piezomagnetic phononic crystal with functionally graded interlayers

Ultrasonics 70 (2016) 158–171

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Dispersion relations of elastic waves in one-dimensional piezoelectric/ piezomagnetic phononic crystal with functionally graded interlayers Xiao Guo a, Peijun Wei a,⇑, Man Lan b, Li Li c a

Department of Applied Mechanics, University of Science and Technology Beijing, Beijing 100083, China Department of Mathematics and Science, Luoyang Institute of Science and Technology, Luoyang 471000, China c Department of Mathematics, Qiqihar University, Qiqihar 161006, China b

a r t i c l e

i n f o

Article history: Received 20 December 2015 Received in revised form 25 March 2016 Accepted 30 April 2016 Available online 2 May 2016 Keywords: Dispersion relations Piezoelectric slab Piezomagnetic slab Phononic crystal Functionally graded interlayer Bloch wave Transfer matrix

a b s t r a c t The effects of functionally graded interlayers on dispersion relations of elastic waves in a onedimensional piezoelectric/piezomagnetic phononic crystal are studied in this paper. First, the state transfer equation of the functionally graded interlayer is derived from the motion equation by the reduction of order (from second order to first order). The transfer matrix of the functionally graded interlayer is obtained by solving the state transfer equation with the spatial-varying coefficient. Based on the transfer matrixes of the piezoelectric slab, the piezomagnetic slab and the functionally graded interlayers, the total transfer matrix of a single cell is obtained. Further, the Bloch theorem is used to obtain the resultant dispersion equations of in-plane and anti-plane Bloch waves. The dispersion equations are solved numerically and the numerical results are shown graphically. Five kinds of profiles of functionally graded interlayers between a piezoelectric slab and a piezomagnetic slab are considered. It is shown that the functionally graded interlayers have evident influences on the dispersion curves and the band gaps. Ó 2016 Elsevier B.V. All rights reserved.

1. Introduction As a kind of artificial periodic composite materials or structures, phononic crystal is of characteristic of the band gap. Due to the convenience of adjusting band gap, the phononic crystal consisting of piezoelectric materials and piezomagnetic materials has attracted attentions of many researchers in recent years. AlvarezMesquida et al. [1] discussed the shear horizontal wave in layered piezoelectric composites in terms of a recursive system of equations involving the piezoelectric impedance. Qian [16,17] studied the dispersion relation of the SH-waves in a periodic piezoelectric-polymeric layered structure and the influences of initial stress on the stop bands and pass bands of anti-plane waves. Monsivais et al. [12] studied the surface and shear horizontal waves in a piezoelectric composite media consisting of piezoelectric layers of hexagonal 6 mm symmetry. It is considered that the finite systems including Fibonacci sequences, the systems with a linear perturbation in the piezoelectric parameters and the periodic systems. Chen et al. [5,6] considered harmonic waves propagating in magneto-electro-elastic multilayered plates made of orthotropic elastic (graphite–epoxy), transversely isotropic, piezoelectric and magnetostrictive materials. In the former papers, dis⇑ Corresponding author. E-mail address: [email protected] (P. Wei). http://dx.doi.org/10.1016/j.ultras.2016.04.025 0041-624X/Ó 2016 Elsevier B.V. All rights reserved.

persion relation and the model shape of wave propagation along multilayered plate with traction-free surfaces at top and bottom was studied. In the latter paper, the reflection and transmission coefficients through the multilayered plate were calculated. Pang et al. [14,15] studied further the dispersion relations of Lamb waves and SH waves in layered periodic composites consisting of piezoelectric and piezomagnetic phases. The elastic wave propagation in two-dimensional and three-dimensional phononic crystals with piezoelectric and piezomagnetic inclusions was investigated by Wang et al. [20,21]. In their investigation, the magnetoelectro-elastic coupling effects and the initial stress effects were taken into account. The band gap characteristics for three kinds of lattice arrangements were investigated by the plane wave expansion (PWE) method. Sun et al. [18] studied the propagation of SH wave in a cylindrically multiferroic composite consisting of a piezoelectric layer and a piezomagnetic central cylinder in which the interface was damaged mechanically, magnetically or electrically. The dispersion relations of SH wave were obtained for two kinds of electric–magnetic boundary conditions at the free surface. Lan and Wei [10] studied the influence of imperfect interfaces, which was modeled as thin membranes with elasticity and inertial even but without thickness, on the dispersion characteristics and the band gaps of SH waves propagating through a laminated piezoelectric phononic crystal.

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In the investigations mentioned above, only the homogeneous material or the piece-wise homogenous material is involved. The wave propagation in the functionally graded material with material composition and properties varying continuously in certain directions has not been included. Due to the spatial-varying of material constants, the wave propagation in the functionally graded slab is more complicated than in the homogeneous slab. Wang and Rokhlin [19] presented the differential equations governing the transfer and stiffness matrices for a functionally graded generally anisotropic magneto-electro-elastic medium. The wave propagation solution for a thick layer or a multilayered structure of inhomogeneous layers was obtained recursively from the thin layer solutions. Pan and Han [13] presented an exact solution for the multilayered rectangular plate made of functionally graded, anisotropic, and linear magneto-electro-elastic materials. The functionally graded material was assumed to be exponential varying in the thickness direction and the homogeneous solution in each layer was obtained based on the pseudo-Stroh formalism. Wu et al. [23] studied the propagation of elastic waves in one-dimensional (1D) phononic crystals (PCs) with functionally graded materials (FGMs) using the spectral finite elements and transfer matrix methods. Cai and Wei [4] investigated propagation characteristics of elastic waves in two-dimensional (2D) phononic crystal consisting of parallel cylinders or cylindrical shells with varying material parameters along the radial direction embedded periodically in a homogeneous host. The influences of the graded medium with different gradient profiles upon dispersion curves and the band gaps were discussed. Golub et al. [8] discussed time-harmonic plane elastic SH-waves propagating in periodically laminated composites with functionally graded (FG) interlayers. A finite stack of periodic layers between two identical elastic half-planes was considered. The functionally graded (FG) interlayers was treated by two different models, i.e. the explicit FG model and the multilayer model. The reflection and transmission coefficients and band gaps of the periodic laminates are calculated. Fomenko et al. [7] considered the in-plane wave propagation in layered phononic crystals composed of functionally graded interlayers arisen from the solid diffusion of homogeneous isotropic materials of the crystal. Lan and Wei [11] discussed the influences of the graded interlayer with different gradient profiles on the anti-plane elastic wave propagating through a laminated piezoelectric/piezomagnetic phononic crystal. The graded interlayer was modeled as a system of homogenous sublayers with both piezoelectric and piezomagnetic effects simultaneously. The effect of the graded interlayer on the band gap was introduced by inserting an additional transfer matrix of interlayer in the calculation of the total transfer matrix. In this paper, the one-dimensional phononic crystal composed of the piezoelectric slabs, the piezomagnetic slabs and the functionally graded interlayers are considered. First, the state transfer differential equations in the homogeneous piezoelectrical and pizomagnetical slabs and in the functionally graded interlayer are derived. The transfer matrixes of the homogeneous piezoelectric slab and the pizomagnetical slab are obtained by solving the state transfer equation with constant coefficient while the transfer matrix of the functionally graded interlayer is obtained by solving the state transfer equation with spatial-varying coefficient. Then, the total transfer matrix of one typical single cell of the periodical structure is obtained by the combination of these transfer matrixes. Finally, the Bloch theorem is used to obtain the dispersion equations of Bloch waves. Five kinds of profiles of functionally graded layers between piezoelectric slabs and piezomagnetic slabs are considered. The dispersion equations of in-plane Bloch waves and anti-plane Bloch waves are both solved and the numerical results are shown graphically. Based on these numerical results, the influences of functionally graded interlayers on the dispersion curves and band gaps are discussed.

2. Transfer matrix of coupled waves in each slab 2.1. Linear vector differential equation for the state vector Consider a one-dimensional phononic crystal which is formed by periodically repeating four different transversely isotropic slabs, i.e. a homogeneous piezoelectric slab (PE), a homogeneous piezomagnetic slab (PM) and two different functionally graded layers (FGLI and FGLII Þ, as shown in Fig. 1. We establish four local Cartesian coordinate systems, which are indicated by the superscripts 0’; 00’; I’ and II’, respectively. In latter formulation, if the physical quantity doesn’t have any superscript, it will be appropriate for all local Cartesian coordinate systems. Let the x3 -axis is the poling direction and the slab is transversely isotropic in the ox1 x2 coordinates plane. In each slab, cijmn ; emij ; qmij ; emi ; lmi and ami are the elastic, piezoelectric, piezomagnetic, dielectric, magnetic permeability and magnetoelectric parameters, respectively. q and d are the mass density and thickness. The piezomagnetic coefficient and the magnetoelectric coefficient are zero (q0mij ¼ 0 and a0mi ¼ 0) for the piezoelectric slab while the piezoelectric coefficient and the magnetoelectric coefficient are zero (e00mij ¼ 0 and a00mi ¼ 0) for the piezomagnetic slab. For the piezoelectric slab and the piezomagnetic slab, all material parameters are constant quantities. The material parameters are only functions of only x3 -axis coordinate for two different functionally graded interlayers. The constitutive equations of the transversely isotropic magnetic-electro-elasto medium are [11]

8 > < rij ¼ cijkl Skl  emij Em  qmij Hm Dm ¼ emij Sij þ emi Ei þ ami Hi > : Bm ¼ qmij Sij þ ami Ei þ lmi Hi

ð1Þ

where rij and Smn are the stress and strain tensors, respectively. Dm ; Em ; Bm and Hm are the electric displacement, electric field, magnetic induction and magnetic field vectors, respectively. The physical constants of the transversely isotropic piezoelectric slab and piezomagnetic slab are given in Appendix A. The strain tensor Smn , the electric field Em and the magnetic field Hm are related with the displacement un , the electric potential u and the magnetic potential w by

Smn ¼

  1 @un @um ; þ 2 @xm @xn

Em ¼ 

@u ; @xm

Hm ¼ 

@w @xm

ð2Þ

in the quasi static electric/magnetic field approximation. The mechanical, electrical and magnetic governing equations can be expressed as

@ rij @ 2 ui ¼q 2 ; @xj @t

@Dm ¼ 0; @xm

@Bm ¼0 @xm

ð3Þ

Fig. 1. A typical single cell of one-dimensional piezoelectric/piezomagnetic phononic crystal with functionally graded interlayers.

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For convenience of the statement of the interface conditions and the periodic boundary condition equations, we define the state vector of the physical fields in each slab as

Vðxi ; t Þ ¼ fu1 ; u3 ; u; w; r33 ; r31 ; D3 ; B3 gT

ð4aÞ

for in-plane waves and

Vðxi ; t Þ ¼ fu2 ; r32 g

T

ð4bÞ

for anti-plane waves in the oblique propagation situation while

Vðxi ; t Þ ¼ fu3 ; u; w; r33 ; D3 ; B3 gT

ð4cÞ

for P wave, and

Vðxi ; t Þ ¼ fu1 ; r31 gT

ð4dÞ

for SV wave in the normal propagation situation. In either the plane strain or the anti-plane strain case, the displacement and the electric/magnetic potential are only the functions of x1 and x3 which can be assumed as:

fu1 ; u2 ; u3 ; u; wg ¼ fU 1 ; U 2 ; U 3 ; U; Wg exp ½ik1 ðx1 þ nx3  ctÞ

ð5Þ

where the wavenumber k ¼ ðk1 ; k2 ; k3 Þ ¼ ðk1 ; 0; k3 Þ and k1 is the apparent wavenumber. cð¼ x=k1 Þ is the apparent wave speed. x is the angular frequency. nð¼ k3 =k1 Þ is the ratio of wave number. It is noticed that the well-known Snell’s law, namely, the apparent wavenumbers k1 , the apparent wave speeds c and the angular frequencies x are same for various coupled waves, is assumed in Eq. (5). Inserting Eqs. (2) and (5) into Eq. (1) leads to

r11 ¼ ik1 c1111 u1 þ c1133

@u3 @u @w þ e311 þ q311 ; @x3 @x3 @x3

r33 ¼ ik1 c3311 u1 þ c3333

@u3 @u @w þ e333 þ q333 ; @x3 @x3 @x3

@u2 ; @x3

@u1 þ ik1 ðe131 u3  e11 u  a11 wÞ; @x3

@u3 @u @w  e33  a33 ; @x3 @x3 @x3   @u1 þ ik1 q131 u3  a11 u  l11 w ; B1 ¼ q131 @x3 B3 ¼ ik1 q311 u1 þ q333

fu1 ; u2 ; u3 g ¼ fU 1 ; U 2 ; U 3 g exp ½iðk3 x3  xtÞ

2

@u3 @u @w þ e311 þ q311 ; @x3 @x3 @x3 @u3 @u @w ¼ c2233 þ e311 þ q311 ; @x3 @x3 @x3

r11 ¼ c1133 r22

@ r32 ¼ qx2 u2 ; @x3

@u3 @u @w  a33  l33 @x3 @x3 @x3

ik1 c3131

@u1 @x3

 k1 ðc3131 u3 þ e131 u þ q131 wÞ þ 2

¼ qx2 u3 ;

@u3 @u @w þ e333 þ q333 ; @x3 @x3 @x3 @u1 ¼ c3131 ; @x3

r33 ¼ c3333 r13 ¼ r31

r32 ¼ r23 ¼ c2323

D1 ¼ e131

@u1 ; @x3

D2 ¼ e223

@u2 ; @x3

D3 ¼ e333

B1 ¼ q131

@u1 ; @x3

B2 ¼ q223

@u2 ; @x3

B3 ¼ q333

@u2 ; @x3

@u3 @u @w  e33  a33 ; @x3 @x3 @x3 @u3 @u @w  a33  l33 @x3 @x3 @x3 ð9Þ

@ r32 ¼ qx2 u2 ; @x3

@ r33 ¼ qx2 u3 ; @x3 ð10Þ

It is noticed from Eq. (10) that QSV and QP waves in the oblique propagation situation reduce to SV wave and P wave in the normal propagation situation. In other word, SV wave and SH wave are both decoupled with P wave. According to Eqs. (6), (7), (9) and (10), we get the linear vector differential equation of the state vector of the physical fields in each slab

Vðx3 ¼ 0; t Þ ¼ V0

ð11Þ

and the explicit expressions of matrix G and H in each slab are given in Appendix B.

ð6Þ

  @u3 @u @w @ r31 þ þ ik1 c1133 þ e311 þ q311 ¼ qx2 u1 ; @x3 @x3 @x3 @x3

k1 c1212 u2 þ

ð8Þ

where the wave number k ¼ ðk1 ; k2 ; k3 Þ ¼ ð0; 0; k3 Þ. Inserting Eqs. (2) and (8) into Eq. (1) leads to

2.2. Solution of linear vector differential equations

Inserting Eq. (6) into Eq. (3) leads to 2 k1 c1111 u1

ð7Þ

It is noticed from Eq. (7) that the displacement component u2 and the stress component r32 are independent of other components which means that SH wave is decoupled with other waves. In the case of normal propagation situation (k1 ¼ 0 or c ¼ 1Þ, the displacement components in each slab are only the functions of x3 and can be assumed as:

@Vðx3 ; tÞ ¼ G1  H  Vðx3 ; t Þ; @x3

D3 ¼ ik1 e311 u1 þ e333

@u2 ; @x3

 @B3 @u1 2  k1 q131 u3  a11 u  l11 w þ ¼0 @x3 @x3

@ r31 ¼ qx2 u1 ; @x3 @D3 @B3 ¼ ¼0 @x3 @x3

@u1 þ ik1 ðc3131 u3 þ e131 u þ q131 wÞ; @x3 @u2 ¼ c2323 ; @x3

r21 ¼ r12 ¼ ik1 c1212 u2 ; D1 ¼ e131

B2 ¼ q223

ik1 q131

Inserting Eq. (9) into Eq. (3) leads to

r13 ¼ r31 ¼ c3131

D2 ¼ e223

@u1 @D3 2  k1 ðe131 u3  e11 u  a11 wÞ þ ¼ 0; @x3 @x3

@u3 @u @w þ e311 þ q311 ; @x3 @x3 @x3

r22 ¼ ik1 c2211 u1 þ c2233

r32 ¼ r23

ik1 e131

The transfer matrix T of coupled waves in each slab is defined by

Vðx3 ¼ d; tÞ ¼ Tðx3 ¼ dÞ  Vðx3 ¼ 0; tÞ

ð12Þ

Inserting Eq. (12) into Eq. (11) leads to

@ r33 @x3

@Tðx3 Þ ¼ A  Tðx3 Þ; @x3

Tðx3 ¼ 0Þ ¼ I

ð13Þ

where A ¼ G1  H and I is a unit matrix. After obtaining the matrix function Tðx3 Þ from Eq. (13) and letting x3 ¼ d, we can get the transfer matrix of coupled waves in each slab

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For the piezoelectric slab and the piezomagnetic slab, all constitutive parameters are constant quantities, and the solution of Eq. (13) is

Tðx3 Þ ¼ exp ðA  x3 Þ ¼ Q A  diag ½expðkA1 x3 Þ; . . . ; expðkAm x3 Þ  Q 1 A ð14Þ where m is the order of matrix A; kAi is the eigenvalues of matrix A; Q A is the eigenvector matrix of matrix A. For two different functionally graded layers, the elements of matrix A are functions of x3 -axis coordinate, the solution of Eq. (13) is

Tðx3 Þ ¼ exp ½Xðx3 Þ

ð15Þ

where Xðx3 Þ is an infinite series (Magnus series) whose terms involve multiple integrals and nested commutators,

Xðx3 Þ ¼

1 X Xn ðx3 Þ

ð16Þ

n¼1

where

Z X1 ðx3 Þ ¼

x3

0

Aðx0 Þ  dx ;

X2 ðx3 Þ ¼

0

X3 ðx3 Þ ¼

1 3!

Z

x3

0

Z

x0

dx 0

00

Z

x00

dx 0

1 2!

Z

x3

0

Z

dx 0

x0

xiþ1 3

where kXi is the eigenvalues of matrix P vector matrix of matrix 6n¼1 Xn ðx3 Þ.

P6

n¼1 Xn ðx3 Þ; Q X

is the eigen-

3. Dispersion equations If any two slabs are bonded together perfectly, therefore, only the perfect interface conditions need to be considered. The perfect interface conditions require that the displacement components, the traction components, the electric potential, the magnetic potential, the normal components of electric displacement and magnetic induction, are continuous across the interface. In other word, the transfer matrix of the prefect interface is a unit matrix. After obtain the transfer matrix of coupled waves in each slab, the state vectors at the left and the right boundaries of a single cell, which consists of the piezoelectric slab, the piezomagnetic slab and two different functionally graded interlayers, can related by

000

dx  f½ Aðx0 Þ; ½ Aðx00 Þ; Aðx000 Þ   þ ½ Aðx000 Þ; ½ Aðx00 Þ; Aðx0 Þ  g

ð18Þ

 x3  xi  A x þ  dx 2 x3 =2 x3 =2

The 8th order approximation to P6 n¼1 Xn ðx3 Þ, where

ð19Þ

P1

n¼1 Xn ðx3 Þ

given by Blanes et al.

  þ   0 I 00 II V0L x03 ¼ d þ d þ d þ d ; t 0 ¼ TII T00 TI T0 V0L x03 ¼ 0þ ; t 0 According to Bloch theorem,

  þ 0 I 00 II V0L x03 ¼ d þ d þ d þ d ; t0 h  i   0 I 00 II ¼ exp iK d þ d þ d þ d V0L x03 ¼ 0þ ; t0

ð24Þ

 38 Q 1 ¼  Bð0Þ þ 24Bð2Þ ; Bð3Þ ; 5  63 ð0Þ 5 B  84Bð2Þ ;  Bð1Þ þ Bð3Þ ; Q2 ¼ 5 28 19 ð0Þ 15 ð2Þ B  B ; 28 7

 20 Q 1 þ 10Q 2 ; Q 4 ¼ Bð3Þ ; 7  6025 ð0Þ 2875 ð2Þ B þ B ; Q5 ¼  4116 343

Bð2Þ þ x3

h

Bð2Þ ;

ð23Þ

where K is the Bloch wave vector. Inserting Eq. (23) into Eq. (22) leads to

¼ x33 ðQ 3 þ Q 4 Þ þ x43 ðQ 5 þ Q 6 Þ þ x53 Q 7 ;

 Bð0Þ ;

ð22Þ

n h  io   0 I 00 II Tcell ðk1 ; xÞ  I exp iK d þ d þ d þ d V0L x03 ¼ 0þ ; t0 ¼ 0

X1 ¼ x3 Bð0Þ ; X2 ¼ x23 ðQ 1 þ Q 2 Þ; X3 þ X4 þ X5 þ X6

Q3 ¼

ð17Þ

0

[2,3] is



ð21Þ

00

In order to simplify the calculation of multiple integrals, let us introduce the univariate integrals [2,3]

Z

Xn ðx3 Þ ¼ Q X  diag½expðkX1 Þ; . . . ; expðkXm Þ  Q 1 X

n¼1

dx  ½ Aðx0 Þ; Aðx00 Þ ;

½ Aðx0 Þ; Aðx00 Þ  ¼ Aðx0 Þ  Aðx00 Þ  Aðx00 Þ  Aðx0 Þ

1

#

0

and so on. The matrix commutator is defined as

BðiÞ ðx3 Þ ¼

exp

" 6 X

where Tcell ðk1 ; xÞ ¼ TII T00 TI T0 is the transfer matrix of one typical single cell with the perfect interfaces. The condition of existing nontrivial solution leads to



61 1 Q  Q 588 1 12 2

Q1

i

h  i



0 I 00 II

Tcell ðk1 ; xÞ  I exp iK d þ d þ d þ d

¼ f ðk1 ; x; K Þ ¼ 0

 ;

;

 20 820 x3 Q 5 ; ðQ 3 þ Q 4 Þ þ Q 6 ¼ Bð3Þ ; 7 189   1 1 ð0Þ ð0Þ B ; B ; Q 3  Q 4 þ x3 Q 5 ; Q7 ¼  42 3 The matrix exponential can be calculated by

ð25Þ

Eq. (25) gives the dispersion relation of Bloch waves in the onedimensional piezoelectric/piezomagnetic phononic crystal with functionally graded interlayers. The coefficient determinant is a function of the apparent wavenumber k1 , the angular frequency x and the Bloch wave vector K. For given k1 and K, more than one x can be solved from Eq. (25). Therefore, a group of dispersion curves is obtained in the considered frequency range. 4. Numerical results and discussions

ð20Þ

In this section, dispersion curves and band gaps of in-plane Bloch waves and anti-plane Bloch waves propagating in the one-dimensional piezoelectric/piezomagnetic phononic crystal are calculated numerically. The material constants of the

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Table 1 Material constants of LiNbO3 and Terfenol-D [9]. Mat

Name

c11

c12

c13

c33

c44

q

e15

e31

A(PE) B(PM)

LiNbO3 Terfenol-D

203 8:541

52:9 0:654

74:9 3:91

243 28:3

59:9 5:55

4700 9250

3.7 0

0.19 0

e33

e11

e33

q15

q31

q33

l11

l33

A(PE) B(PM)

LiNbO3 Terfenol-D

1.31 0

0:39 0:05

0:26 0:05

0 155.56

0 5.7471

0 270.1

5 8:644

10 2:268

cij : GPa; q : Kg m3 ; eij : C m2 ; qij : N A1 m1 ;

eij : nC2 N1 m2 ; lij : lN A2 .

piezoelectric solid LiNbO3 and the piezomagnetic solid Terfenol-D [9] are listed in Table 1. The length ratio of the piezoelectric slab, the piezomagnetic slab and two functionally graded layers in a 0

I

00

II

I

00

single cell is d : d : d : d ¼ 2 : 1 : 2 : 1 and the total length of 0

      PIi xI3 ¼ P0 F i xI3 þ P00 1  F i xI3

ð26aÞ

where

  F 0 xI3 ¼ 1

ð26bÞ

  xI F 1 xI3 ¼ 1  3I d

ð26cÞ

II

single cell is a ¼ d þ d þ d þ d . In order to investigate the influences of functionally graded interlayers, the dispersion curves corresponding with different change profile of functionally graded interlayers are shown in same figure to facilitate the comparison. The dispersion curves are drawn in first Brillouin zone, namely, the dimensionless Bloch wavenumber Ka=p 2 ½1; 1. The dimensionless angular frequencies are xa=ð2pcP Þ and xa=ð2pcSH Þ for in-plane Bloch waves and anti-plane Bloch waves, respectively, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi e02 where the P wave speed cP ¼ q10 c033 þ e15 and the SH wave speed 0 33 qffiffiffiffiffi c044 cSH ¼ q0 . In order to investigate the influences of gradient profile on the dispersion curves, we consider five gradient profiles in the graded interlayer I,

  F 2 xI3 ¼   F 3 xI3 ¼

 1  1

xI3

15

I

d

xI3

ð26dÞ

5

I

d

8  1 > xI3 5 I 1 I >1 1  I  < 2 þ 2 1  2 d I ; x3 6 2 d F 4 x3 ¼  1 > I > : 1  1 1 þ 2 x3I 5 ; xI P 1 dI 3 2 2 2 d

Fig. 2. The gradient profiles in the graded interlayer I and II. (a and b) The symmetrical distribution; (c and d) the asymmetrical distribution.

ð26eÞ

ð26fÞ

X. Guo et al. / Ultrasonics 70 (2016) 158–171

163

Fig. 3. Effects of the 1st gradient profile (linear profile) of symmetrical distributed graded interlayers on dispersion curves ( F 0 , F 1 ) and band gaps ( F0 ; F 1 ) of in-plane and anti-plane Bloch waves. (a–c) Dispersion curves of in-plane Bloch wave; (d–f) dispersion curves of out-plane Bloch wave; (g) band gaps of in-plane Bloch wave; (h) band gaps of out-plane Bloch wave.

8  5 xI3 1 I I >1 1  I  < 2 þ 2 1  2 dI ; x3 6 2 d F 4 x3 ¼  5 > : 1  1 1 þ 2 xI3 ; xI P 1 dI 2

2

d

I

3

ð26gÞ

2

P 0 and P00 represent the material parameter of the piezoelectric solid and the piezomagnetic solid, respectively. PIi represents the material parameter in the graded interlayer I. These gradient profiles are shown in Fig. 2a. The magnetoelectric coefficients of the piezoelectric slab and the piezomagnetic slab are both zero. The presence of the ME coefficient in the graded interlayer can

be taken as the consequence of forming a composite with piezoelectric LiNbO3 and piezomagnetic Terfenol-D. The specific values of the effective ME coefficients are dependent upon the volume fraction and the effective ME coefficients should be determined by the homogenization method. Wang and Pan [22] studied the calculation of effective ME coefficients of a multiferroic fibrous composite by the Mori–Tanaka mean-field method. However, how to obtain the effective ME coefficients is not the focus of this paper and thus the magnetoelectric (ME) coefficients of the functionally graded layers are directly defined as

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Fig. 4. Comparisons of the 1st, 2nd and 3rd kinds of gradient profiles of symmetrical distributed functionally graded interlayers on dispersion curves ( F 1 , —— F 2 , —— —— F 3 ), and band gaps (==== F 2 ; n n n n F 3 ) of in-plane and anti-plane Bloch waves. (a–c) Dispersion curves of in-plane Bloch wave; (d–f) dispersion curves of out-plane Bloch wave; (g) band gaps of in-plane Bloch wave; (h) band gaps of out-plane Bloch wave.















g g a11 x3g ¼ 4a11 F x3g 1  F x3g ; a33 x3g  g   g  ¼ 4a33 F x3 1  F x3



In the case of asymmetrical distribution (see Fig. 2d),

ð27Þ

where ‘‘g” represents ‘‘I” and ‘‘II”. a11 ¼ 5  10 Ns V C and a33 ¼ 3  1012 Ns V1 C1 are prescribed [20,11]. The material 12

1

ð28bÞ

1

parameters in the graded interlayer II can be obtained with consideration of symmetrical and asymmetrical distribution. In the case of symmetrical distribution (see Fig. 2b),

  h  i   II II PIIi xII3 ¼ P0 F i d  xII3 þ P00 1  F i d  xII3

      PIIi xII3 ¼ P 0 1  F xII3 þ P00 F xII3

ð28aÞ

In the numerical example, five kinds of symmetrical distribution and two kinds of asymmetrical distributions are considered. The five kinds of symmetrical distribution are shown in Fig. 2a and b. The two kinds of asymmetrical distributions are shown in Fig. 2c and d. The influences of the linear profile of symmetrical distributed graded interlayers on dispersion curves and band gaps are shown

X. Guo et al. / Ultrasonics 70 (2016) 158–171

165

Fig. 5. Comparisons of the 1st, 4th and 5th kinds of gradient profiles of symmetrical distributed functionally graded interlayers on dispersion curves ( F 1 , –  –  F 4 ,      F 5 ) and band gaps (==== F 5 ; n n n n F 4 ) of in-plane and anti-plane Bloch waves. (a–c) Dispersion curves of in-plane Bloch wave; (d–f) dispersion curves of out-plane Bloch wave; (g) band gaps of in-plane Bloch wave; (h) band gaps of out-plane Bloch wave.

in Fig. 3. By comparison with the homogeneous interlayer, evident differences on the curves and gaps are observed. The dispersion curves corresponding with the graded interlayer shift toward low frequency range, no matter that it is in-plane Bloch waves or out-plane Bloch wave. Moreover, the dispersion curves at high frequency range shift more evident than that at low frequency range. Consequently, the band gaps also shift toward low frequency range and some band gaps become narrow. The observation shows that the existence of graded interlayer has evident influences on the dispersion properties of Bloch waves propagating through the phononic crystal. The width and location of pass bands and stop bands

can also be adjusted by the graded interlayer, see Fig. 3g and f. It is also observed that the Bloch waves have a cut-off frequency at oblique propagation situation. The cut-off frequency increases monotonously with the increasing apparent wavenumber. Another phenomenon is that the dispersion curves will become flat when the apparent wavenumber increases gradually. This means that the dispersion properties of Bloch waves disappear and the speeds of Bloch waves tend to zero. In other word, the localization phenomenon of Bloch wave, namely, the wave mode reduces to the vibration mode, takes place. This can be understandable by considering the situation of all waves propagating along interface.

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Fig. 6. Comparisons of the 2nd kind of gradient profile of symmetrical and asymmetrical distributed functionally graded interlayers on dispersion curves (—— sym, asym) and band gaps (==== symmetrical, asymmetrical) of in-plane and anti-plane Bloch waves. (a–c) Dispersion curves of in-plane Bloch wave; (d–f) dispersion curves of outplane Bloch wave; (g) band gaps of in-plane Bloch wave; (h) band gaps of out-plane Bloch wave.

In order to make a comparison of the influences of the 1st, 2nd and 3rd kinds of profiles, the dispersion curves and band gaps are calculated in the case of symmetrical distributed graded interlayers and shown in Fig. 4. It is observed that the dispersion curves corresponding with the 2nd kind of profile are always above that of linear profile while the dispersion curves corresponding with the 3rd kind of profile are always below that of linear profile, no matter that the Bloch waves is in-plane or out-plane. This phenomenon can be explained as following. The 2nd kind of profile can be approximately taken as the increase of the thickness of the piezoelectric slab while the 3rd kind of profile as the increase

of the thickness of the piezomagnetic slab. Because the elastic constants of the piezoelectric slab are much larger than that of the piezomagnetic slab, the global rigidness of the phononic crystal increases for the 2nd kind of profile while decreases for the 3rd kind of profile. This results in the upper shift and down shift of the dispersion curves. The opposite shifts of dispersion curves for the 2nd kind of profile and the 3rd kind of profile further result in the change of band gaps. In general, band gaps corresponding with the 3rd kind of profile is below the band gaps corresponding with the 2nd kind of profile, see Fig. 4g and h, no matter that the Bloch wave is in-plane or out-plane.

X. Guo et al. / Ultrasonics 70 (2016) 158–171

167

Fig. 7. Comparisons of the 3rd kind of profile of symmetrical and asymmetrical distributed functionally graded interlayers on dispersion curves (—— —— sym, asym) and band gaps (==== symmetrical, anti-symmetrical) of in-plane and anti-plane Bloch waves. (a–c) Dispersion curves of in-plane Bloch wave; (d–f) dispersion curves of outplane Bloch wave; (g) band gaps of in-plane Bloch wave; (h) band gaps of out-plane Bloch wave.

In order to make a comparison of the influences of the 1st, 4th and 5th kinds of profiles, the dispersion curves and band gaps are calculated in the case of symmetrical distributed graded interlayers and shown in Fig. 5. It is observed that the dispersion curves corresponding with the 5th kind of profile are above that of linear profile while the dispersion curves corresponding with the 4nd kind of profile are below that of linear profile. However, by compared with Fig. 4, the dispersion curves corresponding with the 4th and 5th kinds of profiles have much smaller deviation from that of linear profile. In fact, the 4th kind of profile can be approximately taken as increasing simultaneously the thick of the piezo-

electric slab and the piezomagnetic slab. The 5th kind of profile can be approximately taken as a homogenous interlayer with the averaged material constants of the piezoelectric and piezomagnetic slabs. Because the elastic constants of piezomagnetic slab are much smaller than that of piezoelectric slab, the global rigidness of the phononic crystal corresponding with the 5th kind of profile is slight larger than that of the phononic crystal corresponding with the 4th kind of profile. This can explain the opposite deviations of the 4th and 5th kinds of profiles from that of linear profile. It is known that the gradient profiles in two interlayers within one typical single cell can be symmetrical or asymmetrical. The

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X. Guo et al. / Ultrasonics 70 (2016) 158–171

Fig. 8. Comparisons of the piece-wise homogenization method (—— ——) and the direct integration method ( ) for the effects of the 3rd kind of gradient profile of symmetrical distributed functionally graded interlayers on dispersion curves of in-plane and anti-plane Bloch waves. (a–c) dispersion curves of in-plane Bloch wave; (d–f) dispersion curves of out-plane Bloch wave.

symmetrical situations are discussed above. Let us turn to the asymmetric situations and make a comparison with the symmetric situation. Fig. 6 shows the influences of the 2nd kind of profile in both symmetrical and asymmetrical situations on dispersion curves and band gaps. It is observed that the dispersion curves corresponding with the 2nd kind of profile in symmetrical situation are evident larger than that in asymmetric situation. Fig. 7 shows the influences of the 3rd kind of profile in both symmetrical and

asymmetrical situations on dispersion curves and band gaps. The opposite situation is observed, namely, the dispersion curves corresponding with the 3rd kind of profile in symmetrical situation are evident smaller than that in asymmetric situation. This phenomenon can be explained by that the thickness of piezoelectric slab enlarge approximately for the 2nd kind of profile in symmetrical situation while the thickness of piezomagnetic slab enlarge approximately for the 3rd kind of profile in symmetrical situation.

Fig. 9. Effects of the magnetoelectric coefficients of the 1st gradient profile (linear profile) of symmetrical distributed graded interlayers (—— aij ¼ 0, —— —— a11 ¼ 107 Ns V1 C1 ;       a11 ¼ 106 Ns V1 C1 Þ on dispersion curves of in-plane Bloch waves.

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X. Guo et al. / Ultrasonics 70 (2016) 158–171

Because the dispersion curves are evident different for the gradient profile in symmetric and asymmetric situations, the band gaps corresponding with the symmetric and asymmetric situations are also evident different, see Figs. 6g–h and 7g–h. The functionally graded interlayer is often treated by the piecewise homogenization method [23,4,8,7,11] instead of the direct integration method used in present work for convenience. In order to make a comparison of two methods, the dispersion curves of inplane and anti-plane Bloch waves for the 3rd kind of gradient profile of symmetrical distributed functionally graded interlayers are shown in Fig. 8. It is found that the numerical results obtained by two methods have good consistence at low frequency range. However, the evident deviations are observed at the high frequency range. The dispersion curves obtained by the piece-wise homogeneous method are beneath that obtained by the direct integration method. The piece-wise homogeneous method is performed by dividing the graded interlayer into 50 homogeneous sublayers in the present work. But it is expected more sublayers are necessary for the dispersion curves at high frequency range to guarantee convergence. Certainly, the cost is the prominent increase of the calculation-time. The influences of the magnetoelectric coefficients of the functionally graded interlayers on dispersion curves of in-plane waves are shown in Fig. 9. It is found that the dispersion curves shift slightly toward the low frequency range with the increasing magnetoelectric coefficients. Usually, the magnetoelectric coefficients of the magnetoelectric composite are not too large and their influence on the dispersion curves are not remarkable.

5. Conclusions The influences of graded interlayer on the dispersion curves and band gaps of Bloch waves in the one-dimensional piezoelectric/ piezomagnetic phononic crystal are the main concerns of the present work. Five kinds of typical gradient profile are considered. Because the gradient profiles in two graded interlayers within one typical single cell of phononic crystal can be symmetrical or asymmetrical. The symmetrical situations and the asymmetric situation are both considered and a comparison is made. The dispersion curves and band gaps of in-plane and anti-plane Bloch waves are calculated numerically for the symmetrical and asymmetrical situation. Based on these numerical results, the following conclusions can be drawn: (1) The gradient profile of graded interlayers has evident influences on the dispersion curves and therefore can be designed elaborately to adjust the width and the central frequency of band gaps of phononic crystal. In general, the gradient profile has more evident influences on the dispersion curves at high frequency range than that at low frequency range, no matter that the Bloch wave is in-plane or outplane. (2) Compared with the linear gradient profile, the dispersion curves corresponding with 2nd and 4th gradient profiles are above that of linear profile while the dispersion curves corresponding with 3rd and 5th gradient profiles are below that of linear profile. The difference of dispersion curves between the 2nd gradient profile and 3rd gradient profile are largest. The difference of dispersion curves between the 4th gradient profile and 5th gradient profiles are much smaller. (3) Not only the dispersion curves corresponding with different gradient profiles have evident deviations but also the dispersion curves corresponding with same gradient profile in symmetrical and asymmetric situation have evident devia-

tions. The dispersion curves corresponding with 2nd gradient profile in symmetrical situation are evident larger than that in asymmetric situation. However, the dispersion curves corresponding with 3rd gradient profile in symmetrical situation are evident smaller than that in asymmetric situation. (4) The influences of gradient profiles on the dispersion curves and band gaps are similar for the in-plane Bloch waves and out-plane Bloch wave. These dispersion curves become flat gradually when the apparent wavenumber increases gradually. The phenomenon represents that the wave mode reduces gradually the vibration mode when the propagation direction of Bloch waves is near the interface.

Acknowledgments The work is supported by Fundamental Research Funds for the Central Universities (FRF-BR-15-026A) and National Natural Science Foundation of China (No. 10972029). Appendix A The material constants of transversely isotropic piezoelectric slab and piezomagnetic slab are

2

c1111

6c 6 2211 6 6 c3311 6 6c 6 2311 6 4 c3111

c1122

c1133

c1123

c1131

c2222 c3322

c2233 c3333

c2223 c3323

c2231 c3331

c2322

c2333

c2323

c2331

c3122

c3133

c3123

c3131

c1211 c1222 c1233 c1223 2 c11 c12 c13 0 6c 6 12 c11 c13 0 6 6 c13 c13 c33 0 ¼6 6 0 0 0 c44 6 6 4 0 0 0 0 0

0

0

0

c1112

3

c2212 7 7 7 c3312 7 7 c2312 7 7 7 c3112 5

c1231 c1212 3 0 0 0 0 7 7 7 0 0 7 7 0 0 7 7 7 c44 0 5 0

c66

where c66 ¼ 12 ðc11  c12 Þ; cijmn ¼ cijmn ¼ cijnm ¼ cmnij ;

2

e111 e122 e133 e123 e131 e112

2

3

0

0

0

0

6 7 6 0 0 e15 4 e211 e222 e233 e223 e231 e212 5 ¼ 4 0 e31 e31 e33 0 e311 e322 e333 e323 e331 e312 where emij ¼ emji ;

3 2 0 q111 q122 q133 q123 q131 q112 6 7 6 4 q211 q222 q233 q223 q231 q212 5 ¼ 4 0 2

2

3

2

3

e11 e12 e13 e11 0 0 6 7 6 7 4 e21 e22 e23 5 ¼ 4 0 e11 0 5 0 0 e33 e31 e32 e33 where

2

emn ¼ enm ; 3

2

3

l11 l12 l13 l11 0 0 7 6 l22 l23 5 ¼ 4 0 l11 0 7 5 0 0 l33 l31 l32 l33

6 4 l21

where

lmn ¼ lnm .

0 0

q31 q31 q33

q311 q322 q333 q323 q331 q312 where qmij ¼ qmji ;

0 0

e15 0

3

0

7 05

0

0

3 0 q15 0 7 q15 0 0 5 0 0 0

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X. Guo et al. / Ultrasonics 70 (2016) 158–171

2

0 6 cg 6 3131 6 6 0 6 6 0 6 Gg ¼ 6 6 0 6 6 ik c g 6 1 3131 6 g 4 ik1 e131

Appendix B The explicit expressions of matrix G and H for each slab in Eq. (11) are

2

0

6 c0 6 3131 6 6 0 6 6 0 6 G0 ¼ 6 6 0 6 6 ik c0 6 1 3131 6 4 ik1 e0131

e0333

0

0

0 0

0

0

0

0

0 0

e

0

0

0

0

0

e0333

0 33

0

l033

ik1 c01133

ik1 e0311

0

0 1

0

0

0

1 0

0

0

0

0

0

0

0

0

0

0 0

g ik1 c1133

6 0 6 6 g 6 ik 1 e311 6 6 g ik1 q311 6 6 g H ¼6 2 g 6 k1 c1111  q g x2 6 6 0 6 6 6 0 4 0

g ik1 c3131

g ik1 e131

g ik1 q131

0

1 0

0

0

0

0

0 1

0

0

0

0

0

0

0

0

0

0

0

2 g k1 e131 2 g k1 11 2 g k1 11

2 g k1 q131 2 g k1 11 2 g k1 11

0

0

0

0

0

0

0

0

0

0

q x

2

e a

0

2

0 0 0 0

l011 0 0 0 0 3

q00333

0

0

0

0 0

0 e0033

0 0

0 0

0 0

q00333

0

l0033

0

0

ik1 c001133

0

ik1 q00311

0 1

0

0

0

1 0

0

0

0

0

0

07 7 7 07 7 0 07 7 7; 0 07 7 0 07 7 7 1 05

0

0

0

0

0

0 1

ik1 c001133 0

0 ik1 c003131

0 0

6 6 6 6 0 0 0 6 6 6 ik1 q00311 0 0 6 H00 ¼ 6 2 00 0 0 6 k1 c1111  q00 x2 6 2 00 6 00 2 0 0 k1 c3131  q x 6 6 2 6 0 0 k1 e0011 4 2 00 0 0 k1 q131

3

0 1 0 07 7 7 0 0 1 07 7 0 0 0 17 7 7; 0 0 0 07 7 7 0 0 0 07 7 7 0 0 0 05

0 0

0

ik1 q00131

1 0

1 0 0 0

c003333

0

6 c00 6 3131 6 6 0 6 6 0 6 00 G ¼6 6 0 6 6 ik c00 6 1 3131 6 4 0

0

a l

0

2 k1

g q333

0

0

0

0 g e33

0

0 0

0 0

0

0

0

0

3

g ik1 e311

g  33 g  33 g ik1 q311

0

0

0

0

0

0

0

0

07 7 7 07 7 0 07 7 7; 0 07 7 0 07 7 7 1 05

0

0

0

0

0

0 1

g a33

a l

0 1 1 0

0 0

1

0

2 g k1 c3131  g 2 g k1 e131 2 g k1 q131

g e333

g e333 g q333 g ik1 c1133

g ik1 q131

0

0 0 ik1 c01133 6 ik1 e0131 0 ik1 c03131 6 6 6 ik1 e0311 0 0 6 6 0 0 0 6 H0 ¼ 6 6 k21 c01111  q0 x2 0 0 6 6 2 2 6 0 k1 c03131  q0 x2 k1 e0131 6 6 2 2 k1 e011 0 k1 e0131 4

2

3

0

2

0

0

07 7 7 0 07 7 0 07 7 7; 0 07 7 0 07 7 7 1 05

0

0 2

c03333

g c3333

0

0 0

3 0 1 0 0 0 7 ik1 q00131 0 1 0 0 7 7 0 0 0 1 07 7 7 0 0 0 0 17 7 ; 0 0 0 0 07 7 7 2 k1 q00131 0 0 0 0 7 7 7 0 0 0 0 07 5 2 k1 l0011 0 0 0 0

0

0

3

7 07 7 07 7 7 17 7 7 07 7 07 7 7 07 5 0

for in-plane elastic waves in oblique propagation situation (the superscript ‘‘0”, ‘‘00” and ‘‘g” represents the piezoelectric slab, the piezomagnatic slab and the graded interlayer, respectively);



G¼

c2323 0

0 ; 1



H¼

0

1

 qx

2 k1 c1212

2



0

for anti-plane elastic waves in oblique propagation situation;

2

6 6 6 6 0 G ¼6 6 6 6 4

c03333 e0333

e0333

e033

0

0

l

0

0

0

0

0

0

6 6 6 6 00 G ¼6 6 6 6 4

0 c003333 0

e0033

0

0

0

0

3

0

q00333

0

0

0

0

0

0

0 0

l 0

0

0

0

q00333

0

0 0 0 1 3 0 1 0 0 0 0 1 07 7 7 0 0 0 17 7; 0 0 0 07 7 7 0 0 0 05

0 0 0 0

0

07 7 7 0 0 07 7; 1 0 07 7 7 0 1 05

0 0 33

0 0 2 0 0 6 0 0 6 6 6 0 0 H0 ¼ 6 6 q0 x2 0 6 6 4 0 0 2

0

00 33

0

07 7 7 0 0 07 7; 1 0 07 7 7 0 1 05

0

0 0 0 0 1 3 2 0 0 0 1 0 0 6 0 0 0 0 1 07 7 6 7 6 6 0 0 0 0 0 17 00 7 H ¼6 6 q00 x2 0 0 0 0 0 7; 7 6 7 6 4 0 0 0 0 0 05 0

0 0

0

3

0

0

X. Guo et al. / Ultrasonics 70 (2016) 158–171

2 6 6 6 6 Gg ¼ 6 6 6 6 4

g c3333

g e333

g q333

e a

a l

0

0

0

0

0

0

g e333 g q333

g 33 g 33

g 33 g 33

0

0

0 0

0 0

0

3

07 7 7 07 7; 1 0 07 7 7 0 1 05

0

0 0 0 0 1 3 0 0 1 0 0 6 0 0 0 0 1 07 7 6 7 6 6 0 0 0 0 0 17 g 7 H ¼6 6 q g x2 0 0 0 0 0 7; 7 6 7 6 4 0 0 0 0 0 05 2

0

0

0 0 0

0

0

for P wave in normal propagation situation;

 G¼

c3131

0

0

1

;

 H¼

0

1

qx2

0



for SV wave in normal propagation situation;

 G¼

c2323

0

0

1

;

 H¼

0

1

qx2

0



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