Critical Temperatures for Vibrations and Buckling of Magneto-Electro-Elastic Nonlocal Strain Gradient Plates

An analytical method is presented in this work for the linear vibrations and buckling of nano-plates in a hygro-thermal environment. Nonlinear von Kármán terms are included in the plate kinematics in order to consider the instability phenomena. Strain gradient nonlocal theory is considered for its simplicity and applicability with respect to other nonlocal formulations which require more parameters in their analysis. Present nano-plates have a coupled magneto-electro-elastic constitutive equation in a hygro-thermal environment. Nano-scale effects on the vibrations and buckling behavior of magneto-electro-elastic plates is presented and hygro-thermal load outcomes are considered as well. In addition, critical temperatures for vibrations and buckling problems are analyzed and given for several nano-plate configurations.


Introduction
In recent years research has focused heavily on MEMS (Micro-Electro-Mechanical-System) and NEMS (Nano-Electro-Mechanical-System). This interest is mainly due to the wide variety of applications in which these devices could be used [1][2][3][4]. These structures, such as nanoplates, nanorods, and nanobeams [5], can be used in medicine [6], electronics [7], aerospace [8] and in civil construction [9], where linear and nonlinear theories are generally needed [10]. The behavior of this type of structures cannot be well described through the classical theories of continuous mechanics, as they are based on the principle of location of stresses. Due to the size of these devices, the effects induced by nanoscales must be taken into account [11,12]. Then to improve the ability of new devices and systems made with these smart materials, it is necessary to accurately investigate the mechanical behavior of these advanced structures [13,14]. Non-local theories have been widely used for the study of nanostructures since Eringen developed his theory of non-local elasticity [15]. These theories consider the nano-scale effects thanks to the introduction of one or more length scale parameters in addition to the well known linear elastic Lamé parameters [16][17][18][19]. The classification of nonlocal theories is generally presented as: strain gradient [20][21][22][23], stress gradient [24], modified strain gradient [25][26][27], couple stress [28], modified couple stress [29,30], integral type [31,32] and micropolar [33][34][35]. Article [36] offers a overview on unified continuous/reduced-order modeling and non-linear dynamic theories for thermomechanical plates. Kim in [37] developed a matrix method for evaluating effective elastic constants of generally anisotropic multilayer composites with various coupled physical effects including piezoelectricity, piezomagnetism and thermoelasticity. In [38][39][40] a nonlocal nonlinear first-order shear theory is used for investigating the buckling and free vibration of magneto-electro-thermo elastic (METE) nanoplates under magneto-electrothermo-mechanical loadings. Mota in [41] investigated the influence of shear factor used

Theoretical Background
As show in Figure 1, consider a METE thin nanoplate with length a, width b and thickness h, in a Cartesian reference system (x, y, z). The METE nanoplate is in a hygro-thermal environment and is subjected to an electric potential V 0 and to a magnetic potential Ω 0 between the upper and lower surfaces. In this study, classic laminate plate theory is considered. We can define the displacement field of a generic point of the solid by means of the triad of displacement components U, V, W, which are functions of the coordinates (x, y, z) [10].
where u,v and w are the displacements along the x, y and z axis of the point on the middle surface and ∂w/∂x and ∂w/∂y are the corresponding rotation. The constitutive equations for a METE material are: (2) in which σ is the classical stress vector, D E = [D x , D y , D z ] and B M = [B x , B y , B z ] are respectively the vector of stresses, electrical displacement and magnetic flux. ε, E and H are the vector of strain, electric field and magnetic field. C, ξ and χ represent the rigidity matrix, the electrical permittivity matrix and the magnetic permittivity matrix. Finally, e, q, ζ, p, λ, h and η are respectively the piezo-electric, piezo-magnetic, magneto-electro-elastic (MEE), pyro-electric, pyro-magnetic, hygro-electric and hygro-magnetic coefficients. For the stress plane state (σ 3 = 0) the matrices can be reduced by carrying out Therefore the constitutive equations can be rewritten as follows similarly for σ 2 it will be finally, D z can written and similarly for B M,3 it will be So it is possible to writẽ By introducing second order strain gradient theory in the constitutive equations and by considering the mechanical properties variable with respect to the thickness direction we have (the dependency on the time t is omitted for the sake of simplicity) (9) where is the nonlocal parameter and the operator ∇ 2 = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 is the second order gradient operator. For the hygro-thermal loads a linear variation is considered along the thickness as

Electric and Magnetic Potentials
To satisfy Maxwell's equations [73] the electrical and magnetic potential are approximated along the thickness with a linear and cosinusoidal combination. The first amends for the open-circuit condition and the latter for the closed-circuit one in which V 0 represents the difference in electrical potential between the two faces of the plate and Ω 0 represents the difference in magnetic potential. The relationships between electric field and electric potential can be written in accordance with the above that in matrix notation can be rewritten in this form Similarly for the magnetic field we can write as which in matrix notation becomes

Equations of Motion
The equations of motion are derived through Hamilton's principle Writing the variation of enthalpy δH ent by introducing the classical stress resultants N xx , N yy , N xy and M xx , M yy , M xy and the piezo and magneto resultants as The definition of the integrated elastic properties are given in the Appendix A in Equations (A1)-(A3). Thus, it is obtained Integrating by parts Equation (21) is obtained N xx n x + N xy n y δu + N xy n x + N yy n y δv + N xx n x + N xy n y ∂w ∂x + N xy n x + N yy n y ∂w ∂y The external work due to the external boundary loads (where mechanical, electrical and magnetic loads are neglected) can be written as δV = Γ N xx n x +N xy n y δu + N xy n x +N yy n y δv+ − M xx n x +M xy n y ∂δw ∂x − M xy n x +M yy n y ∂δw ∂y + Q x n x +Q y n y δw dΓ Variation of the Kinetic energy can be written as Introducing N (w) and P (w) as defined below and relative boundary conditions become δu = 0 or N xx −N xx n x + N xy −N xy n y = 0 δv = 0 or N yy −N yy n y + N xy −N xy n x = 0

Navier Solution
Analytical solution is obtained using Navier's expansion. This type of solution allow to solve simply supported plate case. Navier expansion for the displacements take the form whereas, the electric and magnetic potentials are both approximated with a double sinusoidal trigonometric expansion.

Buckling Analysis
Replacing the displacement field in the motion equations and performing the derivates the algebraic system is obtained The coefficientsĉ ij ands 33 are defined in Appendix A at Equation (A4). By introducing the quantities N 0 = −N xx , κ =N yy /N xx and a mn as we can write the solution of the eigenvalue problem as The load that buckles the plate depends on m and n and in particular the critical load is the lowest of the buckling loads. The termsN E xx ,N E yy ,N H xx ,N H yy are defined beloŵ Note that the electric and magnetic in-plane loads have the same intensity since in the following applications the material is isotropic in-plane and anisotropic out-of-plane.

Thermal Free Vibration
In this paragraph it will be treated the problem of free vibrations of the FG plate simply supported. For this problem it is necessary to rewrite the solving system in a homogeneous form, and the rotational inertia I 1 are neglected. The system becomes then withm 11 =m 22 = I 0 em 33 = I 0 + I 2 (α 2 + β 2 ). This system can then be rewritten in a more compact matrix form as follows Rewriting the matrix system by applying static condensation we get whereK isK = (K uu − K uφ (K φφ ) −1 K φu ) (37) and U represents the ways of vibrating and ω natural frequencies.

Numerical Results
In this paper for the numerical solution has been considered a FG nanoplate composed of CoFe 2 O 4 and BaTiO 3 , the properties of the materials are shown in Table 1. Since it has not been possible to find in literature the hygrometric coefficients of the materials, the applications presented foresee only the case of thermal environment. The variation of the material properties along the thickness is regulated by the following relationship where P t and P b represent the properties of the material placed on the top and bottom of the plate, respectively. Note that if p n = 0 the plate will be composed entirely of the material of the top side while if p n → ∞ the plate will be composed entirely of the material of the bottom side.

Buckling
In the following applications the values of the critical load will be presented in dimensionless form through the following relationship N cr = N cr a 2 C 11,m h 3 (39) where C 11,m is the average stiffness value of the two materials. As a first result the comparison with Li [74], Park and Han [75] is reported. The plate considered is isotropic with a/h = 1000 and the properties of the material used are as reported in the cited works (which are obtained as average values of two isotropic constituents, not as a functionally graded composite). Figure 2 shows the critical buckling load by varying the electric and magnetic potentials. The present results agree well with the ones presented in the mentioned papers [74,75]. It is emphasized that the slight difference in the results is due to the fact that in the cited studies the potential is approximated using three contributions: one parabolic, one linear and one constant unlike the present study in which the potentials are approximated with a cosine function and a linear part. In addition, in the cited studies, the in-plane components E x , E y , H x and H y of electric and magnetic fields are null. In the applications below a/h = 1000 and n p = 1 are always considered. Table 2 shows the dimensionless critical load of a square plate FG for different values of externally applied potentials and non-local parameter. It can be seen that as the non-local parameter increases, the value of the critical load increases. It can also be seen that by increasing the magnetic potential the critical load increases while the electric potential has the opposite effect. This last phenomenon is also clearly visible in Figure 3, where the value of the critical load is reported as the external potentials applied vary and for different values of the non-local parameter. Finally it is remarked that for same values of the electric and magnetic potentials the critical load takes a negative value, thus, buckling occurs for traction loads instead of compression. Figure 4 shows the dimensional critical load when the aspect ratio varies and for different values of the non-local parameter. As expected the critical load increases as the plane becomes of rectangular shape and as the nonlocal parameter increases.

Thermal Free Vibration
As first application for the free vibration of piezo-electro-magnetic-thermal plates a comparison with [76] is reported ( Table 3). The plate is composed of BaTiO 3 /CoFe 2 O 4 and is of rectangular shape with a = 2 and b = 1, ratio h/a = 0.1 and properties vary linearly along the thickness (n p = 1). The results are written in dimensionless form through the following formulaω = ω a 2 h ρ C 11 (40) where ρ and C 11 respectively represent the density and the (1, 1) position element of the material stiffness matrix on the underside of the plate. It should be noted that in the study just mentioned the Mindlin's moderately thick plates theory (FSDT) is used, and so the results deviate slightly and this difference increases as the vibration mode increases as the effects of shear become more relevant.  Table 4 shows a comparison with article [57] for a thick magneto-electro-elastic square plate. The thickness of the plate is constant and the laminae all have the same thickness. The properties of the material are those reported in Table 1, except the density which is assumed constant for the two materials and equal to 1600 kg/m 3 . The values calculated in this study differ slightly from those in the literature because in [57] a third order plate theory is used while in this study it is used thin plate theory.  Table 5 analyzes the influence of the non-local parameter on the natural frequencies of a square plate with a = b = 1 m and ratio a/h = 100 and n p = 1. The results show an increase in natural frequencies, due to the stiffening of the plate, as the non-local parameter increases. Figure 5 shows the influence of temperature and externally applied potentials on the natural frequency of a nanoplate composed by BaTiO 3 /CoFe 2 O 4 . The plate considered is a simply supported square plate and the thickness ratio is a/h = 100. In particular, the critical temperature of each structure can be identified when the frequency becomes zero.

Conclusions
In this paper the dynamic and buckling problems of METE nanoplates have been analyzed. In particular, the interest focused on the coupling of magnet-electro-thermoelastic effects and the influence that the external potentials applied to the plate have on the critical load and natural vibration frequencies. Through Hamilton's principle motion equations for FG METE thin plates are derived and analytical solution using Navier method is obtained. The materials that have been used in the simulations are BaTiO 3 and CoFe 2 O 4 and the properties of the materials used are reported in the article. The results show that increasing the non-local parameter increases the critical load and natural vibration frequencies. For external potentials instead it was seen that the critical load increases with the increase of the negative electrical potential and the positive magnetic potential. Finally, from the graphs of the natural frequency of vibration it can be seen that the frequencies tend to increase by subjecting the plate to a positive magnetic potential and to decrease by subjecting it to a positive electrical potential. For what concerns the temperature instead we see how an increase of the latter leads to a reduction of the natural frequencies.