1 Introduction

Quasicrystals (QCs) have many excellent properties such as high strength, corrosion resistance, low thermal conductivity, low adhesion and low level of porosity due to their quasiperiodic structures [1]. Different from ordinary crystal materials, there are two elementary excitations, i.e., phonon and phason, in the study of QCs based on the Landau phenomenological theory [2]. The former describes the classical motion of atoms in crystals, whereas the latter corresponds to the spatial rearrangement of atoms [3]. However, QCs are brittle at room temperature, which limits their application as structural components in engineering. Thus, QCs are commonly used as surface coatings and reinforcement phases of composites to enhance the mechanical properties of soft metal matrices and alloys [4,5,6].

A layered plate or laminate as an important composite structure has the advantages of small specific gravity, large specific strength and specific modulus. Duguet et al. [7] demonstrated the possibility of using complex metallic surface alloys as interface layers to enhance the adhesion between QCs and simple metal substrates. Chang et al. [8] formed multilayered coatings by ductile vanadium layers and brittle Al-Cu-Fe-based QC layers and found that the annealed multilayered structure showed a high flow stress due to QC strengthening. Ali et al. [9] investigated the interfacial reaction between Al matrix and QC reinforcing particles to enhance the strength of the Al/QC composites. By using a facile method of heat treatment, Wei and He [10] synthesized a multilayered sandwich-like three-dimensional (3D) structure containing large-scale faceted Al-Cu-Fe QC grains. The pseudo-Stroh formalism is an effective method to obtain the exact solutions for a simply supported and multilayered QC plates. For example, by using the pseudo-Stroh formalism, Yang et al. [11, 12] derived exact solutions for one-dimensional (1D) orthorhombic and two-dimensional (2D) decagonal QC multilayered plates. Waksmanski et al. [13] addressed an exact closed-form solution of free vibration of a simply-supported and multilayered 1D QC plate. With the rapid development of nanotechnology and advanced materials, the mechanical behaviors of multilayered QC nanoplates have been widely investigated. Pan and Waksmanski [14] presented an exact closed-form solution for the 3D static deformation and free vibrational response of a simply-supported and multilayered QC nanoplate with nonlocal effect. Based on the modified couple-stress theory, Li et al. [15] analyzed the static bending deformation of multilayered 1D hexagonal QC nanoplates under surface loadings. Furthermore, Guo et al. [16] investigated the free and forced vibration of layered 1D QC nanoplates with modified couple-stress effect. Recently, the nonlocal strain gradient theory was adopted to investigate the static bending deformation of a functionally graded multilayered nanoplate made of 1D hexagonal piezoelectric QC materials subjected to mechanical and electrical surface loadings [17]. For the other boundary conditions, it is difficult to obtain the general solutions of layered structures by using the pseudo-Stroh formalism. More recently, the state space and differential quadrature approach was adopted to solve the free vibration and bending of 1D QC layered composite beams with various boundary conditions [18].

Most of the previous work mentioned above on multilayered plates is based on the assumption that all the interfaces between different layers are perfect bonding. In other words, the extended displacement and stress are continuous across the interfaces of layered plates. However, due to manufacturing or interlaminar aging and other reasons, microcracks or cavities may occur at the interfaces, which could lead to interface slip or debonding. Cheng et al. [19, 20] presented a linear theory underlying elastostatics and kinetics of multilayered anisotropic plates and laminated shells with imperfect interfaces. This theory has the same advantages as conventional higher-order theories over the classical and the first-order theories. Based on the self-consistent scheme, Fan and Sze [21] proposed a micro-mechanical model to account for the imperfect interface in dielectric materials. Bui et al. [22] used the interface finite elements to investigate the effects of imperfect interlaminar interfaces on the overall mechanical behavior of composite laminates. This numerical approach is an efficient tool for treating laminated composite structures with complicated geometry and/or loading. The state space method and the extended pseud-Stroh formalism are employed to obtain the exact solutions of 3D anisotropic elasticity, which canaccurately predict the mechanical behaviors of thick plates. Chen et al. [23, 24] analyzed the bending and free vibration of a simply-supported, cross-ply laminated rectangular plate featuring interlaminar bonding imperfections. Wang and Pan [25] derived 3D exact solutions for simply-supported and multilayered rectangular plates with imperfect interfaces under static thermo-electro-mechanical loadings. Chen et al. [26] used the state space formulations to consider the bending problem of a multiferroic rectangular plate with magnetoelectric coupling and imperfect interfaces. Based on the extended pseud-Stroh formalism and the dual variable and position method, Kuo et al. [27] derived analytical solutions of static bending for a simply supported, anisotropic and multilayered magneto-electro-elastic plate with imperfect interfaces. Recently, Vattré and Pan [28] derived 3D exact solutions of temperature and thermoelastic stresses in multilayer anisotropic plates and free vibration of fully coupled thermoelastic multilayered composites with imperfect interfaces. Besides, López-Realpozo et al. [29] used the two scales asymptotic homogenization method to determine the effective coefficients of laminated piezoelectric composite with periodic structure under nonuniform electrical and mechanical imperfect contact conditions. The meshless method has the advantage in the field of computational mechanics over the above methods. For instance, a regularized method of moments and the modified multilevel algorithm proposed by Li et al. [30,31,32] show the high precision, the reduction of computational time and the rapid iteration in the calculation, etc.

However, no work on the static deformation and vibrational response of 1D hexagonal layered QC plates with imperfect interfaces has been reported so far. Therefore, this paper focuses on the static bending, free vibration and forced vibration of 1D hexagonal layered QC plates with imperfect interfaces. Exact solutions of the extended stress and displacement (displacements and stresses of phonon and phason fields) of multilayered QC plates are derived by the propagation matrix method and pseudo-Stroh formulism. The effect of imperfect interface parameter and stacking sequence on the static deformation and vibrational response of two sandwich plates made up of crystal and QC materials is analyzed in the numerical examples.

2 Problem description and basic equations

According to the quasiperiodic directions of QCs, there are three kinds of QCs which are classified as 1D, 2D and 3D QCs [2]. The 1D QCs considered in this work are in a 3D structure in which its atomic arrangement is quasi-periodic in one direction and periodic in the plane perpendicular to that direction. We consider an anisotropic and N-layer rectangular 1D hexagonal QC plates with imperfect interfaces, with horizontal dimensions L1 and L2 and total thickness H, and its four sides being simply supported. As shown in Fig. 1, the Cartesian coordinate system (x, y, z) corresponds to (x1, x2, x3) in the figure, and the positive x3-direction is along the thickness of the plate. The quasi-periodic direction is along the positive x3-axis direction. For any layer j (1 ≤ j ≤ N), layer j is bonded by its lower interface \(x_{3-}^{(j)}\) and upper interface \(x_{3+} ^{(j)}\) such that its thickness is \(h_{j} = x_{3 + }^{(j)} - x_{3 - }^{(j)}\) and H = h1 + h2 + … + hN, where + and−represent the upper and lower surfaces of each interface, respectively. It is assumed that the interface between two layers is imperfect or has some defects due to improper preparation or use. Based on the linear elastic theory of 1D QCs, the basic equations include [1]

$$\begin{gathered} \sigma_{11} = C_{11} \varepsilon_{11} + C_{12} \varepsilon_{22} + C_{13} \varepsilon_{33} + R_{1} \omega_{{{33}}}, \hfill \\ \sigma_{22} = C_{12} \varepsilon_{11} + C_{11} \varepsilon_{22} + C_{13} \varepsilon_{33} + R_{1} \omega_{{{33}}}, \hfill \\ \sigma_{33} = C_{13} \varepsilon_{11} + C_{13} \varepsilon_{22} + C_{33} \varepsilon_{33} + R_{2} \omega_{{{33}}}, \hfill \\ \sigma_{23} = \sigma_{32} = 2C_{44} \varepsilon_{23} + R_{3} \omega_{{3{2}}}, \hfill \\ \sigma_{31} = \sigma_{13} = 2C_{44} \varepsilon_{31} + R_{3} \omega_{{{31}}}, \hfill \\ \sigma_{12} = 2C_{66} \varepsilon_{12}, \hfill \\ H_{{{31}}} = 2R_{3} \varepsilon_{31} + K_{2} \omega_{{{31}}}, \hfill \\ H_{{{32}}} = 2R_{3} \varepsilon_{23} + K_{2} \omega_{{3{2}}}, \hfill \\ H_{{{33}}} = R_{1} \left( {\varepsilon_{11} + \varepsilon_{22} } \right) + R_{2} \varepsilon_{33} + K_{1} \omega_{{{33}}}, \hfill \\ \end{gathered}$$
(1)
$$\begin{gathered} \sigma_{ij,i} = \rho \frac{{\partial^{2} u_{i} }}{{\partial t^{2} }}, \hfill \\ H_{3j,j} = \rho \frac{{\partial^{2} w_{3} }}{{\partial t^{2} }}, \hfill \\ \end{gathered}$$
(2)
$$\varepsilon_{ij} = \frac{1}{2}(u_{i,j} + u_{j,i} ),\quad \omega_{3j} = w_{3,j},\quad i,\;j = 1,\;2,\;3$$
(3)
Fig. 1
figure 1

An N-layer QC plate with imperfect interfaces

Full size image

In Eqs. (1) –(3), a comma denotes differentiation with respect to xi (i = 1, 2, 3), repeated indices imply summation. σij, εij and ui are the components of the stress, strain and displacement of the phonon field, respectively; H3j, ω3j and w3 are the components of the stress, strain and displacement of the phason field, respectively; Cij and Ki are the elastic constants of phonon and phason fields, respectively; Ri are the phonon–phason coupling elastic constants.

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

$$\begin{gathered} x_{1} = 0,\;x_{1} = L_{1} :\;\;\,u_{2} = u_{3} = w_{3} = \sigma_{11} =0, \hfill \\ x_{2} = 0,\;x_{2} = L_{2} :\;\;\,u_{{1}} = u_{3} = w_{3} = \sigma_{22} = 0. \hfill \\ \end{gathered}$$
(4)

Due to improper preparation or use of layers, the layered composite structures cannot be perfectly bonded, which results in delamination or slip phenomenon at interfaces [33]. Two imperfect and realistic interface models, i.e., the generalized interface stress type and generalized linear spring type models, are commonly used to simulate the discontinuities of field quantities at interfaces [34,35,36,37]. In the former, the stress field is discontinuous while the displacement field is continuous across the interface. In the latter, the displacement field is discontinuous but the stress field is continuous at the interface. In the current work, the generalized linear spring type model is selected to study the static 3D deformation of layered QC composite plate. Thus, for the imperfect interfaces of QCs, we have the following boundary conditions at the interfaces [26]:

$$\begin{gathered} \sigma_{13}^{(j + 1)} = \sigma_{13}^{(j)} = \left[ {u_{1}^{(j + 1)} - u_{1}^{\left( j \right)} } \right]/R_{1}^{(j)}, \hfill \\ \sigma_{23}^{(j + 1)} = \sigma_{23}^{(j)} = \left[ {u_{2}^{(j + 1)} - u_{2}^{\left( j \right)} } \right]/R_{2}^{(j)}, \hfill \\ \sigma_{33}^{(j + 1)} = \sigma_{33}^{(j)} = \left[ {u_{3}^{(j + 1)} - u_{3}^{\left( j \right)} } \right]/R_{3}^{(j)}, \hfill \\ H_{33}^{(j + 1)} = H_{33}^{(j)} = \left[ {w_{3}^{(j + 1)} - w_{3}^{\left( j \right)} } \right]/R_{4}^{(j)}, \hfill \\ \end{gathered}$$
(5)

where \(R_{i}^{(j)}\)(i = 1 ~ 4) are the imperfect interface parameters. Equation (5) can be also expressed in matrix form as

$$\left[ {\begin{array}{*{20}c} {{\mathbf{u}}\left( {x_{3 - }^{(j + 1)} } \right)} \\ {{\mathbf{t}}\left( {x_{3 - }^{(j + 1)} } \right)} \\ \end{array} } \right]_{{}}^{{}} = {\mathbf{M}}_{j} \left[ {\begin{array}{*{20}c} {{\mathbf{u}}\left( {x_{3 + }^{(j)} } \right)} \\ {{\mathbf{t}}\left( {x_{3 + }^{(j)} } \right)} \\ \end{array} } \right]_{{}}^{{}},$$
(6)

where u = [u1, u2, u3, w3]T and t = [σ13, σ23, σ33, H33]T represent the generalized displacement and stress of layer j, respectively, where the superscript T denotes the transpose of a vector or matrix, and the imperfect interface propagator matrix Mj is given as

$${\mathbf{M}}_{j} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & {\overline{R}_{1}^{\left( j \right)} } & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & {\overline{R}_{2}^{\left( j \right)} } & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & {\overline{R}_{3}^{\left( j \right)} } & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & {\overline{R}_{4}^{\left( j \right)} } \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} } \right],$$
(7)

where

$$\overline{R}_{i}^{(j)} = \frac{{R_{i}^{(j)} H}}{{C_{44} }},\;\;(i = 1,\,\;2,\;\,3), \,\,\,\overline{R}_{4}^{(j)} = \frac{{R_{4}^{(j)} H}}{{K_{2} }}.$$
(8)

If \(R_{i}^{(j)}\) is zero, Mj become the identity matrices. Thus, all the interfaces of QC layered plates become perfect ones. In general, it is assumed that \(R_{3}^{(j)} = 0\) to avoid the impossible phenomenon of material interpenetration [33].

3 Analytical solutions

For a homogeneous 1D hexagonal QC plate with simply-supported on its four sides, the general solution of the extended displacement vector u under time-dependent harmonic motion can be expressed as [12, 38]

$${\mathbf{u}} \equiv \left[ {\begin{array}{*{20}c} {u_{1} } \\ {u_{2} } \\ {u_{3} } \\ {w_{3} } \\ \end{array} } \right] = {\mathrm{e}}^{{sx_{3} + {\mathrm{i}}\omega t}} \left[ {\begin{array}{*{20}c} {a_{1} \cos \left( {px_{1} } \right)\sin \left( {qx_{2} } \right)} \\ {a_{2} \sin \left( {px_{1} } \right)\cos \left( {qx_{2} } \right)} \\ {a_{3} \sin \left( {px_{1} } \right)\sin \left( {qx_{2} } \right)} \\ {a_{4} \sin \left( {px_{1} } \right)\sin \left( {qx_{2} } \right)} \\ \end{array} } \right],$$
(9)

where p = nπ/L1, q = mπ/L2, s the eigenvalue to be determined, ω being the angular frequency, imaginary i = \(\sqrt{-1}\), n and m are two positive integers, and a1 ~ a4 are unknown constants to be determined. By taking the summation over all values of n and m we can derive the general solution in terms of 2D Fourier series expansions.

Then, we can assume the extended stress vector as follows:

$${\mathbf{t}} \equiv \left[ {\begin{array}{*{20}c} {\sigma_{13} } \\ {\sigma_{23} } \\ {\sigma_{33} } \\ {H_{33} } \\ \end{array} } \right] = {\mathrm{e}}^{{sx_{3} + {\mathrm{i}}\omega t}} \left[ {\begin{array}{*{20}c} {b_{1} \cos \left( {px_{1} } \right)\sin \left( {qx_{2} } \right)} \\ {b_{2} \sin \left( {px_{1} } \right)\cos \left( {qx_{2} } \right)} \\ {b_{3} \sin \left( {px_{1} } \right)\sin \left( {qx_{2} } \right)} \\ {b_{4} \sin \left( {px_{1} } \right)\sin \left( {qx_{2} } \right)} \\ \end{array} } \right].$$
(10)

According to the constitution equations, the relationship between vector b = [b1, b2, b3, b4]T and vector a = [a1, a2, a3, a4]T yields

$${\mathbf{b}} = \left( { - {\mathbf{R}}^{T} + s{\mathbf{T}}} \right){\mathbf{a}} = - \frac{1}{s}\left( {{\mathbf{Q}} + s{\mathbf{R}}} \right){\mathbf{a}},$$
(11)

where

$${\mathbf{R}} = \left[ {\begin{array}{*{20}c} 0 & 0 & {C_{13} p} & {R_{1} p} \\ 0 & 0 & {C_{13} q} & {R_{1} q} \\ { - C_{44} p} & { - C_{44} q} & 0 & 0 \\ { - R_{3} p} & { - R_{3} q} & 0 & 0 \\ \end{array} } \right], \quad {\mathbf{T}} = \left[ {\begin{array}{*{20}c} {C_{44} } & 0 & 0 & 0 \\ 0 & {C_{44} } & 0 & 0 \\ 0 & 0 & {C_{33} } & {R_{2} } \\ 0 & 0 & {R_{2} } & {K_{1} } \\ \end{array} } \right],$$
(12)
$${\mathbf{Q}} = \left[ {\begin{array}{*{20}c} { - C_{11} p^{2} - C_{66} q^{2} + \rho \omega^{2} } & { - pq(C_{12} + C_{66} )} & 0 & 0 \\ { - pq(C_{12} + C_{66} )} & { - C_{66} p^{2} - C_{11} q^{2} + \rho \omega^{2} } & 0 & 0 \\ 0 & 0 & { - C_{44} (q^{2} + p^{2} ) + \rho \omega^{2} } & { - R_{3} (q^{2} + p^{2} )} \\ 0 & 0 & { - R_{3} (p^{2} + q^{2}) } & { - K_{2} (p^{2} + q^{2} ) + \rho \omega^{2} } \\ \end{array} } \right].$$
(13)

Substituting Eqs. (9) and (10) into Eq. (2), and using the relation between vectors a and b, the following governing equation is obtained:

$$\left[ {{\mathbf{Q}} + s\left( {{\mathbf{R}}{ - }{\mathbf{R}}^{T} } \right) + s^{2} {\mathbf{T}}} \right]{\mathbf{a}} = {\mathbf{0}}.$$
(14)

It is found that Eq. (14), called the pseudo-Stroh formalism [38], includes eight eigenvalues s and four pairs of opposite numbers.

Using Eq. (11), Eq. (14) is then recast into an 8 × 8 linear eigensystem

$${\mathbf{N}}\left[ {\begin{array}{*{20}c} {\mathbf{a}} \\ {\mathbf{b}} \\ \end{array} } \right] = s \left[ {\begin{array}{*{20}c} {\mathbf{a}} \\ {\mathbf{b}} \\ \end{array} } \right],$$
(15)

where

$${\mathbf{N = }}\left[ {\begin{array}{*{20}c} {{\mathbf{T}}^{ - 1} {\mathbf{R}}^{T} } & {{\mathbf{T}}^{ - 1} } \\ { - {\mathbf{Q}} - {\mathbf{RT}}^{ - 1} {\mathbf{R}}^{T} } & {{\mathbf{RT}}^{ - 1} } \\ \end{array} } \right].$$
(16)

We distinguish the corresponding eight eigenvectors by attaching a subscript to a and b. Then the general solution for the extended displacement and stress vectors is derived as

$$\left[ {\begin{array}{*{20}c} {\mathbf{u}} \\ {\mathbf{t}} \\ \end{array} } \right]{ = }\left[ {\begin{array}{*{20}c} {{\mathbf{A}}_{{\mathbf{1}}} } & {{\mathbf{A}}_{{\mathbf{2}}} } \\ {{\mathbf{B}}_{{\mathbf{1}}} } & {{\mathbf{B}}_{{\mathbf{2}}} } \\ \end{array} } \right]\left\langle {{\mathrm{e}}^{{s^{*}x_{3} }} } \right\rangle \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{{\mathbf{1}}} } \\ {{\mathbf{K}}_{{\mathbf{2}}} } \\ \end{array} } \right],$$
(17)

where

$$\begin{gathered} {\mathbf{A}}_{1} = \left[ {{\varvec{a}}_{1},\;{\varvec{a}}_{2},\;{\varvec{a}}_{3},\;{\varvec{a}}_{4} } \right],\;{\mathbf{A}}_{2} = \left[ {{\varvec{a}}_{5},\;{\varvec{a}}_{6},\;{\varvec{a}}_{7},\;{\varvec{a}}_{8} } \right], \hfill \\ {\mathbf{B}}_{1} = \left[ {{\varvec{b}}_{1},\;{\varvec{b}}_{2},\;{\varvec{b}}_{3},\;{\varvec{b}}_{4} } \right],\;{\mathbf{B}}_{2} = \left[ {{\varvec{b}}_{5},\;{\varvec{b}}_{6},\;{\varvec{b}}_{7},\;{\varvec{b}}_{8} } \right], \hfill \\ \left\langle {{\mathrm{e}}^{{s^{*}x_{3} }} } \right\rangle = {\mathrm{diag}}\left[ {{\mathrm{e}}^{{s_{1} x_{3} }},\;{\mathrm{e}}^{{s_{2} x_{3} }},\;...,\;{\mathrm{e}}^{{s_{8} x_{3} }} } \right]. \hfill \\ \end{gathered}$$
(18)

Both K1 and K2 are 4 × 1 vectors to be determined by the external loads on the surfaces of plates, diag[] is a diagonal matrix.

According to the boundary condition at the bottom surface of plate, we can obtain the constant vectors K1 and K2 and substituting them into Eq. (17), the general solutions of the extended displacement and stress become

$$\left[ {\begin{array}{*{20}c} {{\mathbf{u}}(x_{3} )} \\ {{\mathbf{t}}(x_{3} )} \\ \end{array} } \right]_{{}} = {\mathbf{P}}_{j} (x_{3} )\left[ {\begin{array}{*{20}c} {{\mathbf{u}}\left( {x_{{_{3 - } }}^{(j)} } \right)} \\ {{\mathbf{t}}\left( {x_{{_{3 - } }}^{(j)} } \right)} \\ \end{array} } \right],$$
(19)

where Pj(x3) is the propagator matrix of layer j, and \(x_{3 - }^{\left( j \right)} \le x_{3} \le x_{3 + }^{\left( j \right)}\), i.e.,

$${\mathbf{P}}_{j} (x_{3} ) = \left[ {\begin{array}{*{20}c} {{\mathbf{A}}_{{\mathbf{1}}} } & {{\mathbf{A}}_{{\mathbf{2}}} } \\ {{\mathbf{B}}_{{\mathbf{1}}} } & {{\mathbf{B}}_{{\mathbf{2}}} } \\ \end{array} } \right]\left\langle {{\mathrm{e}}^{{s^{ * } (x_{3} - x_{3 - }^{(j)} )}} } \right\rangle \left[ {\begin{array}{*{20}c} {{\mathbf{A}}_{{\mathbf{1}}} } & {{\mathbf{A}}_{{\mathbf{2}}} } \\ {{\mathbf{B}}_{{\mathbf{1}}} } & {{\mathbf{B}}_{{\mathbf{2}}} } \\ \end{array} } \right]^{ - 1}.$$
(20)

Using the imperfect boundary conditions (5) at partial interfaces, the propagator relation from x3 = 0 to x3 = H of QC layered plate is derived as

$$\left[ \begin{gathered} {\mathbf{u}}(H) \hfill \\ {\mathbf{t}}(H) \hfill \\ \end{gathered} \right] = {\mathbf{Y}}\left[ \begin{gathered} {\mathbf{u}}(0) \hfill \\ {\mathbf{t}}(0) \hfill \\ \end{gathered} \right],$$
(21)

where the propagator matrix Y is defined as

$${\mathbf{Y}} = {\mathbf{P}}_{N} (h_{N} ){\mathbf{M}}_{N - 1} {\mathbf{P}}_{N - 1} (h_{N - 1} ){\mathbf{M}}_{N - 2} \cdots {\mathbf{P}}_{2} (h_{2} ){\mathbf{M}}_{1} {\mathbf{P}}_{1} (h_{1} ),$$
(22)

where Mj (j = 1, 2, …, N–1) are the propagator matrices of imperfect interfaces and Pj (j = 1, 2, …, N) are seen in Eq. (20). When \(R_{i}^{(j)} = 0\), all Mj become identity matrices in Eq. (22). Thus, Eq. (21) is completely consistent with the results of perfect case [12].

3.1 Static deformation

In this section, the general solution of static deformation for a simply-supported multilayered 1D QC plate with imperfect interfaces is presented. The general solution of the extended displacement and stress of layered plate can be solved by omitting the time-dependent terms, i.e., ω = 0.

If the top surface of plate is only subjected to the phonon mechanical loads and the bottom surface is free of traction, the boundary conditions are

$${\mathbf{t}}(H) = [0,\;0,\;\sigma_{0} \sin px_{1} \sin qx_{2},\;0]^{T},\quad {\mathbf{t}}(0) = [0,\;0,\;0,\;0]^{T}.$$
(23)

Thus, Eq. (21) becomes

$$\left[ {\begin{array}{*{20}c} {{\mathbf{u}}(H)} \\ {{\mathbf{t}}(H)} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{\mathbf{C}}_{11} } & {{\mathbf{C}}_{12} } \\ {{\mathbf{C}}_{21} } & {{\mathbf{C}}_{22} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\mathbf{u}}(0)} \\ {\mathbf{0}} \\ \end{array} } \right],$$
(24)

where Cij is the submatrices of Y in Eq. (22). Substituting Eq. (23) in Eq. (24), the unknown displacements at the bottom and top surfaces are obtained as

$${\mathbf{u}}(0) = {\mathbf{C}}_{21}^{ - 1} {\mathbf{t}}(H),\quad {\mathbf{u}}(H) = {\mathbf{C}}_{11} {\mathbf{C}}_{21}^{ - 1} {\mathbf{t}}(H).$$
(25)

Finally, the extended displacements and extended stresses at any position of 1D hexagonal QC layered plates with imperfect interfaces can be obtained as

$$\left[ {\begin{array}{*{20}c} {{\mathbf{u}}(x_{3}^{(j)} )} \\ {{\mathbf{t}}(x_{3}^{(j)} )} \\ \end{array} } \right] = {\mathbf{P}}_{j} (x_{3}^{(j)} - x_{3 - }^{(j)} ){\mathbf{M}}_{j - 1} {\mathbf{P}}_{j - 1} (h_{j - 1} ){\mathbf{M}}_{j - 2} \cdots {\mathbf{P}}_{2} (h_{2} ){\mathbf{M}}_{1} {\mathbf{P}}_{1} (h_{1} )\left[ {\begin{array}{*{20}c} {{\mathbf{u}}(0)} \\ {\mathbf{0}} \\ \end{array} } \right].$$
(26)

3.2 Free vibration

For free vibration analysis, both the top and bottom surfaces of the plate are traction-free, then Eq. (21) is simplified as

$$\left[ {\begin{array}{*{20}c} {{\mathbf{u}}(H)} \\ {\mathbf{0}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{\mathbf{C}}_{11} } & {{\mathbf{C}}_{12} } \\ {{\mathbf{C}}_{21} } & {{\mathbf{C}}_{22} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\mathbf{u}}(0)} \\ {\mathbf{0}} \\ \end{array} } \right].$$
(27)

There are non-zero solutions in Eq. (27), and the submatrix C21 must satisfy

$$\det \left[ {{\mathbf{C}}_{21} } \right] = 0,$$
(28)

where det[] denotes the determinant of a matrix.

3.3 Forced vibration

In this section, the response of 1D hexagonal QC layered plate is considered subject to a harmonic excitation on the top surface, i.e.,

$${\mathbf{t}}(H) = [0,\;0,\;\sigma_{0} {\mathrm{e}}^{{{\mathrm{i}}\omega t}} \sin px_{1} \sin qx_{2},\;0]^{T},\quad {\mathbf{t}}(0) = [0,\;0,\;0,\;0]^{T}.$$
(29)

Substituting Eq. (29) into Eq. (21), the unknown displacement vectors at the top and bottom surfaces can be derived as

$${\mathbf{u}}(0) = {\mathbf{C}}_{21}^{ - 1} {\mathbf{t}}(H),\quad {\mathbf{u}}(H) = {\mathbf{C}}_{11} {\mathbf{C}}_{21}^{ - 1} {\mathbf{t}}(H).$$
(30)

4 Numerical examples

In the numerical analysis, we consider two sandwich C/QC/C and QC/C/QC plates made up of Al–Ni–Co (QC) and BaTiO3 (C) materials, where the material properties of these two materials are listed in Table 1. Each adjacent two layers of plates are imperfect bonding. Referring to the previous work [12, 13, 28, 38], the dimension of the plate is taken as L1 × L2 × H = 1 × 1 × 0.3 m and three layers have equal thickness of 0.1 m. In general, an interface can be modeled as a thin layer with certain material properties in which displacements and stresses are discontinuous across the interface layer [39]. As an extreme case, the spring model without any thickness of interface layer allows a jump of displacements but continuous stresses across the interface. Liu et al. [40, 41] introduced several imperfect interface models including the direct thin-layer, spring and density models to investigate the effect of these models on the response of the layered half-space under time-harmonic surface loadings and found that under either vertical or horizontal load, the thin-layer model with a very small thickness predicts nearly the same result as that by the spring model without any thickness. Thus, the thickness of interface layers is neglected in our analysis. Furthermore, we assume \(R_{3}^{(j)} = 0\) to avoid material embeddedness that is physically impossible, and also assume \(R_{1}^{(j)} = R_{2}^{(j)} = R_{4}^{(j)} = R_{{}}^{(j)}\), where \(R_{{}}^{(j)}\) denotes the imperfect interface parameter at the j-th interface.

Table 1 Material properties of QC [42] and C [43] materials (Cij, Ki and Ri in 109 N/m2 and ρ in 103 kg/m3)
Full size table

4.1 Effect of imperfect interface on static deformation

We assume that the imperfect degree at two imperfect interfaces is the same, i.e., R(1) = R(2) = R(Im). It is assumed that the mechanical loads of phonon field act on the top surface of two sandwich plates, i.e., σ0 = 1 N/m2 (m = n = 1). The responses at (x1, x2) = (0.75L1, 0.25L2) are discussed in the numerical analysis.

Figure 2 shows the effect of imperfect interface parameter R(Im) on the phonon displacements, phason displacement of two sandwich plates. It can be observed that the phonon and phason displacements on the top and bottom surfaces of two sandwich plates always increase with increasing R(Im). The phonon displacement u1 jumps across the imperfect interfaces. When R(Im) = 0, it corresponds to the perfect interface case, which agrees well with the previous results [12]. The stacking sequence has a great influence on the phason displacement.

Fig. 2
figure 2

Variation of phonon and phason displacements of two sandwich plates along the thickness direction

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Figure 3 plots the effect of R(Im) on the phonon stresses and phason stress of two sandwich plates. It can be found that the imperfect interface parameter has a large influence on the phonon stresses of the middle layer. It is interesting to further note that increasing R(Im) can reduce the shear stress σ13 at the interfaces, which can provide a way for defect optimization and help us introduce the imperfect interface to reduce stress. The phason stress H33 in QC plate is greatly affected by the imperfect interface parameter.

Fig. 3
figure 3

Variation of phonon and phason stresses of two sandwich plates along the thickness direction

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Table 2 lists the phonon displacements for two sandwich plates with different R(Im) between the layers for the point (x1, x2) = (0.75L1, 0.25L2) at the bottom of plate. It can be observed that the magnitude of phonon displacements of two sandwich plates always increases with increasing R(Im). The phonon displacement of QC/C/QC plate is smaller than that of C/QC/C plate. When the imperfect interface parameter of upper or lower interface becomes small, the phonon displacement u3 of two sandwich plates always decreases, however, the phonon displacement u1 of two sandwich plates displays a different trend.

Table 2 Phonon displacements of two sandwich plates with imperfect interfaces (ui in 10–12 m)
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4.2 Effect of imperfect interface on free vibration

In this section, the effects of different R(Im) on the natural frequencies of two sandwich plates are analyzed. Since the lowest frequencies are of most importance in free vibration of any system, the fixed values m = n = 1 are selected and the first four natural frequencies of two sandwich plates are listed in Table 3. The natural frequencies are normalized as

$$\Omega { = }{{\omega L_{{1}} } \mathord{\left/ {\vphantom {{\omega L_{{1}} } {\sqrt {{{C_{\max } } \mathord{\left/ {\vphantom {{C_{\max } } {\rho_{\max } }}} \right. \kern-\nulldelimiterspace} {\rho_{\max } }}} }}} \right. \kern-\nulldelimiterspace} {\sqrt {{{C_{\max } } \mathord{\left/ {\vphantom {{C_{\max } } {\rho_{\max } }}} \right. \kern-\nulldelimiterspace} {\rho_{\max } }}} }},$$
(31)
Table 3 Normalized natural frequencies Ω of two sandwich plates
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where Cmax and ρmax represent the maximum value of phonon elastic constant and density in the entire plate, respectively. From Table 3, it can be obviously seen that the natural frequencies of two sandwich plates always increase with increasing mode while they decrease with increasing R(Im) of two imperfect interfaces. It is interesting to note that the decrease R(Im) of upper or lower interface results in an increase of the natural frequency of two sandwich plates. The natural frequency of sandwich C/QC/C plate is greatly dependent on the lower interface while the natural frequency of sandwich QC/C/QC plate greatly depends on the upper interface. Therefore, the vibrational characteristics of two sandwich plates can be optimized by imposing imperfect interfaces and controlling the stacking sequence artificially.

4.3 Effect of imperfect interface on forced vibration

In this section, to investigate the forced vibration response of two sandwich plates, the top surface of plate is subjected to a harmonic excitation. We assumed that the amplitude σ0 = 1 N/m2 and the normalized input frequencies Ω = 0.1, 0.2, …, 0.8, are smaller than the first natural frequency of vibration to avoid resonance. The forced vibration responses of two C/QC/C and QC/C/QC sandwich plates are considered below.

Figure 4 illustrates the variation of phonon displacements of two sandwich C/QC/C and QC/C/QC plates with harmonic excitation and imperfect interface parameter R(Im). The results show that the phonon displacements of two sandwich plates always increase with increasing R(Im) and forcing frequency. It can be further seen that the C/QC/C sandwich plate is more sensitive to imperfect interfaces in forced vibration, which indicates the stiffness of sandwich QC/C/QC plate is higher than that of sandwich C/QC/C plate. This also confirms that QC materials are suitable for the surface coatings of new composites in engineering practice.

Fig. 4
figure 4

Effect of input frequency on phonon displacements u1, u3 of two sandwich C/QC/C and QC/C/QC plates with different R(Im)

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4.4 Application of the present 3D plate model to QC coated Al-based composites

Due to their brittle at room temperature, QCs are difficult to be used directly as structural parts, but they can be commonly used as hard and abrasion-resistant coatings on softer metals or surface modification of soft metallic materials [44]. Duguet et al. [7] demonstrated the possibility of using complex metallic surface alloys as interface layers to enhance the adhesion between QCs and simple metal substrates. By using the thermal spraying technology, QC coatings and QC-reinforced composites can be produced [45]. From the experimental point of view, the interfacial reaction between Al matrix and QC reinforcing particles was considered to enhance the strength of the Al/QC composites [9]. By ductile vanadium layers and brittle Al-Cu-Fe-based QC layers, the multilayered coatings were formed and the annealed multilayered structure showed a high flow stress due to QC strengthening [8]. To reveal the mechanical behavior of QC coated Al-based composites in engineering practice, we consider the effect of thickness of QC coatings and imperfect interface parameter on bending deformation and vibrational response of QC coated Al-based composites (i.e., QC/Al/QC plate) based on the present 3D layered plate model.

As shown in Fig. 5, h1 and h2 denote the thickness of QC coatings and core material Al, respectively, and H (= 2h1 + h2) denotes the total thickness of QC/Al/QC plate. The material properties of QC are listed in Table 1 and the material properties of metallic Al [46] are C11 = C22 = C33 = 108.2 × 109 N/m2, C12 = C13 = C23 = 61.3 × 109 N/m2, C44 = C55 = C66 = 28.5 × 109 N/m2 and ρ = 2699 kg/m3.

Fig. 5
figure 5

A QC-coated Al-based composite plate with imperfect interfaces

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Figure 6 shows the variation of phonon and phason displacements with thickness ratio h1/h2 of QC coatings and core material Al in QC/Al/QC plate for different imperfect interface parameters (R(Im) = 0, 0.3, 0.6, 0.9) under surface loading σ0 = 1 N/m2. It is interesting to note that in general, the phonon displacements decrease with increasing h1/h2 for both perfect and imperfect interfaces, while the magnitude of phason displacement increases with increasing h1/h2 for both perfect and imperfect interfaces. It indicates that an increase in the thickness of QC coatings can greatly enhance the stiffness of sandwich plate as compared to pure Al plate. However, the imperfect interfaces can reduce the stiffness of the plate.

Fig. 6
figure 6

Variation of phonon and phason displacements with thickness ratio of QC coatings and core material Al in QC/Al/QC plate for different R(Im)

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Figure 7 shows the variation of the first two order normalized natural frequency with h1/h2 of QC/Al/QC plate for different R(Im) under free vibration. It can be seen that with increasing thickness of QC coatings, the first order normalized natural frequency always increases for the perfect interfaces, but the first order normalized natural frequency decreases first and then increases for the case of imperfect interfaces. However, the second order normalized natural frequency always increases with increasing thickness of QC coatings for both perfect and imperfect interfaces.

Fig. 7
figure 7

Variation of the first two order normalized natural frequency Ω with thickness ratio of QC coatings and core material Al in QC/Al/QC plate for different R(Im)

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Figure 8 plots the effect of both h1/h2 and R(Im) on phonon displacements u1, u3 and phason displacement w3 of QC/Al/QC plate under the forcing frequencies Ω = 0.4 (see Fig. 8a–c) and Ω = 0.6 (see Fig. 8d–f). It can be observed that the phonon displacements u1 and u3 reach their minimum values for QC plate with perfect bonding. Besides, the phonon displacement u1 reaches its peak for pure Al plate, while the phonon displacement u3 reaches its peak at about h1/h2 = 1 for QC/Al/QC plate with imperfect interfaces. Furthermore, the phason displacement w3 displays a different trend from the phonon displacements due to the special structures of QCs.

Fig. 8
figure 8

Effect of both h1/h2 and R(Im) on phonon displacements u1, u3 and phason displacement w3 of QC/Al/QC plate under the forcing frequencies Ω = 0.4 and Ω = 0.6

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5 Conclusions and outlook

The static deformation and vibrational response of 1D hexagonal QC layered plate with imperfect interfaces are investigated in this work and exact solutions of the extended displacement and stress of 1D hexagonal QC layered plate with imperfect interfaces are derived by using the propagation matrix method and pseudo-Stroh formalism. Numerical examples are provided to show the effects of imperfect interface parameter and stacking sequence on the static deformation and vibrational response of two sandwich plates. Some useful conclusions can be drawn:

  1. (i)

    An increase of the imperfect interface parameter can reduce the shear phonon stress at the interfaces in the bending of QC layered plates with imperfect interfaces. When the imperfect interface parameter of the upper or lower interface becomes small, the phonon displacement u3 of two sandwich plates always decreases, however, the phonon displacement u1 of two sandwich plates displays a different trend.

  2. (ii)

    The natural frequencies of two sandwich plates always decrease with increasing imperfect interface parameter. The decrease of imperfect interface parameter of upper or lower interface results in an increase in the natural frequency of two sandwich plates.

  3. (iii)

    The displacements of phonon and phason fields of two sandwich plates always increase with increasing both imperfect interface parameter and input frequency. In forced vibration, the sandwich C/QC/C plate is more sensitive to imperfect interfaces than the sandwich QC/C/QC plate.

  4. (iv)

    An increase in the thickness of QC coatings can greatly enhance the stiffness of the sandwich plate as compared to the pure Al plate. Thus, QCs are suitable for surface coatings on soft metallic materials in engineering practice.

However, the present study mainly focuses on simply supported boundary conditions at four sides of plate, the analytical method, the phonon-phason couplings and the classical elasticity theory. In the next step, arbitrary boundary conditions, experimental research, various numerical methods such as the molecular dynamics simulation, the finite element method, the differential quadrature approach, the meshless method, etc., the multi-field couplings among phonon, phason, thermal and electric fields, and the size-dependent effect at small scale will be key issues in the further study.