Lamb waves propagation in layered piezoelectric ...

Accepted Manuscript Lamb waves propagation in layered piezoelectric/piezomagnetic plates Hamdi Ezzin, Morched Ben Amor, Mohamed Hédi Ben Ghozlen PII: DOI: Reference:

S0041-624X(16)30416-4 http://dx.doi.org/10.1016/j.ultras.2016.12.016 ULTRAS 5445

To appear in:

Ultrasonics

Received Date: Revised Date: Accepted Date:

15 April 2016 22 December 2016 23 December 2016

Please cite this article as: H. Ezzin, M. Ben Amor, M. Hédi Ben Ghozlen, Lamb waves propagation in layered piezoelectric/piezomagnetic plates, Ultrasonics (2016), doi: http://dx.doi.org/10.1016/j.ultras.2016.12.016

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Lamb waves propagation in layered piezoelectric/piezomagnetic plates Hamdi Ezzin (a), Morched Ben Amor (b)* and Mohamed Hédi Ben Ghozlen(a)

(a) Laboratory of Physics of Materials, Faculty of Sciences of Sfax, BP 1171, 3000 University of Sfax, Tunisia. (b) Sfax Preparatory Engineering Institute, Menzel Chaker Road 0.5 km Sfax Tunisia. BP 1172-3000, Tel: +216 74 241 403, Fax: +216 74 246 347. Reprint requests to Morched Ben Amor, Fax: +21674403934, E-mail: [email protected]

Abstract A dynamic solution is presented for the propagation of harmonic waves in magneto-electroelastic plates composed of piezoelectric BaTiO3(B) and magnetostrictive CoFe2O4(F) material. The state-vector approach is employed to derive the propagator matrix which connects the field variables at the upper interface to those at the lower interface of each layer. The ordinary differential approach is employed to determine the wave propagating characteristics in the plate by imposing the traction-free boundary condition on the top and bottom surfaces of the layered plate. The dispersion curves of the piezoelectric– piezomagnetic plate are shown for different thickness ratios.The numerical results show clearly the influence of different stacking sequences as well as thickness ratio on dispersion curves and on magneto-electromechanical coupling factor. These findings could be relevant to the analysis and design of high-performance surface acoustic wave (SAW) devices constructed from piezoelectric and piezomagnetic materials.

Keys Words: Wave propagation; Laminated piezomagnetic/piezoelectric plates; Dispersion curve; State space approach.

1

1. Introduction Piezoelectric–piezomagnetic composites (PPC) have recently attracted many studies, compared to the one of single phase material. These materials are employed now in a variety of mechanical, civil and aerospace applications as sensors, actuators, and storage devices [1,2] at various scales. The growth of such applications requires the accurate knowledge of elastic wave behaviour to help, to design and optimize PPC devices. For the analysis of elastic waves on multilayered composite structures, the most common method is the matrix formalism and the state-vector (or state space) among others (e.g., Thomson [3]; Haskell [4]; Gilbert and Backus [5]Bahar [6]). Due to their conceptual simplicity and reduced programming effort and computing time, these approaches have now been extended to various complicated layered structures, including, for example, static and free vibration analysis of piezoelectric or magneto-electro-elastic plates [7,9]. The laminated ME materials show a remarkably large ME coefficient, indicating a broad prospect in applications of sensors, transducers and so on. So it is urgent to comprehend the mechanical behaviour of the ME material in depth, like deformation, vibration and wave propagation. Chen et al [10] obtained the general solution to the three dimensional problem of the magneto-electro-elastic materials. Chen et al [11] studied the problem of wave in magneto-electro-elastic multilayered plate using the state space approach. W.X et al [12] have focused on the wave propagating problem, including Lamb waves in the piezoelectric-piezomagnetic plate. Chen et al [13] studied guided waves in a magneto-electro-elastic multilayered structure. Others studies have investigated horizontal shear waves (SH) in two bounded semi-infinite piezoelectric and piezomagnetic materials [14], Love waves in a piezoelectric-piezomagnetic layered structure [15] or layered magnetoelectro-elastic structure with initial stress [16] are also published. Recently, H.Ezzin et al. [17] investigated the Love waves propagation in a transversely isotropic piezoelectric layer on a piezomagnetic half-space by the stiffness matrix method. Using the propagator matrix and state-vector approaches, an analytical treatment is presented for the propagatin of harmonic waves in magneto-electro-elastic plates by Chen et al [18]. However, in the litterature, there were few studies dealing with the propagation of Lamb waves in a magneto-electro-elastic multilayered structure. The objective of this work is to numerically study the Lamb wave propagation in layered piezoelectric/piezomagnetic plates. For that purpose the stiffness matrix method (SMM) and the ordinary differential equation (ODE) [19,20] have been extended to study the magneto-electro-elastic multilayered structure.The constitutive relation used is of general anisotropy with piezoelectric and piezomagnetic coupling. The global 2

stiffness matrix of the multilayer is obtained using the continuity condition at the interface of the layers. Numerical examples are presented to show the features of dispersion curves as well as the effect of the thickness ratio (R=he/hm) on phase velocity and on magnetoelectromechanical coupling factor of the first mode . This paper is organized as follows: In Section 2, we present the basic equations in terms of the state-vector approach. In Section 3, we derive the dispersion relation for the layered plate. While numerical examples are given in Section 4, conclusions are drawn in Section 5.

2. Formulation of the problem. Basic equations:

Let us consider an anisotropic magneto-electro-elastic layered plate which is infinite in the (x1,x2)-plane but finite in the vertical direction with a total thicknesses h=he +hm as shown in Fig. 1. he and hm are the thickness of piezoelectric and the piezomagnetic layers respectively. Both the layers are perfectly bonded along the interface x3 = 0. For such material, the coupled constitutive equations are given by [21]:

CoFe2O4 plate

hm he

BaTiO3 plate

x1

0 x3 Fig.1: skeleton of multiferroic laminate

uk  ekij Ek  f kij H k xl

(1)

Di  eikl

uk   ik Ek xl

(2)

Bi  f ikl

uk  ik H k xl

(3)

 ij  Cijkl

3

where  ij , Di , Bi , Ei , H i and u i are the components of the stress, electric displacement,the magnetic induction, the electric field, the magnetic field and the particule displacement respectively. Here and throughout the article, Einstein’s summation convention are used, and i,j,k and l = 1, 2, 3 or equivalently x, y, z. Cijkl , eijk f ijk ,  ij and ik are the elastic, the piezoelectric, the piezomagnetic, the dielectric permittivity and the magnetic permeability constants, respectively. Here the magneto-electric coupling coefficient is not considered. The dynamic equation for a magneto-electro-elastic medium, when body force, current density, and variations of electric charges density are disregarded, is governed by:



 2ui  ij ,  t 2 x j

(4)

Di 0 xi

(5)

Bi 0 xi

(6)

For an anisotropic piezoelectric-piezomagnetic system as shown in Fig. 1.    iU , T T denotes a state vector .The general solution for the state vector can be represented in the form

 ( x3 )   ( x3 ) exp i ( k1 x1  t ) where  is the angular frequency and k1 is the projection of the wave vector along x1 axis . The wave equation is written under the Thomson-Haskell parameterization of the Stroh formalism [22,23]: d  ( x3 )   i A ( x3 ) dx3

(7)

Where A is the fundamental acoustic tensor, also called state or Stroch matrix, a square matrix dimensioned (8x8), which depends mainly on the physical properties and the guiding slowness component S1 , A can be written as [24,25]:  S133131 A  i  2 1 S1 (1333 31  11 )  I 4

331   S113331 

(8)

I4 is the (4 x 4) identity matrix but with zero (4, 4) element, S 1 denote the first component of the slowness vector. The parameter ρ is the density of the material and ik are the (4x 4)

4

matrices formed from the elastic constants Cijkl , piezoelectric constant ekij , piezomagnetic constants f kij , dielectric permittivity  ij and magnetic permeability constants ij .

C1i1k C1i 3k ek1i f k1i  C ek 3i f k 3i  3i1k C3i 3k  ik   f i1k f i 3k   ik  ik  i,k= 1, 3    f i1k f i 3k  g ik  ik 

(9)

It is obvious on the basis of (Eq.7) that tensor A makes up an eigenvector equation where the eigenvalues are the set of S 3 third component of the slowness vectors. From this perspective

ik submatrices establish a relationship between mechanical displacements , electrical potential

and

magnetic

potential. ik

matrix

form

informs

about

the

magneto-

electromechanical coupling. When the partial waves of Lamb modes are decoupled from the SH modes, the state vector is eight dimensioned and it includes the stress vectors T   13, 33 , D3 , B3  and the particle displacement vector U  iu1 , u3 ,  ,  , where  and T

T

 are the electric and magnetic potentials, respectively,  i 3 are the components of the normal stress vector and D3 and B3 are the normal electric displacement and magnetic induction, respectively: The boundary and continuity conditions for a multilayered piezoelectric-piezomagnetic structure require that: (1) the mechanical displacement, the normal component of stress, electric and magnetic displacement, magnetic and electric potential should be continuous at the interfaces and (2) the normal component of the stress, electric and magnetic displacement, magnetic and electric potential should be zero at the upper and bottom surfaces. The surfaces of piezoelectric/piezomagnetic layers are assumed to be mechanically free (33=13= 0) are automatically incorporated in the constitutive relations of the plate: Electrically and magnetically open conditions (denoted by “oo”) [26]: At x3=  ( h m  he )

D3  B3  0 and  i 3  0 ( i =1,3)

(10)

Electrically and magnetically shorted conditions (denoted by “ss”) [26]:

 = 0,  = 0 and  i 3  0 ( i =1,3)

(11)

At x3=- h e

5

The continuity conditions at the interface of the layer are

u3m  u3e , m   e , m   e

(12)

13m  13e , 33m   33e , D3m  D3e , B3m  B3e

(13)

Where the superscript “m” and “e” indicate the quantities in the piezomagnetic layer and piezoelectric layer respectively. The dispersion equations can be obtained from the conditions that the determinant of the coefficient matrix of the 8 equations vanishes. For the sake of brevity, we shall not list these equations any further. Anyhow, these equations can yield the wave number k for a given value of phase velocity v. However because of the complexity, the dispersion equations can only be numerically solved. Applying the traction-free boundary (33=13= 0) and the electrically and magnetically open

( D3  B3  0 ) or electrically and magnetically shorted ( = = 0), two cases must be distinguished in this study: the electro-magnetically open surface (oo) or the electrooo ss magnetically shorted surface (ss) case. Accordingly, (8x8) characteristic matrix Q and Q

can be constructed, where the supertscript "o" and "s" denotes electro-magnetically open and electro-magnetically shorted surface respectively:

  iU e   Q11oo Q12oo   iU m        T e    Q oo Q oo  T m   0  21  h 22    iU e   Q11ss     T e    Q ss  0  21

Q12ss   iU m    Q22ss  T m  h

(14)

(15)

Specifically, the dispersion equations (14)–(15) for the plate for both cases has non-trivial oo,ss solutions when the determinant ( Q21 ) = 0.

3. Numerical results and discussion a. BaTiO3/CoFe2O4 layered plates. In this section, we present some numerical results by using the formulation presented in this paper. Numerical calculation is completed by using Matlab software. The material properties are listed in Tables 1 and 2 [26]. First, to verify our approach, we will consider an example taken from literature [26]. In the following figures, the horizontal axis represents the non-

6

dimensional wavenumber Kh, and the vertical axis represents the non-dimensional frequency (1) /  (1) .  defined as follows:   h / C44

Fig.1 shows the dispersion curves of Lamb waves modes propagating in a homogeneous plate made of piezomagnetic and piezoelectric layer (CoFe2O4/BaTiO3) without initial stress (=0). The piezomagnetic and piezoelectric axis are oriented along x1. Comparing the solution obtained by our method with the result obtained by Zhou et al [26], we find that they are in agreement.

15



10

5

0

0

1

2

3

4

5 kh

6

7

8

9

Fig.2: Dispersion curves of F/B laminate, without intial stress ( = 0): (a) result from [26], (b): author result.

b. Effect of thickness ratio on dispersion curves.

In this study, a two-layered multiferroic laminate with the stacking sequences F/F, B/B, and F/B (B and F represent, respectively, BaTiO3 and CoFe2O4) are investigated in this section. For comparison, results for the homogeneous plate made of piezoelectric BaTiO 3 (i.e., with a B/B stacking) and magnetostrictive CoFe2O4(i.e., with an F/F stacking) are also presented. The material properties are listed in Tables 1 and 2 [26]. The dispersion curves of F/F and B/B coupled plates are shown in Fig. 3(a-b). It can be seen from Fig. 3(a-b) that the phase velocity approaches the bulk shear wave velocity of the piezoelectric (piezomagnetic) material with the increase in the wave number for different modes. It means that the phase velocity approaches the smaller bulk shear wave velocity of the material in the system. It is observed that the phase velocity has great discrepancy for the 7

10

smaller wave number, and when the wave number becomes larger, the discrepancy will decrease, which is due to the fact that the bulk shear wave velocity of BaTiO 3 is larger than that of CoFe2O4. It is clear from Fig. 3 and 4 that the phase velocity corresponding to the F only (i.e., the F/F stacking sequence) is much larger than those corresponding to the other stacking sequences. Besides, comparing Fig. 3(a-b), one would notice that A2 and S2 modes are separated for the F/F plate but in the case of B/B plate, the two modes approach closely to each other in the kh range between 4 and 6. Fig. 4(a-d) show that the (dispersion curves) phase velocity of the fourth first modes are approximately independent of thickness ratio. However, for the fifth and sixth modes, the phase velocities depend on thickness ratio and when thickness ratio R =0.125 these two modes are coupled.

Table 1: Material coefficients of the piezomagnetic CoFe2O4 [26]. fij (NA-1 m-1)

 ij (10-9 Fm-1 )  ij (10-6 Ns2 C-2)

C11 286.0

f15 550.0

 11 0.08

C33 269.5

f31 580.3

 33

C12 173.0

f33 699.7

C ij (109 Nm-2)

0.093

11 590.0

ρ (103 Kg/m3) 5.3

 33 157.0

C13 170.0 C44 45.3

Table 2: Material coefficients of the piezoelectric BaTiO 3 [26]. C ij (109 Nm-2)

eij (C m-2)

 ij (10-6 Ns2 C-2)

ij (10-11 Fm-1)

C11 166.0

e15 11.6

 ij 5.0

11 11.2

C33 162

e3 1 -4.4

 ij 10.0

33 12.6

C12 77

ρ (103 Kg/m3) 5.800

e33 18.6

C13 170 C44 45.3

In addition,  0  8.8510 12 Fm 1 , and  0  4 .10 7 Am 1 are the dielectric constant and permeability of vacuum respectively.

8

10000

9000

9000

(a)

F/F

(b) B/B

8000

8000

7000

Phase velocity (m/s)

7000 6000 5000 4000

6000 5000 4000 3000

3000

2000

2000

1000

1000 0

0

0

1

2

3

4

5

6

7

8

9

0

1

2

3 4 5 6 7 Dimensionless wave number (kh)

10

8

9

10

Dimensionless wave number (kh)

Fig.3 (a): dispersion curves of F/F plate

Fig.3 (b): dispersion curves of B/B plate 15000

15000

(c)

F/B: R= 1

Phase velocity (m/s)

Phase velocity (m/s)

(a)

10000

5000

0

1

2

3 4 5 6 7 Dimensionless wave number (kh)

8

9

5000

0

1

2

10

3 4 5 6 7 Dimensionless wave number (kh)

8

9

10

8

9

10

15000

15000

(d)

F/B: (R = 0.25)

Phase velocity (m/s)

(b)

10000

5000

0

F/B: (R=0.5)

10000

0

0

Phase velocity (m/s)

Phase velocity (m/s)

10000

10000

5000

0

0

1

2

3 4 5 6 7 Dimensionless wave number (kh)

8

9

10

F/B: (R= 0.125)

0

1

2

3 4 5 6 7 Dimensionless wave number (kh)

Fig.4 (a-d): dispersion curves of for the F/B plates with different thickness ratios: (a) R=1, (b): R = 0.5, (c) R= 0.25 (d) and R= 0.125.

9

In the other hand, in order to provide the effect of thickness ratio on the phase velocity of the F/B plate, we present the phase velocity for the S0 mode for different thickness ratios (R=he/hm). Fig. 5 shows that the phase velocity decreases with the increase of the thickness ratio and when the wave number rises, all the curves converge to the bulk shear wave velocity of the system. This effect is due to the fact that the wave length of the Lamb wave is comparable to the thickness of the layer at the higher ratio, so the phase velocity vary for different thickness ratio of the layer.

5500 R=0.25 R= 0.5 R=1 R=2

F/B

c.

5500

R=0.25 R= 0.5 R=1 R=2

F/B

5450

4500

5400

Phase velocity (m/s)

Phase velocity (m/s)

5000

5350 5300 5250 5200 5150 5100

4000

5050 0.2

0.4

0.6

0.8 1 1.2 1.4 1.6 1.8 Dimensionless wave number (kh)

2

2.2

3500

3000

0

1

2

3 4 5 6 7 Dimensionless wave number (kh)

8

9

10

Fig. 5: variation of phase velocity of S0 mode for the F/B plate with different thickness ratio.

c. Magneto-electromechanical coupling factor. Another significant parameter is the coupled magneto-electromechanical factor which plays an essential role in the design of SAW devices. Using the velocities V00 anod Vss it is possible to evaluate the coefficient of magneto-electromechanical coupling factor K m2 for the F/B plate.Where V00 and Vss are the phase velocities of the Lamb mode for the magnetoelectrically open (oo) and magneto-electrically shorted (ss) cases, respectively. Arguably, it is 10

possible to use the well-known formula that is used for piezoelectric.Therefore, the magnetoelectromechanical coupling factor can be evaluated with the following formula [27, 28]:

K m2  2

( Voo  Vss ) Voo

(17)

Fig.6 associated with S0 Lamb mode represent the variation of magneto-electromechanical coupling factor. One can easily see that the coefficient of magneto-electromechanical coupling factor increases when the thickness ratio increases. The higher thickness ratio is associated with the higher magneto-electromechanical coupling factor, it is about 17% for S0 at low frequency. In the figure 6 we represent the coupling coefficients of magnetomechanical, electro-mechanical, and magneto-electromechanical for different stacking sequence F/F, B/B and F/B. The first view note that the coefficient of magneto-mechanical couplingfor the sequence F/F is very low, and almost zero. Second the electro-mechanical coupling factor of the stacking sequence B/B is the most important it reach a maximum of 22.2% at kh=0.2 compared to magneto-electromechanical coupling factor for the stacking sequence F/B when it reach a maximum of 10.78% at kh=1.2.

Magneto-electro-mechanical coupling factor (%)

18 16 14

F/B

R= R= R= R=

4 2 1 0.5

12 10 8 6 4 2 0

0

1

2

3 4 5 6 7 Dimensionlesswave number (kh)

8

9

10

Fig.6: Magneto-electromechanical coupling factor of the first mode for the F/B plate with different thickness ratio.

11

Magneto-electro-mechanical coupling factor (%)

25 B/B F/F F/B

Thickness ratio (F/B): R=1 20

15

10

5

0

0

1

2

3 4 5 6 7 Dimensionless wave number (kh)

8

9

10

Fig.7: Magneto-electromechanical coupling factor of the first mode for the F/F, B/B and F/B plate.

4. Wave structure analysis In order to analyze the dispersion feature in more detail, we have also studied the modal analysis in the two-layered multiferroic laminate associated with S0 modes. The distributions of some physical quantities through thickness are shown in Figs. 8. The non-dimensional wave numbers is taken to be kh= 2. In these figures, elastic displacements (u1,u3), mechanical stress (13,33) are normalized by the maximum value in the whole thickness region for the two components, and the electric potential  and magnetic potential  are normalized by their corresponding maximum value if they are different from zero. Our main goal of this part is to study the modal shape of the S 0 mode. The physical quantities profiles are given for two types of thickness ratio: R=1 and R= 4. Figs 8 (a,b) show that the in-plane displacement u1 varies along the thickness direction in a nearly anti-symmetric manner, with respect to the geometric middle surface, while the vertical displacement (deflection) u3 changes slightly. Additionally, Figs. 8 (c,d) show that the components 13 and

12

33 vanishes on the free surfaces, which is compatible with the mechanical boundary conditions, it is apparent that the components of stress decreases with the increase of thicknesss ratio. The distributions of the electric potential and magnetic potential of the first mode for magneto-electrically open case (oo) for R = 1 and R= 4 are illustrated as Figs. 8 (e) and (f). The results indicate that the electric and magnetic potential decrease with the increase of the piezomagnetic layer thickness. Likewise, Fig. 8 (g) and (h) show the effect of thickness ratio on the electric and magnetic induction. It can be clearly seen that the thickness ratio has remarkable effect on the electric and magnetic induction of the S0 mode. It can be seen that the electric and magnetic induction are unimportant when the thickness ratio R = 1, but become significant when increasing thickness ratio. 1

2 R=1 R=4

1.8

(a) Normalized Displacement (u )

(a)

3

1

Normalized Displacement (u )

0.9

(b)

0.8

(b)

0.7

0.6

R=1 R=4

(b)

1.6

1.4

1.2

1

0.5

0.8 0.4

0

0.1

0.2

0.3

0.4

0.5 x3/h

0.6

0.7

0.8

0.9

1

0

1 R=1 R=4

(c)

0.3

0.4

0.5 x3/h

0.6

0.7

0.8

0.9

1

R=1 R=4

(d)

0.4

0.2

33

Normalized stress (  )

13

Normalized stress ( )

0.2

0.6

0.8

0.6

0.4

0.2

0

-0.2

0.1

0

-0.2

-0.4

-0.6

0

0.1

0.2

0.3

0.4

0.5 x3/h

0.6

0.7

0.8

0.9

1

-0.8

0

0.1

0.2

0.3

0.4

0.5 x3/h

0.6

0.7

0.8

0.9

13

1

1

1 R=1 R=4

0.9

(f)

(e)

0.85 0.8 0.75 0.7 0.65

0.1

0.2

0.3

0.4

0.5 x3/h

0.6

0.7

0.8

0.9

0.5 0.4 0.3

0

0.1

0.2

0.3

0.4

0.5 x3/h

0.6

0.7

0.8

0.9

1

0.7

0.8

0.9

1

0.8 0.7

R=1 R=4

(g)

3

-0.1

Normalized magnetic induction (B )

3

0.6

1

0 Normalized electric induction (D )

0.7

0.1

0

0.1

-0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8

0.8

0.2

0.6 0.55

R=1 R=4

0.9

Normalized magnetic potential ( )

Normalized electric potential ( )

0.95

R=1 R=4

(h)

0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5 x /h 3

0.6

0.7

0.8

0.9

1

-0.1

0

0.1

0.2

0.3

0.4

0.5 x3/h

0.6

Fig.8: Profiles of some physical quantities of F/B plate, for S0 mode at kh =2. 5. Conclusion The propagation of the Lamb wave in the piezoelectric/piezomagnetic layered plates is studied using the Ordinary differential equation (ODE) approach with the stiffness matrix method (SMM) to determine the wave propagating characteristics in the plate. The dispersion curves for difference sequences (F/F, B/B and F/B) and thickness ratios are discussed.The numerical results show that the phase velocity of S0 mode decreases with the increase of the thickness ratio and when the wave number rises, all the curves converge to the bulk shear wave velocity of the system. Similarly, the magneto-electromechanical coupling factor 14

increases when the thickness ratio increases.The higher thickness ratio is associated with higher magneto-electromechanical coupling factor, it is about 17% for S0 at low frequency.These significant and interesting features will be particularly useful in the analysis and design of magneto-electro-elastic composite laminates, and for a given stacking sequence of the composite laminate, the present numerical solution offers a simple and accurate tool for the prediction, identification, and study of these features.

Acknowledgments The authors are grateful for the funding provided to LPM laboratoryby the Tunisian Ministry of Higher Education, Scientific Research.

References [1] Spaldin NA, Fiebig M. The renaissance of magnetoelectric multiferroics. Science 2005;309 (5733):391–2. [2] Tiercelin N, Dusch Y, Preobrazhensky V, Pernod P. Magnetoelectric memoryusing orthogonal magnetization states and magnetoelastic switching. J Appl Phys 2011;109 (7):07D726. [3] Thomson, W. T. Transmission of elastic waves through a stratified solid medium. J. Appl. Phys. 21 (1950) 89-93. [4] Haskell, N. A. The dispersion of surface waves on multilayered media. Bull. Seism.Soc. Am. 43 (1953) 17-34. [5] Gilbert, F. and G. Backus .Propagator matrices in elastic wave and vibration problem, Geophysics 31, (1966) 326-332. [6] Bahar, L.Y .A state space approach to elasticity. Journal of the Franklin Institute 299, (1975) 33–41. [7] E. Pan.Exact solution for simply supported and multilayered magneto-electro-elastic plates, Journal of Applied Mechanics ,68 (2001) 608–618. [8] E. Pan,P. R. Heyliger.Free vibrations of simply supported and multilayered magnetoelector-elastic plates .Journal of Sound and Vibration ,252 (2002) 429–442. [9] Y.Benveniste. Magneto-electric effect in fibrous composites with piezoelectric and piezomagnetic phases. Phys Rev B ,51 (1995) 16424–7.

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[10] Jiangyi Chen, E. Pan ,Hualing Chen. Wave propagation in magneto-electro-elastic multilayered plates.International Journal of Solids and Structures 44 (2007) 1073–1085. [11] Jiangyi Chen,Hualing Chen, E. Pan, P.R. Heyligerc. Modal analysis of magneto-electroelastic plates using the state-vector approach. Journal of Sound and Vibration ,304 (2007) 722–734. [12]

Xiao-Hong

Wu,

Ya-PengShen,

Qing

Sun.

Lamb

wave

propagation

in

magnetoelectroelastic plates Applied Acoustics 68 (2007) 1224–1240. [13] Jiangyi Chen , E. Pan , Hualing Chen. Wave propagation in magneto-electro-elastic multilayered plates. International Journal of Solids and Structures 44 (2007) 1073–1085. [14] D. Piliposyan. Shear surface waves at the interface of two magneto-electro-elastic media. Multidiscipline Modelling in Materials and Structures, 8 (2012) 417–426. [15] Jin-xi Liu, Dai-Ning Fang, Wei-Yi Wei, Xiao-Fang Zhao. Love waves in layered piezoelectric/piezomagnetic structures, Journal of Sound and Vibration, 315 (2008) 146–156. [16] J.Du, K, X.Jin, J.Wang. Love wave propagation in layered magneto-electro-elastic structures with initial stress, Acta Mechanica , 192 (2007) 169–189 . [17] Hamdi Ezzin , Morched Ben Amor

, Mohamed Hédi Ben Ghozlen. Love waves

propagation in a transversely isotropic piezoelectric layer on

a piezomagnetic half-

space. Ultrasonics 69 (2016) 83–89. [18] J.Y. Chen, E. Pan, H.L. Chen, Wave propagation in magneto-electro-elastic multilayered plates, International Journal of Solids and Structures ,44 (2007) 1073–1085. [19] Issam Ben Salah, Morched Ben Amor, Mohamed Hédi Ben Ghozlen, Effect of a functionally graded soft middle layer on Love waves propagating in layered 465 piezoelectric systems, Ultrasonics 61 (2015) 145–150. [20] Issam Ben Salah , Anouar Njeh , Mohamed Hédi Ben Ghozlen. A theoretical study of the propagation of Rayleigh waves in a functionally graded piezoelectric material (FGPM). Ultrasonics 52 (2012) 306–314. [21] Libo Xin, Hu Zhendong, Free vibration of simply supported and multilayered magnetoelectroelastic plates, Compos. Struct. 121 (2015) 344–350. [22] W.T. Thomson, Transmission of elastic waves through a stratified solid medium, J. Appl. Phys. 21 (1950) 89–93. [23] N.A. Haskell, The dispersion of surface waves on multilayered media, Bull. Seism. Soc. Am. 43 (1953) 17–34.

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[24] E.L. Adler, SAW and pseudo-SAW properties using matrix methods, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 41 (6) (1994) 876–882. [25] S.I. Rokhlin, L. Wang, Modelling of wave propagation in layered piezoelectric media by a recursive asymptotic method, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 51 (2004) 1060–1071. [26]

Y.Y.

Zhou,

C.F.

Lü,

W.Q.

Chen,

Bulk

wave

propagation

in

layered

piezomagnetic/piezoelectric plates with initial stresses or interface imperfections, Compos. Struct. 94 (2012) 2736–2745. [27] Du Jianke, Kai Xian, Ji Wang, SH surface acoustic wave propagation in a cylindrically layered piezomagnetic/piezoelectric structure, Ultrasonics 49 481 (2009) 131–138. [28] Morched Ben Amor, Mohamed Hédi Ben Ghozlen, Lamb waves propagation in functionally graded piezoelectric materials by Peano-series method Ultrasonics, Volume 55, January 2015, Pages 10-14.

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Research highlights ▶ The first order differential equation and Stiffness matrix method are developed. ▶ The phase velocity of the Lamb wave is calculated for the magneto-electrically open and short cases. ▶ the influence of different stacking sequences and thickness ratio on dispersion curves and on magneto electro-mechanical coupling factor are presented.

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