Free vibration analysis of functionally graded magneto-electro-elastic plates with in-plane material heterogeneity

Abstract

A three dimensional elasticity analysis for the transverse free vibration characteristics of functionally graded magneto-electro-elastic plates based on the scaled boundary finite element method (SBFEM) incorporated with the precise integration algorithm (PIA) is presented. The material properties of magneto-electro-elastic plates are changing along the in-plane direction with arbitrary mathematical functions. In the proposed methodology, the strategies of only discretizing the in-plane surface and utilizing the two-dimensional spectral elements to construct diagonal coefficient matrices are adopted, which contributes to decreasing the calculation effort. The derivation process begins with the three dimensional governing equations of magneto-electro-elastic materials. Neither the plate mechanical kinematics nor invoking assumptions on the spatial distributions of electric and magnetic quantities are adopted. Built upon the introduced scaled boundary coordinates, the principle of virtual work and the technique of dual vectors, a first order ordinary differential SBFEM matrix equation for the in-plane functionally graded magneto-electro-elastic plates is obtained. Its general solution is analytically denoted as the matrix exponent. To improve the computation accuracy of the matrix exponent, the PIA is utilized to form the stiffness matrix. By virtue of the kinetic energy technique, it is convenient to construct the mass matrix of the in-plane functionally graded magneto-electro-elastic plates based on the SBFEM for the first time. Finally, comparisons of flexural frequency parameters with those from exact solutions and other numerical methods are provided. The accuracy, effectiveness, and versatility of the employed technique are validated. Moreover, additional numerical exercises are conducted to exhibit the influences of boundary conditions, material gradation functions, and aspect ratios on the free vibration behaviors of in-plane functionally graded magneto-electro-elastic rectangular, circular, and perforated plates.

Keywords

In-plane functionally graded magneto-electro-elastic materials plate structures free vibration characteristics scaled boundary finite element method the precise integration algorithm

1. Introduction

In recent few decades, functionally graded and magneto-electro-elastic materials are incorporated with each other to form a new-type functionally graded magneto-electro-elastic composite material. In practical applications, the advanced functionally graded magneto-electro-elastic materials are usually fabricated as the plate structures to create the so-called functionally graded magneto-electro-elastic plates. This novel kind of plates simultaneously possess the characteristics of two constitute materials, which means that material properties of the produced plates are characterized by continuous and gradual distribution in the spatial domain and present the coupling effect of mechanical, electric, and magnetic fields. The functionally graded magneto-electro-elastic plates can be manufactured by the chemical vapor deposition, powder metallurgy, self-propagating high temperature synthesis, electrochemical method, and so on. Owing to the unique features, functionally graded magneto-electro-elastic plates are applied in plenty of fields such as smart sensors, actuators, vibration and stability control, health monitoring, medical devices, ultrasonic imaging transducers, and other utilizations.

The functionally graded materials with the gradation functions in terms of the transverse coordinate attract most of the researchers’ and engineers’ attention. However, it is noteworthy to point out that the in-plane gradation of material parameters is also able to produce better performance. To excellently design and optimize the relevant intelligent devices and equipment, it is necessary to carry out a good understanding on the free vibration responses of the functionally graded magneto-electro-elastic plates with in-plane material heterogeneity.

Due to the distinctive characteristics of multi-physical field coupling and smooth distributions of material coefficients, many explorations corresponding to the flexural vibration behaviors of functionally graded magneto-electro-elastic plates are conducted by lots of scientists. The exact solutions served as the benchmarks are important, which can be applied to verify the numerical approaches. Ootao et al. (2011) developed the exact analysis for variations of temperature change, magnetic potential, thermal stress, and electric potential in functionally graded smart magneto-electro-thermo-elastic material strips subjected to thermal loads. However, the exact results are only available for specific geometrical shapes and boundary conditions. To solve more complicated problems and simplify the three dimensional plates into two dimensional structures, scholars developed a set of plate theories, such as the first order shear deformation theory (Arshid et al., 2019), the high order shear deformation theory (Ebrahimi and Dabbagh, 2017; Ebrahimi et al., 2017; Shooshtari and Mantashloo, 2018; Zhang and Li, 2013), the layer-wise shear deformation plate theory (Kattimani and Ray, 2015), and so on. Chen et al. (2005) divided the simply supported functionally graded magneto-electro-elastic plates into several thin layers with equal thickness and introduced an approximate laminate model to reveal the free vibration behaviors. Ootao and Ishihara (2011, 2012) recommended the laminated composite model to study the transient structural responses of functionally graded magneto-electro-thermo-elastic hollow sphere and cylinder subjected to uniform temperature fields.

In two dimensional plate theories, the changing rules of elastic displacements are assumed to follow some functions according to the transverse coordinates. To model the plate structures more accurately, the finite element method (FEM) as a most widely used technology is introduced. Bhangale and Ganesan (2005) made utilization of the FEM to simulate the free vibration responses of functionally graded magneto-electro-elastic material finitely long cylindrical thin shells. With the aid of the FEM, Zheng et al. (2012) provided closed solutions to the free vibration behaviors of simply supported functionally graded magneto-electro-elastic plates and shells. Milazzo (2014) proposed a finite element formulation based on the refined equivalent single layer plate theory to solve the natural frequencies and modal variables of functionally graded magneto-electro-elastic rectangular and square plates. Depended on the FEM, Xie et al. (2016) conducted the sensitivity analysis of the dynamic behaviors for functionally graded magneto-electro-elastic square plates with clamped boundary condition. Cai et al. (2018) and Zhou et al. (2019) utilized the inhomogeneous cell-based smoothed FEM to examine the free vibration and non-linear transient behaviors of functionally graded smart magneto-electro-elastic beams under various boundary constraints. Meanwhile, a series of numerical methodologies including the asymptotic approach associated with the multiple scales method (Tsai and Wu, 2008; Wu and Tsai, 2010), Legendre orthogonal polynomial series expansion method (Wu et al., 2008; Yu and Ma, 2010), the propagator matrix incorporated with the successive approximation approach (Wu and Lu, 2009), the power series approach (Cao et al., 2012), Chebyshev spectral element approach (Xiao et al., 2016), and the polynomial expansions (Zhang et al., 2018) are also employed to investigate the dynamic responses of functionally graded magneto-electro-elastic plates.

Through the literature analysis, it can be found that a large number of investigations are paid attention to predicting the dynamic deformations of functionally graded magneto-electro-elastic plates with material parameters graded along the thickness direction; whilst few explorations are conducted on the in-plane functionally graded materials. Similar with the review mentioned above, two dimensional plate theorems are presented to examine the free vibration behaviors of functionally graded plates with in-plane stiffness. Built upon the classical plate theory, Liu et al. (2010) calculated the natural frequencies of the functionally graded rectangular plates with material coefficients varying along the y direction under simply supported, clamped, and free boundary conditions. Uymaz et al. (2012) took advantage of the Ritz method combined with the high order shear deformation plate theory to seek the solutions of modal frequencies and shapes for the in-plane functionally graded square and rectangular plates. Based on the higher order shear deformation plate theory, Amirpour et al. (2018) provided the analytical solutions to the free vibration behaviors of the 3D printed polymetric rectangular plates with in-plane functionally graded materials and compared the results with those obtained by the commercial finite element software and experimental tests.

Built upon the non-uniform rational B-spline, the isogeometric analysis as a useful tool to explore the vibration characteristics of in-plane functionally graded plates was conducted by many scientists. Yin et al. (2017) incorporated the isogeometric analysis with the Kirchhoff-Love theory to present the natural frequencies, mode shapes, and critical buckling loads of in-plane functionally graded square and circular thin plates. Lieu et al. (2018) performed the isogeometric analysis to examine the free vibration responses and deformable behaviors of functionally graded square and circular plates with varied thickness, in which the stiffness of the plate is changed with the volume fraction distribution obeying two kinds of power law functions. With the aid of the isogeometric analysis and the refined plate theory, Xue et al. (2018) investigated the free vibration characteristics of functionally graded square, skew, elliptical plates with material properties distributed as power law and exponential functions of in-plane coordinates. Lieu et al. (2019) carried out the isogeometric analysis incorporated with the generalized shear deformation theory to study the free vibration and buckling responses of functionally graded square plates with in-plane material inhomogeneity along x and y coordinates. Son and Huu-Tai (2019) conducted the isogeometric analysis to investigate the free vibration behaviors of multi-directional functionally graded rectangular, square, perforated, circular, and annular plates under simply supported and clamped boundary constraints. By means of the isogeometric approach, Xue et al. (2019) acquired the natural frequencies and mode shapes of porous functionally graded plates with the shapes of square, circular, rectangular with a circular cutout, and variations of material parameters in terms of the trigonometric functions.

At the same time, other numerical techniques such as the differential quadrature method (Sharma et al., 2012), the meshfree Hermite radial basis collocation technique (Chu et al., 2014), the Chebyshev collocation approach in conjunction with the differential quadrature method (Kumar, 2015), the finite Taylor series expansion approach (Boreyri et al., 2016), the Fourier series combined with the boundary smoothed auxiliary polynomials (Lyu et al., 2017), two dimensional Chebyshev spectral approach (Huang et al., 2019), the pseudo spectral approach (Heshmati and Jalali, 2019), and so on were applied to gain the vibration frequencies and mode shapes of in-plane functionally graded plates under various boundary conditions.

As far as the articles listed above are concerned, it is evident that there are no papers related to the examination on the free vibration responses of the in-plane functionally graded magneto-electro-elastic plates, which provides a main motivation to fill this void. By virtue of the SBFEM and PIA, the present work makes the first attempt to replenish the fundamental frequency solutions of functionally graded magneto-electro-elastic plates with the in-plane stiffness gradation.

When utilizing the SBFEM developed by Wolf and Song (2000) and Song and Wolf (2000) to investigate the structural responses of plates (Man et al., 2012, 2013, 2014; Xiang et al., 2014; Lin et al., 2018; Zhang et al., 2019, 2020a, 2020b), an arbitrary surface parallel with the top-plane is required to be discretized only, which helps to reduce the computational cost and increase the calculation efficiency. Comparing with the FEM (Fu et al., 2020; Li and Yu, 2018; Li et al., 2017a, 2018a, 2018b, 2018d, 2019a, 2020; Yu et al., 2018), the analytical formulations of the stiffness matrix along the thickness direction can be given out. Owing to these unique characteristics of the semi-analytical scheme and only discretization of the boundary, the SBFEM has been adopted in many engineering fields, such as modelling both the unbounded and bounded domains (Bazyar and Song, 2008; Birk et al., 2012; Chen et al., 2014; Song, 2009), voided materials (Sladek et al., 2015, 2016), parametrization (Arioli et al., 2019), weak discontinuities (Natarajan et al., 2020), the interaction of acoustic-structures (Li et al., 2018c; Liu et al., 2019b), uniform beams (Li et al., 2017b), cylindrical shells (Li et al., 2019b), and so on. The general solution of the SBFEM governing equation for plates is in the analytical form of a matrix exponent corresponding to the transverse coordinate. To improve the computation accuracy of fundamental frequencies, this paper employs the PIA proposed by Zhong (2004) to construct the stiffness matrix. As a kind of 2^N algorithm incorporated with the methodology of only tracking the incremental part, the PIA is accurate and stable to solve the matrix exponent and the precision of the obtained results can reach up to the full computer precision.

Aided by the SBFEM and PIA, the first attempt is carried out to assess the free vibration characteristics of in-plane functionally graded magneto-electro-elastic plates in this paper. The structure of the present paper is displayed in the following. Section 2 derives the SBFEM governing equation; the stiffness matrix acquired by the PIA and the mass matrix from the kinetic energy are given out in Section 3; Section 4 illustrates several numerical exercises; the final conclusion is drawn in Section 5.

2. The SBFEM governing equation

In this section, the governing equations for the in-plane functionally graded magneto-electro-elastic plates with the length l, width w, and thickness t are derived based on the SBFEM, as plotted in Figure 1. Only a surface of the plate parallel with the top-plane is discretized and the computational expense can be reduced a lot. Moreover, the stiffness matrix is formulated analytically. Aid by the kinetic energy method, the mass matrix of the in-plane functionally graded magneto-electro-elastic plates is constructed based on the SBFEM for the first time. The SBFEM can be seamlessly coupled with the FEM and the surface elements adopted in the FEM can also be applied in the SBFEM. In this paper, the 2D high-order spectral elements based on the Gauss-Lobatto-Legendre integral, as illustrated in Figure 2, are employed to mesh the plate with 605 degrees of freedom, which helps to simplify some coefficient matrices into diagonal ones and better display the curved boundaries. The derivation of the SBFEM governing equations rigorously starts from the three dimensional basic equations of magneto-electro-elastic materials without importing assumptions on the variations of magnetic, electric, and mechanical fields. Only the elastic displacements along the x, y, and z axes named as u_x = u_x(x,y,z), u_y=u_y(x, y, z), u_z = u_z(x, y, z), the electric potential Φ=Φ(x, y, z), and the magnetic potential Ψ = Ψ(x, y, z) are employed as the unknown variables. To facilitate the following derivation, a variable ${\hat{u}} = {\hat{u} (x, y, z)}$ formulated as ${\hat{u}} = {\hat{u} (x, y, z)} = {[\begin{matrix} \begin{matrix} u_{z} u_{x} \end{matrix} u_{y} Φ Ψ \end{matrix}]}^{T}$ is developed.

Figure 1.

The in-plane functionally graded magneto-electro-elastic plate model.

Figure 2.

A square plate discretized with four fifth order spectral elements.

As for the in-plane functionally graded magneto-electro-elastic plates, the constitutive equations from the reference Pan (2001) are denoted as in equations (1) to (3) and the detailed formulae are listed in Appendix 1.

σ_{ij} = c_{ijlk} (x, y) ε_{lk} - e_{ijk} (x, y) E_{k} - q_{ijk} (x, y) H_{k}

(1)

D_{i} = e_{ilk} (x, y) ε_{lk} + ν_{ik} (x, y) E_{k} + d_{ik} (x, y) H_{k}

(2)

B_{i} = q_{ilk} (x, y) ε_{lk} + d_{ik} (x, y) E_{k} + μ_{ik} (x, y) H_{k}

(3)

in which σ_ij, ε_lk, D_i, B_i, E_k, and H_k (i, j, l, k = x, y, z) represent the components of stress, strain, electric displacement, magnetic induction field, electric field, and magnetic field, respectively.

Meanwhile, in the above formulae c_ijlk(x,y), ν_ik(x,y), μ_ik(x,y), e_ijk(x,y), d_ik(x,y), q_ilk(x,y) stand for the coefficients of elastic stiffness, dielectric constants, magnetic permeabilities, piezoelectric constants, magnetoelectric, and piezomagnetic coupling constants respectively, which are mathematically modeled as the functions in terms of the in-plane x and y coordinates. As for the transversely isotropic magneto-electro-elastic material of Pan (2001), the material parameters are simplified as c_ij(x,y), ν_ij(x,y), μ_ij(x,y), e_ij(x,y), d_ij(x,y), q_ij(x,y). In this paper, the coefficient expressions for the in-plane functionally graded magneto-electro-elastic plates are selected as $c_{ij} (x, y) = c_{ij}^{0} h (x, y)$ , $ν_{ij} (x, y) = ν_{ij}^{0} h (x, y)$ , $μ_{ij} (x, y) = μ_{ij}^{0} h (x, y)$ , $e_{ij} (x, y) = e_{ij}^{0} h (x, y)$ , $d_{ij} (x, y) = d_{ij}^{0} h (x, y)$ , $q_{ij} (x, y) = q_{ij}^{0} h (x, y)$ , in which parameters $c_{ij}^{0}$ , $ν_{ij}^{0}$ , $μ_{ij}^{0}$ , $e_{ij}^{0}$ , $d_{ij}^{0}$ , and $q_{ij}^{0}$ are set as the values at the location x = y = 0 and h(x,y) can be defined as the exponential, power-law, or trigonometric functions of the in-plane coordinates.

Considering the magneto-electro-elastic material, groups of the geometric equation are expressed as

\begin{array}{l} ε_{i j} = \frac{1}{2} (u_{j, i} + u_{i, j}) \\ E_{i} = - Φ_{, i} \\ H_{i} = - Ψ_{, i} \end{array}

(4)

To facilitate the derivation, the differential operator [L] is introduced

[L] = {[\begin{matrix} \frac{\partial}{\partial z} & 0 & 0 & 0 & \frac{\partial}{\partial y} & \frac{\partial}{\partial x} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{\partial}{\partial x} & 0 & \frac{\partial}{\partial y} & 0 & \frac{\partial}{\partial z} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{\partial}{\partial y} & \frac{\partial}{\partial x} & \frac{\partial}{\partial z} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{\partial}{\partial z} & \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\partial}{\partial z} & \frac{\partial}{\partial x} & \frac{\partial}{\partial y} \end{matrix}]}^{T}

(5)

The geometric equations in equation (4) can be rewritten as

\begin{matrix} {\hat{ε}} = [ε_{zz} ε_{xx} ε_{yy} 2 ε_{xy} 2 ε_{yz} 2 ε_{xz} - E_{z} - E_{x} - E_{y} - H_{z} \\ - H_{x} - H_{y}]^{T} = [L] {\hat{u} (x, y, z)} \end{matrix}

(6)

With regard to the in-plane functionally graded magneto-electro-elastic plates, the multi-physical coupled constitutive equation is denoted as

{\hat{σ}} = {[\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} σ_{zz} & σ_{xx} & σ_{yy} \end{matrix} & τ_{xy} & τ_{yz} & τ_{xz} \end{matrix} & D_{z} & D_{x} & D_{y} \end{matrix} & B_{z} & B_{x} & B_{y} \end{matrix}]}^{T} = [H (x, y)] {\hat{ε}}

(7)

where the constitutive matrix is formulated as

\begin{matrix} [H (x, y)] = [\begin{matrix} c_{33}^{0} & c_{13}^{0} & c_{13}^{0} & 0 & 0 & 0 & e_{33}^{0} & 0 & 0 & q_{33}^{0} & 0 & 0 \\ c_{13}^{0} & c_{11}^{0} & c_{12}^{0} & 0 & 0 & 0 & e_{31}^{0} & 0 & 0 & q_{31}^{0} & 0 & 0 \\ c_{13}^{0} & c_{12}^{0} & c_{11}^{0} & 0 & 0 & 0 & e_{31}^{0} & 0 & 0 & q_{31}^{0} & 0 & 0 \\ 0 & 0 & 0 & c_{66}^{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & c_{44}^{0} & 0 & 0 & 0 & e_{15}^{0} & 0 & 0 & q_{15}^{0} \\ 0 & 0 & 0 & 0 & 0 & c_{44}^{0} & 0 & e_{15}^{0} & 0 & 0 & q_{15}^{0} & 0 \\ e_{33}^{0} & e_{31}^{0} & e_{31}^{0} & 0 & 0 & 0 & - ν_{33}^{0} & 0 & 0 & - d_{33}^{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & e_{15}^{0} & 0 & - ν_{11}^{0} & 0 & 0 & - d_{11}^{0} & 0 \\ 0 & 0 & 0 & 0 & e_{15}^{0} & 0 & 0 & 0 & - ν_{11}^{0} & 0 & 0 & - d_{11}^{0} \\ q_{33}^{0} & q_{31}^{0} & q_{31}^{0} & 0 & 0 & 0 & - d_{33}^{0} & 0 & 0 & - μ_{33}^{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & q_{15}^{0} & 0 & - d_{11}^{0} & 0 & 0 & - μ_{11}^{0} & 0 \\ 0 & 0 & 0 & 0 & q_{15}^{0} & 0 & 0 & 0 & - d_{11}^{0} & 0 & 0 & - μ_{11}^{0} \end{matrix}] \\ h (x, y) = [\begin{matrix} [c^{0}] & {[e^{0}]}^{T} & {[q^{0}]}^{T} \\ [e^{0}] & - [ν^{0}] & - {[d^{0}]}^{T} \\ [q^{0}] & - [d^{0}] & - [μ^{0}] \end{matrix}] h (x, y) = [H^{0}] h (x, y) \end{matrix}

(8)

The basic equations of equilibrium without enforcing ad hoc assumptions on the plate kinematics are formulated as

\frac{\partial σ_{xx}}{\partial x} + \frac{\partial τ_{xy}}{\partial y} + \frac{\partial τ_{xz}}{\partial z} = 0

(9)

\frac{\partial τ_{xy}}{\partial x} + \frac{\partial σ_{yy}}{\partial y} + \frac{\partial τ_{yz}}{\partial z} = 0

(10)

\frac{\partial τ_{xz}}{\partial x} + \frac{\partial τ_{yz}}{\partial y} + \frac{\partial σ_{zz}}{\partial z} = 0

(11)

\frac{\partial D_{x}}{\partial x} + \frac{\partial D_{y}}{\partial y} + \frac{\partial D_{z}}{\partial z} = 0

(12)

\frac{\partial B_{x}}{\partial x} + \frac{\partial B_{y}}{\partial y} + \frac{\partial B_{z}}{\partial z} = 0

(13)

With the help of the differential operator [L], the equilibrium equations are simplified as

{[L]}^{T} {\hat{σ}} = 0

(14)

To achieve the transformation from the key partial differential equilibrium equations in equation (14) into the ordinary differential ones, a set of the local scaled boundary coordinates η and ζ is introduced, as revealed in Figure 3. The lines perpendicular to the top-plane are selected as the study object. By virtue of moving the discretized plane along the thickness direction, the whole plate can be obtained, which means that the scaling center is set at the infinity. The in-plane coordinates at the mesh nodes of the spectral elements are indicated as {x} and {y}. The locations of any node on the plane can be acquired with the aid of the shape functions [N] = [N(η,ζ)] = [N₁(η,ζ) N₂(η,ζ) …].

\begin{matrix} x (η, ζ) = [N (η, ζ)] {x} \\ y (η, ζ) = [N (η, ζ)] {y} \end{matrix}

(15)

Figure 3.

The scaled boundary coordinate system.

It is essential to point out that the two dimensional interpolation function [N(η,ζ)] in equation (15) is the product of one dimensional shape function [N(η)] and [N(ζ)].

N_{i} (η, ζ) = N_{k} (η) N_{l} (ζ) (i = 1, 2 . . .)

(16)

In equation (16), the subscripts k and l represent orders of the one dimensional parent element related to the scaled boundary coordinates η and ζ respectively. Lagrange polynomials are utilized to construct the one dimensional Gauss-Lobatto-Legendre shape functions N_k(η) and N_l(ζ).

At the same time, by dint of the scaled boundary coordinates the constitutive matrix can be rewritten as

[H (x, y)] = [H (η, ζ)] = [H^{0}] h ([N] {x}, [N] {y})

(17)

The relationship of the local scaled boundary coordinate and the global Cartesian coordinate system is expressed as

\begin{matrix} {\begin{matrix} \partial / \partial η \\ \partial / \partial ζ \end{matrix}} = J (η, ζ) {\begin{matrix} \partial / \partial x \\ \partial / \partial y \end{matrix}} \\ {\begin{matrix} \partial / \partial x \\ \partial / \partial y \end{matrix}} = J (η, ζ)^{- 1} {\begin{matrix} \partial / \partial η \\ \partial / \partial ζ \end{matrix}} \end{matrix}

(18)

with the Jacobian matrix and its determinant

J (η, ζ) = [\begin{matrix} x_{, η} & y_{, η} \\ x_{, ζ} & y_{, ζ} \end{matrix}]

(19)

| J (η, ζ) | = x_{, η} y_{, ζ} - y_{, η} x_{, ζ}

(20)

In accordance with the equation (19), the differential operator [L] can be rewritten as

[L] = [b^{1}] \frac{\partial}{\partial z} + [b^{2}] \frac{\partial}{\partial η} + [b^{3}] \frac{\partial}{\partial ζ}

(21)

with the coefficients matrices [b¹], [b²], and [b³] formulated in Appendix 2.

Same as the coordinate interpolation, the mechanical, electric, and magnetic variables ${\hat{u} (z, η, ζ)}$ of any position on the discretized plane can be denoted as

\begin{matrix} {\hat{u} (z, η, ζ)} = [\begin{matrix} [N] & [0] & [0] & [0] & [0] \\ [0] & [N] & [0] & [0] & [0] \\ [0] & [0] & [N] & [0] & [0] \\ [0] & [0] & [0] & [N] & [0] \\ [0] & [0] & [0] & [0] & [N] \end{matrix}] \\ [\begin{matrix} {u_{z} (z)} \\ {u_{x} (z)} \\ {u_{y} (z)} \\ {Φ (z)} \\ {Ψ (z)} \end{matrix}] = [N] {\hat{u} (z)} \end{matrix}

(22)

With the help of equations (21) and (22), the vector including strains, electric, and magnetic fields in equation (6) is formulated as

{\hat{ε}} = [B^{1}] {\hat{u} (z)}_{, z} + [B^{2}] {\hat{u} (z)}

(23)

in which [B¹]=[b¹][N] and [B²]=[b²][N]_,η+[b³][N]_,ζ.

Accordingly, equation (7) can be expressed as

{\hat{σ}} = [H (η, ζ)] ([B^{1}] {\hat{u} (z)}_{, z} + [B^{2}] {\hat{u} (z)})

(24)

Aided by the virtual work principle, integration by parts and corresponding boundary conditions, the second-order ordinary differential SBFEM governing equation of the in-plane functionally graded magneto-electro-elastic plate is denoted as

[E^{0}] {\hat{u} (z)}_{, zz} + ([E^{1}]^{T} - [E^{1}]) {\hat{u} (z)}_{, z} - [E^{2}] {\hat{u} (z)} = 0

(25)

with the coefficient matrices [E⁰], [E¹], and [E²]

[E^{0}] = \int_{- 1}^{1} \int_{- 1}^{1} {[B^{1}]}^{T} [H (η, ζ)] [B^{1}] | J | d η d ζ

(26)

[E^{1}] = \int_{- 1}^{1} \int_{- 1}^{1} {[B^{2}]}^{T} [H (η, ζ)] [B^{1}] | J | d η d ζ

(27)

[E^{2}] = \int_{- 1}^{1} \int_{- 1}^{1} {[B^{2}]}^{T} [H (η, ζ)] [B^{2}] | J | d η d ζ

(28)

The aforementioned derivations are depicted only for an element of the SBFEM. With regard to the plate discretized with more elements, it is essential to conduct the process of element assembling for simulating the whole plate structure.

3. The stiffness and mass matrix

In this section, the stiffness and mass matrices for the in-plane functionally graded magneto-electro-elastic plates are constructed. To make sure the high accuracy, the PIA is utilized to create the stiffness matrix from the SBFEM governing equation. What’s more, the general solution of the SBFEM equation can be analytically formulated. For the first time, the mass matrix is given out based on the SBFEM.

To facilitate the analysis, it is necessary to introduce the internal nodal force served as the dual vector of ${\hat{u} (z)}$ .

{\hat{q} (z)} = [E^{0}] {\hat{u} (z)}_{, z} + {[E^{1}]}^{T} {\hat{u} (z)}

(29)

Built upon equation (29), a vector {X(z)} is developed

{X (z)} = {\begin{matrix} {\hat{u} (z)} & {\hat{q} (z)} \end{matrix}}^{T}

(30)

With the help of equation (30), the SBFEM governing equation in equation (25) is simplified into a first-order ordinary differential matrix equation.

{X (z)}_{, z} = - [Z] {X (z)}

(31)

with the coefficient matrix [Z]

[Z] = [\begin{matrix} {[E^{0}]}^{- 1} {[E^{1}]}^{T} & - {[E^{0}]}^{- 1} \\ - [E^{2}] + [E^{1}] {[E^{0}]}^{- 1} {[E^{1}]}^{T} & - [E^{1}] {[E^{0}]}^{- 1} \end{matrix}]

(32)

The general solution to the first-order ordinary differential equation is expressed as

{X (z)} = e^{- [Z] z} {c}

(33)

with the matrix exponent e^-[Z]z and the integration constant vector {c}.

From equation (33), it is apparent that the mechanical, electric, and magnetic quantities across the thickness direction can be analytically formulated.

Substituting the relevant z-coordinates of the bottom and top planes, the following formulae are acquired

{X_{B}} = {c} {X_{T}} = e^{- [Z] t} {c} = \exp (- [Z] t) {c}

(34)

with the bottom and top planes abbreviated by B and T.

With the aid of the relation between the external and internal nodal forces, the expression in equation (34) is rewritten as

{\begin{matrix} {{\hat{u}}_{B}} \\ {{\hat{F}}_{B}} \end{matrix}} = {\begin{matrix} {c_{1}} \\ - {c_{2}} \end{matrix}} {\begin{matrix} {{\hat{u}}_{T}} \\ {{\hat{F}}_{T}} \end{matrix}} = \exp (- [Z] t) {\begin{matrix} {c_{1}} \\ {c_{2}} \end{matrix}}

(35)

with the external force vectors ${{\hat{F}}_{B}}$ and ${{\hat{F}}_{T}}$ applied on the bottom and top planes, respectively.

To improve the computational accuracy of the matrix exponent, the technique of PIA is adopted. The PIA needs to divide the thickness of the plate t into 2^N layers with equal thickness ξ.

\begin{matrix} {X_{T}} = \exp {(- [Z] t / 2^{N})}^{2^{N}} {c} = \exp {(- [Z] ξ)}^{2^{N}} {c} \\ = T^{2^{N}} {c} = \hat{T} {c} \end{matrix}

(36)

in which coefficient matrices T and $\hat{T}$ are denoted as $T = \exp (- [Z] ξ)$ and $\hat{T} = T^{2^{N}}$ respectively.

As usual, the matrix exponent T is calculated by the Taylor series

\begin{matrix} T = \exp (- [Z] ξ) = I_{n} + (- [Z] ξ) + \frac{1}{2!} {(- [Z] ξ)}^{2} \\ + \frac{1}{3!} {(- [Z] ξ)}^{3} + \frac{1}{4!} {(- [Z] ξ)}^{4} + \frac{1}{5!} {(- [Z] ξ)}^{5} + \dots \end{matrix}

(37)

with the unit matrix $I_{n}$ .

It is clear that ξ, which is equal to ξ = t/2^N, is an extremely small thickness. Therefore, the truncated Taylor’s expansion to the fourth order is carried out to promote the evaluation of the matrix exponent, which can insure to obtain enough accuracy.

\begin{matrix} T = \exp (- [Z] ξ) \approx I_{n} + T_{a} T_{a} = (- [Z] ξ) + \frac{1}{2!} {(- [Z] ξ)}^{2} \\ + \frac{1}{3!} {(- [Z] ξ)}^{3} + \frac{1}{4!} {(- [Z] ξ)}^{4} \end{matrix}

(38)

By dint of equations (36) and (38), $\hat{T}$ is estimated through performing the successive factorization N times

\hat{T} = {(I + T_{a})}^{2^{N}} = {(I + T_{a})}^{2^{N - 1}} \times {(I + T_{a})}^{2^{N - 1}} = I + {\hat{T}}_{r}^{N}

(39)

{\hat{T}}_{r}^{i} = 2 T_{r}^{i - 1} + T_{r}^{i - 1} \times T_{r}^{i - 1} (i = 1, 2, . . ., N) T_{r}^{0} = T_{a}

(40)

The foregoing computations are on the basis of the standard matrix algebraic operations, which is favorable to conduct the numerical calculation.

After solving the matrix exponent, equations (35) and (36) are converted into the following form

\begin{matrix} {\begin{matrix} {{\hat{u}}_{T}} \\ {{\hat{F}}_{T}} \end{matrix}} = \hat{T} {\begin{matrix} {c_{1}} \\ {c_{2}} \end{matrix}} = [\begin{matrix} {\hat{T}}_{11} & {\hat{T}}_{12} \\ {\hat{T}}_{21} & {\hat{T}}_{22} \end{matrix}] {\begin{matrix} {c_{1}} \\ {c_{2}} \end{matrix}} \\ = [\begin{matrix} {\hat{T}}_{11} & {\hat{T}}_{12} \\ {\hat{T}}_{21} & {\hat{T}}_{22} \end{matrix}] {\begin{matrix} {{\hat{u}}_{B}} \\ - {{\hat{F}}_{B}} \end{matrix}} \end{matrix}

(41)

Transforming the unknown variable ${\hat{u}}$ and the known external forces ${\hat{F}}$ into each side of equation (41), the stiffness equation for the magneto-electro-elastic plate can be denoted as

[K] {\begin{matrix} {{\hat{u}}_{B}} \\ {{\hat{u}}_{T}} \end{matrix}} = {\begin{matrix} {{\hat{F}}_{B}} \\ {{\hat{F}}_{T}} \end{matrix}}

(42)

with the stiffness matrix [K]

[K] = [\begin{matrix} {\hat{T}}_{12}^{- 1} {\hat{T}}_{11} & - {\hat{T}}_{12}^{- 1} \\ {\hat{T}}_{21} - {\hat{T}}_{22} {\hat{T}}_{12}^{- 1} {\hat{T}}_{11} & {\hat{T}}_{22} {\hat{T}}_{12}^{- 1} \end{matrix}]

(43)

To examine the flexural free vibration behaviors of the in-plane functionally graded magneto-electro-elastic plates, the mass matrix is provided based on the SBFEM for the first time in the present paper. It is noteworthy to indicate that the mass density ρ(x,y) of the plate is also graded along x and y directions. By dint of the scaled boundary coordinates, the density can be reformulated as

ρ (x, y) = ρ (η, ζ) = ρ^{0} h ([N] {x}, [N] {y})

(44)

with the constant ρ⁰ at the location x = y = 0.

Firstly, $δ K$ named as the kinetic energy is expressed as

δ K = \int_{V} δ \hat{u} {(z)}^{T} {[N]}^{T} ρ (η, ζ) [\begin{matrix} 1 & \begin{matrix}  \end{matrix} \\ 1 & \begin{matrix}  \end{matrix} \\ 1 & \begin{matrix}  \end{matrix} \\ \begin{matrix}  \end{matrix} & \begin{matrix}  \end{matrix} & \begin{matrix}  \end{matrix} & \begin{matrix} \begin{matrix} 0 \end{matrix} \\ \begin{matrix} 0 \end{matrix} \end{matrix} \end{matrix}] [N] \overset{\cdot\cdot}{\hat{u}} (z) dV

(45)

Relying on the Fourier transform into the frequency domain and the infinitesimal volume $dV = | J | d η d ζ dz$ , the kinetic energy is rewritten as

\begin{matrix} δ K = \int δ \hat{u} {(z)}^{T} \\ (\int_{S} {[N]}^{T} ρ (η, ζ) [\begin{matrix} 1 & \begin{matrix}  \end{matrix} \\ 1 & \begin{matrix}  \end{matrix} \\ 1 & \begin{matrix}  \end{matrix} \\ \begin{matrix}  \end{matrix} & \begin{matrix}  \end{matrix} & \begin{matrix}  \end{matrix} & \begin{matrix} \begin{matrix} 0 \end{matrix} \\ \begin{matrix} 0 \end{matrix} \end{matrix} \end{matrix}] [N] | J | d η d ζ) ω^{2} \hat{u} (z) dz \end{matrix}

(46)

From equation (46), the mass matrix for the magneto-electro-elastic plate is constructed as

[M] = [\begin{matrix} M_{B} & 0 \\ 0 & M_{T} \end{matrix}]

(47)

with M_B and M_T corresponding to the mass matrices of the bottom and top planes

\begin{matrix} M_{B} = M_{T} = \frac{t}{2} \int_{S} {[N]}^{T} ρ (η, ζ) [\begin{matrix} 1 & \begin{matrix}  \end{matrix} \\ 1 & \begin{matrix}  \end{matrix} \\ 1 & \begin{matrix}  \end{matrix} \\ \begin{matrix}  \end{matrix} & \begin{matrix}  \end{matrix} & \begin{matrix}  \end{matrix} & \begin{matrix} \begin{matrix} 0 \end{matrix} \\ \begin{matrix} 0 \end{matrix} \end{matrix} \end{matrix}] \\ [N] | J | d η d ζ \end{matrix}

(48)

As observed in equations (43) and (47), the stiffness matrix [K] and mass matrix [M] are provided. Hence, variations of the fundamental flexural frequency parameters for the in-plane functionally graded magneto-electro-elastic plates are investigated by the following eigenvalue equation.

[K] - ω^{2} [M] = 0

(49)

In the above formulation, ω is the eigensolutions and ω² is the unknown variable. To obtain the natural frequencies, the determinant $| [K] - ω^{2} [M] | = 0$ is solved and then the eigenvalues are given out.

4. Numerical examples

From the literature review, it is apparent that there is no reports on the free vibration analysis of in-plane functionally graded magneto-electro-elastic plates. On the basis of several numerical examples, this section will exhibit that the present technique is developed as an available tool to assess the dynamic structural responses of magneto-electro-elastic plates, in-plane functionally graded plates and in-plane functionally graded magneto-electro-elastic plates with arbitrary gradation functions. The first three examples are provided to demonstrate the accuracy and applicability of the proposed approach for investigating the distributions of natural frequencies in magneto-electro-elastic and in-plane functionally graded plates. The next three numerical exercises will discuss the effect of thickness-to-length ratios, gradient functions, boundary conditions and constituted materials on the flexural vibration behaviors of in-plane functionally graded magneto-electro-elastic rectangular, circular and perforated plates. All the magneto-electro-elastic material constants listed in Table 1 are from the references Wu and Lu (2009) and Alaimo et al. (2013). But it is essential to state that the superscripts a and b symbolize the different values of $c_{12}^{0}$ , $c_{13}^{0}$ , and $c_{66}^{0}$ obtained from the two articles. However, other material coefficients are set as the same values.

Table 1.

Magneto-electro-elastic material parameters.

	CoFe₂O₄	BaTiO₃
$c_{11}^{0}$	286.0	166.0
$c_{12}^{0}$	173.0^a (170.5)^b	77.0^a (78.0)^b
$c_{13}^{0}$	170.5^a (173.0)^b	78.0^a (77.0)^b
$c_{33}^{0}$	269.5	162.0
$c_{44}^{0}$	45.3	43.0
$c_{55}^{0}$	45.3	43.0
$c_{66}^{0}$	56.5^a (57.8)^b	44.5^a (44.0)^b
$ν_{11}^{0}$	0.08	11.2
$ν_{33}^{0}$	0.093	12.6
$μ_{11}^{0}$	−590.0	5.0
$μ_{33}^{0}$	157.0	10.0
$d_{11}^{0}$	0.0	0.0
$d_{33}^{0}$	0.0	0.0
$e_{15}^{0}$	0.0	11.6
$e_{31}^{0}$	0.0	−4.4
$e_{33}^{0}$	0.0	18.6
$q_{15}^{0}$	550.0	0.0
$q_{31}^{0}$	580.3	0.0
$q_{33}^{0}$	699.7	0.0
$ρ^{0}$	5300	5800

From Wu and Lu (2009).

From Alaimo et al. (2013).

In Table 1, it is necessary to indicate that the unit of the elastic stiffness coefficients $c_{ij}^{0}$ is GPa, the dielectric constants $ν_{ij}^{0}$ 10⁻⁹ F/m, magnetic permeabilities $μ_{ij}^{0}$ 10⁻⁶ Ns²/C², the piezoelectric constants $e_{ij}^{0}$ C/m², the magnetoelectric coupling coefficients $d_{ij}^{0}$ 10⁻¹² Ns/(VC), the piezomagnetic coupling coefficients $q_{ij}^{0}$ N/(Am), and the mass density $ρ^{0}$ kg/m³.

4.1. Verification

4.1.1. Magneto-electro-elastic plates

The first numerical example will pay attention to displaying the effectiveness of the introduced methodology in calculating the natural frequencies of magneto-electro-elastic square plates constituted by materials CoFe₂O₄ and BaTiO₃ under different boundary conditions and thickness-to-length ratios. Figure 4 illustrates that a surface of the plate is discretized by only one spectral element with third, fourth, and sixth orders. The length of the plate is set as l = w = 1 m. The simply supported and grounded boundary conditions are prescribed on the four lateral edges of the plate, that is, u_z = u_x = Φ = Ψ = 0 at y = 0 and y = w, u_z = u_y = Φ = Ψ = 0 at x = 0 and x = l. Three different thickness-to-length ratios t/l = 0.5, 0.3, and 0.025 corresponding to the thick, middle thick, and thin plates are under consideration. As to the top and bottom planes, there are three types of surface constraints: Case 1, σ_zz = τ_xz = τ_yz = Φ = Ψ = 0 at z = 0 and t; Case 2, σ_zz = τ_xz = τ_yz = D_z = B_z = 0 at z = 0 and t; Case 3, σ_zz = τ_xz = τ_yz = Φ = Ψ = 0 at z = t; and σ_zz = τ_xz = τ_yz = D_z = B_z = 0 at z = 0. In this section, the calculated results are normalized as $\hat{ω} = ω l \sqrt{ρ^{0} / c_{11}^{0}}$ . The natural frequency parameters under various boundary constraints and thickness-to-length ratios are listed in Tables 2 to 7. Tables 2 to 4 present the vibration frequencies of the BaTiO₃ plate with the material type b acquired by the developed method utilizing different spectral element orders, the 3D solutions and Milazzo (2012) based on the equivalent single layer theory. It is clear that eigensolutions from the proposed technique converge to the 3D results according to increasing the element orders and better than those from Milazzo (2012) especially for the thick plate. From Tables 2 to 7, it is believed that the tabulated results obtained by the SBFEM and PIA exhibit an excellent agreement with those from the 3D benchmarks and the numerical solutions. In all cases, the largest error is less than 2.8%. At the same time, it can be found that the thick BaTiO₃ plate will obtain the maximal vibration frequencies comparing with the middle thick and thin plates. What’s more, the changing rules of vibration frequencies for the CoFe₂O₄ plate under three different surface boundary conditions are in accordance with the following pattern: Case 3 > Case 2 > Case 1.

Figure 4.

The square plate is meshed by only one spectral element with different orders: (a) third order, (b) fourth order, and (c) sixth order.

Table 2.

Natural frequencies of the BaTiO₃ plate with type b and t/l = 0.5 under Case 1.

Mode	t/l = 0.5
	Third order	Fourth order	Sixth order	3D (Pan and Heyliger, 2002)	Milazzo (2012)	Error (%)
(1,1)	1.0703	1.1062	1.1312	1.1534	1.2255	−1.9239
(1,2)	2.0927	2.1245	2.1436	2.1114	2.2908	1.5251
(1,3)	3.2175	3.1944	3.2094	3.1955	3.5071	0.4365
(2,2)	2.8767	2.8295	2.8054	2.8063	3.0694	−0.0316
(2,3)	3.7188	3.7633	3.7300	3.7109	4.0860	0.5146
(3,3)	4.1390	4.4442	4.4512	4.4464	4.9128	0.1071

Table 3.

Natural frequencies of the BaTiO₃ plate with type b and t/l = 0.3 under Case 1.

Mode	t/l = 0.3
	Third order	Fourth order	Sixth order	3D (Pan and Heyliger, 2002)	Milazzo (2012)	Error (%)
(1,1)	0.8589	0.8711	0.8959	0.8883	0.9203	0.8569
(1,2)	1.8973	1.8308	1.7380	1.7842	1.8908	−2.5886
(1,3)	3.1156	3.0215	2.7793	2.8576	3.0787	−2.7400
(2,2)	2.4636	2.5111	2.4896	2.4684	2.6451	0.8578
(2,3)	3.5287	3.3724	3.3780	3.3772	3.6589	0.0251
(3,3)	4.0283	4.2627	4.1284	4.1256	4.4962	0.0686

Table 4.

Natural frequencies of the BaTiO₃ plate with type b and t/l = 0.025 under Case 1.

Mode	t/l = 0.025
	Third order	Fourth order	Sixth order	3D (Pan and Heyliger, 2002)	Milazzo (2012)	Error (%)
(1,1)	0.1019	0.954	0.0954	0.0949	0.0952	0.4774
(1,2)	0.2471	0.2850	0.2384	0.2382	0.2374	−0.2504
(1,3)	0.3600	0.5606	0.4743	0.4713	0.4724	0.2469
(2,2)	1.1850	0.4466	0.3799	0.3772	0.3787	0.3680
(2,3)	1.6647	0.6836	0.6118	0.6103	0.6123	0.2119
(3,3)	2.0524	0.8726	0.8369	0.8412	0.8437	0.0500

Table 5.

Fundamental frequencies with the material type a and t/l = 0.3 under Case 1.

	1	2	3	4	5
BaTiO₃
Wu and Lu (2009)	1.2523	2.3003	3.8314	5.8050	6.7066
SBFEM	1.2282	2.3002	3.8306	5.8062	6.7070
Error (%)	−1.9271	−0.0045	−0.0212	0.0206	0.0061
CoFe₂O₄
Wu and Lu (2009)	1.0212	1.9747	3.3905	4.6118	5.5175
SBFEM	1.0011	1.9408	3.3901	4.6157	5.5185
Error (%)	−1.9724	−1.7156	−0.0131	0.0843	0.0184

Table 6.

Fundamental frequencies with the material type a and t/l = 0.3 under Case 3.

	1	2	3	4	5
CoFe₂O₄
Wu and Lu (2009)	1.2555	2.3003	3.9650	5.8050	6.8292
SBFEM	1.2840	2.3237	3.9584	5.8020	6.8294
Error (%)	2.2709	1.0152	−0.1672	−0.0523	0.0036

Table 7.

Fundamental frequencies with the material type a and t/l = 0.3 under Case 2.

	1	2	3	4	5	6	7	8
BaTiO₃
Chen et al. (2007)	1.2660	2.3003	4.0111	5.8050	7.2465	9.7703	10.9049	12.9050
SBFEM	1.2410	2.3313	3.9756	5.8047	7.2497	9.7744	10.9032	12.9040
Error (%)	−1.9778	1.3477	−0.8862	−0.0049	0.0450	0.0422	−0.0153	−0.0074
CoFe₂O₄
Chen et al. (2007)	1.0181	1.9747	3.3917	4.6118	5.4544	8.2670	8.5661	11.4523
SBFEM	1.0128	1.9512	3.3921	4.6057	5.4472	8.2653	8.5661	11.4563
Error (%)	−0.5173	−1.1925	0.0108	−0.1314	−0.1328	−0.0211	−0.0006	0.0347

Model 1:

V_{c} (x) = {(\frac{x}{l})}^{k_{x}}

(50)

\begin{matrix} Model 2 : \\ V_{c} (x, y) = {(\frac{x}{l})}^{k_{x}} {(\frac{y}{w})}^{k_{y}} \end{matrix}

(51)

\begin{matrix} Model 3 : \\ V_{c} (x, y) = V_{c, x} (x) V_{c, y} (y) \\ V_{c, x} (x) = {\begin{matrix} \begin{matrix} {(\frac{2 x}{l})}^{k_{x}} & 0 \leq x < \frac{l}{2} \end{matrix} \\ \begin{matrix} {(2 - \frac{2 x}{l})}^{k_{x}} & \frac{l}{2} \leq x \leq l \end{matrix} \end{matrix} V_{c, y} (y) \\ = {\begin{matrix} \begin{matrix} {(\frac{2 y}{w})}^{k_{y}} & 0 \leq y < \frac{w}{2} \end{matrix} \\ \begin{matrix} {(2 - \frac{2 y}{w})}^{k_{y}} & \frac{w}{2} \leq y \leq w \end{matrix} \end{matrix} \end{matrix}

(52)

4.1.2. In-plane functionally graded plates

In this section, two numerical examples are released to prove the high accuracy of the proposed approach when exploring the transverse vibration behaviors of in-plane functionally graded plates. The square and circular plates are taken as examples.

4.1.2.1. Square plates

As a common structural component, square plates are extensively used in engineering applications. The length of the plate is selected as l = w = 1 m and four kinds of thickness-to-length ratios t/l = 0.2, 0.02, 0.01, and 0.001 are discussed. The functionally graded plate is composed of ceramic and metallic phase materials. The relevant Young’s modulus, Poisson’s ratio and mass density of ceramic and metallic compositions are set as E_c = 348.43 GPa, υ_c = 0.24, ρ_c = 2370 kg/m³, and E_m = 201.04 GPa, υ_m = 0.3262, ρ_m = 8166 kg/m³ from Lieu et al. (2018), in which the subscripts of c and m stand for the ceramic and metallic materials respectively. The material constants for the functionally graded plates are computed by E=E_cV_c+E_mV_m, υ=υ_cV_c+υ_mV_m, ρ=ρ_cV_c + ρ_mV_m, and V_c + V_m = 1, in which V_c and V_m represent the volume fractions of ceramic and metallic materials in Lieu et al. (2018). In this example, the parameter V_c obeys the following three models, as exhibited in equations (50) to (52). Three different boundary constraints CSFS, SSSS, and CCCC are applied, in which C, S, and F symbolize the clamped, simply-supported, and free boundary conditions respectively. The vibration parameters normalized as $\hat{ω} = ω {(l / π)}^{2} \sqrt{ρ_{c} t / D_{c}}$ and D_c = E_ct³/12(1 − υ_c²) are listed in Tables 8 to 11 under different boundary conditions, thickness-to-length ratios, and gradient models. It can be indicated from Tables 8 to 9 that through the increment of element orders the eigenvalues calculated by the introduced methodology are convergent to the numerical results provided by Lieu et al. (2018) and even solutions of the fourth order spectral element can insure enough accuracy. From all tables, it is obvious that the non-dimensional frequencies calculated by the employed methodology match well with those from Lieu et al. (2018) based on the isogeometric analysis. Contrary to the magneto-electro-elastic plates, the thicker the functionally graded plate is, the smaller the normalized natural frequencies are. The reason for this phenomenon maybe is that the non-dimensional formula for the square plate takes into the effect of the length and thickness. From Tables 10 to 11, it is observed that the plates with the elastic material varied as the Model 3 produce larger eigensolutions than those with Model 2 when the boundary condition, gradation parameter, and thickness-to-length ratio are same, which means that the gradient functions significantly influence the dynamic responses of in-plane functionally graded plates.

Table 8.

Eigenvalues of the CCCC square plate with Model 1.

t/l	k_x = 0.0	k_x = 0.5	k_x = 1.0	k_x = 2.0	k_x = 5.0	k_x = 10.0
0.02
Fourth order	3.5970	2.5447	2.1673	1.8485	1.6212	1.5734
Fifth order	3.6353	2.5311	2.1464	1.8409	1.6292	1.5747
Sixth order	3.6342	2.5325	2.1469	1.8412	1.6264	1.5721
IGA (Lieu et al., 2018)	3.6311	2.5294	2.1445	1.8417	1.6287	1.5735
Error (%)	0.0864	0.1220	0.1125	−0.0286	−0.1431	−0.0901
0.01
Fourth order	3.6055	2.5515	2.1732	1.8534	1.6255	1.5776
Fifth order	3.6445	2.5380	2.1523	1.8459	1.6336	1.5790
Sixth order	3.6435	2.5393	2.1528	1.8462	1.6308	1.5764
IGA (Lieu et al., 2018)	3.6423	2.5379	2.1518	1.8481	1.6342	1.5789
Error (%)	0.0327	0.0543	0.0464	−0.1014	−0.2069	−0.1561
0.001
Fourth order	3.6083	2.5537	2.1751	1.8551	1.6269	1.5790
Fifth order	3.6475	2.5403	2.1542	1.8476	1.6351	1.5804
Sixth order	3.6466	2.5415	2.1548	1.8479	1.6323	1.5779
IGA (Lieu et al., 2018)	3.6461	2.5407	2.1543	1.8502	1.6361	1.5808
Error (%)	0.0127	0.0330	0.0212	−0.1240	−0.2324	−0.1846

Table 9.

Eigenvalues of the SSSS square plate with Model 1.

t/l	k_x = 0.0	k_x = 0.5	k_x = 1.0	k_x = 2.0	k_x = 5.0	k_x = 10.0
0.02
Fourth order	1.9970	1.3980	1.1904	1.0243	0.9025	0.8670
Fifth order	1.9979	1.3952	1.1850	1.0201	0.9017	0.8660
Sixth order	1.9976	1.3955	1.1850	1.0200	0.9010	0.8654
IGA (Lieu et al., 2018)	1.9974	1.3935	1.1848	1.0215	0.9033	0.8677
Error (%)	0.0108	0.1450	0.0139	−0.1507	−0.2593	−0.2695
0.01
Fourth order	1.9987	1.3994	1.1916	1.0253	0.9034	0.8679
Fifth order	1.9997	1.3966	1.1862	1.0211	0.9025	0.8668
Sixth order	1.9994	1.3969	1.1861	1.0210	0.9018	0.8662
IGA (Lieu et al., 2018)	1.9993	1.3950	1.1861	1.0226	0.9043	0.8687
Error (%)	0.0051	0.1345	0.0020	−0.1613	−0.2734	−0.2874
0.001
Fourth order	1.9993	1.3999	1.1920	1.0257	0.9037	0.8682
Fifth order	2.0003	1.3970	1.1865	1.0214	0.9028	0.8671
Sixth order	2.0000	1.3973	1.1865	1.0213	0.9021	0.8665
IGA (Lieu et al., 2018)	2.0000	1.3955	1.1865	1.0230	0.9046	0.8690
Error (%)	−0.0003	0.1307	0.0006	−0.1683	−0.2747	−0.2897

Table 10.

Eigensolutions of the CSFS plate with Model 2.

t/l	k_y	k_x = 0			k_x = 0.5			k_x = 1			k_x = 2
		Lieuet al.(2018)	SBFEM	Error(%)	Lieuet al.(2018)	SBFEM	Error(%)	Lieuet al.(2018)	SBFEM	Error(%)	Lieuet al.(2018)	SBFEM	Error (%)
0.2	0.0	1.1845	1.1925	0.6780	0.9554	0.9747	2.0209	0.8402	0.8582	2.1452	0.7274	0.7417	1.9661
	0.5	0.8076	0.8256	2.2331	0.7306	0.7492	2.5440	0.6847	0.7013	2.4275	0.6316	0.6451	2.1341
	1.0	0.6825	0.6973	2.1672	0.6405	0.6553	2.3032	0.6146	0.6279	2.1710	0.5831	0.5942	1.9077
	2.0	0.5861	0.5964	1.7651	0.5659	0.5761	1.8023	0.5534	0.5628	1.6919	0.5377	0.5449	1.5186
0.02	0.0	1.3038	1.3041	0.0215	1.0605	1.0718	1.0644	0.9369	0.9473	1.1143	0.8123	0.8199	0.9391
	0.5	0.8949	0.9084	1.5045	0.8114	0.8252	1.6948	0.7616	0.7734	1.5481	0.7032	0.7119	1.2356
	1.0	0.7568	0.7675	1.4076	0.7109	0.7214	1.4703	0.6828	0.6918	1.3176	0.6480	0.6549	1.0660
	2.0	0.6496	0.6558	0.9584	0.6274	0.6334	0.9566	0.6137	0.6190	0.8565	0.5964	0.6005	0.6886
0.01	0.0	1.3057	1.3058	0.0070	1.0620	1.0731	1.0434	0.9382	0.9485	1.0949	0.8135	0.8209	0.9106
	0.5	0.8964	0.9097	1.4793	0.8126	0.8263	1.6822	0.7628	0.7744	1.5237	0.7043	0.7128	1.2116
	1.0	0.7580	0.7686	1.3953	0.7121	0.7224	1.4428	0.6839	0.6928	1.2966	0.6491	0.6558	1.0358
	2.0	0.6507	0.6568	0.9379	0.6284	0.6343	0.9438	0.6147	0.6199	0.8396	0.5974	0.6014	0.6673

Table 11.

Natural frequencies of the CSFS plate with Model 3.

t/l	k_y	k_x = 0			k_x = 0.5			k_x = 1			k_x = 2
		Lieuet al.(2018)	SBFEM	Error(%)	Lieuet al.(2018)	SBFEM	Error(%)	Lieuet al.(2018)	SBFEM	Error(%)	Lieuet al.(2018)	SBFEM	Error(%)
0.2	0.0	1.1845	1.1925	0.6790	0.7410	0.7612	2.7261	0.6437	0.6569	2.0484	0.5746	0.5820	1.2909
	0.5	0.9007	0.9192	2.0513	0.6607	0.6795	2.8418	0.5965	0.6090	2.0935	0.5485	0.5559	1.3431
	1.0	0.7824	0.7969	1.8523	0.6205	0.6353	2.3782	0.5722	0.5824	1.7909	0.5349	0.5413	1.1910
	2.0	0.6753	0.6832	1.1681	0.5782	0.5879	1.6771	0.5460	0.5532	1.3136	0.5200	0.5249	0.9502
0.02	0.0	1.3038	1.3041	0.0215	0.8182	0.8346	2.0044	0.7118	0.7210	1.2919	0.6365	0.6400	0.5432
	0.5	1.0038	1.0191	1.5195	0.7322	0.7484	2.2189	0.6607	0.6700	1.4147	0.6077	0.6116	0.6471
	1.0	0.8740	0.8856	1.3270	0.6883	0.7006	1.7908	0.6342	0.6412	1.1083	0.5928	0.5957	0.4825
	2.0	0.7542	0.7587	0.5987	0.6418	0.6486	1.0673	0.6054	0.6092	0.6321	0.5765	0.5778	0.2272
0.01	0.0	1.3057	1.3058	0.0070	0.8196	0.8357	1.9597	0.7131	0.7221	1.2678	0.6377	0.6410	0.5198
	0.5	1.0055	1.0205	1.4936	0.7335	0.7494	2.1718	0.6619	0.6711	1.3965	0.6088	0.6126	0.6322
	1.0	0.8755	0.8870	1.3086	0.6895	0.7016	1.7477	0.6354	0.6423	1.0831	0.5939	0.5966	0.4623
	2.0	0.7555	0.7599	0.5822	0.6430	0.6495	1.0115	0.6065	0.6102	0.6140	0.5775	0.5788	0.2176

4.1.2.2. Circular plates

Circular plates are widely applied in various engineering practices. In this subsection, the geometric parameters of the circular plate are the radius R = 1.0 m and the thickness t = 0.01 m, as shown in Figure 5. The simply supported and clamped boundary conditions are imposed on the side edges of the plate. The material coefficients are graded along the radial direction following the two cases: Case 1, E = E₀e^rγ, ρ = ρ₀e^rγ with E₀ = 206 GPa, ρ₀ = 8000 kg/m³, and the Poisson’s ratio υ = 0.3 which is assumed to keep a constant; Case 2, E = E_cV_c + E_mV_m, υ = υ_cV_c + υ_mV_m, ρ = ρ_cV_c + ρ_mV_m, V_c + V_m = 1, and V_c = (r/R)ⁿ, (0 < r < R) in which the elastic parameters are the same as those in Section 4.1.2.1. The dimensionless flexural frequencies are expressed as $\hat{ω} = ω R^{2} \sqrt{ρ_{0} t / D_{0}}$ and D₀ = E₀t³/12(1 − υ²) for Case 1; what’s more, $\hat{ω} = ω R^{2} \sqrt{ρ_{c} t / D_{c}}$ and D_c = E_ct³/12(1 − υ_c²) for Case 2. The evaluated eigensolutions of the in-plane functionally graded circular plate from the SBFEM and rotation-free isogeometric analysis (Yin et al., 2017) are displayed in Tables 12 to 15. From the presented results, it is clear that the good reliability of the introduced technique to predict the flexural vibration responses of functionally graded circular plate with in-plane material inhomogeneity is established. From the tabulated results, it is seen that increasing the gradient index γ leads to the increment of vibration frequencies for the plate with Case 1 irrespective of boundary conditions. However, the changing patterns of natural frequencies for the plate with Case 2 present the opposite trend to that with Case 1. The conclusion that gradation functions play an important role in distributions of eigenvalues from the foregoing example is confirmed.

Figure 5.

The in-plane functionally graded circular plate: (a) the plate model and (b) meshed with twelve fourth-order spectral elements.

Table 12.

Fundamental frequencies of the SSSS plate for Case 1.

Mode		γ = 0.0	γ = 0.5	γ = 1.0	γ = 2.0	γ = 5.0	γ = 8.0
1	IGA (Yin et al., 2017)	4.9351	5.0283	5.1187	5.3040	6.0969	7.4300
	SBFEM	4.9346	5.0277	5.1181	5.3030	6.0947	7.4261
	Error (%)	−0.0105	−0.0116	−0.0125	−0.0180	−0.0368	−0.0523
2	IGA (Yin et al., 2017)	13.9004	14.0187	14.1131	14.2448	14.5016	15.2781
	SBFEM	13.8926	14.0101	14.1033	14.2313	14.4667	15.2186
	Error (%)	−0.0559	−0.0616	−0.0697	−0.0951	−0.2407	−0.3895
3	IGA (Yin et al., 2017)	13.9004	14.0187	14.1131	14.2448	14.5016	15.2781
	SBFEM	13.8926	14.0101	14.1033	14.2313	14.4667	15.2186
	Error (%)	−0.0559	−0.0616	−0.0697	−0.0951	−0.2407	−0.3895
4	IGA (Yin et al., 2017)	25.6229	25.6826	25.7517	25.9149	26.6688	28.1530
	SBFEM	25.5942	25.6510	25.7167	25.8712	26.5847	28.0378
	Error (%)	−0.1122	−0.1229	−0.1360	−0.1688	−0.3154	−0.4092

Table 13.

Dimensionless frequency parameters of the SSSS plate for Case 2.

Mode		γ = 0.0	γ = 0.5	γ = 1.0	γ = 2.0	γ = 5.0	γ = 8.0
1	IGA (Yin et al., 2017)	4.8448	3.3379	2.8560	2.4831	2.2177	2.1584
	SBFEM	4.8443	3.3376	2.8558	2.4828	2.2174	2.1581
	Error (%)	−0.0106	−0.0095	−0.0087	−0.0102	−0.0132	−0.0126
2	IGA (Yin et al., 2017)	13.8243	10.2762	8.7858	7.4588	6.3758	6.1147
	SBFEM	13.8166	10.2693	8.7796	7.4536	6.3713	6.1102
	Error (%)	−0.0588	−0.0675	−0.0708	−0.0694	−0.0708	−0.0736
3	IGA (Yin et al., 2017)	13.8243	10.2762	8.7858	7.4588	6.3758	6.1147
	SBFEM	13.8166	10.2693	8.7796	7.4536	6.3713	6.1102
	Error (%)	−0.0588	−0.0675	−0.0708	−0.0694	−0.0708	−0.0736
4	IGA (Yin et al., 2017)	25.5513	19.7272	16.9998	14.4161	12.1030	11.4768
	SBFEM	25.5226	19.7000	16.9750	14.3951	12.0861	11.4608
	Error (%)	−0.1123	−0.1380	−0.1460	−0.1454	−0.1394	−0.1398

Table 14.

Fundamental frequencies of the CCCC plate for Case 1.

Mode		γ = 0.0	γ = 0.5	γ = 1.0	γ = 2.0	γ = 5.0	γ = 8.0
1	IGA (Yin et al., 2017)	10.2165	11.0009	11.8612	13.8340	22.1114	34.4054
	SBFEM	10.2131	10.9969	11.8562	13.8264	22.0885	34.3457
	Error (%)	−0.0329	−0.0364	−0.0420	−0.0552	−0.1034	−0.1736
2	IGA (Yin et al., 2017)	21.2715	22.0498	22.8693	24.6634	31.8491	42.8225
	SBFEM	21.2487	22.0238	22.8385	24.6177	31.6993	42.4338
	Error (%)	−0.1071	−0.1181	−0.1347	−0.1853	−0.4704	−0.9078
3	IGA (Yin et al., 2017)	21.2715	22.0498	22.8693	24.6634	31.8491	42.8225
	SBFEM	21.2487	22.0238	22.8385	24.6177	31.6993	42.4338
	Error (%)	−0.1071	−0.1181	−0.1347	−0.1853	−0.4704	−0.9078
4	IGA (Yin et al., 2017)	34.9104	35.5999	36.3486	38.0345	44.8989	55.1416
	SBFEM	34.8459	35.5270	36.2653	37.9236	44.6289	54.6848
	Error (%)	−0.1849	−0.2049	−0.2292	−0.2916	−0.6015	−0.8285

Table 15.

Dimensionless frequency parameters of the CCCC plate for Case 2.

Mode		γ = 0.0	γ = 0.5	γ = 1.0	γ = 2.0	γ = 5.0	γ = 8.0
1	IGA (Yin et al., 2017)	10.2165	6.8437	5.9037	5.2313	4.7694	4.6408
	SBFEM	10.2132	6.8415	5.9016	5.2292	4.7671	4.6378
	Error (%)	−0.0321	−0.0323	−0.0348	−0.0401	−0.0481	−0.0646
2	IGA (Yin et al., 2017)	21.2715	15.2397	12.9954	11.1630	9.8535	9.5662
	SBFEM	21.2491	15.2193	12.9772	11.1477	9.8390	9.5492
	Error (%)	−0.1055	−0.1337	−0.1403	−0.1373	−0.1475	−0.1774
3	IGA (Yin et al., 2017)	21.2715	15.2397	12.9954	11.1630	9.8535	9.5662
	SBFEM	21.2491	15.2193	12.9772	11.1477	9.8390	9.5492
	Error (%)	−0.1055	−0.1337	−0.1403	−0.1373	−0.1475	−0.1774
4	IGA (Yin et al., 2017)	34.9104	26.0080	22.2423	18.9242	16.2905	15.6824
	SBFEM	34.8467	25.9468	22.1868	18.8778	16.2522	15.6449
	Error (%)	−0.1824	0.2352	0.2496	−0.2451	−0.2352	−0.2391

4.2. The magneto-electro-elastic rectangular plate

This example pays attention to analyzing the structural vibration responses of functionally graded magneto-electro-elastic rectangular plates with material constants varied along the x and y directions in terms of the exponential law. The gradient function is expressed as h(x,y) = e^αx+βy. Figure 6 demonstrates the geometry parameters of the plate l = 1.5 m, w = 1 m, and t = 0.03 m. Moreover, this simulation selects the magneto-electro-elastic materials CoFe₂O₄ and BaTiO₃ of type b. The eigensolutions of the magneto-electro-elastic plates under four different boundary constraints CCFF, CFSF, SSFF, and SSSF are estimated, as depicted in Figure 7. Additionally, the zero electric and magnetic potentials are applied around the sides and on the top and bottom surfaces. Meanwhile, the computed results are normalized as $\hat{ω} = ω l \sqrt{ρ^{0} / c_{11}^{0}}$ , in which ρ⁰ and $c_{11}^{0}$ stand for the density and elastic stiffness coefficient of CoFe₂O₄ respectively. Comparisons of the dimensionless fundamental frequencies for the in-plane functionally graded magneto-electro-elastic rectangular plates with various boundary conditions and gradation parameters are exhibited in Figures 8 and 9 and Tables 16 and 17. In Figure 8, the parameter α is set as α = 0.0, 1.0, 2.0, 4.0, and 8.0. Meanwhile, four different values β = 0.0, 1.0, 2.0, 4.0, and 8.0 are applied in Figure 9. From these figures, it is apparent that with increasing the gradient exponent the non-dimensional vibration frequencies for all types of boundary constraints decrease. Additionally, it can be observed from Figure 8 that the reduction rate of eigensolutions in the SSSF plate is faster than that under other boundary conditions. Irrespective of the constituted materials, the first order natural frequencies of the CFSF and SSFF plates will reach the highest and lowest values, respectively. As for the rest boundary constraint CCFF, the eigenvalues decrease faster with the increment of β than that of α in Tables 16 and 17. Therefore, it can be concluded that the boundary conditions and gradient parameters play a significant role on evaluating the frequency parameters of in-plane functionally graded magneto-electro-elastic rectangular plates.

Figure 6.

The in-plane functionally graded magneto-electro-elastic plate: (a) the rectangular plate and (b) discretized with six spectral elements.

Figure 7.

The four kinds of boundary conditions: (a) CCFF, (b) CFSF, (c) SSFF, and (d) SSSF.

Figure 8.

Natural frequencies of the BaTiO₃ plate with β = 1 and different α.

Figure 9.

Natural frequencies of the BaTiO₃ plate with α = 1 and different β.

Table 16.

Fundamental frequencies of the CoFe₂O₄ plate with β = 1 and different α.

	CCFF	CFSF	SSFF	SSSF
α = 0.0	0.0412	0.1432	0.0193	0.1067
α = 1.0	0.0354	0.1431	0.0152	0.1032
α = 2.0	0.0314	0.1427	0.0117	0.1006
α = 5.0	0.0261	0.1411	0.0045	0.0784
α = 8.0	0.0252	0.1400	0.0012	0.0132

Table 17.

Fundamental frequencies of the CoFe₂O₄ plate with α = 1 and different β.

	CCFF	CFSF	SSFF	SSSF
β = 0.0	0.0442	0.1536	0.0171	0.1040
β = 1.0	0.0354	0.1431	0.0152	0.1032
β = 2.0	0.0293	0.1327	0.0136	0.1010
β = 5.0	0.0184	0.0999	0.0099	0.0876
β = 8.0	0.0133	0.0652	0.0066	0.0740

4.3. The magneto-electro-elastic circular plate

This section explores the free vibration behaviors of the functionally graded magneto-electro-elastic circular plates with material properties inhomogeneity along the radius direction. The semi-diameter of the circular plate is defined as R = 1 m and a characteristic length l = 2R is introduced. The effect of t/l = 0.1, 0.05, 0.02, 0.01, and 0.001 on the variations of natural frequencies is presented. Similar with the example in Section 4.2, the materials CoFe₂O₄ and BaTiO₃ of type b and the same normalized formulation are adopted. The simply-supported boundary condition, that is, u_z = Φ = Ψ = 0, is prescribed on the side of the circular plate. Meanwhile, the top and bottom planes are grounded, which means that the electric and magnetic potentials are set to zero. In this example, the eigenvalues of the circular plates with the gradation expression h(r) = e^γr are predicted. The dimensionless flexural vibration parameters of the functionally graded magneto-electro-elastic circular plate with the in-plane heterogeneous material are performed in Tables 18 and 19. To demonstrate the free vibration behaviors of the magneto-electro-elastic plate with the power-law gradation function, a circular plate composed by the CoFe₂O₄ and BaTiO₃ of type b is cited as an example. The material parameters are denoted as [H(r)]= [H⁰]_BaTiO3(r/R)ⁿ+[H⁰]_CoFe2O4[1− (r/R)ⁿ], in which [H⁰]_BaTiO3 and [H⁰]_CoFe2O4 symbolize the elastic matrix of BaTiO₃ and CoFe₂O₄. The clamped boundary condition u_z = u_x = u_y = Φ = Ψ = 0 is applied. But no boundary constraints are enforced on the top and bottom surfaces. The non-dimensional eigenvalues of the circular plate with the power-law gradient function are listed in Table 20. From the presented results in Tables 18 and 19, the phenomenon that the normalized fundamental frequencies obtained from the CoFe₂O₄ plate are larger than those of BaTiO₃ can be found in this example. What’s more, it can be seen that increasing the gradient index γ results in the increase of frequency parameters, which is due to a decrease in the flexural rigidity of the circular plate with a smaller γ. While keeping the gradation parameter γ fixed, it is revealed that the thicker the plate is, the bigger its corresponding non-dimensional vibration frequencies are. Variations of eigenvalues for thin plates, especially with the ratio t/l = 0.001, are less sensitive to the material gradation index than the thicker ones. It can be found from Table 20 that with increasing the gradient parameter n, the vibration frequencies of the circular plate will increase, which is parallel with the former discover. When the index n gets larger, values of the volume fraction (r/R)ⁿ decrease, which means that the proportion of the material CoFe₂O₄ in the composite plate increases. Meanwhile, the CoFe₂O₄ plates obtain larger frequencies than those with BaTiO₃. Therefore, the bigger the coefficient n is, the larger the eigensolutions are. A conclusion can be drawn that the thickness-to-length ratios and gradation functions have an important role on the free vibration responses of in-plane functionally graded magneto-electro-elastic circular plates.

Table 18.

Normalized frequencies of the circular BaTiO₃ plate.

t/l	0.1	0.05	0.02	0.01	0.001
γ = 0.0	0.3774	0.1940	0.0783	0.0392	0.0040
γ = 0.5	0.3836	0.1973	0.0796	0.0398	0.0041
γ = 1.0	0.3892	0.2003	0.0808	0.0405	0.0042
γ = 2.0	0.3991	0.2060	0.0832	0.0417	0.0044
γ = 5.0	0.4349	0.2297	0.0935	0.0469	0.0054

Table 19.

Vibration frequencies of the circular CoFe₂O₄ plate.

t/l	0.1	0.05	0.02	0.01	0.001
γ = 0.0	0.4333	0.2236	0.0903	0.0452	0.0046
γ = 0.5	0.4413	0.2276	0.0919	0.0460	0.0047
γ = 1.0	0.4487	0.2315	0.0935	0.0468	0.0048
γ = 2.0	0.4623	0.2391	0.0967	0.0484	0.0051
γ = 5.0	0.5127	0.2710	0.1103	0.0554	0.0063

Table 20.

Eigensolutions of the circular plate.

t/l	0.1	0.05	0.02	0.01	0.001
n = 0.0	2.0428	1.2352	0.5357	0.2715	0.0273
n = 0.5	2.0624	1.2490	0.5421	0.2749	0.0276
n = 1.0	2.0789	1.2606	0.5476	0.2777	0.0279
n = 2.0	2.1620	1.2873	0.5601	0.2841	0.0286
n = 5.0	2.1564	1.3159	0.5857	0.2911	0.0293
n = 8.0	2.1987	1.3785	0.5883	0.2986	0.0300

4.4. The magneto-electro-elastic circular plate with a circular cutout

To demonstrate the serviceability of the proposed technique, a functionally graded magneto-electro-elastic perforated circular plate with material properties graded along the radial direction is shown as an example. The outer radius of the perforated plate is R_O = 1.0 m and the inner one is R_i = 0.4R_O, as displayed in Figure 10. The outer boundary is fully clamped, which means that u_z = u_x = u_y = Φ = Ψ = 0; However, the inner one is completely free. In this example, the same characteristic length l = 2R_O, thickness-to-length ratios t/l = 0.1, 0.05, 0.02, 0.01, and 0.001, the materials CoFe₂O₄ and BaTiO₃ of type b, the normalized expression of natural frequencies and the grounded boundary conditions on the top and bottom surfaces as the above numerical example are adopted. To reveal the influence of gradient functions, variations of material constants obey the following two cosine formulae: Type 1, h(r) =1−αcos(πr/R_O−π/2) and Type 2, h(r) = 1−αcos(πr/2R_O). The non-dimensional fundamental frequencies of the in-plane functionally graded magneto-electro-elastic perforated plate under various aspect ratios and graded indexes are described in Tables 21 to 24. As thickness-to-length ratios of the perforated circular plates become smaller, its relevant flexural vibration frequencies decrease, which is the same as the foregoing finding. Similarly, it can be noticed that regardless of forms of the gradation functions, the increase in the value of the gradient parameter α leads to increasing the dimensionless eigenvalues. When the composed materials and gradation indexes are same, it is essential to point out that eigensolutions of the plates with Type 1 are larger than those with Type 2. So to design and apply the in-plane functionally graded magneto-electro-elastic perforated circular plates in practical engineering, the influence of gradation functions on distributions of the transverse vibration frequencies should not be neglected.

Figure 10.

The in-plane functionally graded magneto-electro-elastic perforated plate: (a) the circular plate with a circular cutout and (b) meshed with four spectral elements.

Table 21.

Eigensolutions of the perforated BaTiO₃ plate with Type 1.

t/l	0.1	0.05	0.02	0.01	0.001
α = 0.0	0.8260	0.4805	0.2055	0.1040	0.0104
α = 0.1	0.8484	0.4937	0.2111	0.1068	0.0107
α = 0.2	0.8744	0.5092	0.2177	0.1101	0.0111
α = 0.3	0.9052	0.5276	0.2255	0.1140	0.0114
α = 0.4	0.9425	0.5501	0.2351	0.1189	0.0119
α = 0.5	0.9889	0.5783	0.2472	0.1250	0.0125

Table 22.

Eigensolutions of the perforated BaTiO₃ plate with Type 2.

t/l	0.1	0.05	0.02	0.01	0.001
α = 0.0	0.8260	0.4805	0.2055	0.1040	0.0104
α = 0.1	0.8427	0.4906	0.2099	0.1061	0.0107
α = 0.2	0.8612	0.5018	0.2147	0.1086	0.0109
α = 0.3	0.8820	0.5145	0.2202	0.1114	0.0112
α = 0.4	0.9055	0.5290	0.2265	0.1145	0.0115
α = 0.5	0.9325	0.5456	0.2337	0.1182	0.0119

Table 23.

Eigenvalues of the perforated CoFe₂O₄ plate with Type 1.

t/l	0.1	0.05	0.02	0.01	0.001
α = 0.0	0.9518	0.5619	0.2438	0.1237	0.0124
α = 0.1	0.9775	0.5773	0.2503	0.1270	0.0128
α = 0.2	1.0074	0.5954	0.2580	0.1309	0.0132
α = 0.3	1.0428	0.6169	0.2671	0.1355	0.0136
α = 0.4	1.0856	0.6431	0.2784	0.1411	0.0142
α = 0.5	1.1388	0.6760	0.2925	0.1483	0.0149

Table 24.

Eigenvalues of the perforated CoFe₂O₄ plate with Type 2.

t/l	0.1	0.05	0.02	0.01	0.001
α = 0.0	0.9518	0.5619	0.2438	0.1237	0.0124
α = 0.1	0.9709	0.5736	0.2489	0.1263	0.0127
α = 0.2	0.9922	0.5868	0.2546	0.1292	0.0130
α = 0.3	1.0161	0.6016	0.2610	0.1325	0.0133
α = 0.4	1.0430	0.6184	0.2684	0.1362	0.0137
α = 0.5	1.0739	0.6378	0.2769	0.1405	0.0142

5. Conclusions

In this paper, the SBFEM in conjunction with the PIA is employed to investigate the transverse free vibration characteristics of in-plane functionally graded magneto-electro-elastic plates with arbitrary gradation functions. The general solution of the ordinary differential SBFEM governing equation is analytically expressed as the matrix exponent. The approach of PIA is applied to create the stiffness matrix, which can make sure high accuracy. Aided by the kinetic energy method, the mass matrix of the in-plane functionally graded magneto-electro-elastic plates based on the SBFEM for the first time is constructed.

According to the first three numerical examples, the high accuracy, serviceability, and effectiveness of the proposed technique to examine the free vibration behaviors of magneto-electro-elastic and in-plane functionally graded plates are verified. As for the in-plane functionally graded magneto-electro-elastic rectangular plates, with increasing the gradient exponents the dimensionless natural frequencies decrease. The vibration frequencies of the CFSF and SSFF plates will reach the highest and lowest values. With respect to the circular plate, increasing the gradient index γ results in the increase of frequency parameters. The thicker the plate is, the bigger its fundamental frequencies are. When the plate is under the power-law gradient function, with increasing the gradient parameter n, the vibration frequencies will increase. Regarding the perforated circular plate, eigensolutions of the plates with Type 1 are larger than those with Type 2. Therefore, a conclusion that boundary conditions, gradient indexes, thickness-to-length ratios, and forms of gradation functions play a significant role on the free vibration responses of in-plane functionally graded magneto-electro-elastic plates can be drawn. More meaningful results will be provided in the future.

Footnotes

Appendix 1

The detailed expressions for the constitutive relation between the stresses, electric displacements, magnetic induction field and strains, electric field, magnetic field from Guan and He (2006) are formulated as

(1.1)

\begin{matrix} σ_{zz} = c_{33} (x, y) ε_{zz} + c_{13} (x, y) (ε_{xx} + ε_{yy}) - e_{33} (x, y) E_{z} \\ - q_{33} (x, y) H_{z} \end{matrix}

(1.2)

\begin{matrix} σ_{xx} = c_{13} (x, y) ε_{zz} + c_{11} (x, y) ε_{xx} + c_{12} (x, y) ε_{yy} \\ - e_{31} (x, y) E_{z} - q_{31} (x, y) H_{z} \end{matrix}

(1.3)

\begin{matrix} σ_{yy} = c_{13} (x, y) ε_{zz} + c_{12} (x, y) ε_{xx} + c_{11} (x, y) ε_{yy} \\ - e_{31} (x, y) E_{z} - q_{31} (x, y) H_{z} \end{matrix}

(1.4)

τ_{xy} = 2 c_{66} (x, y) ε_{xy}

(1.5)

τ_{yz} = 2 c_{44} (x, y) ε_{yz} - e_{15} (x, y) E_{y} - q_{15} (x, y) H_{y}

(1.6)

τ_{xz} = 2 c_{44} (x, y) ε_{xz} - e_{15} (x, y) E_{x} - q_{15} (x, y) H_{x}

(1.7)

\begin{matrix} D_{z} = e_{33} (x, y) ε_{zz} + e_{31} (x, y) (ε_{xx} + ε_{yy}) + ν_{33} (x, y) E_{z} \\ + d_{33} (x, y) H_{z} \end{matrix}

(1.8)

D_{x} = 2 e_{15} (x, y) ε_{xz} + ν_{11} (x, y) E_{x} + d_{11} (x, y) H_{x}

(1.9)

D_{y} = 2 e_{15} (x, y) ε_{yz} + ν_{11} (x, y) E_{y} + d_{11} (x, y) H_{y}

(1.10)

\begin{matrix} B_{z} = q_{33} (x, y) ε_{zz} + q_{31} (x, y) (ε_{xx} + ε_{yy}) + d_{33} (x, y) E_{z} \\ + μ_{33} (x, y) H_{z} \end{matrix}

(1.11)

B_{x} = 2 q_{15} (x, y) ε_{xz} + d_{11} (x, y) E_{x} + μ_{11} (x, y) H_{x}

(1.12)

B_{y} = 2 q_{15} (x, y) ε_{yz} + d_{11} (x, y) E_{y} + μ_{11} (x, y) H_{y}

Appendix 2

The coefficient matrices [b¹], [b²], and [b³] are expressed as

(2.1)

\begin{matrix} [b^{1}] = [\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}] [b^{2}] = \frac{1}{| J |} [\begin{matrix} 0 & 0 & 0 & 0 & 0 \\ 0 & y_{, ζ} & 0 & 0 & 0 \\ 0 & 0 & - x_{, ζ} & 0 & 0 \\ 0 & - x_{, ζ} & y_{, ζ} & 0 & 0 \\ - x_{, ζ} & 0 & 0 & 0 & 0 \\ y_{, ζ} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & y_{, ζ} & 0 \\ 0 & 0 & 0 & - x_{, ζ} & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & y_{, ζ} \\ 0 & 0 & 0 & 0 & - x_{, ζ} \end{matrix}] \\ [b^{3}] = \frac{1}{| J |} [\begin{matrix} 0 & 0 & 0 & 0 & 0 \\ 0 & - y_{, η} & 0 & 0 & 0 \\ 0 & 0 & x_{, η} & 0 & 0 \\ 0 & x_{, η} & - y_{, η} & 0 & 0 \\ x_{, η} & 0 & 0 & 0 & 0 \\ - y_{, η} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - y_{, η} & 0 \\ 0 & 0 & 0 & x_{, η} & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & - y_{, η} \\ 0 & 0 & 0 & 0 & x_{, η} \end{matrix}] \end{matrix}

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by Grants 2018M641168 from China Postdoctoral Science Foundation, Grants 51908022, 2015CB57805, and 51774018 from the National Natural Science Foundation of China, Grant JDYC20200309 from the Pyramid Talent Training Project of Beijing University of Civil Engineering and Architecture, for which the authors are gratefully acknowledged.

ORCID iD

Pengchong Zhang

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