1 Introduction

Sandwich structures used in construction of building, vehicle, and airplane usually consist of two relatively thin, stiff, and strong faces separated by a relatively thick core. Therefore, a large number of research works have been carried out on the mechanical behavior of sandwich structures^[1-8]. A generalized finite element modeling of recently developed secant function based on a shear deformation theory was formulated and implemented for free vibration and buckling characteristics of laminated-composite and sandwich plates by Grover et al.^[1]. A new improved high-order theory was presented for a biaxial buckling analysis of sandwich plates with soft orthotropic cores by Kheirkhah et al.^[2]. They used a third-order plate theory for face sheets and quadratic and cubic functions for transverse and in-plane displacements for the core, respectively. Du and Ma^[3] investigated the nonlinear vibration fundamental equation of a circular sandwich plate under the uniform load and circumjacent load using a von Karman plate theory. They concluded that when the circumjacent load makes the lowest natural frequency zero, the critical load can be obtained. Improved transverse shear stiffness for vibration and buckling analysis of functionally graded sandwich plates based on the first-order shear deformation theory (FSDT) was proposed by Nguyen et al.^[4]. The transverse shear stress obtained from the in-plane stress and equilibrium equation was used to analytically derive improved transverse shear stiffness and the associated shear correction factor of a sandwich plate.

In order to investigate the mechanical responses of sandwich structures, the equivalent single layer (ESL), layer wise (LW) and zigzag theories may be utilized. In the ESL theory, the stress field at layer interfaces is discontinuous since the displacements are assumed to be continuous through the thickness of the sandwich structure, and the elastic constants of the adjacent layers are different from each other. To overcome the mentioned problem, the LW theory may be used. However, the problem of the LW theory is that, the theory is inefficient analyzing the multilayered laminates since the number of unknowns depends on the number of layers. In order to overcome the defects of ESL and LW theories, the zigzag theory is introduced^[9]. Cho and Oh^[10] used the higher order zigzag plate theory to refine the prediction of the mechanical, thermal, and electric behaviors of smart composite plates. Ren^[11-12] applied the zigzag theory for solving a plate. Among different types of the zigzag theory used for analyzing sandwich structures, the refined zigzag theory (RZT) is simpler and also more accurate than the others^[13-21]. A mixed-field RZT for laminated plates was presented by Iurlaro et al.^[13]. In another attempt, Sciuva et al.^[14] formulated a class of efficient higher-order C⁰ continuous beams based on the RZT. A third-order RZT for the multilayered composite and sandwich beams was developed by Iurlaro et al.^[15]. Therefore, Chakrabarti et al.^[16] studied vibration and buckling characteristics of sandwich plates with stiff layers for different degrees of imperfections at the layer interface using a refined plate theory. Iurlaro et al.^[17] presented the derivation of the nonlinear equations of motion and consistent boundary conditions of the RZT for multilayer plates. In another attempt, the original formulation of the RZT was extended to the bending and free vibration analysis of sandwich plates embedding functionally graded layers by Iurlaro et al.^[18].

In order to study the mechanical behaviors of nanostructures accurately, the small scale effects should be considered. Therefore, using a classical theory which is a scale independent theory yields inaccurate results. On the other hand, among size dependent theories such as nonlocal elasticity, modified couple stress and modified strain gradient theories, the nonlocal elasticity theory has received increasing attention^[22-33]. Based on the nonlocal elasticity theory, Daneshmehr et al.^[22] investigated free vibration behaviors of the nanoplate made of functionally graded materials using Eringen's nonlocal elasticity and higher order shear deformation plate theories. A free vibration analysis of thick circular/annular functionally graded Mindlin nanoplates was investigated using Eringen's nonlocal elasticity theory by Hosseini-Hashemi et al.^[23]. Also, in another paper, Hosseini-Hashemi et al.^[24] used Eringen's nonlocal and Mindlin plate theories for free vibration of rectangular nanoplates. They investigated effects of the nonlocal parameter on the natural frequency of the nanoplate for different boundary conditions. The analytical solution for bending of a simply supported rectangular grapheme sheet based on three-dimensional theories was studied using nonlocal continuum mechanics by Alibeigloo and Pasha-Zanoosi^[25]. Shen and Li^[26] developed a modified semi-continuum Euler beam model with a relaxation phenomenon and also presented a bending deformation of the extreme-thin beam with micro/nano scale thickness. They compared the semi-continuum, classical continuum and nonlocal continuum models and showed good agreement among mentioned models. Two kinds of torsional models were constructed by Li^[27] to investigate the nonlocal torsional vibration of carbon nanotubes. Li et al.^[28] derived the governing equation of nanobeams by introducing certain simplifying assumptions, based on the two-dimensional differential constitutive relations of nonlocal elasticity in the plane. They applied the equation to the nano-cantilever beam for several typical external forces, and the nonlocal effect on the bending behavior was thus revealed.

Recently, application of smart materials such as piezoelectric and magnetoelectroelastic (MEE) in the smart structure has been the subject of intense interest research. The specific characteristic of piezoelectric material is its ability to produce an electric field when it is subject to deformation, and vice versa while MEE materials are sensitive to both electric and magnetic fields^[34]. These materials are applicable to electro-mechanical and electric devices, such as actuators, sensors, and transducers. On the other hand, after Pan et al.^[35] presented ZnO piezoelectric nanobelts characteristics, various piezoelectric and MEE nanostructures such as nanobeams, nanowires, nanorings, nanotubes, and nanoplates have attracted many researchers^[36-44]. Li et al.^[40] investigated buckling and free vibration of MEE nanoplates on the Pasternak foundation based on the nonlocal Mindlin theory. Also, Li^[41] studied a buckling analysis of MEE plates embedded in the Pasternak foundation. An exact solution was presented for the multilayered rectangular plate made of functionally graded, anisotropic and linear MEE materials by Pan and Han^[42]. A nonlocal geometrically nonlinear beam model was developed for magneto-electro-thermo-elastic nanobeams subject to external electric voltage, external magnetic potential, and uniform temperature rise by Ansari et al.^[43]. Ke and Wang^[44] studied the free vibration of MEE nanobeams based on nonlocal and Timoshenko beam theories. Based on the nonlocal Love's shell theory, Ke et al.^[45] developed vibration of an embedded MEE cylindrical nanoshell model.

However, motivated by these considerations and due to the best of literature, to date, no report has been found on the electro-magneto-buckling of an embedded sandwich nanoplate (SNP). Since upper and lower layers of the SNP are integrated with smart materials, the SNP can be used in nano-electro-mechanical systems (NEMS). Therefore, analyzing the buckling behavior of SNP becomes more prominent in designing smart nanostructures. In order to have the optimum design of SNP, buckling behaviors of SNP subject to external electric and magnetic fields are studied. The formulation presented here is based on the RZT and nonlocal magnetoelectroelasticity theory. By applying the energy method and Hamilton's principle, the governing motion equations are obtained which are then analytically solved to obtain critical buckling loads. The influences of the elastic medium, small scale parameter, thickness of MEE layers, external electric and magnetic loads, and mode numbers on the buckling behavior of the SNP are taken into account.

2 Governing equations

A schematic diagram of an SNP embedded on the Pasternak foundation and subject to electromagnetic loads is depicted in Fig. 1. The geometrical parameters of length a, width b, core thickness h_c, and MEE layers thicknesses h_t, are also shown in Fig. 1.

2.1 Nonlocal magnetoelectroelasticity theory

Based on the nonlocal theory which is presented by Eringen^[46], the stress tensor at a point depends not only on the strains at that point. In fact, it is considered to be a function of strains at all points of the body. The constitutive equations for the MEE material can be expressed as follows^[40]:

(1a)

(1b)

(1c)

in which σ_ij^nl, σ_ij^l, D_k^nl, D_k^l, B_k^nl, and B_k^l are the nonlocal stress, the local stress tensor, nonlocal electric, local electric, nonlocal magnetic and local magnetic displacements, respectively. α(|x-x'|, τ) is the kernel function and is considered as the nonlocal modulus, where the term |x-x'| shows the distance in the Euclidean norm. Also, τ represents the material constant and is defined as τ =e₀a/l, where a and l describe internal characteristic length and external characteristic length, respectively, and e₀ is Eringen's nonlocal parameter. α(|x'-x|) which is proposed by Eringen is considered as a Green's function of a linear differential operator L,

(1d)

Substituting Eqs. (1a), (1b), and (1c) into Eq. (1d), the integral form of nonlocal stress can be reduced as follows:

(1e)

By matching the Fourier transforms of the kernel in the wave number space with the dispersion curves of lattice dynamics, the linear differential operator can be achieved. For curve-fitting at low wave numbers pertinent to the internal length scale, Eq. (1e) can be written as^[46]

(1f)

in which γ represents a small parameter proportional to the integral length scale, such as (e₀a). If a first-order approximation is to be purposed, the Laplacian form of the operator in Eq. (1f) is just maintained. Thus, for the two-dimensional instance, one can rewrite Eqs. (1a), (1b), and (1c) as follows:

(2a)

(2b)

(2c)

where ▽² is the Laplacian operator.

2.2 RZT

Based on the RZT, the kth (k=a, b, c) layer components of displacement vector of the SNP, U₁^k, V₁^k, and W₁^k along the coordinate directions x, y, and z, respectively, are written as^[19]

(3a)

(3b)

(3c)

where u, v, and w are the components of mid-plate. The superscript k represents quantities corresponding to the plate layer, and t represents the time.ϕ_α^k(z) is the zigzag function which depends on the thickness, layers material, and changes for each layer. It is worth mentioning that, setting the zigzag function to zero, the results are obtained based on the FSDT. θ₁ and θ₂ describe average bending rotations of the transverse normal about the positive y-and the negative x-directions, respectively. ψ₁ and ψ₂ represent the spatial amplitudes of the zigzag rotation.

The strain components of an arbitrary point on the SNP are defined as

(4a)

(4b)

(4c)

(4d)

(4e)

Since the SNP consists of two magnetostrictive layers, applying external electric and magnetic potentials to each of these layers causes the deformation of SNP. Therefore, by assuming a half-sinus and linear distribution of the electric and magnetic potentials through the thickness direction, one can write as^[47]

(5a)

(5b)

(5c)

(5d)

where Ω_i and φ_i(i=1, 2) are the spatial variations of the magnetic and electric potentials, respectively. Ω_0i and ϕ_0i (i=1, 2) describe the initial external magnetic and electric potentials, respectively.

The constitutive equations for local stresses, electric displacement and magnetic induction for the kth layers of sandwich structure can be expressed as^[40-41]

(6a)

(6b)

(6c)

where Q_ij^k, e_ij^k, q_ij^k, _ij^k, d_ij^k, and μ_ij^k are the transformed elastic stiffness, piezoelectric, piezomagnetic, dielectric, magnetoelectric and magnetic coefficients for the kth layer, respectively. It should be mentioned that, for the core layer, the elastic stiffness is defined as . Also, electric and magnetic fields are obtained as

(7a)

(7b)

2.3 Energy method and Hamilton's principle

The strain energy of the SNP can be calculated as follows:

(8)

Using Eqs. (5) and (7) and substituting Eq. (4) into Eq. (8), the strain energy of the SNP can be rewritten as

(9)

where

(10a)

(10b)

(10c)

(10d)

(10e)

(10f)

(10g)

(10h)

in which h_k+1 and h_k are the top and the bottom of the kth layer, respectively. In order to obtain Eq. (10) based on the FSDT, instead of using Eq. (10c), the following relations should be used:

(10i)

where k_x and k_y are shear correction factors.

The work done by elastic foundation and external electric and magnetic potentials is written as^{[32, 47]}

(11)

in which k_w is the spring constant of Winkler type, and k_g denotes the shear constant of the Pasternak type, respectively. N_xm, N_ym, N_xe, and N_ye describe the magnetic and electrical forces, respectively, which can be written as follows:

(12a)

(12b)

The equation of motion for the SNP is derived with Hamilton's principle. It can be expressed as

(13)

Substituting Eqs. (9) and (11) into Eq. (13) and with respect to Hamilton's principle, the equations of motion can be written as

(14a)

(14b)

(14c)

(14d)

(14e)

(14f)

(14g)

(14h)

(14i)

(14j)

(14k)

It is convenient to define the following dimensionless parameters:

(15)

Substituting stress, moment and shear resultants into Eq. (14) and using Eq. (15), the dimensionless governing motion equations based on the RZT can be obtained which are written in Appendix A.

3 Solution method

In this paper, the buckling characteristics of simply supported SNPs are investigated using the nonlocal magnetoelectroelasticity theory. Based on the Navier solution and boundary conditions, the following buckling modes are assumed to obtain the critical buckling load:

(16a)

(16b)

(16c)

in which m and n are the half wave numbers. Also, U, V, W, Θ₁, Θ₂, Ψ₁, Ψ₂, Φ₁, Φ₂, Λ₁, and Λ₂ are the amplitude constants. Substituting Eq. (16) into Eqs. (A1)-(A11) leads to

(17)

where K is the coefficient matrix, and

(18a)

(18b)

In order to obtain the non-zero solutions for both RZT and FSDT, the determinant of the coefficient matrix obtained from each theory must be equal to zero,

(19)

Solution to Eq. (19) yields the dimensionless critical buckling loads.

4 Numerical results and discussion

In this section, the influences of Winkler and Pasternak coefficients, small scale parameter, MEE layers thickness, mode numbers, and external electric and magnetic potentials on the critical electric and magnetic loads are investigated numerically. For this purpose, the core is assumed to be made of metal with E_b =70 GPa and υ=0.3, and the material properties of MEE layers are listed in Table 1^[41].

The developed nonlocal theory to date is incapable of determining the small scaling effect (e₀a). However, Eringen^[46] proposed e₀a=0.39 nm by matching the dispersion curves with the nonlocal theory for plane waves and the Born-Karman model of lattice dynamics. For carbon nanotubes, (e₀a) is inducted to be less than 2 nm^[28].

To the best of the authors' knowledge, no published paper is available in the literature covering the same scope of the problem, so one cannot directly validate the presented results. However, the obtained results can be validated partially with a simplified analysis presented by Hosseini-Hashemi and Tourki-Samaei^[48] and Pradhan^[49]. For this purpose, the MEE layers are neglected, and the core is assumed as a graphene sheet. Therefore, the dimensionless buckling load based on the FSDT is obtained and compared with those of Refs.[48] and [49] in Table 2. Clearly, the presented results closely match those reported by Hosseini-Hashemi and Tourki-Samaei^[48] and Pradhan^[49].

A comparison between the results obtained based on the FSDT and the RZT is presented in Table 3 to show the accuracy and significance of the RZT. It is seen from Table 3 that the dimensionless critical buckling load obtained from the FSDT depends on the shear correction factor values (k_x², k_y²). However, for k_x² =0.76 and k_y² =0.64, the difference between results obtained based on the FSDT becomes almost close to those obtained from the RZT. Therefore, for investigating the mechanical response of sandwich structure, neglecting zigzag kinematics in Eq. (3) yields inaccurate results.

Figures 2(a) and 2(b) show the effects of Winkler and Pasternak coefficients on the critical buckling external magnetic and electric loads, respectively. It can be observed that, in both Figs. 2(a) and 2(b), as the Winkler and Pasternak constants are increased, the dimensionless critical magnetic and electric buckling loads applied to the upper layer increase. It is clear that increasing Winkler and Pasternak constants makes the structure stiff. Therefore, as seen in Figs. 2(a) and 2(b), the critical buckling loads increase.

Figures 3(a) and 3(b) show the effects of external magnetic and electric loads imposed to the lower layer on the dimensionless critical magnetic and electric loads applied to the upper layer, respectively. It is seen from Figs. 3(a) and 3(b) that, as the small scale coefficient increases, the dimensionless critical buckling magnetic load decreases. It is due to the fact that, according to Eq. (2), the stress and stiffness of the SNP decrease with the increasing small scale parameter. Consequently, the critical buckling load should be decreased, too. As seen from Figs. 3(a) and 3(b), increasing the external load applied to the lower layer decreases the critical buckling load applied to the upper layer. Therefore, one can say that applying both electric and magnetic loads to each layer makes the system loose.

Figures 4(a) and 4(b) illustrate the effects of the MEE layers thickness on the dimensionless critical buckling magnetic and electric loads versus the small scale parameter, respectively. It is found that both critical magnetic and electric buckling loads are increased with increasing the thickness of MEE layers. It should be noted that, for higher small scale values, the influence of the MEE layers thickness on the critical buckling loads is decreased.

Effects of mode numbers on the dimensionless critical buckling magnetic and electric loads are also presented in Figs. 5(a) and 5(b), respectively. As seen from Figs. 5(a) and 5(b), the critical buckling loads increase as the mode numbers are increased. Meanwhile, the effect of mode numbers on the critical buckling loads becomes more distinguished for the lower small scale coefficient values.

5 Conclusions

Electro-magneto buckling behaviors of SNPs are investigated since they can be applied in designing NEMS. In order to model the SNP, the RZT is taken into account. The effect of surrounding elastic medium, such as the shear constant of the Pasternak type, is taken into account. The nonlocal magneto electro elasticity is used to consider the small scale effects. Hamilton's principle as well as an energy method is used to obtain governing motion equations. By using an analytical method, the critical buckling magnetic and electric loads are calculated. The presented results indicate that increasing the thickness of MEE layers decreases the critical buckling electric and magnetic loads. Furthermore, since increasing Winkler and Pasternak coefficients makes the system stiff, increasing Winkler and Pasternak coefficients increases both critical magnetic and electric buckling loads. Also, considering small scale effects yields lower critical buckling load values. As an important result, imposing electric and magnetic loads to the lower MEE layer decreases the critical buckling loads. Moreover, the dimensionless critical buckling loads obtained based on the RZT are higher than those predicted by the FSDT. Therefore, neglecting the zigzag function yields inaccurate results. Finally, it is hoped that the obtained results would be helpful for design and control of NEMS and sandwich structures.

Appendix A

Dimensionless governing motion equations based on the RZT are written as follows:

(A1)

(A2)

(A3)

(A4)

(A5)

(A6)

(A7)

(A8)

(A9)

(A10)

(A11)

Acknowledgements The author would like to thank the reviewers for their valuable comments and suggestions to improve the clarity of this study.