundary conditions should be considered. If the top surface of the plate is only subject to the electrical-mechanical loads and the bottom surface is free of traction, we have

(25)

From (23) and (25), u(0) can be obtained as

(26)

where Q₂₁ is the submatrix of the propagator matrix Q in (24).

Thus, the general solutions of the extended displacement and traction are solved completely from (20), (25), and (26).

5 Numerical examples

In the numerical analysis, we consider several sandwich plates made up of PQC1, PQC2, QC (Al-Ni-Co), and PE (BaTiO₃) materials, where the material properties of these materials are given in Tables 1 and 2. The effect of stacking sequence for different materials on the phonon, phason, and electric fields is analyzed, which is very useful in designing new laminate composites for engineering structures. Yang et al.^[32] comprehensively presented an exact electroelastic solution of a multilayered 2D decagonal QC plate subject to three different cases of surface patch loading, such as transverse shear force, normal force, and electric potential. In their analysis, the dimensions of the sandwich plates were taken as L_x=L_y=3×10^-2 m and H=3× 10^-3 m, and the responses for fixed horizontal coordinates (x, y)=(0.5L_x, 0.5L_y) were considered. In our numerical analysis, the dimensions of the plate are L_x=L_y=1 m and H=0.3 m. The three layers have equal thickness of 0.1 m. We take σ₀=1 N/m² and D₀= 1 C/m² with n=m=1, and consider the responses for fixed horizontal coordinates (x, y)=(0.75L_x, 0.25L_y).

It should be noted that in QC materials, only phonon and phason fields exist and there is no electric field. Thus, in PQC1/QC/PQC1, QC/PQC1/QC, PE/QC/PE, and QC/PE/QC plates, the dielectric constants λ_ij should be zero in the QC layer. To avoid the singularity of matrices in calculation, we use a small λ_ij value (10^-11 of the corresponding λ_ij) in the QC layer. Similarly, the elastic constants K_ij of phason field in PE materials are taken in this way.

Example 1 The plate subject to mechanical loads on the top surface

Figures 2 and 3 show the variation of displacement of phonon field for four different sandwich plates along the thickness direction of the plates under mechanical loads. It is clear that the stacking sequence of the plates greatly affects the displacement. If the upper and lower layers are QCs, the deformation of all sandwich plates can be reduced. Thus, QCs are commonly used as the thermal barrier coatings, wear resistant coatings, and reinforcements in composites to enhance the mechanical property of the other composite materials^[6-9]. However, PE can only improve the mechanical property of PQC plates.

Figure 4 shows the variation of displacement of the phason field for four different sandwich plates along the thickness direction of the plate under mechanical loads. We find from Fig. 4 that the displacement of the phason field in the PQC2 and QC layers is almost constant. The displacement of phason field is kinked across the interfaces, which is different from the displacement of phonon field. It should be pointed out that the phason displacement in the non-QC layer is set to be zero^[13-15] since there is no physical meaning for it in the non-QC layer. Thus, we also set the phason displacement in the PE layer to be zero. However, for the phason stress in the PE layer, it is always zero in the PE layer as expected.

Figure 5 shows the electric potential ϕ for four different sandwich plates along the thickness direction of the plate under mechanical loads. It is observed that the electric potential for the sandwich plates QC/PE/QC and PE/QC/PE reaches its maximum at the interfaces, while the electric potential of the other sandwich plates reaches its maximum on the surfaces of plates.

Figure 6 shows the variation of the shear stress σ_xz of phonon field for four different sandwich plates along the thickness direction of the plate under mechanical loads. The stacking sequence nearly has no effect on the shear stress except for the middle layer. For the normal stress σ_zz of phonon field (see Fig. 7), there is little difference for four different sandwich plates, which is also independent of the stacking sequence.

Figure 8 shows the variation of the normal stress H_zz of phason field for four different sandwich plates along the thickness direction of the plate under mechanical loads. The stress of the phason field shows a very different variation as compared with the stress of the phonon field (see Fig. 7). Meanwhile, the stacking sequence greatly affects the stress of the phason field. The following features can be further found from Fig. 8. (ⅰ) The stress of the phason field is sharply kinked across the interfaces. (ⅱ) The stress of the phason field in the PE layer is always zero as expected. (ⅲ) The stress of the phason field for PQC1/PQC2/PQC1 and PQC2/ PQC1/PQC2 reaches its maximum at the interfaces, which is fully different from the other sandwich plates.

Figure 9 shows the variation of the electric displacement D_z for four different sandwich plates along the thickness direction of the plate under mechanical loads. It can be observed that the variation of the electric displacement is almost similar for four different plates. The electric displacement is always zero in the QC layer as expected.

Example 2 The plate subject to electrical loads on the top surface

Figures 10-12 show the variation of displacements of phonon and phason fields for four different sandwich plates along the thickness direction of the plates under electrical loads, respectively. It can be observed from Figs. 10 and 11 that the stacking sequence of the plates has a great effect on the displacements of phonon field under electrical loads and the displacements are kinked at the interfaces, which are different from those for mechanical loads (see Figs. 2 and 3). The variation of displacement of phason field under electrical loads (see Fig. 12) is similar to that for mechanical loads (see Fig. 4).

Figure 13 shows the variation of electric potential for four different sandwich plates along the thickness direction of the plates under electrical loads. It can be found that the stacking sequence has a great effect on the electric potential in the upper layers under electrical loads, which is also different from that for mechanical loads shown in Fig. 5.

Figures 14 and 15 illustrate the variation of shear and normal stresses of phonon field for four different sandwich plates along the thickness direction of the plates under electrical loads, respectively. It is interesting to note that the variations of the stresses of phonon field under electrical loads show the opposite trends when the stacking sequence is changed.

Figure 16 shows the variation of stress of phason field for four different sandwich plates along the thickness direction of the plates under electrical loads. The stress of phason field in the PE layer is always zero, while it strongly depends on the stacking sequence in the other layers.

Figure 17 shows the variation of electric displacement for four different sandwich plates along the thickness direction of the plates under electrical loads. It can be seen that the stacking sequence has a large effect on the electric displacement in the upper two layers, while it has little influence on the electric displacement in the lower layer. The variation of electric displacement in all sandwich plates is similar except for PQC1/PE/PQC1 and PE/PQC1/PE plates.

As a special case, the present results can be reduced to the purely QC case^[15] when the PE coefficients are zero and they can be reduced to the purely PE case^[32] when the elastic constants of phason field are zero. Furthermore, the variations of u_z, σ_zz, σ_xz, D_z, and ϕ of sandwiches QC/PE/QC and PE/QC/PE under normal mechanical loads in this work are consistent with those in Fig. 13 by Yang et al.^[32], which justifies correction of the present method.

6 Conclusions

By using the pseudo-Stroh formulism and propagator matrix method, a static deformation of a multilayered 1D hexagonal PQC plate is studied and the exact closed-form solutions of the extended displacement and traction for multilayered PQC plates are derived under surface mechanical and electrical loads. Numerical examples for four sandwich plates made of PQC, PE, and QC are provided to show the effect of stacking sequence on the phonon, phason, and electric fields. Some useful conclusions can be drawn.

(ⅰ) The stacking sequence of plates greatly affects the static deformation of the sandwich composite plates.

(ⅱ) QCs are suitable for coatings or reinforcements in composites to enhance the mechanical property of the other composite materials as compared with PE materials.

(ⅲ) The deformable responses of the layered plates under mechanical loads are fully different from those under electrical loads.

(ⅳ) The electric potential for the sandwich plates QC/PE/QC and PE/QC/PE reaches its maximum at the interfaces under mechanical loads, while the electric potential of the other sandwich plates reaches its maximum on the surfaces.