Lamb wave propagation in magnetoelectroelastic plates

Applied Acoustics 68 (2007) 1224–1240 www.elsevier.com/locate/apacoust

Lamb wave propagation in magnetoelectroelastic plates Xiao-Hong Wu a

a,*

, Ya-Peng Shen a, Qing Sun

b

The Key Laboratory of Mechanical Structural Strength and Vibration, School of Aerospace, Xi’an JiaoTong University, Xi’an 710049, China b Department of Civil Engineering, Xi’an Jiao Tong University, Xi’an 710049, China Received 11 October 2005; received in revised form 10 March 2006; accepted 20 July 2006 Available online 25 October 2006

Abstract Based on the three-dimensional linear elastic equations and magnetoelectroelastic constitutive relations, propagation of symmetric and antisymmetric Lamb waves in an infinite magnetoelectroelastic plate is investigated. The coupled differential equations of motion are solved, and the phase velocity equations of symmetric and antisymmetric modes are obtained for both electrically and magnetically open and shorted cases. The dispersive characteristic of wave propogation is explored. The mechanical, electric and magnetic responses of the lowest symmetric and antisymmetric Lamb wave modes are discussed in detailed. Obtained results are valuable for the analysis and design of broadband magnetoelectric transducer using composite materials.  2006 Elsevier Ltd. All rights reserved. Keywords: Magnetoelectroelastic materials; Symmetric lamb waves; Antisymmetric lamb waves; Phase velocity

1. Introduction As one of the elastic waves frequently encountered in acoustic devices, Lamb waves can propagate in a solid plate with free boundaries. Recent structural health monitoring technologies based on Lamb wave propagation feature have received intensive interests *

Corresponding author. Tel.: +86 29 8266 8753; fax: +86 29 8266 7910. E-mail address: [email protected] (X.-H. Wu).

0003-682X/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2006.07.013

X.-H. Wu et al. / Applied Acoustics 68 (2007) 1224–1240

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[1–3], in which understanding the propagation characteristics of Lamb waves in platelike structures is important. Numerous researchers have studied the vibration and wave propagation in elastic and piezoelectric plates. Among them, most work are concerned about deriving the general solutions of wave propagations in a single layer infinite piezoelectric plate [4–6]. Vibration and harmonic wave propagation in a plate consisting of two layers of different piezoelectric materials were investigated by Cheng and Sun [7]. Jin et al. [8] studied Lamb wave propagation in a metallic semi-infinite medium covered with a piezoelectric layer. The effect of a bias electric field on the propagation of different Lamb wave modes in an infinite piezoelectric plate was discussed by Liu et al. [9,10]. Recently, composites materials consisting of piezoelectric and piezomagnetic phases have drawn considerable attention [11–13]. A micromechanics approach was developed by Li and Dunn [14] to analyze the average fields and effective moduli of heterogeneous magnetoelectroelastic solids that exhibit full coupling between stationary elastic, electric, and magnetic fields. A number of inclusion-related problems in these materials have been studied [15–19] and some static and free vibration analyses for magnetoelectroelastic plates and shells were published [20–24]. However, studies on the wave propagation in plates made of magnetoelectroelastic materials are still limited. In this paper, the solution to the problem of symmetric and antisymmetric Lamb wave propagation in an infinite magnetoelectroelastic plate is derived. The electrically and magnetically open case and shorted case for both types of wave mode are discussed. The dispersive characteristic of the wave problem is presented and compared with that of the piezoelectric materials. Furthermore, the behavior of the mechanical displacement, electric and magnetic potential along thickness direction of the plate is obtained. These new numerical results are useful for the design of a broad-band magnetoelectric transducer using composite materials. 2. General equations Consider the wave propagation in a magnetoelectroelastic plate with thickness being H, or H = 2h (h is the half thickness). The plate is symmetric about the medium plane z = 0 and extends infinitely in the x and y directions. The wave propagates along the positive direction of the x axis with a phase velocity v. The governing field equations of magnetoelectroelastic solid, in the absence of body forces and electric charge density, are rij;j ¼ q€ ui Di;i ¼ 0

ð1Þ

Bi;i ¼ 0 where q is the mass density, ui is the displacement, and rij, Di and Bi are the stresses, electric displacements and magnetic induction, respectively. In Eq. (1), the summation convention over repeated subscripts applies, sup-dot denotes time derivative, and subscript ‘‘,’’ refers to partial differentiation. For a state of plane strain parallel to the x–z plane and the propagation of waves in the x direction, the governing equations can be simplified as

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€ rxx;x þ rxz;z ¼ q  u € rxz;x þ rzz;z ¼ q  w Dx;x þ Dz;z ¼ 0 Bx;x þ Bz;z ¼ 0

ð2Þ

where u and w are the respective displacement components in the x and z directions. The constitutive equations of transversely isotropic magnetoelectroelastic medium can be expressed as rij ¼ cijkl ckl  ekij Ek  fkij H k Di ¼ eikl ckl þ eil El þ gil H l

ð3Þ

Bi ¼ fikl ckl þ gil El þ lil H l where cij ¼ 0:5ðui;j þ uj;i Þ Ei ¼ u;i

H i ¼ w;i

ð4Þ

and u and w are the electric potential and magnetic potential, cij, lijand eij are elastic constants, magnetic permeabilities and dielectric permittivites, respectively. It is seen from Eq. (3) that the elastic fields are coupled to the electric and magnetic fields through the piezoelectric and piezomagnetic coefficients, eij and fij, respectively, while the electric and magnetic fields are coupled through the magnetoelectric constants, gij. Note that for the plane strain problem considered here all quantities are independent of the coordinate y. The constitutive relations of magnetoelectroelastic medium are then given by rxx ¼ c11 u;x þ c13 w;z þ e31 u;z þ f31 w;z ryy ¼ c12 u;x þ c13 w;z þ e31 u;z þ f31 w;z rzz ¼ c13 u;x þ c33 w;z þ e33 u;z þ f33 w;z rxz ¼ c44 ðu;z þ w;x Þ þ e15 u;x þ f15 w;x Dx ¼ e15 ðu;z þ w;x Þ  e11 u;x  g11 w;x

ð5Þ

Dz ¼ e31 u;x þ e33 w;z  e33 u;z  g33 w;z Bx ¼ f15 ðu;z þ w;x Þ  g11 u;x  l11 w;x Bz ¼ f31 u;x þ f33 w;z  g33 u;z  l33 w;z Substitution of Eq. (5) into (2) will lead to the following set of governing equations in terms of displacements, electric potential and magnetic potential c11 u;xx þ c44 u;zz þ ðc13 þ c44 Þw;xz þ ðe15 þ e31 Þu;xz þ ðf15 þ f31 Þw;xz ¼ q€u c44 w;xx þ c33 w;zz þ ðc13 þ c44 Þu;xz þ e15 u;xx þ f15 w;xx þ e33 u;zz þ f33 w;zz ¼ q€ w e15 w;xx þ e33 w;zz þ ðe15 þ e31 Þu;xz  e11 u;xx  g11 w;xx  e33 u;zz  g33 w;zz ¼ 0

ð6Þ

f15 w;xx þ f33 w;zz þ ðf15 þ f31 Þu;xz  g11 u;xx  l11 w;xx  g33 u;zz  l33 w;zz ¼ 0 The potential u0 and w0 in the region outside the plate must satisfy Laplace’s equation

X.-H. Wu et al. / Applied Acoustics 68 (2007) 1224–1240

u0;xx þ u0;zz ¼ 0 w0;xx þ w0;zz ¼ 0

1227

ð7Þ

In the case under investigation, the faces of the infinite plate are traction-free. The electric/magnetic boundary conditions are provided by the continuity of the potential and of the normal component of electric displacement and magnetic induction across the free surface. Thus the boundary conditions for the solutions of Eqs. (6) and (7) are given as follows. (a) At z = ±h, the mechanical traction vanishes, i.e. rzz ðx; hÞ ¼ 0 rxz ðx; hÞ ¼ 0

ð8Þ

(b1) For open case, the electric/magnetic boundary conditions at z = ±h are uðx; hÞ ¼ u0 ðx; hÞ and Dz ðx; hÞ ¼ Dz0 ðx; hÞ wðx; hÞ ¼ w0 ðx; hÞ and Bz ðx; hÞ ¼ Bz0 ðx; hÞ

ð9Þ

(b2) For shorted case, the electric/magnetic boundary conditions at z = ±h are uðx; hÞ ¼ 0 wðx; hÞ ¼ 0

ð10Þ

3. Wave motion equations There are two Lame wave modes, i.e., symmetric and antisymmetric modes, that satisfy the wave equation and boundary conditions of this problem. It should be emphasized that each mode is independent of the other. For the symmetric mode, u of Lamb waves is symmetric while w, u, and w are antisymmetric with respect to the central plane. For the antisymmetric mode, on the contrary, w, u, and w of Lamb waves is symmetric while u are antisymmetric with respect to the central plane. In general, the symmetric mode corresponds to longitudinal vibration modes and the antisymmetric mode related to the flexural vibrations of the plate. For the symmetric mode, solution of Eq. (6) is assumed to have the form of [25] u ¼ A1 cosðakzÞ exp½ikðx  vtÞ w ¼ A2 sinðakzÞ exp½ikðx  vtÞ u ¼ A3 sinðakzÞ exp½ikðx  vtÞ w ¼ A4 sinðakzÞ exp½ikðx  vtÞ

ð11Þ

where k is the wave number, v the phase velocity, A1, A2, A3 and A4 are constants, and a is a parameter to be determined. Substituting Eq. (11) into (6), gives  ðc11 þ c44 a2  qv2 ÞA1 þ iðc13 þ c44 ÞaA2 þ iðe15 þ e31 ÞaA3 þ iðf15 þ f31 ÞaA4 ¼ 0 iðc13 þ c44 ÞaA1 þ ðc44 þ c33 a2  qv2 ÞA2 þ ðe15 þ e33 a2 ÞA3 þ ðf15 þ f33 a2 ÞA4 ¼ 0 iðe15 þ e31 ÞaA1 þ ðe15 þ e33 a2 ÞA2  ðe11 þ e33 a2 ÞA3  ðg11 þ g33 a2 ÞA4 ¼ 0 iðf15 þ f31 ÞaA1 þ ðf15 þ f33 a2 ÞA2  ðg11 þ g33 a2 ÞA3  ðl11 þ l33 a2 ÞA4 ¼ 0

ð12Þ

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which is a set of linear and homogeneous equations in A1, A2, A3 and A4. Non-trivial solution of Eq. (12) implies that the determinant of the coefficient matrix of Eq. (12) vanishes, i.e.    ðc11 þ c44 a2  qv2 Þ ðc13 þ c44 Þai ðe15 þ e31 Þai ðf15 þ f31 Þai    ðc13 þ c44 Þai c44 þ c33 a2  qv2 e15 þ e33 a2 f15 þ f33 a2   ¼0  2 2  ðe15 þ e31 Þai e15 þ e33 a ðe11 þ e33 a Þ ðg11 þ g33 a2 Þ    ðf15 þ f31 Þai f15 þ f33 a2 ðg11 þ g33 a2 Þ ðl11 þ l33 a2 Þ  ð13Þ 2

The above equation is a quadratic equation in a with phase velocity v as the unknown parameter. On account of the form of Eq. (11), for a given v, there are four am (m = 1  4), each of which yields an independent solution. Substitution of am (m = 1  4) into any three of Eq. (12) yields the amplitude ratios A1m/A4m, A2m/A4m and A3m/A4m (m = 1  4), i.e. A1m ¼ L1m A4m A2m ¼ L2m A4m

ðm ¼ 1  4Þ

ð14Þ

A3m ¼ L3m A4m where

L1m

   ðf15 þ f31 Þam i ðc13 þ c44 Þam i ðe15 þ e31 Þam i     e15 þ e33 a2m   ðf15 þ f33 a2m Þ c44 þ c33 a2m  qv2    g11 þ g33 a2 e15 þ e33 a2m ðe11 þ e33 a2m Þ  m   ¼ 2 2 ðc13 þ c44 Þam i ðe15 þ e31 Þam i   ðc11 þ c44 am  qv Þ   ðc13 þ c44 a2m Þ c44 þ c33 a2m  qv2 e15 þ e33 a2m      ðe15 þ e31 Þam i e15 þ e33 a2m ðe11 þ e33 a2m Þ 

L2m

   ðc11 þ c44 a2  qv2 Þ ðf15 þ f31 Þam i ðe15 þ e31 Þam i  m     ðc13 þ c44 a2m Þ ðf15 þ f33 a2m Þ e15 þ e33 a2m      ðe15 þ e31 Þam i g11 þ g33 a2m ðe11 þ e33 a2m Þ   ¼  2 2 ðc13 þ c44 Þam i ðe15 þ e31 Þam i   ðc11 þ c44 am  qv Þ   ðc13 þ c44 a2m Þ c44 þ c33 a2m  qv2 e15 þ e33 a2m      ðe15 þ e31 Þam i e15 þ e33 a2m ðe11 þ e33 a2m Þ 

L3m

   ðc11 þ c44 a2  qv2 Þ ðc13 þ c44 Þam i ðf15 þ f31 Þam i  m    ðc13 þ c44 a2m Þ c44 þ c33 a2m  qv2 ðf15 þ f33 a2m Þ      ðe15 þ e31 Þam i e15 þ e33 a2m g11 þ g33 a2m   ¼  2 2 ðc13 þ c44 Þam i ðe15 þ e31 Þam i   ðc11 þ c44 am  qv Þ   ðc13 þ c44 a2m Þ c44 þ c33 a2m  qv2 e15 þ e33 a2m      ðe15 þ e31 Þam i e15 þ e33 a2m ðe11 þ e33 a2m Þ 

Substituting Eq. (14) into Eq. (11), the displacements, electric potential and magnetic potential are given by

X.-H. Wu et al. / Applied Acoustics 68 (2007) 1224–1240

u¼

4 X

1229

L1m A4m cosðam kzÞ exp½ikðx  vtÞ

m¼1

w¼

4 X

L2m A4m sinðam kzÞ exp½ikðx  vtÞ

m¼1

u¼

4 X

ð15Þ L3m A4m sinðam kzÞ exp½ikðx  vtÞ

m¼1

w¼

4 X

A4m sinðam kzÞ exp½ikðx  vtÞ

m¼1

In addition, the solutions of electric potential and magnetic potential in the vacuum can be obtained from Eq. (7), i.e. u0 ¼ B1 ekz exp½ikðx  vtÞ w0 ¼ B2 e

kz

ðz P hÞ

exp½ikðx  vtÞ

ð16Þ

4. Solutions of phase velocity 4.1. The electrically and magnetically open case Substituting Eqs. (15) and (16) into the boundary conditions Eqs. (8) and (9), and using the required relations in Eq. (5), one can obtain a set of six algebraic equations in the unknown constants A41, A42, A43, A44, B1 and B2. After simplifying, we can obtain the algebraic equations in the unknown constants A41, A42, A43 and A44, i.e. ½H fAg ¼ 0

ð17Þ

where {A} = {A41,A42,A43,A44}T, and [H] is a 4 · 4matrix, with Hlm (l,m = 1  4) given by H 1m ¼ Rm sinðam khÞ ¼ ½c44 ðL2m  i  am L1m Þ þ e15 L3m  i þ f15  i sinðam khÞ H 2m ¼ S m cosðam khÞ ¼ ðc13 L1m  i þ c33 am L2m Þ þ e33 am L3m þ f33 am  cosðam khÞ H 3m ¼ T m cosðam khÞ  L3m sinðam khÞ H 4m ¼ P m cosðam khÞ  sinðam khÞ

ð18Þ

where 1 ðe31 L1m  i þ e33 am L2m  e33 am L3m  g33 am Þ e0 1 P m ¼ ðf31 L1m  i þ f33 am L2m  g33 am L3m  l33 am Þ l0

Tm ¼

ð19Þ

Then the phase velocity is obtained when the determinant of jH j in Eq. (17) vanishes, i.e.     R1 sinða1 khÞ R2 sinða2 khÞ R3 sinða3 khÞ R4 sinða4 khÞ     S 2 cosða2 khÞ S 3 cosða3 khÞ S 4 cosða4 khÞ S 1 cosða1 khÞ    ¼0   H H H H 31 32 33 34    P cosða khÞ  sinða khÞ P cosða khÞ  sinða khÞ P cosða khÞ  sinða khÞ P cosða khÞ  sinða khÞ  1 1 1 2 2 2 3 3 3 4 4 4 ð20Þ

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The above equation is a transcendental equation, the roots of which enable the determination of the dispersion property of the magnetoelectricelastic plate. Eq. (20) contains an infinite number of roots vn, each of which determines a point on the dispersion spectrum. 4.2. The electrically and magnetically shorted case Similar to the electrically open case, one can obtain the phase velocity equation for the electrically shorted case, i.e.    R1 sinða1 khÞ R2 sinða2 khÞ R3 sinða3 khÞ R4 sinða4 khÞ     S cosða khÞ S cosða khÞ S cosða khÞ S cosða khÞ  1 2 2 3 3 4 4   1 ð21Þ ¼0   L31 sinða1 khÞ L32 sinða2 khÞ L33 sinða3 khÞ L34 sinða4 khÞ     sinða khÞ sinða khÞ sinða khÞ sinða khÞ  1

2

3

4

5. Solutions of stress fields After the root of phase velocity is found, the amplitude ratios can be obtained when substituted the root into Eq. (17), as A41 ¼ g1 A44 ; A42 ¼ g2 A44 ; A43 ¼ g3 A44 ; A44 ¼ g4 A44

ð22Þ

where   H 14    H 24   H 34 g1 ¼   H 11   H 21   H 31

   H 11 H 12 H 13     H 22 H 23   H 21    H H 32 H 33  g2 ¼  31  H 11 H 12 H 13     H 22 H 23   H 21     H 31 H 32 H 33

   H 11 H 14 H 13     H 24 H 23   H 21    H H 34 H 33  g3 ¼  31  H 11 H 12 H 13     H 22 H 23   H 21     H 31 H 32 H 33

 H 12 H 14   H 22 H 24   H 32 H 34   g4 ¼ 1 H 12 H 13   H 22 H 23   H 32 H 33 

The value of A44 is determined by the excitation. Substituting Eq. (22) into Eq. (15) gives " # 4 X u¼ L1m gm cosðam kzÞ  A44 exp½ikðx  vtÞ m¼1

" w¼ " u¼

" w¼

4 X

# L2m gm sinðam kzÞ  A44 exp½ikðx  vtÞ

m¼1 4 X m¼1 4 X m¼1

# L3m gm sinðam kzÞ  A44 exp½ikðx  vtÞ # gm sinðam kzÞ  A44 exp½ikðx  vtÞ

ð23Þ

Elastic constant (109 N/m2)

C11 226

C33 218

C12 122

C13 121

Piezoelectric constant (C/m2) C44 48

e31 2.1

e33 6.9

e15 0.08

Piezomagnetic constant (N/Am)

Magnetic permeability (106 Ns2/C2)

Dielectric permittivity (109 C/Vm)

Magnetoelectric constants (109 Ns/VC)

Mass density (Kg/m3)

f31 279

l11 258

e11 0.19

g31 0.005

q 7.5

f33 358

f15 238

l33 98

e33 5.1

g33 2.73

X.-H. Wu et al. / Applied Acoustics 68 (2007) 1224–1240

Table 1 Material properties of the electromagnetic composites

1231

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By substituting Eq. (23) into constitutive equations, the stress fields can be obtained as " # 4 X rxx ¼ ðc11 L1m  i þ c13 am L2m þ e31 am L3m þ f31 am Þbm cosðam kzÞ  k  A44 exp½ikðx  vtÞ "

m¼1

# 4 X ðc12 L1m  i þ c13 am L2m þ e31 am L3m þ f31 am Þbm cosðam kzÞ  k  A44 exp½ikðx  vtÞ ryy ¼ m¼1

a

6.0 5.5

phase velocity (km/s)

5.0

s2

4.5

s1

4.0 3.5

s0

3.0

Share wave velocity

2.5 Rayleigh wave velocity

2.0 0

2

4

6

8

10

12

14

kh

b

6.0

phase velocity (km/s)

5.5

s2

5.0 4.5 4.0

s1

3.5

s0

3.0

Share wave velocity

2.5 Rayleigh wave velocity

2.0 0

2

4

6

8

10

12

14

kh Fig. 1. The dispersion curves of the s0, s1 and s2 modes of magnetoelectroelastic materials; (a) the electrically and magnetically open case; (b) the electrically and magnetically shorted case.

X.-H. Wu et al. / Applied Acoustics 68 (2007) 1224–1240

1233

"

# 4 X rzz ¼ ðc13 L1m  i þ c33 am L2m þ e33 am L3m þ f33 am Þbm cosðam kzÞ  k  A44 exp½ikðx  vtÞ m¼1

( rxz ¼

4 X

) ½c44 ðL2m  i  am L1m Þ þ e15 L3n  i þ f15  ibm sinðam kzÞ  k  A44 exp½ikðx  vtÞ

m¼1

ð24Þ

a

6.0

phase velocity (km/s)

5.5 5.0

s2 4.5

s1

4.0 3.5

s0

3.0

Share wave velocity

2.5 Rayleigh wave velocity

2.0 0

2

4

6

8

10

12

14

10

12

14

kh

b

6.0

phase velocity (km/s)

5.5 5.0

s2 4.5

s1

s0

4.0 3.5 3.0

Share wave velocity

2.5 Rayleigh wave velocity

2.0 0

2

4

6

8

kh Fig. 2. The dispersion curves of the s0, s1 and s2 modes of piezoelectric ceramic; (a) the electrically open case; (b) the electrically shorted case.

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X.-H. Wu et al. / Applied Acoustics 68 (2007) 1224–1240

The aforementioned set of solutions and dispersion curves completely determines the longitudinal modes for the magnetoelectroelastic plate. Another independent set of solutions for antisymmetric modes can be obtained by interchanging cos(akz) and sin(akz) in Eqs. (11), (15), (23) and (24), sin(amkh) and cos (amkh) in Eqs. (18), (20) and (21). This set of solutions yields dispersion curves and all physical fields, which are independent of those of the symmetric mode and determine the flexural modes for the plate. 4.0 3.5

a1 phase velocity (km/s)

3.0 Share wave velocity

2.5

Rayleigh wave velocity

2.0

a0

1.5 1.0 0.5 0.0 0

2

4

6

8

10

12

14

16

kh Fig. 3. The dispersion curves of the a0 and a1 modes of magnetoelectroelastic plate. 4.0 3.5

a1

phase velocity (m/s)

3.0 Share wave velocity Rayleigh wave velocity

2.5 2.0

a0

1.5 1.0 0.5 0.0 0

2

4

6

8

10

12

14

16

kh Fig. 4. The dispersion curves of the a0 and a1 modes of piezoelectric ceramic.

X.-H. Wu et al. / Applied Acoustics 68 (2007) 1224–1240

1235

6. Numerical examples and discussion In the numerical examples, we choose the BaTiO3–CoFe2O4 fibrous composite material as magnetoelectroelastic media. The material contain 40% BaTiO3 and 60% CoFe2O4, and its effective moduli proposed in Li [14] are presented in Table 1. The dielectric constant of

a

imaginary part

1.6 1.4

kh=1.5

1.2 1.0

displacement u (10-8m)

0.8 0.6 0.4 0.2 0.0 -0.2 -0.4

kh=3.0

-0.6 -0.8 -1.0 -1.2 -1.4 -1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

0.75

1.00

z/h

b

2.5

real part

2.0

kh=3.0

displacement w (10-8m)

1.5 1.0 0.5 0.0 -0.5

kh=1.5 -1.0 -1.5 -2.0 -2.5 -1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

z/h Fig. 5. Variations of mechanical displacements of s0 mode along thickness of the magnetoelectroelastic plate; (a) u; (b) w.

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X.-H. Wu et al. / Applied Acoustics 68 (2007) 1224–1240

vacuum is e0 = 8.85 · 1012 F/m, the magnetic permeability of the vacuum is l0 = 4p · 107 H/m. The plate thickness is 0.5 mm. The exact dispersion curves with the phase velocity v of the first several symmetric modes versus the dimensionless wave number kh in an infinite magnetoelectroelastic plate are shown in Fig. 1 for electrically open and shorted cases. It is seen that the phase velocity v of electrically open cases are slightly higher than that of electrically shorted cases. The shear wave velocity vsh in magnetoelectroelastic materials equals

a

14 12

kh=3.0

10 8

electric potential (V)

6 4 2 0 -2

kh=1.5

-4 -6 -8 -10 -12 -14 -1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

0.50

0.75

1.00

z/h

b

0.025

kh=3.0

0.020

magneto potential (C/s)

0.015 0.010 0.005 0.000

kh=1.5

-0.005 -0.010 -0.015 -0.020 -0.025 -1.00

-0.75

-0.50

-0.25

0.00

0.25

z/h Fig. 6. Variations of electric and magnetic potential of s0 mode along thickness of the magnetoelectroelastic plate; (a) electric potential u; (b) magnetic potential w.

X.-H. Wu et al. / Applied Acoustics 68 (2007) 1224–1240

1237

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc44 ðg211  l11 e11 Þ þ 2e15 f15 g11  e215 l11  f152 e11 Þ=ðqðg211  l11 e11 ÞÞ. As the wavelengths become shorter, i.e., k becomes larger, the phase velocities for the lowest mode approaches to the Rayleigh surface wave velocity. The phase velocities for the higher modes become asymptotic to the shear wave velocity. To compare the results for magnetoelectroelastic materials with those for piezoelectric materials, the dispersion curves for wave propagating in a plate of piezoelectric ceramic

a

1.5

displacement u (10-8m)

1.0

kh=1.5

0.5

kh=3.0 0.0

-0.5

-1.0

-1.5 -1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

0.50

0.75

1.00

z/h

b

0.5

kh=3.0 displacement w (10-8m)

0.0

-0.5

-1.0

-1.5

-2.0

kh=1.5 -2.5 -1.00

-0.75

-0.50

-0.25

0.00

0.25

z/h Fig. 7. Variations of mechanical displacements of a0 mode along thickness of the magnetoelectroelastic plate; (a) u; (b) w.

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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi BaTiO3 are presented in Fig. 2. v ¼ ðc44 e11 þ e215 Þ=ðqe11 Þ is the shear wave velocity of this materials. The results in Fig. 2 for the piezoelectric material show similar trend to those for the magnetoelectroelastic material. Nevertheless, the Rayleigh surface wave velocity and shear wave velocity of the magnetoelectroelastic material are lower than those of the piezoelectric material.

a

1.0 0.5 0.0

kh=3.0

-0.5

electric potential (V)

-1.0 -1.5 -2.0 -2.5 -3.0 -3.5 -4.0 -4.5 -5.0 -5.5

kh=1.5

-6.0 -6.5 -7.0 -1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

z/h

b

0.030

kh=1.5

magnetic potential (C/s)

0.025 0.020 0.015 0.010 0.005 0.000

kh=3.0

-0.005 -0.010 -0.015 -1.0

-0.5

0.0

0.5

1.0

z/h Fig. 8. Variations of electric and magnetic potential of a0 mode along thickness of the magnetoelectroelastic plate; (a) electric potential u; (b) magnetic potential w.

X.-H. Wu et al. / Applied Acoustics 68 (2007) 1224–1240

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Velocities of Lamb wave modes a0 and a1 for magnetoelectroelastic and piezoelectric materials as functions of kh are shown in Figs. 3 and 4. For the antisymmetric modes, the phase velocities for electrically open case and shorted case are very close. It is seen that with kh increasing, the velocity of the a0 mode approaches closely to the Rayleigh surface wave velocity, and the velocity of the a1 mode approaches to the shear wave velocity. To display the amplitudes of mechanical displacements, electric and magnetic potential as functions of z, the real parts of their expressions are plotted [26]. Let c denote all these variables, then cðz; x; tÞ ¼ cðzÞ exp½ikðx  ctÞ This lead to Re½cðz; x; tÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 fRe½cðzÞg þ fIm½cðzÞg  cos½kðx  ctÞ þ a

where

  Im½cðzÞ a ¼ arctg Re½cðzÞ

Generally, we plot Im½cðzÞ and Re½cðzÞ to show the z dependence of the amplitude of c. Variations of the amplitude of c of the s0 mode along the thickness of a magnetoelectroelastic plate with kh = 1.5 and kh = 3.0 are shown in Figs. 5 and 6. The horizontal axis is the depth into the material, z/h = 1 corresponds to the top surface of the plate and z/h = 1 the bottom surface of the plate. It can be seen that the real part of the displacement u compared to its imaginary part is negligible whilst the imaginary part of the displacement w, electric potential u and magnetic potential w is negligible in light of their real part for the case kh = 1.5. When the value of the dimensionless wave number kh increases, the behavior of the real and imaginary of the mechanical displacements, electric and magnetic field are different. Analogously, Figs. 7 and 8 presents the amplitude of c of the a0 mode with kh = 1.5 and kh = 3.0. Similar analysis should be done in this case. 7. Conclusions Analytical solutions of the problem for wave propagation in an infinite magnetoelectroelastic plate are obtained. The symmetric and antisymmetric wave modes are discussed. The electrically and magnetically open case and shorted case for both types of wave mode are discussed. The dispersive characteristics of waves is presented and compared with that of the piezoelectric materials. Furthermore, the behavior of mechanical displacements, electric and magnetic potential along thickness of plate is obtained. Acknowledgements This work is supported by the National Natural Science Foundation of China (Nos. 10472088 and 10132010). References [1] Kessler SS, Spearing SM, Soutis C. Damage detection in composite materials using Lamb wave methods. Smart Mater Struct 2002;11:269–79.

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