Vibration Analysis of a Magnetoelectroelastic Rectangular Plate Based on a Higher-Order Shear Deformation Theory

Free vibration of a magnetoelectroelastic rectangular plate is investigated based on the Reddy’s third-order shear deformation theory. The plate rests on an elastic foundation and it is considered to have different boundary conditions. Gauss’s laws for electrostatics and magnetostatics are used to model the electric and magnetic behavior. The partial differential equations of motion are reduced to a single partial differential equation and then by using the Galerkin method, the ordinary differential equation of motion as well as an analytical relation for the natural frequency of the plate is obtained. Some numerical examples are presented to validate the proposed model and to investigate the effects of several parameters on the vibration frequency of the considered smart plate.


Vibration Analysis of a Magnetoelectroelastic Rectangular Plate
Based on a Higher-Order Shear Deformation Theory

INTRODUCTION
Magnetoelectroelastic composite materials are a new class of smart materials which exhibit a coupling between mechanical, electric and magnetic fields and are capable of converting energy among these three energy forms.These materials have direct application in sensors and actuators, control of vibrations in structures, energy harvesting, etc.
Static and dynamic responses of piezoelectric plates have been investigated extensively in the past years (Alibeigloo and Kani, 2010;Behjatet al., 2011;Rezaiee-Pajand and Sadeghi, 2013;Ghashochi-Bargh and Sadr, 2014;Rafiee et al., 2014;Padoina et al., 2015).Moon et al. (2007) designed a linear magnetostrictive actuator using Terfenol-D to control structural vibration.Hong (2007) studied the thermal vibration of magnetostrictive material embedded in laminated plate by using the generalized differential quadrature method.Later, the same author (2010)used the generalized differential quadrature method to compute the transient response of the laminated magnetostrictive plates under thermal vibration.Pan (2001) studied multilayered magnetoelectroelastic plates analytically for the first time and derived exact solutions for three-dimensional magnetoelectroelastic plates.Pan and Heyliger (2002) derived analytical solutions for free vibrations of these smart plates.Pan and Heyliger (2003) studied the response of multilayered magnetoelectroelastic plates under cylindrical bending.Ramirez et al. (2006a) presented an approximate solution for the free vibration problem of two-dimensional magnetoelectroelastic laminated plates.Ramirez et al. (2006b) also determined natural frequencies of orthotropic magnetoelectroelastic graded composite plates by using a discrete layer model.Liu and Chang (2010) derived a closed form expression for the transverse vibration of a magnetoelectroelastic thin plate and obtained the exact solution for the free vibration of a twolayered BaTiO 3 -CoFe 2 O 4 composite.Single-layer approaches to static and free vibration analysis of magnetoelectroelastic laminated plates have also been introduced (Milazzo 2012(Milazzo , 2014a(Milazzo , 2014b;;Milazzo and Orlando, 2012).Chen et al. (2014) studied the free vibration of multilayered magnetoelectroelastic plates under combined clamped/free boundary conditions.Moita et al. (2009) presented a higher-order finite element model for static and free vibration analyses of magnetoelectroelastic plates.Based on the nonlocal Love's shell theory, Ke et al. (2014) developed an embedded magnetoelectroelastic cylindrical nanoshell model to study the vibration response of these structures.Razavi and Shooshtari (2014) used Donnell shell theory to analyze the free vibration of magnetoelectroelastic curved panels.Li and Zhang (2014) studied the free vibration of a magnetoelectroelastic plate resting on a Pasternak foundation based on the Mindlin theory.Piovan and Salazar (2015) presented a one-dimensional model for dynamic analysis of magnetoelectroelastic curved beams.Based on three-dimensional elasticity theory, Xin and Hu (2015) derived semianalytical solutions for free vibration of simply supported and multilayered magnetoelectroelastic plates.Nonlinear free and forced vibration of one-layered and multilayered magnetoelectroelastic rectangular plates based on the classical and first order shear deformation theory have also been investigated (Shooshtari andRazavi 2015a, 2015b;Razavi and Shooshtari, 2015).Li et al. (2014Li et al. ( ,2015) ) investigated dynamic response of magnetoelectroelastic nanoplate and nanobeam based on nonlocal Mindlin theory and nonlocal and Timoshenko beam theories, respectively.Ansari et al. (2015) developed a nonlocal geometrically nonlinear beam model for magnetoelectroelastic nanobeams subjected to external electric voltage, external magnetic potential and uniform temperature rise.Recently, Shooshtari and Razavi (2015c) investigated large amplitude vibration of laminated magnetoelectroelastic doubly-curved panels.
According to the published articles, there is not any study dealing with analytical study of free vibration of these smart plates based on a higher-order shear deformation theory.So, this study fills the gap in the analysis of magnetoelectroelastic rectangular plates.In this paper, free vibration of simply-supported, clamped and simply-supported/clamped magnetoelectroelastic rectangular plates resting on an elastic foundation is investigated based on the Reddy's third-order shear deformation theory.The Galerkin method is implemented to reduce the partial differential equation of motion to anordinary differential equation and then an analytical relation is obtained for the natural frequency.Some numerical examples are presented to validate the proposed model and to investigate the effects of several parameters such as foundation parameters, plate geometry, and the applied electric and magnetic potentials on the natural frequency of the considered smart plate.

THEORETICAL FORMULATION
Consider a rectangular plate resting on an elastic foundation with dimensions of a ×b ×h as shown in Figure 1.Based on the Reddy's third-order shear deformation theory, the displacement field of a composite plate is given as (Reddy, 2004): , , Whereu 0 ,v 0 , andw 0 are the displacements of the mid-surface alongx,y, andz directions, respectively, andθ x andθ y are the rotations of a transverse normal about they andx directions, respectively.The linear strain-displacement relations based on the displacement field given in Eq. ( 1) are (Reddy, 2004): Assuming that the electric and magnetic fields are applied alongz-direction, the constitutive equations of a magnetoelectroelastic material can be written in the following form (Pan, 2001;Li and Zhang, 2014): ) and assuming a constant value for the density of the plate, the equations of motion of a rectangular plate can be expressed in the following form (Reddy, 2004): where k w and k s are spring and shear coefficients of the elastic foundation, respectively and: in which ρ 0 is the density of the material of the plate and the force and moment resultants are obtained by: ( ) To express Eqs.( 6) -( 10) in terms of displacements and rotations, the resultants must be calculated from Eq. ( 12).To this end, σ αβ and σ αz can be substituted from Eq. ( 3).However, since ϕ ,z and ψ ,z are unknown parameters, Eqs. ( 4) and( 5) along with Gauss's laws for electrostatics and magnetostatics, i.e., are used which results in: where ) Integrating the relations of Eq. ( 14) with respect to z, one obtains: Whereϕ 0 , ϕ 1 , ψ 0 and ψ 1 are constants of the integration and are obtained by using the magnetoelectric boundary conditions on the two surfaces of the plate.The magnetoelectroelastic body is poled along the z direction and subjected to an electric potential V 0 and a magnetic potential Ω 0 between the upper and lower surfaces of the plate.So, the magnetoelectric boundary conditions are stated as: Eqs. ( 17) and ( 18) give 0 0 . Then the gradients of electric and magnetic potentials are obtained from Eq. ( 16): Now, the resultants are obtained by Eqs. ( 3), ( 12) and ( 19): where L i (i=1,2,…,37) are constant coefficients which are functions of applied electric and magnetic potentials, foundation parameters, and material and geometrical properties of the plate and are given in Appendix A.
It can be seen that Eqs. ( 24) and ( 25) are decoupled from Eqs. ( 26) -( 28).So, to study the transverse motion ofthe plate, it is sufficient to consider only Eqs. ( 26) -(28).Eqs.( 27) and ( 28) constitute a set of linear equations in terms ofθ x andθ y .Algebraic solution of this equations results in: whereA i (i=1,…,6) are partial differential operators and are defined in Appendix B. Substituting Eq. ( 29) into (26) one obtains the following partial differential equation for the transverse motion of the magnetoelectroelastic plate: ( ) ( ) which is expressed in terms of w 0 .Three boundary conditions are considered in the present study, which are simply-supported, clamped and combination of simply-supported and clamped edges, that is: The transverse displacement for each of these boundary conditions can be obtained by: where (m,n) denotes the mode of vibration and W(t) is unknown function in terms of time (t).Substituting Eqs.(32a) -(32c)into Eq. ( 30) and employing the orthogonality of trigonometric functions, the following ordinary differential equationis obtained for each boundary condition: in which the terms containing d 4 W/dt 4 and d 6 W/dt 6 are neglected.In this equation, M eq and K eq are the equivalent mass and stiffness of the system, respectively.

RESULTS
To validate the present study, some numerical examples are presentedand the results are compared with the published ones.As a first comparison, an isotropic simply-supportedsquare plate is considered and the dimensionless frequencies for different length-to-thickness ratios are obtained.The dimensionless frequencies are obtained by using ( ) , where E is the Young's modulusof the plate and ω 0 = (K eq /M eq ) 1/2 is the circular natural frequency.The results are shown in Table 1 and compared with the results of Vel and Batra (2004)  As a second comparison, a simply-supported isotropic thin plate with different aspect ratios is considered.The dimensionless frequencies are obtained by    As the last comparison, three piezoelectric, piezomagnetic and isotropic square plates with simplysupported boundary condition are considered and two firstdimensionlessfrequencies of these plates are obtained.Table 5 shows the results.The considered piezoelectric, piezomagnetic and isotropic plates are of BaTiO  Effects of aspect ratio, and the applied electric and magnetic potentials on the dimensionless fundamental frequencies of a magnetoelectroelastic plate with different boundary conditions are studied and the results are shown in Table 6.The dimensionless frequencies are obtained by The material properties of the magnetoelectroelastic plate are (Li and Zhang, 2014): C 11 = 226×10 9 Nm -2 , C 12 = 124×10 9 Nm -2 , C 22 = 216×10 9 Nm -2 , C 44 = C 55 = 44×10 9 Nm -2 , C 66 = 51×10 9 Nm -2 , e 32 = e 31 = -2.2Cm -2 , q 32 = q 31 = 290.2NA -1 m -1 , η 33 = 6.35×10 -9 C 2 N -1 m -2 ,d 33 = 2737.5×10 -1 NsV -1 C -1 , μ 33 = 83.5×10 - Ns 2 C -2 , and ρ 0 = 5500 kgm -3 .It is noticed that increasingthe aspect ratio increases the dimensionless frequency of the magnetoelectroelastic plate.Moreover, Table 6 shows that increasing the electric potential decreases the dimensionless frequency of the magnetoelectroelastic plate whereas magnetic potential increases the dimensionless frequency.It is also noticeable that potentials effects on dimensionless frequency are more significant in plates with higher aspect ratios and plates with clamped edges.
Table 7 shows the effects of a/h ratio and foundation parameters on the dimensionless frequencies of a magnetoelectroelastic square plate.In this table, the dimensionless frequencies are obtained by  ( ) The magnetoelectric boundary condition is considered to be closed-circuit meaning that in Eq. ( 18), 0 0 0 V = Ω = is substituted.It is seen that a/h ratio tends to decrease the dimensionless frequency.Foundation parameters increase the natural frequencies because the presence of elastic foundation results in the increase of the stiffness of the system.It is also obvious that the dimensionless shear coefficient ( s K ) has more effect on the natural frequencies.In addition, it is observed that similar to the results of Tables 3 and 4, clamped edges increase the dimensionless frequencies.Figures 2 and 3 show the effects of shear coefficient of foundation and a/h ratio on the natural frequencies of magnetoelectroelastic plates, respectively.It can be seen that for fixed material and geometric properties, clamped plate has the most natural frequency among the considered plates.Moreover, as it was also shown above, foundation parameter increases the natural frequency whereas the a/h ratio decreases it.

CONCLUSIONS
In this study, free vibration of a magnetoelectroelastic rectangular plate with different edge supports was investigated analytically.To this end, Reddy's third-order shear deformation theory and Gauss's laws for electrostatics and magnetostatics were used to model the considered smart plate.Galerkin method was applied to the partial differential equation of motion to reduce it to an ordinary differential equation and then an analytical relation was obtained for the natural frequency.
Some numerical examples were presented and it was shown that: (a) electric potential decreases the dimensionless natural frequency of the magnetoelectroelastic plate while the magnetic potential increases it, (b) clamped edges increase the dimensionless frequencies of magnetoelectroelastic plate so that the clamped plate has the most dimensionless frequency whereas the simply-supported plate has the least one, and (c) elastic foundation increases the stiffness of the system and consequently increases the natural frequency of the magnetoelectroelastic plate.

Appendix
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Figure 1 :
Figure 1: Schematic of a magnetoelectroelastic plate on an elastic foundation.
along x-axis and clamped along y-axis (SCSC) 3 , CoFe 2 O 4 and aluminum materials, respectively.The BaTiO 3 (shown with B) and CoFe 2 O 4 (shown with F) plates are thick with a = b = 1 m and h = 0.3 m and their material properties are given byWu and Lu (2009).However, the aluminum plate (shown with Al) is thin with a = b = 300 mm and h = 1 mm.The dimensionless frequencies of BaTiO 3 and CoFe 2 is the maximum value of the stiffness coefficient of the plate, whereas The dimensionless frequencies of aluminum plate are obtained by

Figure 2 :
Figure 2: Effect of shear coefficient of foundation on the fundamental natural frequency of closed-circuit magnetoelectroelastic square plates (a/h = 25, 0 w K = ).

Figure 3 :
Figure 3: Effect of length-to-thickness on the fundamental natural frequency of closed-circuit magnetoelectroelastic square plates (h = 1 mm, 0 w s K K = = ).

Table 1 :
based on the three-dimensional approach, Hosseini-Hashemiet al. (2011) based on the third-order shear deformation plate theory, and Kianiet al. (2012) based on the first-order shear deformation theory.It is seen that there is Comparison of dimensionless fundamental frequency of asimply-supportedsquare plate (ν = 0.3).

Table 3 :
Table3shows first four dimensionless frequencies of clamped (CCCC) and simplysupported/clamped (SCSC) square thin plates.The frequencies are obtained by First four dimensionless frequencies of square plates with different boundary conditions (ν = 0.3, a/h=1000).

Table 5 :
Dimensionless frequencies of several square plates.