Buckling analysis of piezoelectrically actuated smart nanoscale plates subjected to magnetic field

Abstract

Static stability analysis of a size-dependent magneto-electro-elastic functionally graded nanoplate has an immense contribution in identification and improvement of the performance of nano-electro-mechanical systems. A refined trigonometric plate theory is employed to formulate the magneto-electro-elastic functionally graded nanoplate for the first time. Magneto-electro-elastic properties of nanoplate change in spatial coordinate based on power-law form. Regarding the small-scale effects at nanoscales, the size-dependent nonlocal continuum theory is employed to derive governing equations of the nonclassical magneto-electro-elastic functionally graded nanoplate. Analytical solution possessing functions which satisfy different boundary conditions is adopted to solve the equations. The results illustrate the size-dependent buckling behavior of magneto-electro-elastic functionally graded nanoplate affected by magnetic potential, electric voltage, various boundary conditions, small-scale parameter, material composition, plate side-to-thickness ratio, and aspect ratio.

Keywords

Magneto-electro-elastic nanoplate functionally graded material buckling refined plate theory nonlocal elasticity

Introduction

In the domain of materials science, some recent progressions are the intelligent materials in which piezoelectric and piezomagnetic phases are involved. These materials called magneto-electro-elastic composites have the potency of transforming energy from magnetic, electric, and mechanical forms to the other. Recently, magneto-electro-elastic structures are produced with graded material properties. In such structures, the continuous variation of compositions leads to a smoothing change in mechanical property. Therefore, researchers have carried out a series of in-depth studies on the mechanical analysis of magneto-electro-elastic functionally graded (MEE-FG) structures within the framework of continuum mechanics. Bhangale and Ganesan (2005) investigated free vibration response of functionally graded magneto-electro-elastic finite cylindrical shells. A compressive analysis of the magneto-electro-elastic plates with graded properties is presented by Pan and Han (2005). Also, an analytical investigation of mechanical response of MEE-FG beams is performed by Huang et al. (2007). In another survey, Wu and Tsai (2007) examined the static behavior of a doubly curved MEE-FG shell employing an asymptotic approach. Wu et al. (2010) investigated the static analysis of MEE-FG plates implementing the modified Pagano method. Kattimani and Ray (2015) researched large amplitude vibration responses of MEE-FG plates. Static behavior of a circular MEE-FG plate is analyzed by Sladek et al. (2015) using a meshless method. Bending behavior of layered FG neutral magneto-electro-elastic plates on elastic foundations is analyzed by Lezgy-Nazargah and Cheraghi (2015).

With the trends toward the miniaturization of systems, the MEE nanomaterials (e.g. NiFe₂O₄–PZT and BiTiO₃–CoFe₂O₄) and their nanostructures (e.g. nanoplates and nanofilms) have gained remarkable attention from several investigators. By possessing smaller size and larger surface-to-volume ratio, MEE nanoscale structures provide excellent magneto-electrical coupling and gained potential applications in nano-electromechanical systems (NEMs). Classical magneto-electro-mechanical continuum theories were implemented in the primary investigations (Pan and Han, 2005). Size-dependent material properties which are recognized as unique physical properties of magneto-electro-elastic nanoscale structures exceed the limit of classical theories. Therefore, the classical magneto-electro-mechanical theory would not be adequate to explore the questions raised in nanomechanics. Hence, in order to model the magneto-electro-elastic nanostructures accurately, nonlocal theories are employed in various researches (Eringen, 1983; Eringen and Edelen, 1972). Ke and Wang (2014) explored frequency response of MEE nanoscale beams implementing nonlocal elasticity theory. They supposed that external electric potential, magnetic potential, and uniform temperature rise are exerted to the MEE nanobeam. In another study, Ke et al. (2014) researched vibrational behavior of MEE nanoscale plates based on nonlocal Kirchhoff plate model. Li et al. (2014) investigated the stability and vibration of a MEE homogenous nanoplate in elastic medium based on nonlocal Mindlin plate model. Most recently, Ansari et al. (2015) analyzed the forced vibration responses of size-dependent MEE nanobeams in thermal environment based on the nonlocal third-order beam theory. Also, Jandaghian and Rahmani (2016) examined the free vibration analysis of magneto-electro-thermo-elastic nanobeams resting on a Pasternak foundation.

Nonlocal elasticity theory has been extensively employed to analyze the static and dynamic behaviors of functionally graded nanostructures. In fact, the application of functionally graded materials (FGMs) has become very appealing in the nanosize structures as these materials can be designed for peculiar performance. Stability analysis of FG nanoscale beams using nonlocal elasticity theory is performed by Şimşek and Yurtcu (2013). Also, Ebrahimi and Salari (2015a, 2015c) explored the thermal effects on mechanical response of nonlocal temperature-dependent FGM nanobeams. Zare et al. (2015) investigated the frequency response of a nanoscale FGM plate considering different boundary conditions employing an analytical solution. Recently, due to the impotency of classical beam and plate theories in determining transverse shear deformations, as well as dependency of first-order theory on shear correction factors, some researchers have proposed higher order theories for more accurate modeling of FGM structures. Ebrahimi and Barati (2016a) applied a nonlocal third-order beam model for vibrational analysis of FG nanobeams. Most recently, Zemri et al. (2015) proposed a nonlocal shear deformable refined theory for investigating the mechanical behavior of FG nanobeams. Mahmoud et al. (2015) developed a shear and normal deformable beam model for bending and buckling analyses of FGM nanoscale beams. Also, Barati et al. (2016) employed an inverse cotangential refined theory to study the thermal buckling behavior of nonlocal FG plates on elastic foundation. Barati and Shahverdi (2016) presented an analytical solution for thermal vibration of compositionally graded nanoplates with arbitrary boundary conditions based on physical neutral surface position. In the case of smart size-dependent FGM structures, Ebrahimi and Salari (2015b) investigated the size-dependent thermo-electrical buckling analysis of functionally graded piezoelectric nanobeams. Also, Beni (2016) analyzed the electromechanical bending, buckling, and free vibration analysis of functionally graded piezoelectric nanobeams. Most recently, Ebrahimi and Barati (2016c, 2016d) presented the dynamic modeling of a thermo–piezo-electrically actuated functionally graded nanosize beam subjected to a magnetic field. Ebrahimi and Barati (2016b) presented an exact solution for buckling analysis of embedded piezoelectro-magnetically actuated nanoscale beams.

A brief survey of the literature reveals that the buckling analysis of the MEE-FG nanoplates with arbitrary boundary conditions has not been covered thoroughly via nonlocal continuum mechanics until now. This article studies the buckling analysis of a MEE-FG nanoplate by developing a nonlocal higher order refined plate theory. Spatially graded material properties of the MEE-FG nanoplate are described via power-law function. The equations of motion and boundary conditions for MEE-FG nanoplate are derived using Hamilton’s principle and Eringen’s nonlocal elasticity theory. Analytical solution possessing functions which satisfy different boundary conditions is used to solve the equations. Obtained buckling loads of this article are validated with those of FGM nanoplate available in the literature. It is shown that buckling loads of MEE-FG nanoplates are influenced by magnetic potential, external electric voltage, material gradient index, boundary conditions, and plate geometrical parameters.

Theoretical formulations

The material properties of MEE-FG nanoplates

An MEE-FG nanoplate with length a, width b, and thickness h is considered as indicated in Figure 1. The MEE-FG nanoplate is subjected to a magnetic potential $γ (x, z, t)$ and an electric potential $Φ (x, z, t)$ . The effective material properties of MEE-FG plate change continuously in the thickness direction according to modified power-law distribution. The effective material properties can be expressed as

P = P_{2} V_{2} + P_{1} V_{1}

(1)

where $P_{2}$ and $P_{1}$ are the material properties of the top and bottom sides, respectively, and $V_{2}$ and $V_{1}$ are the volume fraction of the top and bottom surfaces, respectively, and are related by

V_{2} = {(\frac{z}{h} + \frac{1}{2})}^{p}, V_{1} = 1 - V_{2}

(2)

where $(p \geq 0)$ is the power-law exponent which determines the material distribution across the plate thickness.

Finally, the effective material properties of MEE-FG plates take the following form

P (z) = (P_{2} - P_{1}) {(\frac{z}{h} + \frac{1}{2})}^{p} + P_{1}

(3)

It must be noted that, the top surface at $z = + h / 2$ of MEE-FG nanoplate is fully $CoF e_{2} O_{4}$ , whereas the bottom surface ( $z = - h / 2$ ) is fully $BaTi o_{3}$ with the properties presented in Table 1. The magnetic permittivity coefficient value of −590 for $CoF e_{2} O_{4}$ was a long-term period where the negative value was used by various researchers until Pan (2002) approved from the positive energy point of view and corrected it. After that, the positive value of magnetic permittivity ( $χ_{11}$ ) should be considered.

Table 1.

Magneto-electro-elastic coefficients of material properties (Ramirez et al., 2006).

Properties	$BaTi O_{3}$	$CoF e_{2} O_{4}$
$c_{11} = c_{22} (GPa)$	166	286
$c_{33}$	162	269.5
$c_{13} = c_{23}$	78	170.5
$c_{12}$	77	173
$c_{55}$	43	45.3
$c_{66}$	44.5	56.5
$e_{31} (C m^{- 2})$	−4.4	0
$e_{33}$	18.6	0
$e_{15}$	11.6	0
$q_{31} (N / Am)$	0	580.3
$q_{33}$	0	699.7
$q_{15}$	0	550
$s_{11} (10^{- 9} C^{2} m^{- 2} N^{- 1})$	11.2	0.08
$s_{33}$	12.6	0.093
$χ_{11} (10^{- 6} N s^{2} C^{- 2} / 2)$	5	+590
$χ_{33}$	10	157
$d_{11} = d_{22} = d_{33}$	0	0

Theoretical formulation

The displacement field at any point of the plate according to four-unknown refined shear deformation plate model can be expressed as

u_{1} (x, y, z) = u (x, y) - z \frac{\partial w_{b}}{\partial x} - f (z) \frac{\partial w_{s}}{\partial x}

(4)

u_{2} (x, y, z) = v (x, y) - z \frac{\partial w_{b}}{\partial y} - f (z) \frac{\partial w_{s}}{\partial y}

(5)

u_{3} (x, y, z) = w_{b} (x, y) + w_{s} (x, y)

(6)

where u and v are the displacement of mid-plane along x- and y-axis, respectively, and $w_{b}$ and $w_{s}$ are the bending and shear components of transverse displacement of a point on the mid-plane of the plate, respectively. $f (z)$ denotes a shape function estimating the distribution of shear stress across the plate thickness. So, it is not required to use any shear correction factor. The present theory has a function in the form (Mantari et al., 2014)

f (z) = z - \frac{\sin (ξ z)}{ξ}

(7)

The electric potential and magnetic potential distributions across the thickness are approximated via a combination of a cosine and linear variation to satisfy Maxwell’s equation in the quasi-static approximation as follows (Ke and Wang, 2014)

Φ (x, y, z) = - \cos (ξ z) ϕ (x, y) + \frac{2 z}{h} V

(8)

ϒ (x, y, z) = - \cos (ξ z) γ (x, y) + \frac{2 z}{h} Ω

(9)

where $ξ = π / h$ . Also, V and $Ω$ are the external electric voltage and magnetic potential applied to the MEE-FG plate. Nonzero strains of the four-variable plate model are expressed as

{\begin{array}{l} ε_{x} \\ ε_{y} \\ γ_{x y} \end{array}} = {\begin{array}{l} ε_{x}^{0} \\ ε_{y}^{0} \\ γ_{x y}^{0} \end{array}} + z {\begin{array}{l} κ_{x}^{b} \\ κ_{y}^{b} \\ κ_{x y}^{b} \end{array}} + f {\begin{array}{l} κ_{x}^{s} \\ κ_{y}^{s} \\ κ_{x y}^{s} \end{array}}, {\begin{array}{l} γ_{y z} \\ γ_{x z} \end{array}} = g {\begin{array}{l} γ_{y z}^{s} \\ γ_{x z}^{s} \end{array}}, g = 1 - \frac{\partial f}{\partial z}

(10)

where

{\begin{array}{l} ε_{x}^{0} \\ ε_{y}^{0} \\ γ_{x y}^{0} \end{array}} = {\begin{array}{l} \frac{\partial u}{\partial x} \\ \frac{\partial v}{\partial y} \\ \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \end{array}}, {\begin{array}{l} κ_{x}^{b} \\ κ_{y}^{b} \\ κ_{x y}^{b} \end{array}} = {\begin{array}{l} - \frac{\partial^{2} w_{b}}{\partial x^{2}} \\ - \frac{\partial^{2} w_{b}}{\partial y^{2}} \\ - 2 \frac{\partial^{2} w_{b}}{\partial x \partial y} \end{array}}, {\begin{array}{l} κ_{x}^{s} \\ κ_{y}^{s} \\ κ_{x y}^{s} \end{array}} = {\begin{array}{l} - \frac{\partial^{2} w_{s}}{\partial x^{2}} \\ - \frac{\partial^{2} w_{s}}{\partial y^{2}} \\ - 2 \frac{\partial^{2} w_{s}}{\partial x \partial y} \end{array}}, {\begin{array}{l} γ_{y z}^{s} \\ γ_{x z}^{s} \end{array}} = {\begin{array}{l} \frac{\partial w_{s}}{\partial y} \\ \frac{\partial w_{s}}{\partial x} \end{array}}

(11)

According to equation (8), the relation between the electric field ( $E_{x}, E_{y}, E_{z}$ ) and the electric potential $(Φ)$ can be obtained as

E_{x} = - Φ_{, x} = \cos (ξ z) \frac{\partial ϕ}{\partial x}

(12)

E_{y} = - Φ_{, y} = \cos (ξ z) \frac{\partial ϕ}{\partial y},

(13)

E_{z} = - Φ_{, z} = - ξ \sin (ξ z) ϕ - \frac{2 V}{h}

(14)

Also, the relation between the magnetic field $(H_{x}, H_{y}, H_{z})$ and the magnetic potential $(ϒ)$ can be expressed from equation (9) as

H_{x} = - ϒ_{, x} = \cos (ξ z) \frac{\partial γ}{\partial x}

(15)

H_{y} = - ϒ_{, y} = \cos (ξ z) \frac{\partial γ}{\partial y}

(16)

H_{z} = - ϒ_{, z} = - ξ \sin (ξ z) γ - \frac{2 Ω}{h}

(17)

Hamilton’s principle is used to derive the static equilibrium equation

\int_{0}^{t} δ (Π_{S} + Π_{W}) dt = 0

(18)

Here, $Π_{S}$ is the strain energy and $Π_{W}$ is the work done by external forces. The virtual variation of strain energy can be written as

\begin{matrix} δ Π_{S} = \int_{v} σ_{ij} δ ε_{ij} dV \\ = \int_{v} (σ_{x} δ ε_{x} + σ_{y} δ ε_{y} + σ_{xy} δ γ_{xy} + σ_{yz} δ γ_{yz} + σ_{xz} δ γ_{xz} \\ - D_{x} δ E_{x} - D_{y} δ E_{y} - D_{z} δ E_{z} - B_{x} δ H_{x} - B_{y} δ H_{y} - B_{z} δ H_{z}) dV \end{matrix}

(19)

Substituting equations (10) and (11) into equation (19) yields

\begin{matrix} δ Π_{S} = & \int_{0}^{a} \int_{0}^{b} [N_{x} \frac{\partial δ u}{\partial x} - M_{x}^{b} \frac{\partial^{2} δ w_{b}}{\partial x^{2}} - M_{x}^{s} \frac{\partial^{2} δ w_{s}}{\partial x^{2}} + N_{y} \frac{\partial δ v}{\partial y} - M_{y}^{b} \frac{\partial^{2} δ w_{b}}{\partial y^{2}} - M_{y}^{s} \frac{\partial^{2} δ w_{s}}{\partial y^{2}} \\ + N_{xy} (\frac{\partial δ u}{\partial y} + \frac{\partial δ v}{\partial x}) - 2 M_{xy}^{b} \frac{\partial^{2} δ w_{b}}{\partial x \partial y} - 2 M_{xy}^{s} \frac{\partial^{2} δ w_{s}}{\partial x \partial y} + Q_{yz} \frac{\partial δ w_{s}}{\partial y} + Q_{xz} \frac{\partial δ w_{s}}{\partial x}] dxdy \\ + \int_{0}^{a} \int_{0}^{b} \int_{- h / 2}^{h / 2} [- D_{x} \cos (ξ z) δ (\frac{\partial ϕ}{\partial x}) - D_{y} \cos (ξ z) δ (\frac{\partial ϕ}{\partial y}) + D_{z} ξ \sin (ξ z) δ ϕ \\ - B_{x} \cos (ξ z) δ (\frac{\partial γ}{\partial x}) - B_{y} \cos (ξ z) δ (\frac{\partial γ}{\partial y}) + B_{z} ξ \sin (ξ z) δ γ] dzdxdy \end{matrix}

(20)

In which the variables in the last expression are expressed as

(N_{i}, M_{i}^{b}, M_{i}^{s}) = \int_{A} (1, z, f) σ_{i} dA, i = (x, y, xy)

(21a)

Q_{i} = \int_{A} g σ_{i} dA, i = (xz, yz)

(21b)

The first variation of work done by applied forces can be written in the form

\begin{matrix} δ Π_{W} = \int_{0}^{a} \int_{0}^{b} (N_{x}^{0} \frac{\partial (w_{b} + w_{s})}{\partial x} \frac{\partial δ (w_{b} + w_{s})}{\partial x} + N_{y}^{0} \frac{\partial (w_{b} + w_{s})}{\partial y} \frac{\partial δ (w_{b} + w_{s})}{\partial y} \\ + 2 δ N_{x y}^{0} \frac{\partial (w_{b} + w_{s})}{\partial x} \frac{\partial (w_{b} + w_{s})}{\partial y}) d x d y \end{matrix}

(22)

where $N_{x}^{0}, N_{y}^{0}, and N_{xy}^{0}$ are the in-plane applied loads. In this study, it is assumed that the MEE-FG nanoplate is under external electric voltage, magnetic potential, and the shear loading is ignored. So, $N_{xy}^{0} = 0$ and $N_{x}^{0}, N_{y}^{0}$ are the normal forces induced by external electric voltage V and external magnetic potential $Ω$ , respectively, which are defined as

N_{x}^{0} = N_{y}^{0} = N^{b} + N^{E} + N^{H}

(23a)

N^{E} = - \int_{- h / 2}^{h / 2} {\tilde{e}}_{31} \frac{2 V}{h} dz, N^{H} = - \int_{- h / 2}^{h / 2} {\tilde{q}}_{31} \frac{2 Ω}{h} dz

(23b)

The following Euler–Lagrange equations are obtained by substituting equations (20) and (22) in equation (18) when the coefficients of $δ u, δ v, δ w_{b}, δ w_{s}, δ ϕ$ , and $δ γ$ are equal to 0

\frac{\partial N_{x}}{\partial x} + \frac{\partial N_{xy}}{\partial y} = 0

(24)

\frac{\partial N_{xy}}{\partial x} + \frac{\partial N_{y}}{\partial y} = 0

(25)

\begin{array}{l} \frac{\partial^{2} M_{x}^{b}}{\partial x^{2}} + 2 \frac{\partial^{2} M_{x y}^{b}}{\partial x \partial y} + \frac{\partial^{2} M_{y}^{b}}{\partial y^{2}} \\ - (N^{b} + N^{E} + N^{H}) \nabla^{2} (w_{b} + w_{s}) = 0 \end{array}

(26)

\begin{array}{l} \frac{\partial^{2} M_{x}^{s}}{\partial x^{2}} + 2 \frac{\partial^{2} M_{x y}^{s}}{\partial x \partial y} + \frac{\partial^{2} M_{y}^{s}}{\partial y^{2}} + \frac{\partial Q_{x z}}{\partial x} \\ + \frac{\partial Q_{y z}}{\partial y} - (N^{b} + N^{E} + N^{H}) \nabla^{2} (w_{b} + w_{s}) = 0 \end{array}

(27)

\int_{- h / 2}^{h / 2} (\cos (ξ z) \frac{\partial D_{x}}{\partial x} + \cos (ξ z) \frac{\partial D_{y}}{\partial y} + ξ \sin (ξ z) D_{z}) dz = 0

(28)

\int_{- h / 2}^{h / 2} (\cos (ξ z) \frac{\partial B_{x}}{\partial x} + \cos (ξ z) \frac{\partial B_{y}}{\partial y} + ξ \sin (ξ z) B_{z}) dz = 0

(29)

and the associated boundary conditions

u = 0, or N_{x} n_{x} + N_{xy} n_{y} = 0

(30a)

v = 0, or N_{xy} n_{x} + N_{y} n_{y} = 0

(30b)

\begin{matrix} w_{b} = & 0, or n_{x} (\frac{\partial M_{x}^{b}}{\partial x} + \frac{\partial M_{xy}^{b}}{\partial y} - (N^{b} + N^{E} + N^{H}) \frac{\partial (w_{b} + w_{s})}{\partial x}) \\ + n_{y} (\frac{\partial M_{y}^{b}}{\partial y} + \frac{\partial M_{xy}^{b}}{\partial x} - (N^{b} + N^{E} + N^{H}) \frac{\partial (w_{b} + w_{s})}{\partial y}) \\ + (n_{x} \frac{\partial M_{ns}^{b}}{\partial y} - n_{y} \frac{\partial M_{ns}^{b}}{\partial x}) = 0 \end{matrix}

(30c)

\frac{\partial w_{b}}{\partial x} = 0, or M_{x}^{b} n_{x} + M_{xy}^{b} n_{y} = 0

(30d)

\frac{\partial w_{b}}{\partial y} = 0, or M_{xy}^{b} n_{x} + M_{y}^{b} n_{y} = 0

(30e)

\begin{matrix} w_{s} = & 0, or n_{x} (\frac{\partial M_{x}^{s}}{\partial x} + \frac{\partial M_{xy}^{s}}{\partial y} - (N^{b} + N^{E} + N^{H}) \frac{\partial w_{s}}{\partial x} + Q_{xz}) \\ + n_{y} (\frac{\partial M_{y}^{s}}{\partial y} + \frac{\partial M_{xy}^{s}}{\partial x} - (N^{b} + N^{E} + N^{H}) \frac{\partial w_{s}}{\partial y} + Q_{yz}) \\ + (n_{x} \frac{\partial M_{ns}^{s}}{\partial y} - n_{y} \frac{\partial M_{ns}^{s}}{\partial x}) = 0 \end{matrix}

(30f)

\frac{\partial w_{s}}{\partial x} = 0, or M_{x}^{s} n_{x} + M_{xy}^{s} n_{y} = 0

(30g)

\frac{\partial w_{s}}{\partial y} = 0, or M_{xy}^{s} n_{x} + M_{y}^{s} n_{y} = 0

(30h)

ϕ = 0, or \int_{- h / 2}^{h / 2} (\cos (ξ z) D_{x} n_{x} + \cos (ξ z) D_{y} n_{y}) dz = 0

(30i)

γ = 0, or \int_{- h / 2}^{h / 2} (\cos (ξ z) B_{x} n_{x} + \cos (ξ z) B_{y} n_{y}) dz = 0

(30j)

where

M_{ns}^{b} = (M_{y}^{b} - M_{x}^{b}) n_{x} n_{y} + M_{xy}^{b} (n_{x}^{2} - n_{y}^{2})

(31a)

M_{ns}^{s} = (M_{y}^{s} - M_{x}^{s}) n_{x} n_{y} + M_{xy}^{s} (n_{x}^{2} - n_{y}^{2})

(31b)

Nonlocal elasticity theory for the magneto-electro-elastic materials

The main point of the nonlocal elasticity theory is that the nonlocal stress tensor at a reference point relies not only on the strain tensor of the same coordinate but also on other points in the solid. For a nonlocal magneto-electro-elastic plate, the basic equations may be defined as

σ_{ij} = \int_{V} α (| x' - x |, τ) [c_{ijkl} ε_{kl} (x') - e_{mij} E_{m} (x') - q_{nij} H_{n} (x')] dV (x')

(32a)

D_{i} = \int_{V} α (| x' - x |, τ) [e_{ikl} ε_{kl} (x') + s_{im} E_{m} (x') + d_{in} H_{n} (x')] dV (x')

(32b)

B_{i} = \int_{V} α (| x' - x |, τ) [q_{ikl} ε_{kl} (x') + d_{im} E_{m} (x') + χ_{in} H_{n} (x')] dV (x')

(32c)

where $σ_{ij}$ , $D_{i}$ , and $B_{i}$ denote the components of stress, electric displacement, and magnetic induction, respectively; $ε_{kl}$ , $E_{m}$ , and $H_{n}$ are the components of linear strain, electric field, and magnetic field, respectively. Additionally, $c_{ijkl}$ , $s_{im}$ , and $χ_{in}$ are the components of elastic stiffness, dielectric permittivity, and magnetic permittivity coefficients, respectively; Finally, $e_{mij}$ , $q_{nij}$ , and $d_{in}$ are the piezoelectric, piezomagnetic, and magneto-electric-elastic coefficients, respectively. $α (| x' - x |, τ)$ is the nonlocal kernel function and $| x' - x |$ is the Euclidean distance. $τ = e_{0} a / l$ is defined as the scale coefficient, where $e_{0}$ is a material constant which is determined experimentally or approximated by matching the dispersion curves of plane waves with those of atomic lattice dynamics; and a and l are the internal and external characteristic lengths of the nanostructures, respectively. Finally, it is possible to represent the integral constitutive relations given by equation (32) in an equivalent differential form as (Ke et al., 2014)

σ_{ij} - (e_{0} a)^{2} \nabla^{2} σ_{ij} = c_{ijkl} ε_{kl} - e_{mij} E_{m} - q_{nij} H_{n}

(33a)

D_{i} - (e_{0} a)^{2} \nabla^{2} D_{i} = e_{ikl} ε_{kl} + s_{im} E_{m} + d_{in} H_{n}

(33b)

B_{i} - (e_{0} a)^{2} \nabla^{2} B_{i} = q_{ikl} ε_{kl} + d_{im} E_{m} + χ_{in} H_{n}

(33c)

where $\nabla^{2}$ is the Laplacian operator. The constitutive equations can be expressed as

(1 - μ \nabla^{2}) σ_{xx} = {\tilde{c}}_{11} ε_{xx} + {\tilde{c}}_{12} ε_{yy} - {\tilde{e}}_{31} E_{z} - {\tilde{q}}_{31} H_{z}

(34)

(1 - μ \nabla^{2}) σ_{yy} = {\tilde{c}}_{12} ε_{xx} + {\tilde{c}}_{11} ε_{yy} - {\tilde{e}}_{31} E_{z} - {\tilde{q}}_{31} H_{z}

(35)

(1 - μ \nabla^{2}) σ_{xy} = {\tilde{c}}_{66} γ_{xy}

(36)

(1 - μ \nabla^{2}) σ_{xz} = {\tilde{c}}_{55} γ_{xz} - {\tilde{e}}_{15} E_{x} - {\tilde{q}}_{15} H_{x}

(37)

(1 - μ \nabla^{2}) σ_{yz} = {\tilde{c}}_{55} γ_{yz} - {\tilde{e}}_{15} E_{y} - {\tilde{q}}_{15} H_{y}

(38)

(1 - μ \nabla^{2}) D_{x} = {\tilde{e}}_{15} γ_{xz} + {\tilde{s}}_{11} E_{x} + {\tilde{d}}_{11} H_{x}

(39)

(1 - μ \nabla^{2}) D_{y} = {\tilde{e}}_{15} γ_{yz} + {\tilde{s}}_{11} E_{y} + {\tilde{d}}_{11} H_{y}

(40)

(1 - μ \nabla^{2}) D_{z} = {\tilde{e}}_{31} ε_{xx} + {\tilde{e}}_{31} ε_{yy} + {\tilde{s}}_{33} E_{z} + {\tilde{d}}_{33} H_{z}

(41)

(1 - μ \nabla^{2}) B_{x} = {\tilde{q}}_{15} γ_{xz} + {\tilde{d}}_{11} E_{x} + {\tilde{χ}}_{11} H_{x}

(42)

(1 - μ \nabla^{2}) B_{y} = {\tilde{q}}_{15} γ_{yz} + {\tilde{d}}_{11} E_{y} + {\tilde{χ}}_{11} H_{y}

(43)

(1 - μ \nabla^{2}) B_{z} = {\tilde{q}}_{31} ε_{xx} + {\tilde{q}}_{31} ε_{yy} + {\tilde{d}}_{33} E_{z} + {\tilde{χ}}_{33} H_{z}

(44)

where ${\tilde{c}}_{ij}, {\tilde{e}}_{ij}, {\tilde{q}}_{ij}, {\tilde{d}}_{ij}, {\tilde{k}}_{ij}$ , and ${\tilde{χ}}_{ij}$ are the reduced constants for the FG plate under the plane stress state (Ke and Wang, 2014) which are given as

\begin{matrix} {\tilde{c}}_{11} = c_{11} - \frac{c_{13}^{2}}{c_{33}}, {\tilde{c}}_{12} = c_{12} - \frac{c_{13}^{2}}{c_{33}}, \\ {\tilde{c}}_{66} = c_{66}, {\tilde{e}}_{15} = e_{15}, {\tilde{e}}_{31} = e_{31} - \frac{c_{13} e_{33}}{c_{33}} \\ {\tilde{q}}_{15} = q_{15}, {\tilde{q}}_{31} = q_{31} - \frac{c_{13} q_{33}}{c_{33}}, \\ {\tilde{d}}_{11} = {\tilde{d}}_{11}, {\tilde{d}}_{33} = {\tilde{d}}_{33} + \frac{q_{33} e_{33}}{c_{33}} \\ {\tilde{s}}_{11} = {\tilde{s}}_{11}, {\tilde{s}}_{33} = {\tilde{s}}_{33} + \frac{e_{33}^{2}}{c_{33}}, \\ {\tilde{χ}}_{11} = χ_{11}, {\tilde{χ}}_{33} = χ_{33} + \frac{q_{33}^{2}}{c_{33}} \end{matrix}

(45)

By integrating equations (34) to (44) over the area of plate cross section, the following relations for the force–strain and the moment–strain and other necessary relation of the refined FG plate can be obtained

\begin{matrix} (1 - μ \nabla^{2}) {\begin{matrix} N_{x} \\ N_{y} \\ N_{xy} \end{matrix}} = (\begin{matrix} A_{11} & A_{12} & 0 \\ A_{12} & A_{22} & 0 \\ 0 & 0 & A_{66} \end{matrix}) {\begin{matrix} \frac{\partial u}{\partial x} \\ \frac{\partial v}{\partial y} \\ \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \end{matrix}} \\ + (\begin{matrix} B_{11} & B_{12} & 0 \\ B_{12} & B_{22} & 0 \\ 0 & 0 & B_{66} \end{matrix}) {\begin{matrix} - \frac{\partial^{2} w_{b}}{\partial x^{2}} \\ - \frac{\partial^{2} w_{b}}{\partial y^{2}} \\ - 2 \frac{\partial^{2} w_{b}}{\partial x \partial y} \end{matrix}} \\ + (\begin{matrix} B_{11}^{s} & B_{12}^{s} & 0 \\ B_{12}^{s} & B_{22}^{s} & 0 \\ 0 & 0 & B_{66}^{s} \end{matrix}) {\begin{matrix} - \frac{\partial^{2} w_{s}}{\partial x^{2}} \\ - \frac{\partial^{2} w_{s}}{\partial y^{2}} \\ - 2 \frac{\partial^{2} w_{s}}{\partial x \partial y} \end{matrix}} \\ + {\begin{matrix} A_{31}^{e} \\ A_{31}^{e} \\ 0 \end{matrix}} ϕ + {\begin{matrix} A_{31}^{m} \\ A_{31}^{m} \\ 0 \end{matrix}} γ - {\begin{matrix} N_{x}^{E} \\ N_{y}^{E} \\ 0 \end{matrix}} - {\begin{matrix} N_{x}^{H} \\ N_{y}^{H} \\ 0 \end{matrix}} \end{matrix}

(46)

\begin{matrix} (1 - μ \nabla^{2}) {\begin{matrix} M_{x}^{b} \\ M_{y}^{b} \\ M_{xy}^{b} \end{matrix}} = (\begin{matrix} B_{11} & B_{12} & 0 \\ B_{12} & B_{22} & 0 \\ 0 & 0 & B_{66} \end{matrix}) {\begin{matrix} \frac{\partial u}{\partial x} \\ \frac{\partial v}{\partial y} \\ \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \end{matrix}} \\ + (\begin{matrix} D_{11} & D_{12} & 0 \\ D_{12} & D_{22} & 0 \\ 0 & 0 & D_{66} \end{matrix}) {\begin{matrix} - \frac{\partial^{2} w_{b}}{\partial x^{2}} \\ - \frac{\partial^{2} w_{b}}{\partial y^{2}} \\ - 2 \frac{\partial^{2} w_{b}}{\partial x \partial y} \end{matrix}} \\ + (\begin{matrix} D_{11}^{s} & D_{12}^{s} & 0 \\ D_{12}^{s} & D_{22}^{s} & 0 \\ 0 & 0 & D_{66}^{s} \end{matrix}) {\begin{matrix} - \frac{\partial^{2} w_{s}}{\partial x^{2}} \\ - \frac{\partial^{2} w_{s}}{\partial y^{2}} \\ - 2 \frac{\partial^{2} w_{s}}{\partial x \partial y} \end{matrix}} \\ + {\begin{matrix} E_{31}^{e} \\ E_{31}^{e} \\ 0 \end{matrix}} ϕ + {\begin{matrix} E_{31}^{m} \\ E_{31}^{m} \\ 0 \end{matrix}} γ - {\begin{matrix} M_{bx}^{E} \\ M_{by}^{E} \\ 0 \end{matrix}} - {\begin{matrix} M_{bx}^{H} \\ M_{by}^{H} \\ 0 \end{matrix}} \end{matrix}

(47)

\begin{matrix} (1 - μ \nabla^{2}) {\begin{matrix} M_{x}^{s} \\ M_{y}^{s} \\ M_{xy}^{s} \end{matrix}} = (\begin{matrix} B_{11}^{s} & B_{12}^{s} & 0 \\ B_{12}^{s} & B_{22}^{s} & 0 \\ 0 & 0 & B_{66}^{s} \end{matrix}) {\begin{matrix} \frac{\partial u}{\partial x} \\ \frac{\partial v}{\partial y} \\ \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \end{matrix}} \\ + (\begin{matrix} D_{11}^{s} & D_{12}^{s} & 0 \\ D_{12}^{s} & D_{22}^{s} & 0 \\ 0 & 0 & D_{66}^{s} \end{matrix}) {\begin{matrix} - \frac{\partial^{2} w_{b}}{\partial x^{2}} \\ - \frac{\partial^{2} w_{b}}{\partial y^{2}} \\ - 2 \frac{\partial^{2} w_{b}}{\partial x \partial y} \end{matrix}} \\ + (\begin{matrix} H_{11}^{s} & H_{12}^{s} & 0 \\ H_{12}^{s} & H_{22}^{s} & 0 \\ 0 & 0 & H_{66}^{s} \end{matrix}) {\begin{matrix} - \frac{\partial^{2} w_{s}}{\partial x^{2}} \\ - \frac{\partial^{2} w_{s}}{\partial y^{2}} \\ - 2 \frac{\partial^{2} w_{s}}{\partial x \partial y} \end{matrix}} \\ + {\begin{matrix} F_{31}^{e} \\ F_{31}^{e} \\ 0 \end{matrix}} ϕ + {\begin{matrix} F_{31}^{m} \\ F_{31}^{m} \\ 0 \end{matrix}} γ - {\begin{matrix} M_{sx}^{E} \\ M_{sy}^{E} \\ 0 \end{matrix}} - {\begin{matrix} M_{sx}^{H} \\ M_{sy}^{H} \\ 0 \end{matrix}} \end{matrix}

(48)

(1 - μ \nabla^{2}) {\begin{array}{l} Q_{x z} \\ Q_{y z} \end{array}} = (\begin{matrix} A_{44}^{s} & 0 \\ 0 & A_{55}^{s} \end{matrix}) {\begin{array}{l} \frac{\partial w_{s}}{\partial x} \\ \frac{\partial w_{s}}{\partial y} \end{array}} - A_{15}^{e} {\begin{array}{l} \frac{\partial ϕ}{\partial x} \\ \frac{\partial ϕ}{\partial y} \end{array}} - A_{15}^{m} {\begin{array}{l} \frac{\partial γ}{\partial x} \\ \frac{\partial γ}{\partial y} \end{array}}

(49)

\int_{- h / 2}^{h / 2} (1 - μ \nabla^{2}) {\begin{array}{l} D_{x} \\ D_{y} \end{array}} \cos (ξ z) d z = E_{15}^{e} {\begin{array}{l} \frac{\partial w_{s}}{\partial x} \\ \frac{\partial w_{s}}{\partial y} \end{array}} + F_{11}^{e} {\begin{array}{l} \frac{\partial ϕ}{\partial x} \\ \frac{\partial ϕ}{\partial y} \end{array}} + F_{11}^{m} {\begin{array}{l} \frac{\partial γ}{\partial x} \\ \frac{\partial γ}{\partial y} \end{array}}

(50)

(1 - μ \nabla^{2}) \int_{- h / 2}^{h / 2} D_{z} ξ \sin (ξ z) d z = A_{31}^{e} (\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}) - E_{31}^{e} \nabla^{2} w_{b} - F_{31}^{e} \nabla^{2} w_{s} - F_{33}^{e} ϕ - F_{33}^{m} γ

(51)

\int_{- h / 2}^{h / 2} (1 - μ \nabla^{2}) {\begin{array}{l} B_{x} \\ B_{y} \end{array}} \cos (ξ z) d z = E_{15}^{m} {\begin{array}{l} \frac{\partial w_{s}}{\partial x} \\ \frac{\partial w_{s}}{\partial y} \end{array}} + F_{11}^{m} {\begin{array}{l} \frac{\partial ϕ}{\partial x} \\ \frac{\partial ϕ}{\partial y} \end{array}} + X_{11}^{m} {\begin{array}{l} \frac{\partial γ}{\partial x} \\ \frac{\partial γ}{\partial y} \end{array}}

(52)

\begin{matrix} \int_{- h / 2}^{h / 2} (1 - μ \nabla^{2}) B_{z} ξ \sin (ξ z) dz = A_{31}^{m} (\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}) \\ - E_{31}^{m} \nabla^{2} w_{b} - F_{31}^{m} \nabla^{2} w_{s} - F_{33}^{m} ϕ - X_{33}^{m} γ \end{matrix}

(53)

In which the cross-sectional rigidities are defined as follows

{\begin{cases} A_{11}, B_{11}, B_{11}^{s}, D_{11}, D_{11}^{s}, H_{11}^{s} \\ A_{12}, B_{12}, B_{12}^{s}, D_{12}, D_{12}^{s}, H_{12}^{s} \\ A_{66}, B_{66}, B_{66}^{s}, D_{66}, D_{66}^{s}, H_{66}^{s} \end{cases}} = \int_{- h / 2}^{h / 2} {\begin{matrix} {\tilde{c}}_{11} \\ {\tilde{c}}_{12} \\ {\tilde{c}}_{66} \end{matrix}} (1, z, f, z^{2}, z f, f^{2}) d z

(54)

{A_{31}^{e}, E_{31}^{e}, F_{31}^{e}} = \int_{- h / 2}^{h / 2} {\tilde{e}}_{31} ξ \sin (ξ z) {1, z, f} dz

(55)

{A_{31}^{m}, E_{31}^{m}, F_{31}^{m}} = \int_{- h / 2}^{h / 2} {\tilde{q}}_{31} ξ \sin (ξ z) {1, z, f} dz

(56)

{A_{15}^{e}, E_{15}^{e}} = \int_{- h / 2}^{h / 2} {\tilde{e}}_{15} \cos (ξ z) {1, g} dz

(57)

{A_{15}^{m}, E_{15}^{m}} = \int_{- h / 2}^{h / 2} {\tilde{q}}_{15} \cos (ξ z) {1, g} dz

(58)

{F_{11}^{e}, F_{33}^{e}} = \int_{- h / 2}^{h / 2} {{\tilde{s}}_{11} \cos^{2} (ξ z), {\tilde{s}}_{33} ξ^{2} \sin^{2} (ξ z)} dz

(59)

{F_{11}^{m}, F_{33}^{m}} = \int_{- h / 2}^{h / 2} {{\tilde{d}}_{11} \cos^{2} (ξ z), {\tilde{d}}_{33} ξ^{2} \sin^{2} (ξ z)} dz

(60)

{X_{11}^{m}, X_{33}^{m}} = \int_{- h / 2}^{h / 2} {{\tilde{χ}}_{11} \cos^{2} (ξ z), {\tilde{χ}}_{33} ξ^{2} \sin^{2} (ξ z)} dz

(61)

A_{44}^{s} = A_{55}^{s} = \int_{- h / 2}^{h / 2} {\tilde{c}}_{55} g^{2} dz

(62)

Also, normal forces and moments due to magneto-electrical field in equations (46) to (48) can be defined as

N_{x}^{E} = N_{y}^{E} = - \int_{- h / 2}^{h / 2} {\tilde{e}}_{31} \frac{2 V}{h} d z, N_{x}^{H} = N_{y}^{H} = - \int_{- h / 2}^{h / 2} {\tilde{q}}_{31} \frac{2 Ω}{h} d z

(63)

M_{b x}^{E} = M_{b y}^{E} = - \int_{- h / 2}^{h / 2} {\tilde{e}}_{31} \frac{2 V}{h} z d z, M_{b x}^{H} = M_{b y}^{H} = - \int_{- h / 2}^{h / 2} {\tilde{q}}_{31} \frac{2 Ω}{h} z d z

(64)

M_{s x}^{E} = M_{s y}^{E} = - \int_{- h / 2}^{h / 2} {\tilde{e}}_{31} \frac{2 V}{h} f (z) d z, M_{s x}^{H} = M_{s y}^{H} = - \int_{- h / 2}^{h / 2} {\tilde{q}}_{31} \frac{2 Ω}{h} f (z) d z

(65)

The governing equations of refined four-variable shear deformation MEE-FG nanoplate in terms of the displacements and potentials can be derived by substituting equations (46) to (53) into equations (24) to (29) as follows

\begin{matrix} A_{11} \frac{\partial^{2} u}{\partial x^{2}} + A_{66} \frac{\partial^{2} u}{\partial y^{2}} + (A_{12} + A_{66}) \frac{\partial^{2} v}{\partial x \partial y} - B_{11} \frac{\partial^{3} w_{b}}{\partial x^{3}} \\ - (B_{12} + 2 B_{66}) \frac{\partial^{3} w_{b}}{\partial x \partial y^{2}} - B_{11}^{s} \frac{\partial^{3} w_{s}}{\partial x^{3}} \\ - (B_{12}^{s} + 2 B_{66}^{s}) \frac{\partial^{3} w_{s}}{\partial x \partial y^{2}} + A_{31}^{e} \frac{\partial ϕ}{\partial x} + A_{31}^{m} \frac{\partial γ}{\partial x} = 0 \end{matrix}

(66)

\begin{matrix} A_{66} \frac{\partial^{2} v}{\partial x^{2}} + A_{22} \frac{\partial^{2} v}{\partial y^{2}} + (A_{12} + A_{66}) \frac{\partial^{2} u}{\partial x \partial y} \\ - B_{22} \frac{\partial^{3} w_{b}}{\partial y^{3}} - (B_{12} + 2 B_{66}) \frac{\partial^{3} w_{b}}{\partial x^{2} \partial y} - B_{22}^{s} \frac{\partial^{3} w_{s}}{\partial y^{3}} \\ - (B_{12}^{s} + 2 B_{66}^{s}) \frac{\partial^{3} w_{s}}{\partial x^{2} \partial y} + A_{31}^{e} \frac{\partial ϕ}{\partial y} + A_{31}^{m} \frac{\partial γ}{\partial y} = 0 \end{matrix}

(67)

\begin{matrix} B_{11} \frac{\partial^{3} u}{\partial x^{3}} + (B_{12} + 2 B_{66}) \frac{\partial^{3} u}{\partial x \partial y^{2}} + (B_{12} + 2 B_{66}) \frac{\partial^{3} v}{\partial x^{2} \partial y} \\ + B_{22} \frac{\partial^{3} v}{\partial y^{3}} - D_{11} \frac{\partial^{4} w_{b}}{\partial x^{4}} + E_{31}^{e} - A_{15}^{e} (\frac{\partial^{2} ϕ}{\partial x^{2}} + \frac{\partial^{2} ϕ}{\partial y^{2}}) \\ - 2 (D_{12} + 2 D_{66}) \frac{\partial^{4} w_{b}}{\partial x^{2} \partial y^{2}} - D_{22} \frac{\partial^{4} w_{b}}{\partial y^{4}} - D_{11}^{s} \frac{\partial^{4} w_{s}}{\partial x^{4}} \\ - 2 (D_{12}^{s} + 2 D_{66}^{s}) \frac{\partial^{4} w_{s}}{\partial x^{2} \partial y^{2}} + E_{31}^{m} - A_{15}^{e} (\frac{\partial^{2} γ}{\partial x^{2}} + \frac{\partial^{2} γ}{\partial y^{2}}) \\ - D_{22}^{s} \frac{\partial^{4} w_{s}}{\partial y^{4}} + (1 - μ \nabla^{2}) \\ (- (N^{b} + N^{E} + N^{H}) (\frac{\partial^{2} (w_{b} + w_{s})}{\partial x^{2}} + \frac{\partial^{2} (w_{b} + w_{s})}{\partial y^{2}})) = 0 \end{matrix}

(68)

\begin{matrix} B_{11}^{s} \frac{\partial^{3} u}{\partial x^{3}} + (B_{12}^{s} + 2 B_{66}^{s}) \frac{\partial^{3} u}{\partial x \partial y^{2}} + (B_{12}^{s} + 2 B_{66}^{s}) \frac{\partial^{3} v}{\partial x^{2} \partial y} \\ + B_{22}^{s} \frac{\partial^{3} v}{\partial y^{3}} - D_{11}^{s} \frac{\partial^{4} w_{b}}{\partial x^{4}} - A_{15}^{e} (\frac{\partial^{2} ϕ}{\partial x^{2}} + \frac{\partial^{2} ϕ}{\partial y^{2}}) \\ + A_{55}^{s} \frac{\partial^{2} w_{s}}{\partial x^{2}} + A_{44}^{s} \frac{\partial^{2} w_{s}}{\partial y^{2}} - 2 (D_{12}^{s} + 2 D_{66}^{s}) \frac{\partial^{4} w_{b}}{\partial x^{2} \partial y^{2}} \\ - D_{22}^{s} \frac{\partial^{4} w_{b}}{\partial y^{4}} - H_{11}^{s} \frac{\partial^{4} w_{s}}{\partial x^{4}} - 2 (H_{12}^{s} + 2 H_{66}^{s}) \frac{\partial^{4} w_{s}}{\partial x^{2} \partial y^{2}} \\ - H_{22}^{s} \frac{\partial^{4} w_{s}}{\partial y^{4}} + F_{31}^{e} (\frac{\partial^{2} ϕ}{\partial x^{2}} + \frac{\partial^{2} ϕ}{\partial y^{2}}) + F_{31}^{m} (\frac{\partial^{2} γ}{\partial x^{2}} + \frac{\partial^{2} γ}{\partial y^{2}}) \\ + (1 - μ \nabla^{2}) (- (N^{b} + N^{E} + N^{H}) (\frac{\partial^{2} (w_{b} + w_{s})}{\partial x^{2}}) \\ + (\frac{\partial^{2} (w_{b} + w_{s})}{\partial y^{2}}) - A_{15}^{m} (\frac{\partial^{2} γ}{\partial x^{2}} + \frac{\partial^{2} γ}{\partial y^{2}})) = 0 \end{matrix}

(69)

\begin{matrix} A_{31}^{e} (\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}) - E_{31}^{e} (\frac{\partial^{2} w_{b}}{\partial x^{2}} + \frac{\partial^{2} w_{b}}{\partial y^{2}}) \\ - (F_{31}^{e} - E_{15}^{e}) (\frac{\partial^{2} w_{s}}{\partial x^{2}} + \frac{\partial^{2} w_{s}}{\partial y^{2}}) + F_{11}^{e} (\frac{\partial^{2} ϕ}{\partial x^{2}} + \frac{\partial^{2} ϕ}{\partial y^{2}}) \\ + F_{11}^{m} (\frac{\partial^{2} γ}{\partial x^{2}} + \frac{\partial^{2} γ}{\partial y^{2}}) - F_{33}^{e} ϕ - F_{33}^{m} γ = 0 \end{matrix}

(70)

\begin{matrix} A_{31}^{m} (\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}) - E_{31}^{m} (\frac{\partial^{2} w_{b}}{\partial x^{2}} + \frac{\partial^{2} w_{b}}{\partial y^{2}}) \\ - (F_{31}^{m} - E_{15}^{m}) (\frac{\partial^{2} w_{s}}{\partial x^{2}} + \frac{\partial^{2} w_{s}}{\partial y^{2}}) + F_{11}^{m} (\frac{\partial^{2} ϕ}{\partial x^{2}} + \frac{\partial^{2} ϕ}{\partial y^{2}}) \\ + X_{11}^{m} (\frac{\partial^{2} γ}{\partial x^{2}} + \frac{\partial^{2} γ}{\partial y^{2}}) - F_{33}^{m} ϕ - X_{33}^{m} γ = 0 \end{matrix}

(71)

Solution procedure

In the present analytical method, the generalized displacements are expanded in a double Fourier series in terms of unknown parameters. The selection of the functions in these series is associated to those which satisfy the boundary edges of the nanoplate. These boundary edges are given as follows (Sobhy, 2013):

Simply-supported (S)

\begin{matrix} v = w_{b} = w_{s} = N_{x} = M_{x}^{b} = M_{x}^{s} = 0 at x = 0, a \\ u = w_{b} = w_{s} = N_{y} = M_{y}^{b} = M_{y}^{s} = 0 at x = 0, b \end{matrix}

(72)

Clamped (C)

u = v = w_{b} = w_{s} = 0 at x = 0, a and y = 0, b

(73)

Free (F)

\begin{matrix} M_{x}^{b} = M_{x}^{s} = M_{xy}^{b} = M_{xy}^{s} = Q_{xz} = 0 at x = 0, a \\ M_{y}^{b} = M_{y}^{s} = M_{xy}^{b} = M_{xy}^{s} = Q_{yz} = 0 at y = 0, b \end{matrix}

(74)

To satisfy the above-mentioned boundary conditions, the displacement quantities are presented in the following form

u = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} U_{mn} \frac{\partial X_{m} (x)}{\partial x} Y_{n} (y)

(75)

v = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} V_{mn} X_{m} (x) \frac{\partial Y_{n} (y)}{\partial y}

(76)

w_{b} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} W_{bmn} X_{m} (x) Y_{n} (y)

(77)

w_{s} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} W_{smn} X_{m} (x) Y_{n} (y)

(78)

ϕ = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} Φ_{mn} X_{m} (x) Y_{n} (y)

(79)

γ = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} ϒ_{mn} X_{m} (x) Y_{n} (y)

(80)

where $U_{mn}$ , $V_{mn}$ , $W_{bmn}$ , $W_{smn}$ , $Φ_{mn}$ , and $γ_{mn}$ are the unknown coefficients, and the functions $X_{m}$ and $Y_{n}$ are tabulated in detail in Table 2 for different boundary conditions ( $α = m π / a$ , $β = n π / b$ ). Substituting equations (75) to (80) in equations (66) to (71), respectively, leads to

\begin{matrix} [(A_{11} r_{1} + A_{66} r_{2})] U_{mn} + (A_{12} + A_{66}) r_{2} V_{mn} \\ - [B_{11} r_{1} + (B_{12} + 2 B_{66}) r_{2}] W_{bmn} \\ + [- (B_{12}^{s} + 2 B_{66}^{s}) r_{2} - B_{11}^{s} r_{1})] W_{smn} \\ + A_{31}^{e} r_{11} Φ_{mn} + A_{31}^{m} r_{11} γ_{mn} \end{matrix}

(81)

\begin{matrix} [(A_{12} + A_{66}) r_{3}] U_{mn} + [A_{66} r_{3} + A_{22} r_{4}] V_{mn} \\ + [- B_{22} r_{4} - (B_{12} + 2 B_{66}) r_{3}] W_{bmn} \\ + [- B_{22}^{s} r_{4} - (B_{12}^{s} + 2 B_{66}^{s}) r_{1}] W_{smn} \\ + A_{31}^{e} r_{12} Φ_{mn} + A_{31}^{m} r_{12} λ_{mn} = 0 \end{matrix}

(82)

\begin{matrix} [B_{11} r_{5} + (B_{12} + 2 B_{66}) r_{6}] U_{mn} + [(B_{12} + 2 B_{66}) r_{6} \\ + B_{22} r_{7}] V_{mn} + [- D_{11} r_{5} - 2 (D_{12} + 2 D_{66}) r_{6} \\ - D_{22} r_{7}) + (N^{b} + N^{E} + N^{H}) (μ (r_{5} + 2 r_{6} + r_{7}) \\ - (r_{10} + r_{9}))] W_{bmn} \\ + [- D_{11}^{s} r_{5} - 2 (D_{12}^{s} + 2 D_{66}^{s}) r_{6} - D_{22}^{s} r_{7} \\ + (N^{b} + N^{E} + N^{H}) (μ (r_{5} + 2 r_{6} + r_{7}) \\ - (r_{10} + r_{9}))] W_{smn} + [E_{31}^{e} (r_{10} + r_{9})] Φ_{mn} \\ + [E_{31}^{m} (r_{9} + r_{10})] γ_{mn} = 0 \end{matrix}

(83)

\begin{matrix} [B_{11}^{s} r_{5} + (B_{12}^{s} + 2 B_{66}^{s}) r_{6}] U_{mn} \\ + [(B_{12}^{s} + 2 B_{66}^{s}) r_{6} + B_{22}^{s} r_{7}] V_{mn} \\ + [- D_{11}^{s} r_{5} - 2 (D_{12}^{s} + 2 D_{66}^{s}) r_{6} - D_{22}^{s} r_{7}) \\ + (N^{b} + N^{E} + N^{H}) (μ (r_{5} + 2 r_{6} + r_{7}) \\ - (r_{10} + r_{9}))] W_{bmn} + [A_{44}^{s} (r_{10} + r_{9}) - H_{11}^{s} r_{5} \\ - 2 (H_{12}^{s} + 2 H_{66}^{s}) r_{6} - H_{22}^{s} r_{7} \\ + (N^{b} + N^{E} + N^{H}) (μ (r_{5} + 2 r_{6} + r_{7}) - (r_{10} + r_{9})) \\ - μ (r_{5} + 2 r_{6} + r_{7}))] W_{smn} + [(F_{31}^{e} - A_{15}^{e}) (r_{10} + r_{9})] Φ_{mn} \\ + (F_{31}^{m} - A_{15}^{m}) (r_{10} + r_{9}) γ_{mn} = 0 \end{matrix}

(84)

\begin{matrix} A_{31}^{e} r_{10} U_{mn} + A_{31}^{e} r_{9} V_{mn} + [- E_{31}^{e} (r_{10} + r_{9})] W_{bmn} \\ + [(E_{15}^{e} - F_{31}^{e}) (r_{10} + r_{9})] W_{smn} \\ + [F_{11}^{e} (r_{10} + r_{9}) - F_{33}^{e} r_{8}] Φ_{mn} \\ + [F_{11}^{m} (r_{10} + r_{9}) - F_{33}^{m} r_{8}] γ_{mn} = 0 \end{matrix}

(85)

\begin{matrix} A_{31}^{m} r_{10} U_{mn} + A_{31}^{m} r_{9} V_{mn} - E_{31}^{m} (r_{10} + r_{9}) W_{bmn} \\ + (E_{15}^{m} - F_{31}^{m}) (r_{10} + r_{9}) W_{smn} \\ + [F_{11}^{m} (r_{10} + r_{9}) - F_{33}^{m} r_{8}] Φ_{mn} \\ + [X_{11}^{m} (r_{10} + r_{9}) - X_{33}^{m} r_{8}] γ_{mn} = 0 \end{matrix}

(86)

where

{r_{3}, r_{4}, r_{12}} = \int_{0}^{a} \int_{0}^{b} X (x) Y^{'} (y) {X^{″} (x) Y^{'} (y), X (x) Y^{‴} (y), X (x) Y^{'} (y)} d x d y

(87)

\begin{array}{l} {r_{1}, r_{2}, r_{11}} \\ = \int_{0}^{a} \int_{0}^{b} X^{'} (x) Y (y) {X^{‴} (x) Y (y), X^{'} (x) Y^{″} (y), X^{'} (x) Y (y)} d x d y \end{array}

(88)

\begin{array}{l} {r_{5}, r_{6}, r_{7}} \\ = \int_{0}^{a} \int_{0}^{b} X (x) Y (y) {{X^{‴}}^{'} (x) Y (y), X^{″} (x) Y^{″} (y), X (x) {Y^{‴}}^{'} (y)} d x d y \end{array}

(89)

\begin{array}{l} {r_{8}, r_{9}, r_{10}} = \\ \int_{0}^{a} \int_{0}^{b} X (x) Y (y) {X (x) Y (y), X (x) Y^{″} (y), X^{″} (x) Y (y)} d x d y \end{array}

(90)

By finding the determinant of the coefficient matrix of the above equations and setting this multinomial to 0, we can find the buckling loads

{[K]}_{6 * 6} [\begin{matrix} U_{mn} \\ V_{mn} \\ W_{bmn} \\ W_{smn} \\ Ψ_{mn} \\ ϒ_{mn} \end{matrix}] = 0

(91)

Table 2.

Admissible functions $X_{m} (x)$ and $Y_{n} (y)$ (Sobhy, 2013).

	Boundary conditions		Functions $X_{m}$ and $Y_{n}$
	At x = 0, a	At y = 0, b	$X_{m} (x)$	$Y_{n} (y)$
SSSS	$X_{m} (0) = X_{m}^{″} (0) = 0$	$Y_{n} (0) = Y_{n}^{″} (0) = 0$	$Sin (α x)$	$Sin (β y)$
SSSS	$X_{m} (a) = X_{m}^{″} (a) = 0$	$Y_{n} (b) = Y_{n}^{″} (b) = 0$
CSSS	$X_{m} (0) = X'_{m} (0) = 0$	$Y_{n} (0) = Y_{n}^{″} (0) = 0$	$Sin (α x) [Cos (α x) - 1]$	$Sin (β y)$
CSSS	$X_{m} (a) = X_{m}^{″} (a) = 0$	$Y_{n} (b) = Y_{n}^{″} (b) = 0$
CSCS	$X_{m} (0) = X'_{m} (0) = 0$	$Y_{n} (0) = Y'_{n} (0) = 0$	$Sin (α x) [Cos (α x) - 1]$	$Sin (β y) [Cos (β y) - 1]$
CSCS	$X_{m} (a) = X_{m}^{″} (a) = 0$	$Y_{n} (b) = Y_{n}^{″} (b) = 0$
CCSS	$X_{m} (0) = X'_{m} (0) = 0$	$Y_{n} (0) = Y_{n}^{″} (0) = 0$	${Sin}^{2} (α x)$	$Sin (β y)$
CCSS	$X_{m} (a) = X'_{m} (a) = 0$	$Y_{n} (b) = Y_{n}^{″} (b) = 0$
CCCC	$X_{m} (0) = X'_{m} (0) = 0$	$Y_{n} (0) = Y'_{n} (0) = 0$	${Sin}^{2} (α x)$	${Sin}^{2} (β y)$
CCCC	$X_{m} (a) = X'_{m} (a) = 0$	$Y_{n} (b) = Y'_{n} (b) = 0$
CCFF	$X_{m}^{″} (0) = X ″'_{m} (0) = 0$	$Y_{n} (0) = Y'_{n} (0) = 0$	${Sin}^{2} (α x)$	${Cos}^{2} (β y) [{Sin}^{2} (β y) + 1]$
CCFF	$X_{m}^{″} (a) = X ″'_{m} (a) = 0$	$Y_{n} (b) = Y'_{n} (b) = 0$

Numerical results and discussions

The static stability behavior of size-dependent MEE-FG nanoplates under various boundary conditions is studied in this section using a trigonometric refined plate theory. Presented boundary conditions for nanoplate are depicted in Figure 2. The length of the nanoplate is considered to be a = 10 nm. This study is verified by comparing the obtained buckling results with those presented by Sobhy (2015) for a simply-supported higher-order shear deformable (HSDT) FG nanoplate, and a good agreement is observed according to the results presented in Table 3. The material properties for comparison study are considered as E_c = 380 GPa, E_m = 70 GPa, and v_c = v_m = 0.3.

Table 3.

Comparison of dimensionless buckling load ( $\bar{N} = 12 N a^{2} (1 - v^{2}) / E_{c} h^{3}$ ) of simply-supported square FG nanoplates (a/h = 10).

Gradient index, p	µ = 0 nm		µ = 2 nm
Gradient index, p	HSDT (Sobhy, 2015)	Present study	HSDT (Sobhy, 2015)	Present study
0	18.6876	18.6877	10.4425	10.4426
0.5	10.0638	10.0638	5.6235	5.62359
2.5	6.2593	6.25935	3.4976	3.49769
5.5	5.5200	5.52002	3.0845	3.08455
10.5	4.9677	4.96776	2.7759	2.77596

Also, a comparison is carried out for buckling loads of homogenous MEE nanoplates with those presented by Ansari and Gholami (2016) using differential quadrature method (DQM). The results are presented in Table 4 for various voltages when h = 10 nm, a/h = 10, Ω = 0 (A/m), and a good agreement is observed. Then, the effects of gradient index, magnetic and electric fields, different boundary conditions, small-scale parameter, and plate geometrical parameters on the buckling loads of the MEE-FG nanoplate will be explored. The non-dimensional form of buckling load can be defined as

\bar{N} = N^{b} \frac{a^{2}}{D_{c}}, D_{c} = c_{11}^{u} h^{3}

(92)

Table 4.

Comparison of dimensionless buckling load of simply-supported and clamped homogenous magneto-electro-elastic nanoplates (a/h = 10, Ω = 0 A/m, and µ = 0.03 nm²).

V	SSSS		CCCC
V	Ansari and Gholami (2016)	Present study	Ansari and Gholami (2016)	Present study
−0.2	20.0777	20.0653	44.7189	44.7012
−0.1	18.9090	18.8871	43.5502	43.5379
0	17.7403	17.7286	42.3815	42.3691
+0.1	16.5716	16.5574	41.2128	41.2023
+0.2	15.4029	15.3856	40.0441	40.0367

Tables 5 and 6 present numerical examples to show the effect of small-scale parameter (µ), electric voltage (V), magnetic potential (Ω), and various boundary conditions (SSSS, CSSS, CSCS, CCSS, CCCC, and CCFF) on dimensionless buckling loads of square MEE-FG nanoplates at a/h = 100 and p = 1. It is noted that the buckling loads of nonlocal MEE-FG nanoplate are always smaller than that of the classical MEE-FG plate, and it reduces with the rise in the small-scale parameter at a fixed magnetic potential and electric voltage. Such phenomenon is due to the fact that the small-scale influence, which captures the mutual influence of all points in the region, may reduce the stiffness of the nanostructures. In addition, it is observed that negative values of magnetic potential lead to smaller buckling loads of FGM nanoplate than positive magnetic potentials for all boundary conditions. However, smaller values of electric voltage lead to larger buckling loads. Furthermore, for all six types of boundary conditions, the CCFF leads to the highest buckling loads of MEE-FG nanoplates and the SSSS results in the lowest. The reason is that stronger support in boundary conditions tends to make higher buckling loads.

Table 5.

Variation of non-dimensional buckling load of FG nanoplate for various electric voltages, nonlocal parameter, and boundary conditions (a = b = 100 h, p = 1).

µ	V = −0.5	V = 0	V = + 0.5	V = −0.5	V = 0	V = + 0.5	V = −0.5	V = 0	V = + 0.5
µ	SSSS			CSSS			CCSS
0	1.17429	0.920074	0.665856	1.69248	1.43827	1.18405	1.99425	1.74004	1.48582
1	1.02262	0.768398	0.51418	1.35107	1.09685	0.84263	1.51449	1.26027	1.00605
2	0.913871	0.659653	0.405435	1.14064	0.886426	0.632209	1.24211	0.987888	0.73367
3	0.832089	0.577872	0.323654	0.997963	0.743745	0.489527	1.06654	0.812321	0.55810
	CSCS			CCCC			CCFF
0	2.11949	1.86527	1.61106	2.64692	2.39271	2.13849	3.10692	2.8527	2.59848
1	1.58214	1.32792	1.07371	1.82179	1.56757	1.31335	2.00323	1.74901	1.4948
2	1.28515	1.03093	0.776714	1.41982	1.16561	0.911388	1.51532	1.2611	1.00689
3	1.09672	0.842505	0.588287	1.18193	0.92771	0.673499	1.24025	0.986036	0.73181

Table 6.

Variation of non-dimensional buckling load of FG nanoplate for various magnetic potentials, nonlocal parameter, and boundary conditions (a = b = 100 h, p = 1).

µ	Ω = −0.05	Ω = 0	Ω = + 0.05	Ω = −0.05	Ω = 0	Ω = +0.05	Ω = −0.05	Ω = 0	Ω = +0.05
µ	SSSS			CSSS			CCSS
0	0.63258	0.92007	1.20756	1.15078	1.43827	1.72576	1.45254	1.74004	2.02753
1	0.48090	0.76839	1.05589	0.80935	1.09685	1.38434	0.972779	1.26027	1.54776
2	0.37216	0.65965	0.94714	0.59893	0.88642	1.17392	0.700397	0.98788	1.27538
3	0.29038	0.57787	0.86536	0.45625	0.74374	1.03124	0.52483	0.81232	1.09981
	CSCS			CCCC			CCFF
0	1.57778	1.86527	2.15276	2.10522	2.39271	2.6802	2.56521	2.8527	3.14019
1	1.04043	1.32792	1.61541	1.28008	1.56757	1.85506	1.46152	1.74901	2.03651
2	0.74344	1.03093	1.31842	0.87811	1.16561	1.4531	0.973613	1.26110	1.54859
3	0.55501	0.84250	1.1300	0.64022	0.92771	1.21521	0.698546	0.98603	1.27353

Figures 3 and 4 examine the influence of material graduation index (p) on dimensionless buckling loads of MEE-FG nanoplates under various boundary conditions, respectively, for different values of electric voltage (V) and magnetic potential (Ω) at a/h = 100 and µ = 0.5 nm². It can be seen that buckling loads of MEE-FG nanoplate are quite sensitive to the variation of external electric voltage, magnetic potential, and gradient index. Also, smaller values of gradient index have more considerable impact on variation of buckling loads. However, larger values of gradient index lead to a non-obvious variation in buckling loads.

Moreover, the buckling response relies on the sign of magnetic and electric field intensities. In fact, the exerted positive/negative magnetic potentials may produce the axial tensile and compressive forces, while electric field shows an opposite trend. When FG nanoplate is fully $CoF e_{2} O_{4}$ , which means gradient index is zero, buckling loads are identical for various electric voltages since $CoF e_{2} O_{4}$ has a zero piezoelectric coefficient ( $e_{31}$ ). Such behavior cannot be observed for a homogenous MEE nanoplate where the material properties are not graded. Another study is carried out to indicate the small-scale effect on buckling loads of MEE-FG nanoplate versus electric voltage and magnetic potential in Figures 5 and 6 for various boundary conditions when a/h = 100 and p = 1. It is realized that the local theory overestimates the buckling loads of the MEE-FG nanoplates compared to the nonlocal theory which means nonlocality diminishes the rigidity of the nanoplate. Moreover, nonlocal effect is more prominent for a MEE nanoplate with stronger support in boundary conditions. Therefore, buckling loads of nanoplate with CCFF boundary conditions are more affected by small-scale parameter than other boundary conditions. Also, comparison of buckling loads with respect to various boundary conditions reveals that SSSS leads to lowest buckling loads, while CCFF leads to highest buckling loads at a prescribed electric voltage and magnetic potential. Another observation from these figures is that buckling loads increase and decrease linearly with the increase in magnetic potential and electric voltage.

Figures 7 and 8 elucidate that buckling load of MEE-FG nanoplate is significantly affected by side-to-thickness ratio (a/h) and also signs of electric voltage and magnetic potential. In these figures, it is considered that µ = 1 nm² and p = 1. When electric and magnetic field intensities are set to zero, buckling loads are not influenced by the higher side-to-thickness ratios. Also, variation of buckling load is more significant according to larger values of a/h. Moreover, side-to-thickness ratio presents increasing and reducing effect on buckling loads for positive and negative magnetic potentials for all six boundary conditions. While effect of side-to-thickness ratio on buckling loads for positive and negative voltages has, respectively, a reducing and increasing trend. Figure 9 reveals that the buckling loads of MEE-FG naoplate are prominently influenced by plate aspect ratio (a/b) and small-scale parameter. It this figure, a/h = 100, p = 1, V = +1, and Ω = +0.1 A/m are adopted. It is seen that increasing a/b leads to higher buckling loads for all kinds of boundary conditions. Moreover, buckling loads predicted by local and nonlocal models become more distinguished at higher values of aspect ratio. Buckling loads of local plate model have more sensible variation with the increase in aspect ratio than nonlocal plate model.

Conclusion

The buckling problem of a nonlocal MEE-FG nanoplate under different boundary conditions is analyzed via the refined sinusoidal plate model. Magneto-electro-elastic properties of nanoplate change in spatial coordinate based on power-law form. The equations of motion and boundary conditions for an embedded MEE-FG nanoplate modeled via a four-unknown plate theory are derived using Hamilton’s principle and Eringen’s nonlocal elasticity theory. Admissible functions are provided to satisfy various boundary conditions and analytically solve the governing equations. Effect of magnetic potential, electric voltage, various boundary conditions, small-scale parameter, material composition, plate side-to-thickness ratio, and aspect ratio on buckling response of the MEE-FG nanoplate is examined. It is realized that the buckling loads of nonlocal MEE-FG nanoplate are always smaller than that of the classical MEE-FG plate, and it reduces with the increase in the nonlocal parameter. Also, smaller values of gradient index have more considerable impact on variation of buckling loads. However, larger values of gradient index have no important effect on variation of buckling loads. In addition, it is observed that negative values of magnetic potential lead to smaller buckling loads of FGM nanoplate than positive magnetic potentials for all boundary conditions. However, smaller values of electric voltage lead to larger buckling loads. Furthermore, for all six types of boundary conditions, the CCFF leads to the highest buckling loads of MEE-FG nanoplates, and the SSSS results in the lowest. Also, buckling loads predicted by local and nonlocal models become more distinguished at larger values of plate aspect ratio. Side-to-thickness ratio has increasing and reducing influence on buckling loads for positive and negative magnetic potentials, while effect of side-to-thickness ratio on buckling loads for positive and negative voltages has, respectively, a reducing and increasing behavior.

Footnotes

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) received no financial support for the research, authorship, and/or publication of this article.

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