piezomagnetic phononic crystal with initial stresses

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Ultrasonics xxx (2015) xxx–xxx

Contents lists available at ScienceDirect

Ultrasonics journal homepage: www.elsevier.com/locate/ultras

Dispersion relations of elastic waves in one-dimensional piezoelectric/ piezomagnetic phononic crystal with initial stresses Xiao Guo, Peijun Wei ⇑ Department of Applied Mechanics, University of Science and Technology Beijing, Beijing 100083, China

a r t i c l e

i n f o

Article history: Received 30 July 2015 Received in revised form 3 November 2015 Accepted 8 November 2015 Available online xxxx Keywords: Dispersion relations Piezoelectric slab Piezomagnetic slab Phononic crystal Initial stress

a b s t r a c t The dispersion relations of elastic waves in a one-dimensional phononic crystal formed by periodically repeating of a pre-stressed piezoelectric slab and a pre-stressed piezomagnetic slab are studied in this paper. The influences of initial stress on the dispersive relation are considered based on the incremental stress theory. First, the incremental stress theory of elastic solid is extended to the magneto-electroelasto solid. The governing equations, constitutive equations, and boundary conditions of the incremental stresses in a magneto-electro-elasto solid are derived with consideration of the existence of initial stresses. Then, the transfer matrices of a pre-stressed piezoelectric slab and a pre-stressed piezomagnetic slab are formulated, respectively. The total transfer matrix of a single cell in the phononic crystal is obtained by the multiplication of two transfer matrixes related with two adjacent slabs. Furthermore, the Bloch theorem is used to obtain the dispersive equations of in-plane and anti-plane Bloch waves. The dispersive equations are solved numerically and the numerical results are shown graphically. The oblique propagation and the normal propagation situations are both considered. In the case of normal propagation of elastic waves, the analytical expressions of the dispersion equation are derived and compared with other literatures. The influences of initial stresses, including the normal initial stresses and shear initial stresses, on the dispersive relations are both discussed based on the numerical results. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction Phononic crystal as a sort of artificial periodic composite material produces complete acoustic band gaps within which elastic waves can’t propagate while in other frequencies range they can transmit without amplitude fading. As the inevitable existence of initial stresses came from, for example, machining, enhancing fracture toughness, creep deformation and chemical shrinkage, the influences of initial stresses on the propagation characteristic of elastic waves early attracted researcher’s attention. Biot [2] initially developed a theory of incremental stresses superposed on initial stresses within the geophysical context and studied the effect of initial stress on the propagation of small amplitude elastic waves. By using Biot’s incremental stress theory, many researchers investigated the propagation behavior of elastic waves in prestressed mediums. E1-Naggar and Saliem [6] studied the frequency equation for phase velocity of waves under initial stresses propagating in two kinds of laminated mediums. One of them consists of two elastic layers of finite thickness and the other consists of an elastic layer of finite thickness over an elastic half space. It ⇑ Corresponding author. E-mail address: [email protected] (P. Wei).

was shown that the wave approaches Rayleigh waves at the two outer surfaces with the possibility of Stonely waves at the interface when the wavelength becomes very small compared with the thickness of each layer. Chakraborty and Singh [3,4], investigated the reflection and refraction of thermo elastic plane waves at the interface of two dissimilar isotropic and homogeneous generalized thermo elastic half-spaces and the reflection/refraction problem of thermoelastic waves at a solid–liquid interface under initial stress. Gupta et al. [7] studied dispersion equation which determines the velocity of torsional surface waves in a homogeneous layer of finite thickness over an initially stressed heterogeneous half-space. It was found that the velocity of torsional surface waves increases as compressive initial stress increases and decreases in tensile initial stress. Chatterjee et al. [5] studied the reflection of threedimensional plane waves at a traction free boundary of a halfspace composed of triclinic crystalline material under initial stresses. Yu and Zhang [26] investigated the guided wave propagation in FGM plates under gravity homogeneous initial stress in the thickness direction and inhomogeneous initial stress in the wave propagation direction. The effects of the initial stresses on the Lamb-like wave and on the SH wave were both considered. The phononic crystal with the piezoelectric phase or the piezomagnetic phase is especially interesting due to the coupling effects

http://dx.doi.org/10.1016/j.ultras.2015.11.008 0041-624X/Ó 2015 Elsevier B.V. All rights reserved.

Please cite this article in press as: X. Guo, P. Wei, Dispersion relations of elastic waves in one-dimensional piezoelectric/piezomagnetic phononic crystal with initial stresses, Ultrasonics (2015), http://dx.doi.org/10.1016/j.ultras.2015.11.008

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X. Guo, P. Wei / Ultrasonics xxx (2015) xxx–xxx

between the electric, magnetic and mechanical fields. The peculiar characteristics of piezoelectric/piezomagnetic materials make it gaining extensive use in engineering fields, for example, the ultrasonic transducer, actuator and sensor, the acoustic energy harvest in defect modes, the high-efficiency acoustic waveguides, the sound-light interaction devices and so on. For the onedimensional piezoelectric/piezomagnetic phononic crystal, many investigations have been reported, for example, AlvarezMesquida et al. [1], Qian et al. [18], Monsivais et al. [15], Li and Wang [13]. Pang et al. [17,16], Wang et al. [24], Huang et al. [9], Rodríguez-Ramos et al. [21], Lan and Wei [11,12], Kuo and Peng [10]. Liu et al. [14] investigated Bleustein–Gulyaev surface wave (B–G wave) propagation in a pre-stressed layered piezoelectric structure. It was found that the initial stress has evident influences on the phase velocity of B–G wave. For a given value of the ratio of layer thickness to wavelength, the fractional changes in phase velocity are almost linear functions of the initial stress for both electrically open and shorted circuit surfaces. Hsu and Wu [8] presented a study on the existence of Bleustein–Gulyaev–Shimizu piezoelectric surface acoustic waves in a two-dimensional piezoelectric phononic crystal. Vasseur et al. [23] investigated the properties of piezoelectric phononic crystal freestanding plates and a plate supported on a silicon substrate. Qian et al. [20] discussed the influence of initial stresses on the stop band and the dispersion relation of the horizontally polarized shear waves (SH-waves) in a periodic piezoelectric-polymeric layered structure. It was showed that the stop band is dramatically changed with the increase of the absolute value of initial stress. Su et al. [22] studied the propagation of elastic waves in the layered piezoelectric media with inhomogeneous initial stresses by using the method of transfer matrix. Qian et al. [19] investigated the propagation behavior of Love waves in a functionally graded layer over a nonpiezoelectric half-space with initial stress taken into account. It was observed that the phase velocity of the Love waves increases with the increase of the initial tensile stress while decreases with the increase of the initial compression stress. The initial stress effects are negligible as the magnitudes of the initial stress are less than 100 MPa. Wang et al. [25] also investigated the stop band properties of elastic waves in three-dimensional piezoelectric phononic crystals with initial stress by using the plane wave expansion (PWE) method. However, only one component of initial normal stress is considered in their works and the initial shear stresses are not considered. In this paper, we investigate the effects of six kinds initial stresses including the normal initial stresses and the shear initial stresses, i.e. r011 ; r022 ; r033 ; r032 ; r031 and r012 , on the dispersion relations of elastic waves in one-dimensional phononic crystal composed of a piezoelectric slab and a piezomagnetic slab. First, the incremental stress theory proposed by Biot [2] is extended to the magneto-electro-elasto solid. The governing equations, constitutive equations, and boundary conditions related with the incremental stresses are derived. Then, the total transfer matrix of one typical single cell of the periodical structure is obtained by the combination of the transfer matrices of two adjacent slabs. Finally, the Bloch theorem is used to obtain the dispersive equations of Bloch waves. The dispersion equations of in-plane Bloch wave and anti-plane Bloch wave are both considered. The oblique propagation and the normal propagation situations are also considered, respectively.

2. Constitutive and governing equations for a pre-stressed magneto-electro-elasto medium Consider a volume V bounded by a surface S with initial stress

r0ij , initial electrical displacement D0i and initial magnetic induction

B0i . At this initial configuration, these initial mechanical, electrical and magnetic quantities satisfy

r0ij;j þ qf 0i ðxl Þ ¼ 0; D0i;i ¼ 0; B0i;i ¼ 0

ð1Þ

where the coordinates system ðx1 ; x2 ; x3 Þ is established at the initial 0

configuration. q is the initial mass density and f i is the components of body force per unit mass. Now, a small incremental deformation which is due to the small amplitude elastic waves is superposed upon the initial finite deformation in the magneto-electro-elasto medium. The incremental stress rij , the incremental electrical displacement Di , and the incremental magnetic induction Bi which result from the small amplitude elastic waves are superposed upon the initial stress

r0ij , initial electric displacement D0i and the initial

magnetic induction B0i . The total stress components, total electric displacement components and total magnetic induction components which refer to the current coordinates system ðg1 ; g2 ; g3 Þ can be obtained by the coordinate deformation rule of tensor and vector

8 > rtmn ¼ rij þ r0ij cos ðm; iÞ cos ðn; jÞ > > > < Dtm ¼ Di þ D0i cos ðm; iÞ > > > > : Bt ¼ B þ B0 cos ðm; iÞ i m i

ð2Þ

where m; n ¼ g1 ; g2 ; g3 and i; j ¼ x1 ; x2 ; x3 . The directional cosines to the first order approximation can be expressed

3 3 2 cos ðg1 ; x1 Þ cos ðg1 ; x2 Þ cos ðg1 ; x3 Þ 1 x12 x31 7 6 7 6 1 x23 5 4 cos ðg2 ; x1 Þ cos ðg2 ; x2 Þ cos ðg2 ; x3 Þ 5 ¼ 4 x12 cos ðg3 ; x1 Þ cos ðg3 ; x2 Þ cos ðg3 ; x3 Þ x31 x23 1 2

ð3aÞ where the local rigid rotation tensor

xkj ¼

1 uk;j uj;k 2

ð3bÞ

and ui is the incremental displacement. Inserting Eq. (3) into Eq. (2) and leads to

8 t r ¼ r0ij þ rij þ r0kj xik þ r0ik xjk > < mn Dtm ¼ D0i þ Di þ D0k xik > : t Bm ¼ B0i þ Bi þ B0k xik

ð4Þ

After deformation the surface S becomes a surface S0 while the volume V becomes a volume V 0 . The motion equation of any representative volume element can be expressed as RRR 8R R RRR 0 t t t > € q0 dV 0 f ðgl Þq0 dV ¼ u > V0 m V0 m > S0 rmg1 dg2 dg3 þ rmg2 dg3 dg1 þ rmg3 dg1 dg2 þ > S g1 2 3 > > R R > : Bt dg dg þ Bt dg dg þ Bt dg dg ¼ 0 0 S

g1

2

3

g2

3

1

g3

1

2

ð5Þ 0

where q is the current mass density. Consider that

gi ¼ xi þ ui ðxl Þ

ð6aÞ

q0 dV 0 ¼ qdV

ð6bÞ

and by using Green’s theorem, Eq. (5) can be rewritten as

o 8 R R R n RRR > € qdV rtmk ðgl ÞMkj ðxl Þ ;j þ qf m ðgl Þ dV ¼ u > V V m > > < R R R n o t Dm ðgl ÞM kj ðxl Þ ;j dV ¼ 0 V > > n > o R R R > : Btm ðgl ÞM kj ðxl Þ ;j dV ¼ 0 V

ð7aÞ

where

Please cite this article in press as: X. Guo, P. Wei, Dispersion relations of elastic waves in one-dimensional piezoelectric/piezomagnetic phononic crystal with initial stresses, Ultrasonics (2015), http://dx.doi.org/10.1016/j.ultras.2015.11.008

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M kj ðxl Þ ¼ ð1 þ eÞdkj ekj þ xkj ekj ¼

ð7bÞ

1 uk;j þ uj;k 2

ð7cÞ ð7dÞ

e ¼ ui;i

dkj is the Kronecker delta and ekl is the incremental strain tensor. Inserting Eqs. (1) and (4) into Eq. (7a) leads to the governing equations in terms of the incremental stress, the incremental electrical displacement and the incremental magnetic induction

h i 8 0 0 0 0 > € > > rij þ rij e rik ekj þ rkj xik ;j þ q f i ðgl Þ f i ðxl Þ ¼ qum > > < Dj þ D0j e D0k ekj ¼ 0 ;j > > > > > : Bj þ B0j e B0k ekj ¼ 0

ð8Þ

;j

The boundary conditions associated with the incremental stress, incremental electrical displacement and incremental magnetic induction can be expressed as

8 0 0 0 > r þ r e r e þ r x Nj ¼ ti ij kj ik > ij ik kj > > < 0 0 Dj þ Dj e Dk ekj Nj ¼ qu > > > > : B þ B0 e B0 e N ¼ q j j w j k kj

ð9Þ

where t i ; qu and qw are the surface traction, electric charge density and magnetic flux acting on per unit area of S. In order to obtain the constitutive relation between the incremental stress and the incremental strain, we introduce [2]

t ij ¼ rij þ r0ij e r0ik ekj

ð10Þ

t ij ¼ 1 t ij þ t ij ¼ rij þ r0ij e 1 r0ik ekj þ r0jk eki 2 2

ð11Þ

Because tij is the incremental stresses resulting from pure deformation, t ij is related with the small amplitude strain ekl by

t ij ¼ cijmn emn emij Em qmij Hm

ð12Þ

Inserting Eq. (11) into Eq. (12), we get the constitutive equation for the incremental stress tensor

rij ¼ ^cijmn emn r0ij e emij Em qmij Hm

ð13aÞ

1 0 r dmj þ r0im dnj þ r0jn dmi þ r0jm dni 4 in

ð13bÞ

It is noted that the elastic coefficient ^cijmn has the same symmetry properties with cijmn , i.e. ^cijmn ¼ ^cjimn ¼ ^cijnm ¼ ^cmnij . By comparing with the situation without initial stress, we introduce the effective stress tensor, effective electrical displacement and effective magnetic induction vector

8 > r^ ¼ rij þ r0ij e r0ik ekj þ r0kj xik > < ij ^ m ¼ Dm þ D0 e D0 ekm D m k > > :^ Bm ¼ Bm þ B0m e B0k ekm

ð14Þ

Then, the mechanical, electric and magnetic constitutive relations in the magneto-electro-elasto medium with initial stresses can be expressed as

8 r^ ij ¼ ^cijkl uk;l þ r0kj xik r0ik ekj þ emij u;m þ qmij w;m > > < ^ m ¼ emij uj;i emn u amn w D ;n ;n > > :^ Bm ¼ q uj;i amn u l w mij

;n

mn

;n

where emij ¼ emji ; qmij ¼ qmji ; emn lmn and amn are the piezoelectric, piezomagnetic, dielectric, magnetic permeability and magnetoelectric coefficient, respectively. The quasi-static approximation of electric field and magnetic field is used, namely, the electric potential u and the magnetic potential w exists and the electric field Em and the magnetic field Hm can be expressed as Em ¼ u;m and Hm ¼ w;m . 3. Transfer matrix of coupled waves in a single prestressed cell Consider a pre-stressed phononic crystal formed by periodically repeat of a piezoelectric slab and a piezomagnetic slab, as shown in Fig. 1. The piezomagnetic coefficient and the magnetoelectric coefficient are zero (qmij ¼ 0 and amn ¼ 0) for the piezoelectric slab while the piezoelectric coefficient and the magnetoelectric coefficient are zero (emij ¼ 0 and amn ¼ 0) for the piezomagnetic slab. The two slabs labeled A (PE) and B (PM) are bonded together with the perfect interface across which all physical fields are continuous. There are ten kinds of coupled elastic waves in each slab and four of them always propagate along the interface. In order to describe the propagating properties of coupled waves in respective slab, two local Cartesian coordinate systems, which are indicated by the superscript ‘‘0 ” and ‘‘00 ”, respectively, are established. In latter formulation, if the physical quantity doesn’t have superscript ‘‘0 ” or ‘‘00 ”, it will be appropriate for both local Cartesian coordinate systems. The x3 – axis is the poling direction of the piezoelectric slab and the piezomagnetic slab. The physical constants of these slabs are given in Appendix A. d is the thicknesses of slab. Six kind of initial stresses, i.e. r011 ; r022 ; r033 ; r032 ; r031 and r012 , are considered. The incremental displacement, electric potential and magnetic potential are functions of x1 and x3 , and can be expressed as

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

W 11 6 6 W 21 26 k1 6 6 W 31 6 4 W 41

W 12

W 13

W 14

W 22 W 32

W 23 W 33

0 W 34

0

W 43

W 44

0

W 53

0

W 51

3 8 9 8 9 U1 > > 0> > > > > > > > > > 7 > > > > > > 0 7 > U 0> > > > 2> < = < = 7 W 35 7 7 > U3 > ¼ > 0 > > > > > 7 > > > 0 5 > U > > > > > >0> > > : > ; > : > ; W 55 W 0 W 15

ð17Þ

where W 11 ¼ c1111 þ c3131 þ 34 r033 14 r011 n2 þ r031 n qc2 , 1 0 1 1 r r0 n2 r032 n; 2 12 4 12 4 1 1 1 1 W 13 ¼ W 31 ¼ r031 r031 n2 þ c3131 þ c1133 r011 r033 n; 2 2 4 4

W 12 ¼ W 21 ¼

W 14 ¼ W 41 ¼ ðe113 þ e311 Þn; W 15 ¼ W 51 ¼ ðq113 þ q311 Þn; 3 1 3 1 3 W 22 ¼ c1212 þ r011 r022 þ c2323 þ r033 r022 n2 þ r031 n qc2 ; 4 4 4 4 2 1 0 1 0 2 1 0 W 23 ¼ W 32 ¼ r32 r32 n r12 n; 4 2 4 3 1 W 33 ¼ c3131 þ r011 r033 þ c3333 n2 þ r031 n qc2 ; W 34 ¼ W 43 ¼ e131 þ e333 n2 ; 4 4

W 35 ¼ W 53 ¼ q131 þ q333 n2 ,

ð15Þ

ð16Þ

where the wave vector k ¼ ðk1 ; k2 ; k3 Þ ¼ ðk1 ; 0; k1 nÞ; 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 wavenumber and its relationship with the propagation direction angle is h ¼ arccotn. According to the well-known Snell’s law, the apparent wavenumbers and the angular frequencies of various waves are same, thus the apparent wave speeds are also same for various waves. Inserting Eq. (16) into Eq. (8) and ignoring the body force leads to

2

where the effective elastic coefficient [2]

^cijmn ¼ cijmn þ

W 44 ¼

e11 þ e33 n2 , W 55 ¼ l11 þ

2

l33 n Þ. For the piezoelectric slab, the magnetic potential w is independent of u1 ; u2 ; u3 and u due to qmij ¼ 0, which means that the wave arising from variation of magnetic field is decoupled with

Please cite this article in press as: X. Guo, P. Wei, Dispersion relations of elastic waves in one-dimensional piezoelectric/piezomagnetic phononic crystal with initial stresses, Ultrasonics (2015), http://dx.doi.org/10.1016/j.ultras.2015.11.008

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Fig. 1. One-dimensional piezoelectric/piezomagnetic phononic crystal with initial stresses.

other waves. It is the magnetic potential (MP) wave. Similarly, for the piezomagnetic slab, the electric potential u is independent of u1 ; u2 ; u3 and w due to emij ¼ 0, which means that the wave arising from variation of electric field is decoupled with other waves. It is the electric potential (EP) wave. The existence of non-trivial solution of Eq. (17) requires that the coefficient determinant equals to zero,

W ij ¼ f ðn; cÞ ¼ 0

ð18Þ

For a given apparent wave speed c; f ðn; cÞ is a polynomial of five orders about n2 , thus there are five pairs of roots about n. These roots stand for all possible wave modes in these two slabs. The real roots stand for the bulk waves propagating in the ox1 x3 plane. The imaginary roots stand for the surface waves propagating along interface with attenuation vertical to the propagation direction. The complex roots stands for the bulk wave accompanied with attenuation. For the piezoelectric solids, six of them are bulk waves and four of them are the surface wave. Let n01 ; n03 ; n05 ; n07 and rﬃﬃﬃﬃﬃﬃ l0 n09 ¼ i l11 be the forward QPþ0 (quasi-longitudinal), QSVþ0 0 33

(quasi-transverse), QSHþ0 (quasi-shear), EAþ0 (electro-acoustic) waves and MPþ0 (magnetic potential) wave while n02 ; n04 ; n06 ; n08 and rﬃﬃﬃﬃﬃﬃ l0 n010 ¼ i l11 be the backward QP0 ; QSV0 ; QSH0 ; EA0 and MP0 0

n o

u1q ; u2q ; u3q ; uq ; wq ¼ G1q ; G2q ; G3q ; Guq ; Gwq U 1q exp ik1 x1 þ nq x3 ct

ð20Þ

where the subscript q denotes different types of coupled waves. Inserting Eq. (20) into Eq. (15), the effective stress, the effective electric displacement and the effective magnetic induction components can be uniformly expressed as

n

o

r^ 13q ; r^ 23q ; r^ 33q ; D^ 3q ; B^3q ¼ ik1 H1q ; H2q ; H3q ; Jq ; Lq U 1q

exp ik1 x1 þ nq x3 ct

ð21Þ

where

1 0 3 1 1 1 r31 þ c3131 þ r033 r011 nq G1q r032 þ r012 nq G2q 2 4 4 4 4 1 0 1 0 1 0 þ c3131 r33 r11 r31 nq G3q þ e113 Guq þ q113 Gwq ; 4 4 2

H1q ¼

1 0 3 3 1 r12 nq G1q þ r031 þ c2323 þ r033 r022 nq G2q 4 4 4 4 1 0 1 0 r þ r n G3q ; 4 12 2 32 q

H2q ¼

33

waves. Similarly, for the piezomagnetic solids, let n001 ; n003 ; n005 ; rﬃﬃﬃﬃﬃ e00 and n009 be the forward QPþ00 ; QSVþ00 ; QSHþ00 ; MAþ00 n007 ¼ i e11 00

H3q ¼

33

(magneto-acoustic) wave and EPþ00 (electric potential) waves while rﬃﬃﬃﬃﬃ e00 and n0010 be the backward QP00 ; QSV00 ; n002 ; n004 ; n006 ; n008 ¼ i e11 00 33

QSH00 ; MA00 waves and EP00 waves. Define the amplitude ratio of QP, QSV, QSH and EA wave in the piezoelectric solid

U0 U0 U0 U0 W0 G01 ¼ 10 ¼ 1; G02 ¼ 20 ; G03 ¼ 30 ; G0u ¼ 0 ; G0w ¼ 0 ¼ 0 U1 U1 U1 U1 U1

ð19aÞ

Similarly, the amplitude ratio of QP, QSV, SH and MA wave in the piezomagnetic solid are defined as

G001 ¼

U 001 U 002 U 003 U00 W00 00 00 00 00 00 ¼ 1; G2 ¼ 00 ; G3 ¼ 00 ; Gu ¼ 00 ¼ 0; Gw ¼ 00 U1 U1 U1 U1 U1

ð19bÞ

then, G1 ; G2 ; G3 ; Gu ; Gw stands for the coupled relation among the displacement components ðu1 ; u2 ; u3 Þ, the electric potential u and the magnetic potential w, which can be called the vibration mode. By using these definitions, the coupled waves in the piezoelectric or piezomagnetic solid can be uniformly expressed as

1 1 1 0 c1133 r031 nq G1q r032 nq G2q þ r31 þ c3333 nq G3q 2 2 2 þ e333 Guq nq þ q333 Gwq nq ;

J q ¼ e311 G1q þ e333 G3q nq e33 Guq nq ; Lq ¼ q311 G1q þ q333 G3q nq l33 Gwq nq : It should be pointed out that the EP wave in the piezomagnetic solid is decoupled with all other waves, therefore,

n o u001q ; u002q ; u003q ; u00q ; w00q h i n o 00 ¼ G001q ; G002q ; G003q ; G00uq ; G00wq U00q exp ik1 x001 þ n00q x003 c00 t h i 00 ¼ f0; 0; 0; 1; 0gU00q exp ik1 x001 þ n00q x003 c00 t n

r^ 0013q ; r^ 0023q ; r^ 0033q ; D^00 3q ; B^00 3q

ð22aÞ

o

h i n o 00 00 ¼ ik1 H001q ; H002q ; H003q ; J 00q ; L00q U00q exp ik1 x001 þ n00q x003 c00 t n o h i 00 00 ¼ ik1 0; 0; 0; e0033 n00q ; 0 U00q exp ik1 x001 þ n00q x003 c00 t

ð22bÞ

Please cite this article in press as: X. Guo, P. Wei, Dispersion relations of elastic waves in one-dimensional piezoelectric/piezomagnetic phononic crystal with initial stresses, Ultrasonics (2015), http://dx.doi.org/10.1016/j.ultras.2015.11.008

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X. Guo, P. Wei / Ultrasonics xxx (2015) xxx–xxx

where q = 7, 8 stands for forward and backward EP wave, respectively. Similarly, the MP wave in the piezoelectric solid is decoupled with all other waves,

n o u01q ; u02q ; u03q ; u0q ; w0q n o h i 0 ¼ G01q ; G02q ; G03q ; G0uq ; G0wq W0q exp ik1 x01 þ n0q x03 c0 t h i 0 ¼ f0; 0; 0; 0; 1gW0q exp ik1 x01 þ n0q x03 c0 t n

r^ 013q ; r^ 023q ; r^ 033q ; D^0 3q ; B^0 3q

0

V0Lnext x3 ¼ 0þ ; t ¼ T00 T0 VL x3 ¼ 0þ ; t Tcell ¼ T00 T0

ð30Þ

0

00

where T and T are the transfer matrix of slab A and slab B.

ð23aÞ

4. Without initial stresses r032 and r012 situation If shear initial stresses

2

0 ik1

ð23bÞ

where q = 9, 10 stands for forward and backward MP wave, respectively. For convenience of statement of the interface conditions and the periodic condition, we define the state vector of the physical fields as

3; B 3 T r 1 ; u 2 ; u 3 ; u 13 ; r 23 ; r 33 ; D ; w; Vðxi ; t Þ ¼ u

ð29Þ

Therefore, the transfer matrices of a single cell is

o

h i n o 0 H01q ; H02q ; H03q ; J 0q ; L0q W0q exp ik1 x01 þ n0q x03 c0 t ¼ h i n o 0 0 ¼ ik1 0; 0; 0; 0; l033 n0q W0q exp ik1 x01 þ n0q x03 c0 t

ð24Þ

P10 P10 P10 P10 1 ¼ where u q¼1 u1q ; u2 ¼ q¼1 u2q ; u3 ¼ q¼1 u3q ; u ¼ q¼1 uq ; P P P P 10 10 10 10 3 ¼ ^ ^ ^ w ¼ q¼1 wq ; r13 ¼ q¼1 r13q ; r23 ¼ q¼1 r23q ; r33 ¼ q¼1 r33q ; D P10 ^ P10 ^ q¼1 D3q ; B3 ¼ q¼1 B3q . The state vectors at the left and at the right boundaries of slab with thickness d can be expressed as

V0L x03 ¼ 0þ ; t ¼ T0L U 011 ; U 012 ; U 013 ; U 014 ; U 015 ; U 016 ; U 017 ; U 018 ; W09 ; W0A 0 exp ik1 x01 c0 t ð25aÞ

W 11 6 6 0 26 k1 6 6 W 31 6 4 W 41 W 51

r032 ¼ r012 ¼ 0, Eq. (17) reduce to

0

W 13

W 14

W 22

0

0

0

W 33

W 34

0

W 43

W 44

0

W 53

0

3 8 9 8 9 U1 > > 0> > > > > > > > > > > > > 7 > > > 0 7 > U 0> > > > 2> < = < = 7 W 35 7 7 > U3 > ¼ > 0 > > > > > 7 > > > 0 5 > U> > > > > >0> > > : > ; > : > ; W 55 W 0 W 15

It is noticed from Eq. (31) that u2 is independent of u1 ; u3 ; u and w which means that QSH wave reduce to SH wave and is decoupled with all other waves. For in-plane elastic wave, the amplitude ratio ^ 23 ¼ 0. For SH wave, G2 ¼ 0 and effective stress component r

n o u1q ; u2q ; u3q ; uq ; wq ¼ f0; 1; 0; 0; 0gU 2q exp ik1 x1 þ nq x3 ct

ð32aÞ n

o

where H2q ¼ r þ c2323 þ Define the state vector 0 31

3 4

r r 0 33

1 4

0 22

for the piezoelectric solids.

2 ; r 23 gT V ð xi ; t Þ ¼ f u

VL x003 ¼ 0þ ; t ¼ T00L U 0011 ; U 0012 ; U 0013 ; U 0014 ; U 0015 ; U 0016 ; U007 ; U008 ; U 0019 ; U 001A 00 00 exp ik1 x1 c00 t ð25cÞ

00 V00R x003 ¼ d ; t ¼ T00R U 0011 ; U 0012 ; U 0013 ; U 0014 ; U 0015 ; U 0016 ; U007 ; U008 ; U 0019 ; U 001A 00 00 exp ik1 x1 c00 t ð25dÞ for the piezomagnetic solids considered. Wherein, the subscript ‘‘A” denotes ‘‘10”. The explicit expressions of TL and TR are given in Appendix B. The state vectors at the left and at the right boundaries of a slab can be related by the transfer matrix of this slab, namely,

VR ðx3 ¼ d ; t Þ ¼ TVL x3 ¼ 0þ ; t

ð26Þ

where T ij is the transfer matrix of slabs. The transfer matrix is a square matrix of ten order. Inserting Eq. (25) into Eq. (26), the transfer matrices of slab can be obtained by

T ¼ TR T1 L

ð27Þ

For the perfect interface, all physical fields, namely, the displacement components, the effective traction components, the electric potential, the magnetic potential, the normal components of effective electric displacement and effective magnetic induction, are continuous across interface. The interface condition can be expressed as

00 ¼ w 0; 0i ; u 00 ¼ u 0; w u00 i ¼ u 00 0 ; ði ¼ 1; 2; 3Þ ¼B B 3 3

r 00i3 ¼ r 0i3 ; D 003 ¼ D 03 ; ð28Þ

For a single cell composed of slab A and slab B, the state vectors at the left and right boundaries of the single cell are related by

ð32bÞ 3 4

3; B 3 T r 13 ; r 33 ; D 1 ; u 3 ; u ; w; V ð xi ; t Þ ¼ u

r^ 13q ; r^ 23q ; r^ 33q ; D^ 3q ; B^3q ¼ ik1 0; H2q ; 0; 0; 0 U 2q exp ik1 x1 þ nq x3 ct

0 V0R x03 ¼ d ; t ¼ T0R U 011 ; U 012 ; U 013 ; U 014 ; U 015 ; U 016 ; U 017 ; U 018 ; W09 ; W0A 0 exp ik1 x01 c0 t ð25bÞ 00

ð31Þ

nq .

ð33aÞ

for in-plane elastic waves, and

ð33bÞ P7;8;9;10

1 ¼ q¼1;2;3;4 u1q ; u 2 ¼ anti-plane elastic waves, where u P7;8;9;10 P7;8;9;10 P7;8;9;10 q¼5 u2q ; u3 ¼ q¼1;2;3;4 u3q ; u ¼ q¼1;2;3;4 uq ; w ¼ q¼1;2;3;4 wq ; r13 ¼ P6 P7;8;9;10 P7;8;9;10 3 ¼ P7;8;9;10 D ^ ^ ^ ^ r ; r ¼ r ; r ¼ r ; D 13q 23 23q 33 33q q¼1;2;3;4 q¼5 q¼1;2;3;4 q¼1;2;3;4 3q ; P 7;8;9;10 3 ¼ ^ B q¼1;2;3;4 B3q . The state vectors at the left and at the right boundaries of slab with thickness d can be expressed as 0

V0L x03 ¼ 0þ ; t ¼ T0L U 011 ; U 012 ; U 013 ; U 014 ; U 017 ; U 018 ; W09 ; W0A exp ik1 x01 c0 t for P6

ð34aÞ 0

0 V0R x03 ¼ d ; t ¼ T0R U 011 ; U 012 ; U 013 ; U 014 ; U 017 ; U 018 ; W09 ; W0A exp ik1 x01 c0 t ð34bÞ

for the piezoelectric solids, and 00

V00L x003 ¼ 0þ ; t ¼ T00L U 0011 ; U 0012 ; U 0013 ; U 0014 ; U007 ; U008 ; U 0019 ; U 001A exp ik1 x001 c00 t ð35aÞ 00

00

00

00

VR x003 ¼ d ; t ¼ TR U 0011 ; U 0012 ; U 0013 ; U 0014 ; U007 ; U008 ; U 0019 ; U 001A exp ik1 x001 c00 t

ð35bÞ

for the piezomagnetic solids for in-plane elastic waves.

VL x3 ¼ 0þ ; t ¼ TL fU 25 ; U 26 g exp ½ik1 ðx1 ct Þ

VR ½x3 ¼ d ; t ¼ TR fU 25 ; U 26 g exp ½ik1 ðx1 ct Þ

ð36aÞ ð36bÞ

for anti-plane elastic waves. The explicit expressions of TL and TR are given in Appendix C. The transfer matrices of slab and a single cell can also be expressed as Eqs. (27) and (30). Because the state vector Vðxi ; tÞ is different for in-plane elastic waves and the anti-plane waves,

Please cite this article in press as: X. Guo, P. Wei, Dispersion relations of elastic waves in one-dimensional piezoelectric/piezomagnetic phononic crystal with initial stresses, Ultrasonics (2015), http://dx.doi.org/10.1016/j.ultras.2015.11.008

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X. Guo, P. Wei / Ultrasonics xxx (2015) xxx–xxx

the transfer matrices of a single cell Tcell is of 8 8 order for inplane elastic waves and 2 2 order for anti-plane elastic waves. 5. Normal propagation situation In the case of normal propagation situation (k1 ¼ 0), the displacement components in the piezoelectric slab are only the function of x3 and can be assumed as:

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

ð37Þ

8 @ 2 u1 1 0 @ 2 u2 1 0 @ 2 u3 @ 2 u1 3 0 1 0 > > > c3131 þ 4 r33 4 r11 @x23 4 r12 @x23 2 r31 @x23 ¼ q @t2 > > > 2 2 2 2 > > > 14 r012 @@xu21 þ c2323 þ 34 r033 14 r022 @@xu22 12 r032 @@xu23 ¼ q @@tu22 > > 3 3 3 > < 2 2 2 2 2 2 12 r031 @@xu21 12 r032 @@xu22 þ c3333 @@xu23 þ e333 @@xu2 þ q333 @@xw2 ¼ q @@tu23 > 3 3 3 3 3 > > 2 2 > > > e333 @@xu23 e33 @@xu2 ¼ 0 > > 3 3 > > > 2 2 > : q333 @ u23 l33 @ w2 ¼ 0

W 13

W 22

W 23

0 0

W 32

W 33

W 34

0

W 43

W 44

0

W 53

0

ð39Þ

ð40Þ

r^ 13q ; r^ 23q ; r^ 33q ; D^ 3q ; B^3q ¼ ik3q H1q ; H2q ; H3q ; Jq ; Lq U 1q

exp ik3q x3 v q t where

H2q H3q

00 k3A

where ¼ ¼ 1 and q = 9, 10 stands for forward and backward MA wave, respectively. EP wave can be expressed as

o

n

ð44aÞ

o

00 k38

n o n o 0 u01q ; u02q ; u03q ; u0q ; w0q ¼ 0; 0; 0; 0; 1 þ k3q x03 W0q exp ðix0 tÞ n

o

n

ð41Þ

c3131 þ

ð45aÞ

o

r^ 013q ; r^ 023q ; r^ 033q ; D^0 3q ; B^0 3q ¼ 0; 0; 0; 0; l033 k03q W0q exp ðix0 tÞ ð45bÞ 0 k39

0 k3A

where ¼ ¼ 1 and q = 9, 10 stands for forward and backward MP wave, respectively. It is noted from Eq. (41)–(45) that EA wave, MA wave, EP wave and MP wave become into the stationary waves in the normal propagation situation.

r032

In the case of only existence of shear initial stress r032 , Eqs. (38) and (39) reduce to

8 @ 2 u1 @ 2 u1 > > > c3131 @x23 ¼ q @t2 > > > > > c2323 @ 2 u2 1 r0 @ 2 u3 ¼ q @2 u2 > > 2 32 @x2 @x23 @t 2 > 3 > < 2 2 2 2 @ 2 u3 1 0 @ u2 2 r32 @x2 þ c3333 @x2 þ e333 @@xu2 þ q333 @@xw2 ¼ q @@tu23 3 3 3 3 > > > 2 2 > > > e @ u3 e33 @@xu2 ¼ 0 > > 333 @x23 3 > > > 2 2 > : q333 @ u23 l33 @ w2 ¼ 0 @x3

3 0 1 1 1 r33 r011 G1q r012 G2q r031 G3q ; 4 4 4 2 1 0 3 0 1 0 1 ¼ r12 G1q þ c2323 þ r33 r22 G2q r032 G3q ; 4 4 4 2 1 0 1 0 ¼ r31 G1q r32 G2q þ c3333 G3q þ e333 Guq þ q333 Gwq ; 2 2

H1q ¼

ð43bÞ 00 k39

5.1. Only existence of shear initial stress

n o

u1q ; u2q ; u3q ; uq ; wq ¼ 1; G2q ; G3q ; Guq ; Gwq U 1q exp ik3q x3 v q t

o

where ¼ ¼ 1 and q = 7, 8 stands for forward and backward EP wave, respectively. MP wave can be expressed as

W 32 ¼ 12 r032 ,W 33 ¼ c3333 qv 2 , W 34 ¼ W 43 ¼ e333 , W 35 ¼ W 53 ¼ q333 , W 44 ¼ e33 , W 55 ¼ l33 . These coupled waves can be expressed as

o

n

ð44bÞ

where W 11 ¼ c3131 þ 34 r033 14 r011 qv 2 , W 12 ¼ W 21 ¼ 14 r012 , 1 0 3 0 1 0 W 22 ¼ c2323 þ 4 r33 4 r22 qv 2 , W 23 ¼ W 13 ¼ W 31 ¼ 2 r31 ,

n

o

r^ 0013q ; r^ 0023q ; r^ 0033q ; D^00 3q ; B^00 3q ¼ 0; 0; q00333 k003q ; 0; l0033 k003q W00q exp ðix00 tÞ

00 k37

It is noted that QP wave, QSV wave and QSH wave are coupled together even in the normal propagation situation due to the existence of initial stresses. The dispersive equations of various coupled waves can be obtained by requiring the coefficient determinant equal to zero,

W 12

n

ð43aÞ

r^ 0013q ; r^ 0023q ; r^ 0033q ; D^00 3q ; B^00 3q ¼ 0; 0; 0; e0033 k003q ; 0 U00q exp ðix00 tÞ

@x3

0

0

W 35

¼ 0

0

W 55

n o n o 00 u001q ; u002q ; u003q ; u00q ; w00q ¼ 0; 0; 0; 0; 1 þ k3q x003 W00q exp ðix00 t Þ

n

ð38Þ

W 11

W 21

f ðk3 ; xÞ ¼

W 31

0

0

0

o n o n 00 u001q ; u002q ; u003q ; u00q ; w00q ¼ 0; 0; 0; 1 þ k3q x003 ; 0 U00q exp ðix00 t Þ

Inserting Eq. (37) into Eq. (8) leads to

@x3

0

where k37 ¼ k38 ¼ 1 and q = 7, 8 stands for forward and backward EA wave, respectively. MA wave can be expressed as

@x3

and

W 11

0

f ðk3 ; xÞ ¼

0

0

0

ð46Þ

0

0

0

W 22

W 23

0

W 32

W 33

W 34

0 0

W 43 W 53

W 44 0

0

0

W 35

¼ 0

0

W 55

ð47Þ

where W 11 ¼ c3131 qv 2 , W 22 ¼ c2323 qv 2 , W 23 ¼ W 32 ¼ 12 r032 ,

(=1, 2, . . . , 6) stands for various coupled elastic waves. EA wave can be expressed as

W 34 ¼ W 43 ¼ e333 , W 35 ¼ W 53 ¼ q333 , W 33 ¼ c3333 qv 2 , W 44 ¼ e33 , W 55 ¼ l33 . It is noticed from Eq. (46) that u1 is independent of u2 ; u3 ; u and w, namely, QSV wave reduced to SV wave and is decoupled with all other waves. SV wave can be expressed as

ð42aÞ

ð48aÞ

J q ¼ e333 G3q e33 Guq ; Lq ¼ q333 G3q l33 Gwq ; k3q ¼ v q x

and

o n o n 0 u01q ; u02q ; u03q ; u0q ; w0q ¼ 0; 0; 0; 1 þ k3q x03 ; 0 U0q exp ðix0 tÞ n

o

n

q

o

r^ 013q ; r^ 023q ; r^ 033q ; D^0 3q ; B^0 3q ¼ 0; 0; e0333 k03q ;e033 k03q ;0 U0q exp ðix0 tÞ ð42bÞ

n o u1q ;u2q ; u3q ; uq ; wq ¼ f1; 0;0;0; 0gU 1q exp ik3q x3 v q t n

o

r^ 13q ; r^ 23q ; r^ 33q ; D^ 3q ; B^3q ¼ ik3q fc3131 ; 0; 0; 0; 0gU 1q exp ik3q x3 v q t

ð48bÞ

Please cite this article in press as: X. Guo, P. Wei, Dispersion relations of elastic waves in one-dimensional piezoelectric/piezomagnetic phononic crystal with initial stresses, Ultrasonics (2015), http://dx.doi.org/10.1016/j.ultras.2015.11.008

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X. Guo, P. Wei / Ultrasonics xxx (2015) xxx–xxx

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where v 3 ¼ v 4 ¼ c3131 =q; k33 ¼ k34 ¼ x q=c3131 and q = 3, 4 stands for forward and backward SV waves, respectively. All other coupled waves can be expressed as

n o

u1q ; u2q ; u3q ; uq ; wq ¼ 0; 1; G3q ; Guq ; Gwq U 2q exp ik3q x3 v q t ð49aÞ n

o

r^ 13q ; r^ 23q ; r^ 33q ; D^ 3q ; B^3q ¼ ik3q 0; H2q ; H3q ; Jq ; Lq U 2q exp ik3q x3 v q t

ð49bÞ

where H2q ¼ c2323 12 r032 G3q ; H3q ¼ 12 r032 þ c3333 G3q þ e333 Guq þ q333 Gwq , J q ¼ e333 G3q e33 Guq , Lq ¼ q333 G3q l33 Gwq , k3q ¼ vxq and q = 1, 2, 5, 6 stands for different coupled elastic waves, respectively. 5.2. Only existence of shear initial stress

r031

In the case of only existence of shear initial stress r031 , Eqs. (38) and (39) reduce to

8 2 2 2 > c3131 @@xu21 12 r031 @@xu23 ¼ q @@tu21 > > > 3 3 > > 2 2 > > > c2323 @@xu22 ¼ q @@tu22 > > 3 > < 2 2 2 2 2 12 r031 @@xu21 þ c3333 @@xu23 þ e333 @@xu2 þ q333 @@xw2 ¼ q @@tu23 > 3 3 3 3 > > 2 2 > > > e333 @@xu23 e33 @@xu2 ¼ 0 > > 3 3 > > > 2 > : q333 @ u23 l33 @ 2 w2 ¼ 0 @x3

ð50Þ

@x3

In the case of only existence of shear initial stress r012 , Eqs. (38) and (39) reduce to

8 2 2 2 > c3131 @@xu21 14 r012 @@xu22 ¼ q @@tu21 > > > 3 3 > > 2 2 2 > > > 14 r012 @@xu21 þ c2323 @@xu22 ¼ q @@tu22 > > 3 3 > < 2 2 2 2 c3333 @@xu23 þ e333 @@xu2 þ q333 @@xw2 ¼ q @@tu23 > 3 3 3 > > 2 2 > > > e @ u3 e33 @@xu2 ¼ 0 > > 333 @x23 3 > > > 2 > : q333 @ u23 l33 @2 w2 ¼ 0 @x3

@x3

and

W 11

W 12

f ðv Þ ¼

0

0

0

ð54Þ

W 12

0

0

W 22

0

0

0

W 33

W 34

0

W 43

W 44

0

W 53

0

0

0

W 35

¼ 0

0

W 55

ð55Þ

where W 11 ¼ c3131 qv , W 12 ¼ W 21 ¼ 14 r012 , W 22 ¼ c2323 qv 2 , W 33 ¼ c3333 qv 2 W 34 ¼ W 43 ¼ e333 , W 35 ¼ W 53 ¼ q333 , W 44 ¼ e33 , W 55 ¼ l33 . It is noticed that u1 and u2 coupled together while u3 is coupled with u and w. Coupled QSV and QSH waves can be expressed as n o

u1q ; u2q ; u3q ; uq ; wq ¼ 1; G2q ; 0; 0; 0 U 1q exp ik3q x3 v q t ð56aÞ

n

o

r^ 13q ; r^ 23q ; r^ 33q ; D^ 3q ; B^3q ¼ ik3q H1q ; H2q ; 0; 0;0 U 1q exp ik3q x3 v q t

0

W 13

0

W 22

0

0

0

W 33

W 34

0

W 43

W 44

0

W 53

0

ð56bÞ

0

0

W 35

¼ 0

0

W 55

where G2q ¼

ð51Þ

4ðc3131 qv

r012

2 q

Þ

; H1q ¼ c3131 14 r012 G2q ; H2q ¼ 14 r012 þ c2323 G2q ,

k3q ¼ vxq and q = 3, 4, 5, 6 stands for forward and backward QSV and QSH waves, respectively. P wave can be expressed as n o

u1q ; u2q ; u3q ; uq ; wq ¼ 0; 0; 1; Gu ; Gw U 3q exp ik3q x3 v q t

where W 11 ¼ c3131 qv ; W 13 ¼ W 31 ¼ r ¼ c2323 qv ; W 34 ¼ W 43 ¼ e333 ; W 35 ¼ W 53 ¼ q333 ; W 44 ¼ e33 ; W 55 ¼ l33 . It is noticed from Eq. (50) that u2 is independent of u1 ; u3 ; u and w which means that QSH wave reduces SH wave and is decoupled with all other waves. SH wave can be expressed as n o u1q ; u2q ; u3q ; uq ; wq ¼ f0; 1; 0; 0; 0gU 2q exp ik3q x3 v q t ð52aÞ 12

2

n

r012

2

and

W 11

0

f ðv Þ ¼

W 31

0

0

5.3. Only existence of shear initial stress

0 31 ; W 22

o

2

r^ 13q ; r^ 23q ; r^ 33q ; D^ 3q ; B^3q ¼ ik3q f0; c2323 ; 0; 0; 0gU 2q exp ik3q x3 v q t

n

o

ð57aÞ

r^ 13q ; r^ 23q ; r^ 33q ; D^ 3q ; B^3q ¼ ik3q f0; 0; H3 ; 0; 0gU 3q exp ik3q x3 v q t

ð57bÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ v 2 ¼ H3 =q

where v pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ and k31 ¼ k32 ¼ x q=H3 . q = 1, 2 stands for forward and backward P waves, respectively. l e2 þe33 q2333 Gu ¼ ee333 ; Gw ¼ ql333 ; H3 ¼ c3333 þ 33 333 ; 1 e33 l33 33 33

5.4. Only existence of initial normal stresses

ð52bÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where v 5 ¼ v 6 ¼ c2323 =q; k35 ¼ k36 ¼ x q=c2323 and q = 5, 6 stands for forward and backward SH waves, respectively. All other coupled waves can be expressed as

n o

u1q ; u2q ; u3q ; uq ; wq ¼ 1; 0; G3q ; Guq ; Gwq U 1q exp ik3q x3 v q t ð53aÞ n

o

r^ 13q ; r^ 23q ; r^ 33q ; D^ 3q ; B^3q ¼ ik3q H1q ;0;H3q ;Jq ;Lq U 1q exp ik3q x3 v q t

ð53bÞ

where H1q ¼ c3131 G1q 12 r031 G3q ; H3q ¼ 12 r031 G1q þ c3333 G3q þ e333 Guq þq333 Gwq , J q ¼ e333 G3q e33 Guq , Lq ¼ q333 G3q l33 Gwq , k3q ¼ vxq and q = 1, 2, 3, 4 stands for different coupled elastic waves, respectively.

In the case of only existence of normal initial stress, Eq. (38) reduces to

8 2 2 > c3131 þ 34 r033 14 r011 @@xu21 ¼ q @@tu21 > > > 3 > > 2 2 > > > c2323 þ 34 r033 14 r022 @@xu22 ¼ q @@tu22 > > 3 > < 2 2 2 2 c3333 @@xu23 þ e333 @@xu2 þ q333 @@xw2 ¼ q @@tu23 > 3 3 3 > > 2 2 > > > e @ u3 e33 @@xu2 ¼ 0 > > 333 @x23 3 > > > 2 > : q333 @ u23 l33 @2 w2 ¼ 0 @x3

ð58Þ

@x3

It is noted that displacement components u1 ; u2 and u3 are decoupled each other. Only displacement component u3 is coupled with u and w. which means that QP, QSV and QSH wave reduce to P, SV and SH wave, respectively.

Please cite this article in press as: X. Guo, P. Wei, Dispersion relations of elastic waves in one-dimensional piezoelectric/piezomagnetic phononic crystal with initial stresses, Ultrasonics (2015), http://dx.doi.org/10.1016/j.ultras.2015.11.008

8

X. Guo, P. Wei / Ultrasonics xxx (2015) xxx–xxx

6. Dispersion equation of Bloch waves

7. Numerical results and discussion

By using the transfer matrix of coupled waves in each slab, the state vectors at the left and the right boundaries of a typical single cell are related by

In this section, we discuss the dispersion curves of in-plane Bloch waves and anti-plane Bloch waves propagating in the onedimensional piezoelectric/piezomagnetic phononic crystal which is composed of a piezoelectric slab and a piezomagnetic slab repeating alternatively. The material constants of two solids, PZT-5J and CoFe2O4 [10], are listed in Table 1. The length ratio of two slabs 0 00 in a single cell are d =d ¼ 1 and the total length of single cell is 0 00 a ¼ d þ d . In order to investigate the influences of initial stresses, the dispersion curves are calculated for both situations with and without initial stresses and shown in the same figure to facilitate comparison. The dimensionless wavenumber in first Brilouin zone, namely, Ka=p 2 ½1; 1, is considered in calculation due to the periodical dependence of dispersion curves on wavenumber. The dimensionless angular frequencies are xa= 2pv 0P and xa= 2pv 0SH for in-plane Bloch waves and anti-plane Bloch waves rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ e02 and the respectively, where the P wave speed v 0P ¼ q10 c033 þ e15 0 33 qﬃﬃﬃﬃﬃ 0 c SH wave speed v 0SH ¼ q440 in the normal propagation situation.

h i 0 00 þ V0L x3 ¼ d þ d ; t ¼ T00 T0 V0L x3 ¼ 0þ ; t

ð59Þ

According to Bloch theorem of elastic waves through periodical structure,

h i 0 0 00 þ 00 V0L x3 ¼ d þ d ; t ¼ exp iK d þ d V0L x3 ¼ 0þ ; t

ð60Þ

where K is the Bloch wave vector. Inserting Eq. (59) into Eq. (60) leads to

0 00 0 Tcell ðc; xÞ I exp iK d þ d VL x3 ¼ 0þ ; t ¼ 0

ð61Þ

where Tcell ðc; xÞ ¼ T00 T0 is the transfer matrix of one single cell. The condition of existing non-trivial solution leads to

Tcell ðc; xÞ I exp iK d0 þ d00

0

00

¼ f ðx; K; c; d ; d ; r0ij ; cijkl ; Þ ¼ 0

ð62Þ

where I is a unit matrix. Eq. (62) gives the dispersive relation of Bloch waves in the one-dimensional piezoelectric/piezomagnetic phononic crystal. After dimensional analysis, more than one non 0 00 dimensional frequency x d þ d =ð2pv Þ can be obtained from Eq. 0 00 (62) for given non-dimensional wavenumber K d þ d =p and the 0

thickness ratio dd00 in the oblique and normal propagation situations when the materials of two slabs are prescribed. Therefore, a set of normalized dispersive curves is obtained in considered frequency range. For the normal propagation situation and only existence of normal initial stresses, Eq. (62) can be explicitly expressed as

0 0 00 00 cos K d þ d ¼ cos xd =v 0s cos xd =v 00s 0 00 1 H0 v 00s H00 v 0s sin xd =v 0s sin xd =v 00s þ 2 H00 v 0s H0 v 00s ð63Þ 33 333 where s = P, H ¼ c3333 þ 33 333 and vP ¼ e33 l33 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ l33 e2333 þe33 q2333 1 for Bloch P wave; similarly, s = SV, q c3333 þ e33 l33 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ H ¼ c3131 þ 34 r033 14 r011 and v SV ¼ q1 c3131 þ 34 r033 14 r011

l e2 þe q2

for

v SH

Bloch SV wave; s = SH, H ¼ c2323 þ 34 r033 14 r022 and qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 1 3 0 1 0 ¼ q c2323 þ 4 r33 4 r22 for Bloch SH wave. If the initial

stresses are not taken into consideration and the piezomagnetic slab is replaced by an isotropic slab, then, Eq. (63) reduces to Eq. (25) in Qian et al. [20] and reduces to Eq. (24) in Lan and Wei [11]. Further, if these two slabs are both replaced by two isotropic slabs, Eq. (63) reduces to Eq. (22) in Zheng and Wei [27].

Apart from the normal initial stresses, namely, r011 ; r022 and r033 , the shear initial stresses, r031 ; r032 and r012 , are also considered in the numerical calculations but it is assumed that D0i ¼ B0i ¼ 0. 7.1. Influences of normal initial stresses The influences of the initial normal compressive stress r011 on the dispersion curves of in-plane and anti-plane Bloch waves are shown in Figs. 2 and 3, respectively. It is observed that the existence of compressive stress r011 make the dispersion curves of in-plane Bloch waves shifting towards the high frequency while the dispersion curves of anti-plane Bloch waves shifting toward the low frequency. Greater is the absolute value of the normal initial compressive stress r011 , more evident the dispersive curves shift. This can be explained by that the initial normal compressive stress r011 makes the in-plane rigidity of phononic crystal enhanced while the anti-plane rigidity degraded. It is also observed that the initial compressive stress r011 has more evident influences on the QSV wave than on the QP wave. As the apparent wave speed increases, QSV wave decouples with QP wave gradually and the influences of the initial normal stress r011 on the QP wave and SH wave decrease gradually, see Figs. 2(a)–(e) and 3(a) and (b) respectively. In the normal propagation situation, SV wave is decoupled with P wave completely and the initial normal stress r011 has only influences on Bloch SV wave, see Figs. 2(f) and 3(c). This observation can be explained by the fact that the initial normal stress r011 is only coupled with the displacement component u1 of Bloch SV wave in the normal propagation situation, see Eq. (58). In other word, only the Bloch SV wave is affected by the initial stress r011 while the Bloch P wave and the Bloch SH wave are not affected by the initial stress r011 in the normal propagation situation. Different from the initial normal stress r011 , the initial normal stress r022 only

Table 1 Material constants of PZT-5J and CoFe2O4. Mat

Name

c11

c12

c13

c33

c44

q

e15

e31

A(PE) B(PM)

PZT-5J CoFe2O4

82.3 286

34.1 173

30.2 170.5

59.8 269.5

21.3 45.3

7500 5300

14.26 0

10.42 0

Mat A(PE) B(PM)

Name PZT-5J CoFe2O4

e33 16.58 0

e11

e33

q31 0 580.3

q33 0 699.7

l33

10.12 0.093

q15 0 550

l11

14.53 0.08

1.26 590

1.26 157

cij : GPa, q: kg m3, eij : C m2 , qij : N A1 m1 ,

eij : 109 C2 N1 m2 , lij : 106 N A2 .

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9

Fig. 2. Effects of the initial stress r011 (r022 ¼ r033 ¼ r032 ¼ r031 ¼ r012 ¼ 0) on dispersion curves of in-plane Bloch waves (The curves labeled by P in Fig.2f are Bloch P waves which are decoupled to Bloch SV waves in the normal propagation situation) (— r011 ¼ 0, r011 ¼ 5 GPa, r011 ¼ 8 GPa).

has influences on anti-plane Bloch waves, see Fig. 4. It is observed that the existence of the initial normal stress r022 makes anti-plane Bloch waves dispersion curves shifting toward the high frequency. Similar with the initial normal stress r011 , the initial normal stress r033 has influences on both in-plane Bloch wave and anti-plane

Bloch wave. The influences of the initial normal stress r033 on the dispersion curves of in-plane and anti-plane Bloch waves are shown in Figs. 5 and 6, respectively. It is observed that the existence of the initial normal compressive stress r033 makes both in-plane and anti-plane Bloch waves dispersion curves shifting

Fig. 3. Effects of the initial stress r011 (r022 ¼ r033 ¼ r032 ¼ r031 ¼ r012 ¼ 0) on dispersion curves of anti-plane Bloch waves which are decoupled to in-plane Bloch waves in both the oblique and normal propagation situations (— r011 ¼ 0, r011 ¼ 3 GPa, r011 ¼ 5 GPa).

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Fig. 4. Effects of the initial stress r022 (r011 ¼ r033 ¼ r032 ¼ r031 ¼ r012 ¼ 0) on dispersion curves of anti-plane Bloch waves (— r022 ¼ 0,

toward the low frequency. Moreover, the initial normal compressive stress r033 has different influences on the QSV wave and QP wave. As the apparent wave speed increases, the influences on QP wave decreases gradually and disappear completely in the normal propagation situation. However, the initial normal compressive stress r033 has more evident influences on the anti-plane

r022 ¼ 3 GPa,

r022 ¼ 5 GPa).

Bloch wave than the initial normal stress r011 and this can be understood easily by observing Eq. (58) which shows that the initial normal stress r033 is not only coupled with the displacement components u1 of Bloch SV wave but also the displacement component u2 of Bloch SH wave in the normal propagation situation. This different coupling results in the differences between Fig. 3c and c.

Fig. 5. Effects of the initial stress r033 (r011 ¼ r022 ¼ r032 ¼ r031 ¼ r012 ¼ 0) on dispersion curves of in-plane Bloch waves (The curves labeled by P in Fig.5f are Bloch P waves which are decoupled to Bloch SV waves in the normal propagation situation) (— r033 ¼ 0, r033 ¼ 3 GPa, r033 ¼ 5 GPa).

Please cite this article in press as: X. Guo, P. Wei, Dispersion relations of elastic waves in one-dimensional piezoelectric/piezomagnetic phononic crystal with initial stresses, Ultrasonics (2015), http://dx.doi.org/10.1016/j.ultras.2015.11.008

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X. Guo, P. Wei / Ultrasonics xxx (2015) xxx–xxx

Fig. 6. Effects of the initial stress r033 (r011 ¼ r022 ¼ r032 ¼ r031 ¼ r012 ¼ 0) on dispersion curves of anti-plane Bloch waves (— r033 ¼ 0,

No matter that the initial normal stress is r011 or r033 , their influences on the high frequency dispersive curves are more evident than on the low frequency dispersive curves. This implies that the dispersive curves at high frequency are more sensitive to the initial stress than the dispersive curves at low frequency. This

r033 ¼ 3 GPa,

r033 ¼ 5 GPa).

phenomenon can be explained by the high frequency oscillating nature of the cosine and sine functions involved in the dispersive equation. This can be understood easily by observing the explicit expression of the dispersive equation in the normal propagation situation, i.e. Eq. (63).

Fig. 7. Effects of the initial stress r031 (r011 ¼ r022 ¼ r033 ¼ r032 ¼ r012 ¼ 0) on dispersion curves of in-plane Bloch waves (The curves labeled by P in Fig.7f are Bloch P waves which are decoupled to Bloch SV waves in the normal propagation situation) (— r031 ¼ 0, r031 ¼ 3 GPa, r031 ¼ 5 GPa).

Please cite this article in press as: X. Guo, P. Wei, Dispersion relations of elastic waves in one-dimensional piezoelectric/piezomagnetic phononic crystal with initial stresses, Ultrasonics (2015), http://dx.doi.org/10.1016/j.ultras.2015.11.008

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Fig. 8. Effects of the initial stress

X. Guo, P. Wei / Ultrasonics xxx (2015) xxx–xxx

r031 (r011 ¼ r022 ¼ r033 ¼ r032 ¼ r012 ¼ 0) on dispersion curves of anti-plane Bloch waves (— r031 ¼ 0,

7.2. Influences of shear initial stresses Figs. 7 and 8 show the influences of initial shear stress r031 on the dispersion curves of in-plane and anti-plane Bloch waves. Compared with initial normal stresses, the initial shear stress r031 has much smaller influences on the dispersion curves. Furthermore, as the increase of the apparent speed c, the influences of the initial shear stress r031 decreases gradually and eventually disappears completely in the normal propagation situation. Fig. 9 shows the influences of the initial shear stress r032 . In the case of existence of only initial shear stress r032 , QP wave, QSV wave and QSH wave are coupled together in the oblique propagation situation. As the increase of apparent wave speed, QSV wave is decoupled gradually with QP wave and QSH wave and eventually is decoupled completely in the normal propagation situation. But QP wave and QSH wave are still coupled together due to the existence of initial shear stress r032 . However, the influences of the initial shear stress r032 on the dispersive curves are nearly unnoticed in the low frequency range considered. Fig. 10 shows the influences of the initial shear stress r012 . In the case of existence of only initial shear stress r012 , QP wave, QSV wave and QSH wave are coupled together in the oblique propagation situation. As the increase of apparent wave speed, QP wave is

r031 ¼ 3 GPa,

r031 ¼ 5 GPa).

decoupled gradually with QSV wave and QSH wave and eventually is decoupled completely in the normal propagation situation. But QSV wave and QSH wave are still coupled together due to the existence of initial shear stress r012 . Different from the initial shear stress r032 , the initial shear stress r012 has influences on the coupled QSV wave and QSH wave while has nearly unnoticed influences on QP wave. Furthermore, the existences of the initial shear stress r012 makes the dispersive curves of QSV wave and QSH wave shifting toward to opposite direction, namely, one toward high frequency and the other toward low frequency. In order to give a reasonable explanation about this phenomenon, let us consider the following fact. When the initial shear stress r012 doesn’t exist, we can deduce the wave speeds of the decoupled SH and SV waves i.e. qﬃﬃﬃﬃﬃ v SV ¼ v SH ¼ cq44 , from Eq. (55) in the normal propagation situation. As the decoupled Bloch SV and SH waves share the same phase velocity, their dispersive curves coincide. When the influences of the initial shear stress r012 are taken into consideration, the wave speeds of the coupled QSH wave and QSV wave change qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ qﬃﬃﬃﬃﬃ ﬃ qﬃﬃﬃﬃﬃ into v 3 ¼ q1 c44 þ 14 r012 > cq44 and v 5 ¼ q1 c44 14 r012 < cq44 , respectively. This implies that the influences of r012 are opposite for the QSV wave and QSH wave. Thus, the phase velocities of coupled Bloch QSV wave and QSH wave deviates from each other

Fig. 9. Effects of the initial stress r032 (r011 ¼ r022 ¼ r033 ¼ r031 ¼ r012 ¼ 0) on dispersion curves of Bloch waves (QP wave, QSV wave and QSH wave are coupled together in the oblique propagation situation. SV wave is independent in the normal propagation situation) (— r032 ¼ 0, r032 ¼ 3 GPa, r032 ¼ 5 GPa).

Please cite this article in press as: X. Guo, P. Wei, Dispersion relations of elastic waves in one-dimensional piezoelectric/piezomagnetic phononic crystal with initial stresses, Ultrasonics (2015), http://dx.doi.org/10.1016/j.ultras.2015.11.008

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Fig. 10. Effects of the initial stress r012 (r011 ¼ r022 ¼ r033 ¼ r032 ¼ r031 ¼ 0) on dispersion curves of Bloch waves (QP wave, QSV wave and QSH wave are coupled together in the oblique propagation situation. P wave is independent in the normal propagation situation) (— r012 ¼ 0, r012 ¼ 3 GPa, r012 ¼ 5 GPa).

shear stresses r012 on dispersive curves exist not only in the oblique propagation situation but also in the normal propagation situation.

and their dispersive curves shift along opposite direction as shown in Fig. 10(c). 8. Conclusions The influences of initial stresses on dispersive curves of Bloch waves in the one-dimensional piezoelectric/piezomagnetic periodical structure are investigated in the present work. The initial stresses considered in the present work includes initial normal stresses, r011 ; r022 and r033 , and initial shear stress, r031 ; r032 and r012 . In-plane Bloch waves and anti-plane Bloch waves in the oblique propagation and in the normal propagation are both considered. From the numerical results of dispersion curves, the following conclusions can be drawn: (1) No matter that it is in-plane Bloch waves or anti-plane Bloch waves, the initial normal stresses have more evident influences than initial shear stresses. Moreover, the influences of initial stresses, no matter that it is normal or shear initial stresses, are more evident on dispersion curves at high frequencies than that at low frequencies. (2) Both initial normal stresses r011 and r033 have influences on in-plane Bloch waves but the influences of r033 are more evident than that of r011 . Moreover, the influences of r033 and r011 on the in-plane Bloch waves are opposite. r022 has no any influences on the in-plane Bloch wave. All initial normal stresses have influences on the dispersive curves of antiplane Bloch wave. But r022 has the opposite influences compared with other two initial normal stresses. (3) In the normal propagation situation, QP wave, QSV wave and QSH wave are decoupled each other even if the initial normal stress exists. However, the situations are different when the initial shear stresses exist. The initial shear stress r031 makes QP wave and QSV wave coupled together; The initial shear stress r032 makes QP wave and QSH wave coupled together; the initial shear stress r012 makes QSV wave and QSH wave coupled together. (4) The influences of the initial shear stresses r031 and r032 on the dispersive curves decrease gradually as the apparent wave speed increases and are nearly unnoticed in the normal propagation situation. However, the influences of initial

Acknowledgments The work is supported by the National Natural Science Foundation of China (No. 10972029) and Opening fund of State Key Laboratory of Nonlinear Mechanics (LNM). Appendix A The physical constants of piezoelectric slab and piezomagnetic slab are 2

c1111 6c 6 2211 6 6 c3311 6 6 6 c2311 6 6 4 c3111

c2222 c2233 c2223 c2231 c3322 c3333 c3323 c3331 c2322 c2333 c2323 c2331 c3122 c3133 c3123 c3131

3

2 c11 c12 c13 0 0 7 c2212 7 6 0 6 c12 c11 c13 0 7 6 6 c3312 7 0 7 6 c13 c13 c33 0 7¼6 6 0 0 c44 0 c2312 7 7 6 0 7 6 0 0 0 c44 c3112 5 4 0

c1122 c1133 c1123 c1131 c1112

c1211 c1222 c1233 c1223 c1231 c1212

where c66 ¼

1 ðc11 2

0

0

0

0

0

0

3

0 7 7 7 0 7 7 7 0 7 7 7 0 5 c66

c12 Þ; cijmn ¼ cijmn ¼ cijnm ¼ cmnij ;

2

3 2 3 0 0 0 0 e15 0 e111 e122 e133 e123 e131 e112 6 7 6 7 6 e211 e222 e233 e223 e231 e212 7 ¼ 6 0 0 0 e15 0 0 7 4 5 4 5 e311 e322 e333 e323 e331 e312 0 0 e31 e31 e33 0

where emij ¼ emji ; 3 3 2 0 0 0 0 q15 0 q111 q122 q133 q123 q131 q112 7 6 7 6 6 q211 q222 q233 q223 q231 q212 7 ¼ 6 0 0 0 q15 0 0 7 5 4 5 4 q31 q31 q33 0 q311 q322 q333 q323 q331 q312 0 0 2

where qmij ¼ qmji ;

2

e11 e12 e13 3 2 e11 0 0 3 6 7 e22 e23 7 5 ¼ 4 0 e11 0 5 e31 e32 e33 0 0 e33

6 4 e21

where

emn ¼ enm ;

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2

X. Guo, P. Wei / Ultrasonics xxx (2015) xxx–xxx

3

2

3

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

6 4 l21

where

lmn ¼ lnm .

Appendix B The explicit expressions of elements of matrix T L and T R in Eq. (25) are

T L1p ¼ G1q ;

T L2p ¼ G2q ;

T L6p ¼ ik1 H1q ;

T L3p ¼ G3q ;

T L7p ¼ ik1 H2q ;

T L4p ¼ Guq ;

T L8p ¼ ik1 H3q ;

T L5p ¼ Gwq ;

T L9p ¼ ik1 J q ;

T L10p ¼ ik1 Lq ;

T R1p ¼ G1q exp ik1 nq d ; T R2p ¼ G2q exp ik1 nq d ; T R3p ¼ G3q exp ik1 nq d ; T R4p ¼ Guq exp ik1 nq d ; T R5p ¼ Gwq exp ik1 nq d ; T R6p ¼ ik1 H1q exp ik1 nq d ; T R7p ¼ ik1 H2q exp ik1 nq d ; T R8p ¼ ik1 H3q exp ik1 nq d ; T R9p ¼ ik1 J q exp ik1 nq d ; T R10p ¼ ik1 Lq exp ik1 nq d ; where p ¼ q ¼ 1; 2; . . . ; 10. Appendix C The explicit expressions of matrix T L and T R for in-plane elastic waves in Eq. (35) are

T L1p ¼ G1q ;

T L2p ¼ G3q ;

T L3p ¼ Guq ;

T L4p ¼ Gwq ;

T L5p ¼ ik1 H1q ; T R1p T R3p T R4p T R6p T R7p

T L6p ¼ ik1 H3q ; T L7p ¼ ik1 J q ; T L8p ¼ ik1 Lq ; ¼ G1q exp ik1 nq d ; T R2p ¼ G3q exp ik1 nq d ; ¼ Guq exp ik1 nq d ; ¼ Gwq exp ik1 nq d ; T R5p ¼ ik1 H1q exp ik1 nq d ; ¼ ik1 H3q exp ik1 nq d ; ¼ ik1 J q exp ik1 nq d ; T R8p ¼ ik1 Lq exp ik1 nq d ;

where q ¼ p when p ¼ 1; 2; 3; 4; q ¼ 7 when p ¼ 5; q ¼ 8 when p ¼ 6; q ¼ 9 when p ¼ 7 and q ¼ 10 when p ¼ 8. The explicit expressions of matrix T L and T R for anti-plane elastic waves in Eq. (36) are 1 1 exp ðik1 n5 dÞ exp ðik1 n6 dÞ : ; TR ¼ TL ¼ ik1 H25 ik1 H26 ik1 H25 exp ðik1 n5 dÞ ik1 H26 exp ðik1 n6 dÞ

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Please cite this article in press as: X. Guo, P. Wei, Dispersion relations of elastic waves in one-dimensional piezoelectric/piezomagnetic phononic crystal with initial stresses, Ultrasonics (2015), http://dx.doi.org/10.1016/j.ultras.2015.11.008

piezomagnetic phononic crystal with initial stresses

piezomagnetic phononic crystal with initial stresses

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