Magneto-Electro-Elastic 3D Coupling in Free Vibrations of Layered Plates

1  Introduction

Magneto-electro-elastic (MEE) structures are smart structures where the energy between all the involved fields is exchanged interacting each other [1–3]. This ability (called MEE coupling) is employed in smart structures for the health monitoring and for the suppression of unwanted vibrations [4,5]. For these reasons, researchers are interested in analytical and numerical tool developments for the analysis of these types of smart structures. Several analytical and numerical models have been developed in the open literature to understand smart structure behaviors [6,7].

In the framework of numerical models, Carrera et al. [8] developed refined finite elements for multilayered plates subjected to MEE fields. The natural characteristics of a three-layered simply supported MEE multilayered plate were studied by Yang et al. in [9]. In [10], a free vibration study for anisotropic functionally graded MEE plates was carried out using a semi-analytical finite element method based on the series solution in the in-plane directions of the plate and on the finite element approximation through the thickness direction of the structure. Chen et al. [11] presented a free vibration study of multilayered MEE plates, under combined clamped/free lateral boundary conditions, by using a semi-analytical discrete-layer approach. Jiangong et al. [12] studied the dispersion behavior of waves in a layered MEE plate. Legendre orthogonal polynomial series were employed in controlling equations to include the MEE coupling. In [13], a finite element (FE) approach was proposed for free vibrations and transient peculiarities of MEE composite rectangular and elliptical plates resting on the visco-Pasternak medium in the case of a hygro-thermal environment and blast load applications. Kattimani and Ray [14] presented an FE model to analyze the active constrained layer damping for large amplitude vibrations of smart MEE plates. In [15], the nonlinear forced vibration behavior of MEE composite plates involving elastic foundations was studied via Reddy’s third-order shear deformation theory. Vinyas [16] proposed the free vibration study of different carbon nanotube-reinforced MEE plates adopting FE methods and considering the higher-order shear deformation theory. Milazzo [17,18] proposed a variable kinematics approach for moderately large deflection analysis of smart MEE multilayered and functionally graded plates, 2D refined equivalent single-layer models were developed. The same author [19] also proposed an equivalent single-layer model for the free vibration analysis of smart laminated plates by considering quasi-static electric and magnetic fields. Davì and Milazzo [20] developed a regular variational boundary element formulation, based on a hybrid variational principle, for dynamic analyses of 2D MEE domains. Vinyas and Kattimani [21] discussed the effects of the hygrothermal environment on free vibration characteristics of MEE plates, the FE method and a higher order shear deformation theory were employed. Ramirez et al. [22] presented a free vibration approximate solution for the 2D MEE laminates in order to determine their fundamental behavior. In [23], a layerwise FE model was developed by Kiran and Kattimani using the shear deformation theory and coupled constitutive equations, non-dimensional eigenfrequencies of 3D multilayered MEE plates with skewed edges were computed.

In the case of analytical formulations for the free vibration study of MEE structures, Soni et al. [24] proposed a nonlinear analytical classical plate theory for transverse vibrations of cracked MEE thin plates. The material employed in this study was the reinforced BaTiO3–CoFe2O4 composite including a central partial crack. In Xu et al. [25], the nonlinear free vibration response of MEE composite plates was proposed, the von Karman’s nonlinear strain–displacement theory and the high-order shear deformation theory were employed. The bending and free vibrations of an MEE plate with surface effects were proposed in Yang and Li [26]. The governing differential equations for bending and vibrations of MEE plates were derived considering surface effects in the Kirchhoff thin plate theory. Chen et al. [27] presented a state-vector approach to detect free vibrations of MEE laminated plates. The same authors [28,29] developed a 3D analytical solution for the propagation of time-harmonic waves in transversely isotropic and multilayered MEE plates with nonlocal effects. Pan and Heyliger [30] presented an analytical solution for free vibrations of simply supported multilayered MEE rectangular plates; the Stroh formalism was used for the general solution. Dat et al. [31] developed an analytical solution for the nonlinear MEE vibrations of smart sandwich plates made of a carbon nanotube reinforced composite core embedded between two MEE face sheets. In Wang et al. [32], a state variable mixed type formulation was developed for the free vibration of laminated structures embedding actuators/sensors. Razavi and Shooshtari [33] presented the nonlinear free vibration study of symmetric MEE laminated rectangular plates with simply supported boundary conditions, first-order shear deformation theory and von Karman’s nonlinear strains were included in the model. A 3D semi-analytical solution of the elasticity theory was derived by Xin and Hu [34] for free vibrations of simply supported and multilayered MEE plates, the analysis combined the state space approach and the discrete singular convolution algorithm. Kuo and Wang [35] proposed an analysis for the investigation of wave motion characteristics (e.g., dispersion curves and mode shapes) for MEE laminated plates including generalized membrane-type interfacial imperfections. In Dinh and Duc [36], the vibration behavior of a honeycomb composite structure embedding electromagnetic layers was presented. Combined effects of uniform loadings, thermal stresses and electromagnetic potentials were investigated. The effect of the elastic foundation and the viscoelastic interface on the dynamic behaviour of laminated simply supported MEE rectangular plates was investigated by Hamidi et al. [37]. The state space method in the Laplace domain was used. Free vibrations of simply supported rectangular MEE nanoplates were studied in [38], the nonlocal Kirchhoff plate theory was employed. A buckling and free vibration model of MEE nanoplates was presented by Li et al. [39], nonlocal Mindlin theory resting on Pasternak foundation was implemented. Chang [40] investigated the free vibration behavior of fluid-loaded transversely isotropic MEE rectangular plates. In Jamalpoor et al. [41], free vibration and biaxial buckling of MEE microplates were shown. The Kelvin–Voigt visco-Pasternak foundation was implemented when initial external electric and magnetic potentials were applied. In [42], nonlinear forced vibrations of an immovable simply supported MEE rectangular plate were studied considering the first-order shear deformation theory and harmonic excitation forces.

The present 3D coupled MEE analytical approach for the free vibration analysis of open and closed circuit simply supported plates is employed to comprehend the behavior of layered smart structures for health monitoring and suppression of vibrations. The present 3D model consists of a set of five second-order differential equations. Navier harmonic forms in the planar directions and the exponential matrix in the transverse direction are applied. The exponential matrix in the thickness direction allows exact computations of stresses, strains, electric displacement components and magnetic induction components. This 3D formulation includes a layer-wise approach where congruence and equilibrium conditions are imposed at each layer interface. Displacements, electric potential, magnetic potential, stresses, transverse normal electric displacement and transverse normal magnetic induction are correctly evaluated by taking into account the zigzag effect. Brischetto and Cesare proposed 3D coupled electro-elastic free vibrations in [43] and 3D coupled magneto-elastic free vibrations in [44]. 3D pure elastic free vibration analysis of multilayered structures was proposed in [45]. For the first time, a 3D coupled magneto-electro-elastic free vibration analysis of layered plates is proposed considering the exponential matrix method in the transverse direction and the Navier harmonic forms in the planar directions. Results proposed in this paper can be used by scientists involved in the development of other formulations for magneto-electro-elastic analyses and by those researchers interested in the free vibration analysis of smart magneto-electro-elastic structures. The present paper is organized in different sections: in the second section, the 3D coupled magneto-electro-elastic model is presented in terms of governing, geometric and constitutive equations and solution methodology; in the third section, assessments are proposed to validate the model and new benchmark cases are shown to investigate the free vibration behavior of MEE multilayered plates. Some conclusive remarks and future developments are reported in the last section.

2  3D Magneto-Electro-Elastic Analytical Formulation

The present section shows the 3D formulation for coupled magneto-electro-elastic free vibration analyses of flat panels. In the first subsection, the set of five second-order differential governing equations for the 3D magneto-electro-elastic coupling is proposed. Then, constitutive and geometrical relations are shown. In the last subsection discusses the solution method in terms of Navier harmonic forms in the planar directions and the exponential matrix in the transverse direction.

2.1 3D Set of Governing Equations for the Magneto-Electro-Elastic Plate Problem

The five 3D governing equations for the magneto-electro-elastic free vibration analysis can be explicitly written for each k layer included in multilayered flat panels:

σxx,xk+σxy,yk+σxz,zk=ρku¨k,(1a)

σxy,xk+σyy,yk+σyz,zk=ρkv¨k,(1b)

σxz,xk+σyz,yk+σzz,zk=ρkw¨k,(1c)

ℬx,xk+ℬy,yk+ℬzk,z=0,(1d)

𝒟x,xk+𝒟y,yk+𝒟zk,z=0.(1e)

Stresses (σxxk,σxyk,σxzk,σyyk,σyzk,σzzk), magnetic inductions (ℬxk, ℬyk, ℬzk) and electric displacements (𝒟xk, 𝒟yk, 𝒟zk) are dependent from coordinates x, y and z and time t. ρk indicates material mass density. u¨k, v¨k and w¨k are displacements second derivatives in time. Subscripts ,x, ,y and ,z state partial derivatives performed with respect to rectilinear coordinates. Eqs. (1d and 1e) have been derived in an orthogonal reference system using the work by Povstenko [46]; magnetic induction and electric displacement have been properly rewritten. The orthogonal rectilinear reference system and geometry defined for flat panels (plates) are clearly indicated in Fig. 1.

Figure 1: Plate (flat panel) geometry. Dotted green lines represent the Ω0 middle reference surface

2.2 Geometrical and Constitutive Equations for the Magneto-Electro-Elastic Plate Problem

Geometrical equations are employed to link displacements with strains, the electric potential with the electric field and the magnetic potential with the magnetic field. MEE geometrical relations are:

εxxk=u,xk,εyyk=v,yk,εzzk=w,zk,(2a)

γyzk=v,zk+w,yk,γxzk=u,zk+w,xk,γxyk=u,yk+v,xk,(2b)

ℰxk=−ϕ,xk,ℰyk=−ϕ,yk,ℰzk=−ϕ,zk,(3)

ℋxk=−ψ,xk,ℋyk=−ψ,yk,ℋzk=−ψ,zk,(4)

where εxxk, εyyk, εzzk, γyzk, γxzk and γxyk are strain components; ℰxk, ℰyk and ℰzk are electric field components; ϕk is the electric potential; ℋxk, ℋyk and ℋzk are magnetic field components; ψk is the magnetic potential.

Constitutive equations are employed to link stresses, the electric displacement and the magnetic induction with strains, electric field components and magnetic field components. Their explicit form is:

σxxk=C11kεxxk+C12kεyyk+C13kεzzk+C16kγxyk−e31kℰzk−q31kℋzk,(5a)

σyyk=C12kεxxk+C22kεyyk+C23kεzzk+C26kγxyk−e32kℰzk−q32kℋzk,(5b)

σzzk=C13kεxxk+C23kεyyk+C33kεzzk+C36kγxyk−e33kℰzk−q33kℋzk,(5c)

σyzk=C44kγyzk+C45kγxzk−e14kℰxk−e24kℰyk−q14kℋxk−q24kℋyk,(5d)

σxzk=C45kγyzk+C55kγxzk−e15kℰxk−e25kℰyk−q15kℋxk−q25kℋyk,(5e)

σxyk=C16kεxxk+C26kεyyk+C36kεzzk+C66kγxyk−e36kℰzk−q36kℋzk,(5f)

𝒟xk=e14kγyzk+e15kγxzk+ϵ11kℰxk+ϵ12kℰyk+d11kℋxk+d12kℋyk,(6a)

𝒟yk=e24kγyzk+e25kγxzk+ϵ12kℰxk+ϵ22kℰyk+d12kℋxk+d22kℋyk,(6b)

𝒟zk=e31kεxxk+e32kεyyk+e33kεzzk+e36kγxyk+ϵ33kℰzk+d33kℋzk,(6c)

ℬxk=q14kγyzk+q15kγxzk+d11kℰxk+d12kℰyk+μ11kℋxk+μ12kℋyk,(7a)

ℬyk=q24kγyzk+q25kγxzk+d12kℰxk+d22kℰyk+μ12kℋxk+μ22kℋyk,(7b)

ℬzk=q31kεxxk+q32kεyyk+q33kεzzk+q36kγxyk+d33kℰzk+μ33kℋzk,(7c)

where Cijk are elastic coefficients, eijk are piezoelectric coefficients, qijk are piezomagnetic coefficients, ϵijk are electric permittivity coefficients, dijk are electro-magnetic coupling coefficients and μijk are magnetic permittivity coefficients. Eqs. (2a)–(7a) allow the interaction between magnetic, electric and elastic fields.

2.3 Solution Methodology

The solution methodology for the 3D magneto-electro-elastic free vibration problem (Eq. (1a)) is here discussed in depth. The present methodology is based on a closed-form solution. Therefore, the following coefficients have to be imposed to zero:

C16k=C26k=C36k=C45k=0,ϵ12k=0,μ12k=0,d12k=0,(8)

e14k=e25k=e36k=0,q14k=q25k=q36k=0.(9)

The imposition suggested in Eq. (8) permits the analysis of only 0∘/90∘ lamination angles for orthotropic laminae. The introduction of Eqs. (2a)–(4) into Eqs. (5a)–(7a) allows to write Eq. (1a) in terms of displacements (uk, vk and wk), magnetic potential ψk and electric potential ϕk. These are the primary variables of the magneto-electro-elastic free vibration problem.

Navier harmonic forms for primary variables are introduced in the planar directions of the plate as follows:

uk(x,y,z,t)=Uk(z)eiωtcos⁡(α¯x)sin⁡(β¯y),(10a)

vk(x,y,z,t)=Vk(z)eiωtsin⁡(α¯x)cos⁡(β¯y),(10b)

wk(x,y,z,t)=Wk(z)eiωtsin⁡(α¯x)sin⁡(β¯y),(10c)

ϕk(x,y,z,t)=Φk(z)eiωtsin⁡(α¯x)sin⁡(β¯y),(10d)

ψk(x,y,z,t)=Ψk(z)eiωtsin⁡(α¯x)sin⁡(β¯y),(10e)

where terms written in capital letters are primary variable amplitudes. ω=2πf is the circular frequency, i is the imaginary unit and t indicates the time. Terms α¯ and β¯ are defined as:

α¯=mπa,β¯=nπb,(11)

considering a and b as the planar dimensions of flat panels and m and n the half-wave numbers in x and y directions, respectively.

Harmonic forms proposed in Eq. (10a) automatically satisfy simply supported boundary conditions for each side belonging to flat panels:

vk=0,wk=0,ϕk=0,ψk=0,σxxk=0forα=0,a,uk=0,wk=0,ϕk=0,ψk=0,σyyk=0forβ=0,b.(12)

Navier harmonic forms allows to write the 3D set of second-order differential magneto-electro-elastic equations in terms of primary variable amplitudes and related first and second derivatives in z. The compact form of these equations is:

A1kUk+A2kVk+A3kWk+A4kΦk+A5kΨk+A6kU,zk+A7kW,zk+A8kΦ,zk+A9kΨ,zk+A10kU,zzk=0,(13a)

A11kUk+A12kVk+A13kWk+A14kΦk+A15kΨk+A16kV,zk+A17kW,zk+A18kΦ,zk+A19kΨ,zk+A20kV,zzk=0,(13b)

A21kUk+A22kVk+A23kWk+A24kΦk+A25kΨk+A26kU,zk+A27kV,zk+A28kW,zk+A29kΦ,zk++A30kΨ,zk+A31kW,zzk+A32kΦ,zzk+A33kΨ,zzk=0,(13c)

A34kUk+A35kVk+A36kWk+A37kΦk+A38kΨk+A39kU,zk+A40kV,zk+A41kW,zk+A42kW,zzk++A43kΦ,zzk+A44kΨ,zzk=0,(13d)

A45kUk+A46kVk+A47kWk+A48kΦk+A49kΨk+A50kU,zk+A51kV,zk+A52kW,zk+A53kW,zzk++A54kΦ,zzk+A55kΨ,zzk=0,(13e)

where each Aik term includes elastic coefficients, piezoelectric coefficients, magnetic coefficients and α¯ and β¯ parameters.

The exponential matrix method is used as a solution in the z thickness direction. Coefficients Aik are constant in each k physical layer because they depend on well-known coefficients for elastic, electric and magnetic properties and on calculated terms α and β. The number of primary unknowns must be redoubled to obtain a set of first-order differential equations. This procedure has been clearly proposed in [47] where the redoubling of primary unknowns is obtained by considering in the related vector both primary variables and their first derivatives in the z thickness direction. Therefore, the final set of ten first-order differential equations in z is:

A10kU,zk=A10kU,zk,(14a)

A20kV,zk=A20kV,zk,(14b)

P1kW,zk=P1kW,zk,(14c)

P1kΦ,zk=P1kΦ,zk,(14d)

P1kΨ,zk=P1kΨ,zk,(14e)

A10kU,zzk=−A1kUk−A2kVk−A3kWk−A4kΦk−A5kΨk−A6kU,zk−A7kW,zk+−A8kΦ,zk−A9kΨ,zk,(14f)

A20kV,zzk=−A11kUk−A12kVk−A13kWk−A14kΦk−A15kΨk−A16kV,zk−A17kW,zk+−A18kΦ,zk−A19kΨ,zk,(14g)

P1kW,zzk=−P2kUk−P3kVk−P4kWk−P5kΦk−P6kΨk−P7kU,zk−P8kV,zk−P9kW,zk+−P10kΦ,zk−P11kΨ,zk,(14h)

P1kΦ,zzk=−P12kUk−P13kVk−P14kWk−P15kΦk−P16kΨk−P17kU,zk−P18kV,zk−P19kW,zk+−P20kΦ,zk−P21kΨ,zk,(14i)

P1kΨ,zzk=−P22kUk−P23kVk−P24kWk−P25kΦk−P26kΨk−P27kU,zk−P28kV,zk−P29kW,zk+−P30kΦ,zk−P31kΨ,zk.(14j)

Eq. (14a) are solved via the exponential matrix method. Coefficients Pik are explicitly given in the Appendix A section.

Matrix form of Eq. (14a) is:

DkX,zk=AkXk⇒X,zk=A∗jXk,(15)

where Xk and X,zk primary variable amplitudes and related derivatives in z, respectively. The 10×1 vector is Xk={U(z)k,V(z)k,W(z)k,Φ(z)k,Ψ(z)k,U(z),zk,V(z),zk,W(z),zk,Φ(z),zk,Ψ(z),zk}. The solution of Eq. (15) is obtained by applying the exponential matrix method via the Taylor expansion [47]:

Xk(hk)=A∗∗kXk(0)=[∑i=0N(A∗k)ii!hki]Xk(0).(16)

where (A∗k)0=I is the 10×10 identity matrix. k indicates the layer and it goes from 1 (first layer) to M (last layer). hk is the layer thickness for each physical layer.

Interlaminar continuity conditions for primary variables, transverse normal and transverse shear stresses (σzzk, σxzk, σyzk), transverse normal electric displacement component (𝒟zk) and transverse normal magnetic induction component (ℬzk) are fundamental to obtain a layer wise model. Interlaminar continuity conditions consider the top (t) of the k−1 layer and the bottom (b) of the k layer.

Interlaminar continuity conditions are imposed as:

ubk=utk−1,vbk=vtk−1,wbk=wtk−1,ϕbk=ϕtk−1,ψbk=ψtk−1(17a)

σxzbk=σxztk−1,σyzbk=σyztk−1,σzzbk=σzztk−1,𝒟zbk=𝒟ztk−1,ℬzbk=ℬztk−1.(17b)

The introduction of Eqs. (5a)–(7a) and (10a) in Eq. (17) allows interlaminar continuity conditions written in terms of maximum amplitudes:

Ubk=Utk−1,(18a)

Vbk=Vtk−1,(18b)

Wbk=Wtk−1,(18c)

Φbk=Φtk−1,(18d)

Ψbk=Ψtk−1,(18e)

C55kα¯Wbk+C55kU,zbk+e15kα¯Φbk+q15kα¯Ψbk=C55k−1α¯Wtk−1+C55k−1U,ztk−1+e15k−1α¯Φtk−1+q15k−1α¯Ψtk−1,(18f)

C44kβ¯Wbk+C44kV,zbk+e24kβ¯Φbk+q24kβ¯Ψbk=C44k−1β¯Wtk−1+C44k−1V,ztk−1+e25k−1β¯Φtk−1+q25k−1β¯Ψtk−1,(18g)

C13kα¯Ubk−C23kβ¯Vbk+C33kW,zbk+e33kΦ,zbk+q33kΨ,zbk=−C13k−1α¯Utk−1−C23k−1β¯Vtk−1+C33k−1W,ztk−1++e33k−1Φ,ztk−1+q33k−1Ψ,ztk−1,(18h)

−e31kα¯Ubk−e32kβ¯Vbk+e33kW,zbk−ϵ33kΦ,zbk−d33kΨ,zbk=−e31k−1α¯Utk−e32k−1β¯Vtk+e33k−1W,ztk−1+−ϵ33k−1Φ,ztk−1−d33k−1Ψ,ztk−1,(18i)

−q31kα¯Ubk−q32kβ¯Vbk+q33kW,zbk−d33kΦ,zbk−μ33kΨ,zbk=−q31k−1α¯Utk−q32k−1β¯Vtk+q33k−1W,ztk−1+−d33k−1Φ,ztk−1−μ33k−1Φ,ztk−1.(18j)

The matrix form is:

Xbk=Tk,k−1Xtk−1,(19)

where Tk,k−1 is the 10×10 transfer, it allows to link interfaces between two adjacent layers k and k−1. In Eq. (19), Xtk means Xk(hk) and Xbk means Xk(0).

The recursive introduction of Eq. (16) in Eq. (19) allows the solution along the z thickness direction. The final equation has the following form:

XtM=HmXb1,(20)

where Hm=A∗∗MTM,M−1…T2,1A∗∗1. Matrix Hm includes the matrices Tk,k−1 and A∗∗k computed for each k layer of the multilayered plate. M is the last layer at the top and 1 is the first layer at the bottom. These two matrices include elastic, magnetic and electric material coefficients, dimensions for the flat panel and thickness of each layer. The most important feature of matrix Hm is the 10×10 dimension, independently of the number of k layers involved in the multilayered configuration and the order N used in Eq. (16). In this way, a low computational cost is guaranteed.

Load boundary conditions for open-circuit and closed-circuit configurations are:

σzz=0,σxz=0,σyz=0,𝒟z=0,ℬz=0forz=±h/2(open circuit)(21)

σzz=0,σxz=0,σyz=0,ϕ=0,ψ=0forz=±h/2(closed circuit).(22)

The open-circuit configuration may be explicitly stated as:

σzzb1=−C131α¯Ub1−C231β¯Vb1+C331W,zb1+e331Φ,zb1+q331Ψ,zb1=0,(23a)

σyzb1=C441β¯Wb1+C441V,zb1+e241β¯Φb1+q241β¯Ψb1=0,(23b)

σxzb1=C551α¯Wb1+C551U,zb1+e151α¯Φb1+q151α¯Ψb1=0,(23c)

𝒟zb1=e311α¯Ub1+e321β¯Vb1+e331W,zb1−ϵ331Φ,zb1−d331Ψ,zb1=0,(23d)

ℬzb1=q311α¯Ub1+q321β¯Vb1+q331W,zb1−d331Φ,zb1−μ331Ψ,zb1=0,(23e)

σzztM=−C13Mα¯UtM−C23Mβ¯VtM+C33MW,ztM+e33MΦ,ztM+q33MΨ,ztM=0,(23f)

σβztM=C44Mβ¯WtM+C44MV,ztM+e24Mβ¯ΦtM+q24Mβ¯ΨtM=0,(23g)

σαztM=C55Mα¯WtM+C55MU,ztM+e15Mα¯ΦtM+q15Mα¯ΨtM=0,(23h)

𝒟ztM=e31Mα¯UtM+e32Mβ¯VtM+e33MW,ztM−ϵ33MΦ,ztM−d33MΨ,ztM=0,(23i)

ℬztM=q31Mα¯UtM+q32Mβ¯VtM+q33MW,ztM−d33MΦ,ztM−μ33MΨ,ztM=0.(23j)

The closed-circuit configuration can be explicity written as:

σzzb1=−C131α¯Ub1−C231β¯Vb1+C331W,zb1+e331Φ,zb1+q331Ψ,zb1=0,(24a)

σβzb1=C441β¯Wb1+C441V,zb1+e241β¯Φb1+q241β¯Ψb1=0,(24b)

σαzb1=C551α¯Wb1+C551U,zb1+e151α¯Φb1+q151α¯Ψb1=0,(24c)

Φb1=0,(24d)

Ψb1=0,(24e)

σzztM=−C13Mα¯UtM−C23Mβ¯VtM+C33MW,ztM+e33MΦ,ztM+q33MΨ,ztM=0,(24f)

σβztM=C44Mβ¯WtM+C44MV,ztM+e24Mβ¯ΦtM+q24Mβ¯ΨtM=0,(24g)

σαztM=C55Mα¯WtM+C55MU,ztM+e15Mα¯ΦtM+q15Mα¯ΨtM=0.(24h)

ΦtM=0,(24i)

ΨtM=0.(24j)

Both open and closed circuit conditions can be further compacted in matrix form as:

Bb1Xb1=0,(25a)

BtMXtM=0,(25b)

where Bb1 and BtM are the generic 5×10 load boundary condition matrices at the bottom (b) of the first layer (1) and at top (t) of the last layer (M), respectively. Eq. (25) can be rewritten by using Eq. (20):

[BtMHmBb1]Xb1={00}⇒EXb1=0.(26)

The null space of matrix E must be imposed via the computation of eigenvalues. The determinant allows to compute eigenvalues and eigenvectors. Matrix E determinant is a polynomial function in terms of ω2, where ω=2πf is the circular frequency. Solving the equation, the minor ω value satisfying the polynomial equation makes null the space of matrix E. Therefore, the eigenvectors and eigenvalues computation of matrix E by considering the proper ω2 value is possible. Eigenvectors can be evaluated along the thickness direction of the flat panel via the recursive use of Eqs. (16) and (19). In this way, frequency modes along the thickness direction are shown considering displacements, magnetic potential, electric potential, transverse normal magnetic induction/electric displacement and stresses.

3  Results

The results section is made up of two subsections. In the first one, two assessment cases are shown to validate the present 3D coupled magneto-electro-elastic model (called 3D-u-ϕ-ψ). Comparisons are made using a 3D electro-elastic model (3D-u-ϕ) and a 3D magneto-elastic model (3D-u-ψ). Then, a new multilayered plate case is studied considering different thickness ratios.

The 3D-u-ϕ-ψ model is preliminarily confronted with the 3D-u-ϕ model by Brischetto and Cesare [43] and with the 3D-u-ψ model by Brischetto and Cesare [44]. Open-circuit and closed-circuit configurations are validated. For both assessments, different a/h ratios are compared. A complete validation is proposed because magnetic and electric fields are separately validated considering thick and thin flat panels. Further effects involved in the flat panels, such as material and thickness layer effects, are evaluated. Comparisons are proposed in terms of the first three circular frequency values ω¯ for several (m,n). Third and fourth assessments also verify the full electro-elastic-magnetic coupling correctness.

In the benchmark subsection, a multilayered square flat panel is analyzed under open-circuit and closed-circuit conditions. The first five circular frequencies ω¯ for different (m,n) couples are reported in tabular form. First three frequency modes through the z direction are represented in graphical form for different thickness ratios a/h. These frequency modes are given considering normalized displacements u∗, v∗ and w∗, electric potential ϕ∗, magnetic potential ψ∗, transverse normal stress σzz∗, transverse normal electric displacement 𝒟z∗ and transverse normal magnetic induction ℬz∗. Normalization for each variable is performed by dividing each variable by its maximum value.

3.1 Assessments

In the first two proposed assessments, the magnetic permittivity coefficients μ1 and μ2 for the CoFe2O4 are considered equal to −590⋅103 nH/m in Table 1 to be coherent with reference results proposed in [43] and [44]. Elastic material properties in terms of Young modulus (Ei), shear modulus (Gij) and Poisson ratio (νij) have been computed from Cij elastic coefficients by considering relations proposed in [48].

The first assessment (A1) considers a simply supported multilayered square flat panel with in-plane dimensions a=b=1 m and variable thickness h. Multilayered configuration is PZT-4/Al2024/PVDF/Al2024/PZT-4 where hPZT−4=hAl2024=0.1h and hPVDF=0.6h. h is the total thickness of the flat panel. Both closed circuit (A1-CC) and open circuit (A1-OC) configurations are validated. The open circuit configuration for the electro-elastic case states 𝒟zt=𝒟zb=0 C/m2; the closed circuit configuration imposes ϕt=ϕb=0 V. Three different impositions for half-wave number couples are analyzed for open and closed circuit configurations: m=n=1, m=1 and n=2 and m=n=2. The first three frequencies are given for each (m,n) couple. Properties related to mechanical, electric and magnetic material coefficients are reported in Table 1. Comparisons are performed considering as reference solution the 3D-u-ϕ electro-elastic model by Brischetto and Cesare [43]. In Tables 2 and 3, the first three circular frequencies ω¯ for several (m,n) values are compared with the reference solution for thick (a/h=4), moderately thick (a/h=10,20) and thin (a/h=50,100) flat panels in both closed-circuit and open-circuit configurations. A perfect matching is shown when the 3D-u-ϕ-ψ electro-magneto-elastic model is confronted with the 3D-u-ϕ model for each thickness ratio. Results in Tables 2 and 3 allow to state that the 3D-u-ϕ-ψ model opportunely includes the electro-elastic coupling, and related material and thickness layer effects. Differences between results obtained in closed circuit configuration and results obtained in open circuit configuration are very small. These small differences are more evident for thicker plates and higher (m,n) values.

In the second assessment (A2), a simply supported multilayered square flat panel is considered. In-plane dimensions are a=b=1 m and the thickness h is variable. The multilayered configuration is CoFe2O4/Al2024/Foam/Al2024/CoFe2O4 where hCoFe2O4=hAl2024=0.1h and hPVDF=0.6h; h is the total thickness of the multilayered flat panel. Material mass density of the CoFe204 for this assessment is ρCoFe204=5300 kg/m3. The open-circuit configuration (OC) is analyzed. This configuration states ℬzt=ℬzb=0 T. Material properties in terms of elastic, electric and magnetic coefficients are reported in Table 1. The reference solution used is the 3D-u-ψ magneto-elastic model in [44]. In Table 4, the first three circular frequencies ω¯ for (1,1), (2,2) and (3,3) half-wave couples are reported for the same thickness ratios proposed in A1. Even in this case, a perfect matching is highlighted between ω¯ values computed with the 3D-u-ϕ-ψ electro-magneto-elastic model and those calculated with the 3D-u-ψ magneto-elastic model. This feature is confirmed for each thickness ratio. Results presented in Table 4 permit validation of the magneto-elastic coupling and related material and thickness layer effects for thick, moderately thick and thin multilayered magneto-elastic flat panels.