piezomagnetic cylinders

Accepted Manuscript

Effects of functionally graded interlayers on dispersion relations of shear horizontal waves in layered piezoelectric/piezomagnetic cylinders Xiao Guo , Peijun Wei , Li Li , Man Lan PII: DOI: Reference:

S0307-904X(17)30720-5 10.1016/j.apm.2017.11.029 APM 12071

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

13 July 2017 1 November 2017 15 November 2017

Please cite this article as: Xiao Guo , Peijun Wei , Li Li , Man Lan , Effects of functionally graded interlayers on dispersion relations of shear horizontal waves in layered piezoelectric/piezomagnetic cylinders, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.11.029

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Highlights Dispersive relations of SH guide waves in layered PE/PM cylinders are studied.

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Seven kinds of gradient profiles of graded interlayers are considered.

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Four kinds of mechanical and electromagnetic surfaces are considered.

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Direct integration of graded interlayer is used instead of layer-wise homogenization.

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The radial variation of vibration form for low and high order modes is studied.

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Effects of functionally graded interlayers on dispersion relations of shear horizontal waves in layered piezoelectric/piezomagnetic cylinders

a

Department of Applied Mechanics, University of Science and Technology Beijing, Beijing, 100083, China

b

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Xiao Guoa,b, Peijun Weia,e, Li Lic, Man Land

Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China

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Department of Mathematics, Qiqihar University, Qiqihar, 161006, China

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c

Department of Mathematics and Science, Luoyang Institute of Science and Technology, Luoyang, 471000, China e

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Corresponding author. E-mail: [email protected]

Abstract The effects of functionally graded interlayers on dispersion relations of shear

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horizontal waves in layered piezoelectric/piezomagnetic cylinders are studied. First, the basic physical quantities of elastic waves in piezoelectric cylinder are derived by

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assuming that the SH waves propagate along the circumferential direction steadily. Then the transfer matrices of the functional graded interlayer and outer piezomagnetic

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cylinder are obtained by solving the state transfer equations with spatial-varying coefficients. Furthermore, making use of the electro-magnetic surface conditions of the

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outer cylinder, the dispersion relations for the shear horizontal waves in layered piezoelectric/piezomagnetic cylinders are obtained and the numerical results are shown graphically. Seven kinds of functionally graded interlayers and four kinds of electro-magnetic surface conditions are considered. It is found that the functionally graded interlayers have evident influences on the dispersion relations of shear horizontal waves in layered piezoelectric/piezomagnetic cylinders. The high order modes are more sensitive to the gradient interlayers while the low order modes are more sensitive to the 2

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electro-magnetic surface conditions. Keyword: Functionally Graded Interlayer, Piezoelectric, Piezomagnetic, Layered Cylinder, SH Wave, Dispersion Relations. 1. Introduction

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Piezoelectric/Piezomagnetic composites (or Magneto-electro-elastic materials) constituted by piezoelectric (PE) materials and piezomagnetic (PM) materials owning the abilities to convert the energies between mechanical and electro-magnetic fields have attracted attentions of many researchers in recent years. Liu et al discussed the propagation of Love waves in layered structures for two cases: a PM layer on a PE

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half-space and the reverse configuration to illustrate the variations of the phase and group velocities versus the wavenumber for the combinations of different materials [1]. Pang et al analyzed the reflection and refraction of a plane wave incidence obliquely at the interface between PE and PM media and showed that the coupled QP and QSV

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waves, evanescent electroacoustic (EA), magnetic potential (MP), magneto-acoustic (MA) and electric potential (EP) waves construct the reflected and transmitted wave

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fields in the PE/PM medium, in which the EA, MA, MP and EP waves propagate along the interface [2]. Feng et al investigates Rayleigh waves in hexagonal (6mm) symmetry

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magneto-electro-elastic half planes by considering sixteen sets of boundary conditions [3]. Iadonisi et al investigated the propagation of the acoustic and the electro-magnetic

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signals in materials with tetragonal and hexagonal symmetries and devised a method to study separately the acoustic and the electro-magnetic solutions [4]. Sun et al

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investigated the effects of the imperfect interface parameters, the thickness of the PE layer and the wave number on the phase velocity of the SH wave in a cylindrical PE/PM structure [5]. Numerical results showed that the mechanical imperfection may strongly reduce the phase velocity. Piliposyan investigated the problem of the existence and propagation of a surface SH wave at the interface of two magneto-electro-elastic half-spaces [6]. It is found that the existence condition is easier to satisfy for an electrically closed contact or no electro-magnetic contact between two half-spaces. 3

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Elhosni et al developed a theoretical model for prediction of PE/PM sensor sensitivity to an external applied magnetic field in the same direction of the acoustic wave propagation which confirms that the realization of high sensitivity magnetic sensor based on layered Surface Acoustic Wave structure is achievable [7]. Li and Wei investigated the direction dependence of surface wave speed and the influence of electrically and magnetically short/open circuit conditions [8]. It is found that the

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increase of surface wave speed results from the PE enhanced effects, namely, the mechanic–electric coupling makes the effective elastic constants increasing. Yu et al solved the wave propagation problem in a layered PE–PM composites (PPCs) bar with a rectangular cross-section by proposing a double orthogonal polynomial series approach

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[9]. The dispersion curves and mechanical displacement profiles of various layered PPC rectangular bars were calculated which showed that high frequency waves propagate predominantly in the layers with a lower bulk wave speed. Fan et al theoretically studied the size-dependent dispersion relation with interface effect in layered PE/PM

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nano-structure in which the cut-off frequency and phase velocity are quite sensitive to the variations of interface, especially in the case that the interface possesses high

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strength and excellent PE/PM property [10]. Zhao et al proposed an analytical approach to investigate shear horizontal wave propagation in layered pre-stressed cylinder in

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which a PM material thin layer was bonded to a PE cylinder [11]. It is found that the initial stress, the thickness ratio and the material performance have a great influence on

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the phase velocity.

As a new kind of nonhomogeneous composite, functionally graded material is

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applied into layered PE/PM structure realizing smooth transition of the physical constitutive parameters of the PE and PM materials which can improve the materials’ mechanics performance. On account of the spatial-varying of physical constitutive parameters, the wave propagation in the functionally graded material is more complicated than in the homogeneous medium. Wang and Rokhlin presented the differential equations governing the transfer and stiffness matrices for a functionally graded generally anisotropic magneto-electro-elastic medium and calculated the surface 4

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wave velocity dispersion for a functionally graded coating on a semi-space [12]. Pan and Han presented an exact solution for the multilayered rectangular plate made of functionally graded, anisotropic, and linear magneto-electro-elastic materials [13]. Wu et al investigated the propagation of elastic waves in one-dimensional phononic crystals (PCs) with functionally graded materials (FGMs) using the spectral finite elements and transfer matrix methods [14]. It is shown that the PCs with FGMs have a higher

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frequency of band gaps than the PCs without FGMs as the material properties of FGMs show a power law variation. Cao et al studied the propagation behavior of Lamb waves in the functionally graded PE–PM material plate with material parameters varying continuously along the thickness direction, using the power series technique to solve

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these variable coefficient ordinary differential equations [15]. Numerical results showed that the elastic parameters and density varying along the thickness direction obviously affect the variation of phase velocity. Singh and Rokne investigated the propagation of SH waves in two bonded semi-infinite material, one functionally gradient PE and the

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other functionally gradient PM, of which the material properties varied in two directions, one parallel to the interface and the other perpendicular to the interface [16]. Fomenko et

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al considered the in-plane wave transmission and band-gaps due to the material gradation and incident wave-field in layered phononic crystals composed of functionally

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graded (FG) interlayers arisen from the solid diffusion of homogeneous isotropic materials of the crystal [17]. Numerical results revealed that band-gaps shift to higher

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frequencies and pass-bands widen with the increasing thickness of FG interlayers. Lan and Wei discussed the influences of the graded interlayer modeled as a system of

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homogenous sublayers with both PE and PM effects simultaneously on the anti-plane elastic wave propagating through a laminated PE/PM phononic crystal [18]. The gradient profile of the graded interlayer with PE and PM effect have intricate influences on the dispersive curves and the band gaps in which high frequency range are more sensitive to the gradient profiles than those at low frequency range. Guo et al also studied the effects of functionally graded interlayers on dispersion relations of in-plane and anti-plane Bloch waves in a one-dimensional PE/PM phononic crystal [19]. 5

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Different from Lan and Wei [18], the method approximating the gradient interlayer with a system of homogenous sublayers was discarded, instead, a direct integral approach was used to obtain the transfer matrix of the gradient interlayer. In this paper, the effects of electro-magnetic surfaces and functionally graded interlayers on dispersion relations of shear horizontal waves along circumferential directions in layered PE/PM cylinders are studied. Different from the plane layered

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structure in the literature [18] and [19], the layered cylindrical structure is considered in the present work. The state transfer equation for a cylindrical gradient interlayer is much more complicated than the situation of plane layered structure. The state transfer equation is always variable-coefficient no matter for the homogeneous cylindrical layer

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or for the gradient cylindrical layer. Moreover, the bulk waves was studied in [18,19] while the present work is concentrated on the guided waves propagating along circumferential direction of layered cylinders. Apart from the dispersion properties of lower and higher order guided modes, the radial variations of mechanical displacement,

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electric potential and magnetic potential are also considered. The objectives of the present research include the following aspects: (1) to establish the theoretical models of

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layered PE/PM cylinders with consideration of the electro-magnetic surface conditions and the functionally graded interlayers; (2) to establish the state transfer differential

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equation of the cylindrical graded interlayer and the secular equation of the dispersion relation of shear horizontal waves along circumferential directions; (3) to investigate the

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influences of electro-magnetic surfaces and gradient profiles on the dispersion relations. This paper is organized as: in Section 2, a layered PE/PM cylinder is introduced, formed

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by two transversely isotropic PE and PM concentric cylinders. The basic physical quantities of elastic waves in the PE cylinder are derived assuming that the SH waves propagate along the circumferential direction steadily. Then, the transfer matrices of the functional graded interlayer and outer PM cylinder are obtained by solving the state transfer equations with spatial-varying coefficients in Section 3. Furthermore, the transfer matrices of the functionally graded interlayer and the boundary conditions of the surface pave the road for deriving dispersion relations of shear horizontal waves in 6

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layered PE/PM cylinders—the main contents in Section 4. Finally, the numerical results of the influences of electro-magnetic surfaces and gradient profiles are shown graphically in Section 5, after which concluding remarks are made in Section 6. 2. Basic physical quantities of elastic waves in PE cylinder

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Consider the layered PE/PM cylinders formed by two transversely isotropic PE and PM concentric cylinders (labeled PE and PM, respectively). A functionally graded interlayer (FGI) between the inner PE cylinder and outer PM cylinder is transversely isotropic magnetic-electro-elastic medium, as shown in Fig.1. We establish the Cartesian

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coordinate system Ox1 x2 x3 and the cylindrical coordinate system Or z where the

x3 -axis and z -axis share the same direction. Let the x3 -axis is the poling direction and these PE and PM cylinders are transversely isotropic in the Ox1 x2 coordinates plane. In each medium, cijmn , emij , qmij ,  mi , mi and  mi are the elastic, PE, PM, dielectric,

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magnetic permeability and magneto-electric parameters, respectively.

 , re , rf

and

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rm are the mass density and the radii of the cylindrical structure. In latter formulation, the physical quantities of the PE cylinder, the PM cylinder and the FGI are labeled the

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superscript ‘ ' ’, ‘ '' ’ and ‘ f ’, respectively. The PM coefficient and the magneto-electric

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  0 ) for the PE cylinder while the PE coefficient   0 and mi coefficient are zero ( qmij   0 ) for the PM cylinder.   0 and mi and the magneto-electric coefficient are zero ( emij

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For these PE and PM cylinders, all material parameters are constant quantities. The material parameters are functions of only r -axis coordinate for the FGI. Considering the shear horizontal waves propagating along the  direction, the

constitutive equations of the transversely isotropic PE medium under the cylindrical coordinate system Or z are   zr  c44

uz   uz   uz        ,   z  c44 , Dr  e15 ,  e15  e15  11 r r r r r r 7

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 D  e15

    uz   , Br   11 , B   11   11 r r r r

(1)

in the quasi static electric/magnetic approximation.  ij is the stress tensor. Dm , Bm , and um are the electric displacement, the magnetic induction and the mechanical displacement vectors, respectively.   and   are the electric potential and magnetic

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potential, respectively. The mechanical, electrical and magnetic governing equations can be expressed as

 zr  zr   z  2u Dr Dr D Br Br B      2z ,    0,   0 r r r r r r r r r t

(2)

Inserting Eq. (1) into Eq. (2), we can obtain the governing equations

2  2   r 2 rr r 2 2

cylindrical coordinates.

(3)

is the two-dimensional Laplacian operator in the

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where 2 

 2uz  2uz  11 2   0 , 2   0 , e15 2 t

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 2uz  e15 2    c44

As we assume that the SH wave propagates along the circumferential direction

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steadily, the solution of Eq. (3) for the inner PE cylinders can be taken as

u  r, , t  ,    r, , t  ,    r, , t   U   r  ,   r  ,   r exp i  k  t  z

z

(4)

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where k is the angular circular wavenumber.  is the angular frequency. Inserting Eq.

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(4) into Eq. (3) and considering that uz ,  ,   finite value when r  0 lead to



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uz ,  ,    C1J k  r  ,

where, C1 , C2

  H   c44

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and C3

 e15 C1J k  r   C2r k , C3r k  exp i  k  t  11 

are undetermined constants.   

(5)

 H   , cSH ,   cSH

e152  is the wave velocity of SH wave. J k is the k th order Bessel . cSH 11

function of the first kind. Inserting Eq. (5) into Eq. (1) leads to the stress, electric displacement and magnetic induction 8

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 zr , Dr, Br  C1H  

J k  r    C2e15 kr k 1,  C211 kr k 1,  C311 kr k 1  exp i  k  t  r  (6)

3. Transfer matrix of the FGI and outer PM cylinder

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For the FGI, the material parameters are functions of only r -axis coordinate. Inserting the coupled displacement, electric potential and magnetic potential of SH waves which propagates along the circumferential direction

u  r, , t  ,   r, , t  ,   r, , t   U  r  ,   r  ,   r exp i  k  t  f z

f

f

f z

f z

f z

(7)

into the constitutive equations of the transversely isotropic magnetic-electro-elasto

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medium, we can obtain

1 1 1 uzf  f  f ,  fz  ikc44f uzf  ike15f  f  ikq15f  f ,  e15f  q15f r r r r r r

Drf  e15f

1 1 1 uzf  f  f , Df  ike15f uzf  ik11f  f  ik11f  f ,  11f  11f r r r r r r

1 1 1 uzf  f  f , Bf  ikq15f uzf  ik11f  f  ik 11f   11f  11f r r r r r r

f

(8)

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Brf  q15f

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 zrf  c44f

equations

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Inserting Eqs. (7) and (8) into the mechanical, electrical and magnetic governing

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2 f  zrf 1 f  fz Drf 1 f Df Brf 1 f Bf f  uz   zr    Dr   0,  Br   0 (9) , r r r r 2 r r r r r r

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leads to

  zrf  k 2 f 1 k2 k2   2 c44   f  2  uzf   zrf  2 e15f  f  2 q15f  f , r  r r r r  Drf k 2 f f 1 f k 2 f f k 2 f f  2 e15uz  Dr  2 11  2 11 , r r r r r Brf k 2 f f 1 f k 2 f f k 2 f f  2 q15uz  Br  2 11  2 11 r r r r r

(10)

For convenience of the statement of interface conditions, we define the state vector 9

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of physical fields in each region as V  r,  , t   uz ,  ,  ,  zr , Dr , Br  . T

(11)

We assume that the state vectors are continuous along the interface between the inner PE cylinder and the FGI, namely, V  re , , t   V f  re , , t 

(12)

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According to Eqs (8) and (10), we get the linear vector differential equation of the state vector in the FGI

1 V f  r, , t   G f  r   H f  r   V f  r, , t  , re  r  rf r

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where

(13a)

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 c44f  r  e15f  r  q15f  r   f f f  e15  r  11  r  11  r   q15f  r  11f  r   11f  r  f G r   0 0  0  0 0 0  0 0  0

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0   0  0   2 f 2 k f H f  r    r 2 c44  r     r    k2 f  e15  r  r2   k2 f  q15  r  r2 

The transfer matrix T

f

r

0 0

0 0

0 2

0 0 0  0 0 0 0 0 0 , 1 0 0 0 1 0  0 0 1

0 2

k f k f e r q15  r  2 15   r r2 k2 k2  2 11f  r   2 11f  r  r r 2 k k2  2 11f  r   2 11f  r  r r

0  0   0 0 1   1  0 0   r  1 0  0  r  1 0 0   r 1 0

0 1

(13b)

of the FGI is defined by

V f  r, , t   T f  r   V f  re , , t  , re  r  rf

(14)

Inserting Eq. (14) into Eq. (13) leads to

T f  r  f  A f  r   T f  r  , T  re   I , re  r  rf r 10

(15)

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1

where A f  r   G f  r   H f  r  and I is a unit matrix. After obtaining the matrix function T

f

r

from Eq. (15) and letting r  rf , we can get the transfer matrix of the

FGI. Similarly, for the outer PM cylinder, we get the linear vector differential equation of the state vector

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V  r, , t   G1  H  r   V  r, , t  , rf  r  rm r where

q15

11

0

0

 11

0

0

0

0

0

0

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0 0 0 0 0 0  0 0 0 , 1 0 0 0 1 0  0 0 1

0 0

0 0

1 0

0 1

0

0

0

0

0

k2  q15 r2

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0   0  0   2  k     2 H  r    r 2 c44   0   k2   q15 r2 

0

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 c44 0   q G   15 0 0  0



k2 11 r2 0

(16a)



1 r

0

0

k2  2 11 r

0

0 

1 r

0

0  0   1   0    0   1   r

(16b)

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By the introduction of the transfer matrix T  r  of the outer PM cylinder, namely,

V  r,  , t   T  r   V  rf ,  , t  , rf  r  rm

(17a)

Eq.(16a) can be rewritten as

T  r   A  r   T  r  , T  rf   I , rf  r  rm r

(17b)

1 where A  r   G  H  r  and I is a unit matrix. After obtaining the matrix function

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f

r

and A  r  are functions of r -axis

coordinate, let us introduce a linear combination of the univariate integrals and the definition of matrix commutator [20-21]

r 

B

i 

r 

i

1

r  r    f re  r   re  2 e   A  r   dr , re  r  rf r

 r  re 

i

1

 r  rf 

i



r

rf

r  rf    r   rf  2  

i

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B

i  f

(18a)

    A  r   dr , rf  r  rm 

(18b)

Bi  f , Bi    Bi  f  Bi   Bi   Bi  f  

AN US

(18c)

The solutions of Eqs. (15) and (17b) are

T  r   exp  Ω  r    exp  Ω[ n ]  r  

(19)

[20-21]. For n  6 ,

1 1 b3   20b1  b3  s1, b2  r1  12 240

ED

Ω[6]  b1 

M

where Ω[ n ]  r  approximates Ω  r  up to order n based on the Magnus expansion

PT

where

AC







3 1  0  2 0 2 3B   20B  , b2  12B , b3  15 B  12B 4

CE

b1 

s1  b1, b2  , r1  

1 b1, 2b3  s1  60

(20)

 (21)

The matrix exponential can be calculated by

exp Ω6  r   Q  diag exp(1 ), ... ,exp(m )  Q1

(22)

 6 where i is the eigenvalues of matrix Ω  r  , Q is the eigenvector matrix of  6 matrix Ω  r  .

12

ACCEPTED MANUSCRIPT

4. Dispersion equation Making use of the transfer matrices of the FGI and the outer PM cylinder, we can get the relationship

uz ,  ,  ,  zr , Dr, Brrr T

m

 T  rm   T f  rf   uz ,  ,  ,  zr , Dr , Brr r . T

(23)

e

The total transfer matrix is defined as

CR IP T

Tt  T  rm   T f  rf 

(24)

The surface of the outer PM cylinder r  rm is mechanically free and subjected to four kinds of electro-magnetic loads which are electrically open and magnetically open interface (denoted by OO case)

AN US

 zr  rm , , t   0,    rm , , t   0, Dr rm , , t   0

(25a)

electrically open and magnetically short interface (denoted by OS case)

 zr  rm , , t   0, Dr  rm , , t   0, Br  rm , , t   0

(25b)

M

electrically short and magnetically open interface (denoted by SO case)

 zr  rm , , t   0,    rm , , t   0,    rm , , t   0

(25c)

ED

electrically short and magnetically short interface (denoted by SS case) (25d)

PT

 zr  rm , , t   0,    rm , , t   0, Br rm , , t   0

For these four mechanical and electro-magnetic surfaces, the state vectors of physical

AC

CE

fields are

V  rm , , t   uz ,  , 0, 0, 0, Brr r

(26a)

V  rm , , t   uz ,  ,  , 0, 0, 0r r

(26b)

V  rm , , t   uz , 0, 0, 0, Dr, Brr r

(26c)

V  rm , , t   uz , 0,  , 0, Dr, 0r r

(26d)

T

m

T

m

T

m

T

m

Inserting Eqs. (23) and (24) into Eq. (26) leads to

13

ACCEPTED MANUSCRIPT T34t T33t  uz      T43t       T44t t  T53t    r re T54

T35t T36t   zr  0     t t   T45 T46    Dr   0     T55t T56t    Br r re 0 

(27a)

T41t T42t  t t T51 T52 T61t T62t 

T44t T43t  uz      T53t       T54t t  T63t    r re T64

T45t T46t   zr  0     t t   T55 T56    Dr   0     T65t T66t    Br r re 0 

(27b)

T21t T22t  t t T31 T32 T41t T42t 

T24t T23t  uz      T33t       T34t t  T43t    r re T44

T25t T26t   zr  0     t t   T35 T36    Dr   0     T45t T46t    Br r re 0 

(27c)

T21t T22t  t t T41 T42 T61t T62t 

T24t T23t  uz      T43t       T44t t  T63t    r re T64

T25t T26t   zr  0     t t   T45 T46    Dr   0     T65t T66t    Br r re 0 

(27d)

AN US

CR IP T

T31t T32t  t t T41 T42 T51t T52t 

Further, inserting Eqs. (5) and (6) into Eqs. (27) leads to

T45t T46t  C1  0      t t  T55 T56   Tij   C2   0  C   0  T65t T66t   3  

(28b)

T25t T26t  C1  0       T35t T36t   Tij   C2   0  C   0  T45t T46t   3  

(28c)

T25t T26t  C1  0      t t  T45 T46   Tij   C2   0  C   0  T65t T66t   3  

(28d)

T43t T44t

T45t

t 53

T

t 54

T

T43t T44t T53t

T54t

T63t T64t

T21t T22t  t t T31 T32 T41t T42t 

T23t T24t

T21t T22t  t t T41 T42 T61t T62t 

T23t T24t

PT CE

(28a)

T35t

ED

T41t T42t  t t T51 T52 T61t T62t 

T36t  C1  0       T46t   Tij   C2   0  C   0  T56t   3  

T33t T34t

M

T31t T32t  t t T41 T42 T51t T52t 

T33t T34t t 43

T

t 44

T

T43t T44t T63t T64t

t 55

T

AC

where

  J k  re   Tij    0  0  

 e15 J  re  11 k

0

H   J k  r  r  r  re

0

rek 0

0 rek

 krek 1 e15 0

 krek 1 11 0

T

   0  (28e) k 1   kre  11   0

If the nontrivial solutions of C1 , C2 and C3 exist, the determinant of the coefficient 14

ACCEPTED MANUSCRIPT

matrix must be equal to zero. This gives the dispersion relations for shear horizontal waves in layered PE/PM cylinders with FGIs. For the FGI, we consider every physical constitutive parameter changes as

P f  r   PF  r   P 1  F  r   where

(29)

P and P are physical constitutive parameters of the PE and PM cylinders,

the gradient profile functions

F0  r   1, re  r  rf r  re , re  r  rf rf  re

AN US

F1  r   1 

CR IP T

respectively. F  r  is the function of the gradient profile. We consider seven kinds of

(30a)

(30b)

1

 r  re  g F2  r    1   , re  r  rf  rf  re  

(30c)

g

(30d)

M

 r  re  F3  r   1  , r  r  rf  r  r  e f e  

(30e)

g 1 1  r r r  re    1  2 , re  r  f e  rf  re  2  2 2  F5  r    g 1 1  r  re  rf  re  r  rf  ,    1  2 rf  re  2  2 2 

(30f)

AC

CE

PT

ED

1  g    1  1 1  2 r  re , r  r  rf  re  e  2 2  rf  re  2   F4  r    1  g   r r  1  1  1  2 r  re  , f e  r  rf   rf  re  2  2 2 

F6  r   0, re  r  rf

(30g)

The curves of various gradient profiles are shown in Fig.2. F0  r  and F6  r  denote two limit cases, i.e. the homogeneous PE and PM interlayers, respectively. F1  r  15

ACCEPTED MANUSCRIPT

denotes a linear profile between the PE and PM cylinders. g is an index describing the nonlinear properties of profiles. As the magnetoelectric coefficients of the PE cylinder and the PM cylinder are zero, we define the magneto-electric coefficients of the FGI as [18]

11  r   411F  r  1  F  r    33  r   433 F  r  1  F  r  

CR IP T

(31)

where 11  5 1012 NsV-1C-1 and 33  3  1012 NsV-1C-1 are prescribed. 5. Numerical results and discussions

AN US

In this section, dispersion curves of shear horizontal waves in layered PE/PM concentric cylinders with FGI are calculated numerically. The material constants of the PE solid LiNbO3 and the PM solid Terfenol-D [22] are listed in Tab.1. The radius ratio is assumed as re : rf : rm  1:1.2 :1.4 . The non-dimensional phase velocity of SH wave is

M

  . Considering the similar characteristics of different order modes, V  rm   kcSH

ED

only the first three modes of dispersion curves are calculated in the present work.

Tab.1. Material constants of LiNbO3 and Terfenol-D c11

c12

c13

c33

LiNbO3

203

52.9

74.9

243

B(PM) Terfenol-D 8.541 0.654

3.91

28.3

AC

CE

A(PE)

PT Name

Mat



e15

e31

59.9 4700

3.7

0.19

5.55 9250

0

0

c44

Mat

Name

e33

11

 33

q15

q31

q33

11

33

A(PE)

LiNbO3

1.31

0.39

0.26

0

0

0

5

10

B(PM)

Terfenol-D

0

0.05

0.05

155.56

-5.7471 270.1 8.644 2.268

2 1 1 2 1 2 cij : G Pa ,  : Kg  m3 , eij : C  m , qij : N  A  m ,  ij : nC  N  m , ij :  N  A2

It is shown in Fig.3 the influences of the electro-magnetic surfaces on dispersion 16

ACCEPTED MANUSCRIPT

curves of first three modes. The linear profile F1 and two limit case, i.e. F0 and F6 , are considered. It is observed form Fig.3(a), (b) and (c) that dispersive curves corresponding with OO interface and SO interface are approximately identical meanwhile the dispersive curves corresponding with OS interface and SS interface are also approximately identical. This means that the dispersive curves are only sensitive to

CR IP T

the magnetic surface condition while insensitive to the electric surface condition. This phenomenon results from the fact that the outer cylinder is PM cylinder. It is also observed that surface electro-magnetic conditions have larger influences on the low order modes than on the high order modes. Compared with the dispersion curves

AN US

corresponding with F0 and F6 profile, the dispersion curves corresponding with F1 linear profile are always situated between the dispersive curves of

F0 and F6

profiles, shown in Fig.3(d). Moreover, the dispersive curves corresponding with 2nd and 3rd order modes are more sensitive to the gradient profiles than 1st order mode. Disregard

M

of the gradient profiles and the electro-magnetic surface conditions, the non-dimensional speed tends to constant when the wavenumber increases gradually.

ED

In order to make a comparison of the influences of different profiles of FGIs, dispersion curves corresponding with various gradient profiles under electro-magnetic

PT

surface case OO are shown in Fig.4. It is observed that the dispersion curve corresponding with F2 profile is always above that of F5 profile, shown in Figs. 4(a)

CE

and 4(b). Because the elastic constants of the inner PE cylinder are much larger than that

AC

of the outer PM cylinder, F5 profile can be approximately taken as a homogenous interlayer with the averaged material constants of the PE and PM cylinders. But F2 profile can be approximately taken as the increase of the radius of the PE cylinder. The global rigidness of the layered cylinders corresponding with F2 profile is larger than that corresponding with

F5 profile. This can explain why dispersion curve

corresponding with F2 profile is always above that of F5 profile. Similarly, the fact 17

ACCEPTED MANUSCRIPT

that the dispersion curve corresponding with F3 profile is always below that of F4 profile can be understandable. Compared with 1st, 2nd and 3rd order modes are more sensitive to the gradient profile. It is observed from Fig.2 that the index g have evident influences on the gradient profile. The linear profile F1 can be taken as the limiting case

CR IP T

of F2 , F3 , F4 and F5 profiles when g  1 . The increase of index g makes the nonlinear degree of profile increasing and therefore makes the gradient profile exerting a more complicated influence on the dispersion curves.

In order to distinguish the different influences on the low order and the high order modes, the influences of the electro-magnetic surfaces on 1st, 7th and 17th order modes

AN US

are shown in Fig.5(a). The influences of the graded profiles on 1th, 7th and 17th order modes are shown in Fig.5(b). It is observed that low high order modes are more sensitive to the electric-magnetic surface conditions than the high order modes. In contrast, the high order modes are more sensitive to the graded profile than the low order

M

modes. This means that it is fruitless to adjust the dispersive feature of high order modes by the electric magnetic surface conditions. However, we can design the graded profile

ED

elaborately to adjust the dispersive feature of the high order modes. The shear horizontal guided waves of different order modes propagate along

PT

circumferential direction in layered PE/PM concentric cylinders. The angular wavenumber k indicates the number of wave along circumference. When k is integer,

CE

the stationary wave condition holds and the guided waves present the stationary waves. Only when k is not integer, the guided waves present the propagating waves.

AC

Disregard of the stationary waves or the propagating waves, the radial variation of mechanical displacement, the electric potential and the magnetic potential are interesting because the radial variation accounts for the vibration form of different order guide modes. Fig.6 shows the radial variation of dimensionless mechanical displacement, the electric potential and the magnetic potential of 1st and 7th order guide modes for the linear profile F1 of interlayer and the electro-magnetic surface case OO. It is observed 18

ACCEPTED MANUSCRIPT

that the mechanical displacement always increases gradually with the increasing radial coordinate and arrive the maximum at the outer surface. For the low order mode (e.g. 1st mode) the radial variation, disregard of mechanical displacement, the electric potential or the magnetic potential, appears with the flat gradient. However, for the high order modes (e.g. 7th mode), the radial variation fluctuates drastically. At the outer layer, the fluctuations of radial variation are more frequent than the inner layer. The drastic

CR IP T

fluctuation of vibration form is the essential feature of the high order guide modes.

It is the direct integration method that is used to calculate the transfer matrices of FGIs in the present work. In order to make a comparison of the present method with the layer-wise homogenization method often used in the previous literatures, dispersion

AN US

curves corresponding with F3 gradient profile under OO electro-magnetic surface condition are calculated by two methods, respectively, and the results are graphically shown in Fig.5. It is observed that the deviations between two methods are unnoticed for the low order modes. But the deviations are evident for the high order modes. For the

increase of the wavenumber.

M

same order mode, the deviation between two methods increases gradually with the

ED

Finally, in order to validate the numerical results obtained in present work, the comparison between the present results and that in Ref. [22] is performed and is

PT

graphically shown in Fig.6. The layered PE/PM cylinder formed by PZT-4/Terfenol-D is considered instead. Let the thickness of gradient interlayer h decreasing gradually and

CE

compare the results obtained with the situation without interlayer as in the Ref [22]. It is observed that the results obtained in the present work tend to that in Ref [22] when the

AC

thickness of gradient interlayer gradually tends to zero. This validates the present numerical results to some extent. 6. Conclusions The dispersive properties of shear horizontal wave propagating along circumferential directions of the layered PE/PM cylinders with the functional gradient interlayer are studied. The influences of the electro-magnetic surface conditions and the 19

ACCEPTED MANUSCRIPT

gradient profiles of functional gradient interlayer on the dispersion curves of shear horizontal waves are the main concerns of the present work. From the numerical results, the following conclusions can be drawn: (1) Magnetic surface conditions have more evident influences on the dispersion curves of the shear horizontal wave than the electric surface condition. The low order modes are more sensitive to the electro-magnetic surface conditions than the high

CR IP T

order modes. For the same order mode, the electro-magnetic surface conditions have more evident influences at high frequency range than the low frequency range.

(2) The dispersive curves of the high order modes are more sensitive to the gradient profiles of FGI. The nonlinear index g controls the nonlinear degree of the

AN US

gradient profile. Increasing index g makes nonlinear degree of gradient profile increasing and therefore enhances the adjusting ability of FGI. (3) The dispersion curves corresponding with 2nd and 5th gradient profiles are above that of linear profile while the dispersion curves corresponding with 3rd and 4th gradient

M

profiles are below that of linear profile. The differences of dispersion curves between the 2nd and 3rd gradient profile are largest. The differences of dispersion

ED

curves between the 4th and 5th gradient profiles are much smaller. (4) There are two methods to deal with the FGI. The deviations between the direct

PT

integration method and the layer-wise homogenization method are much smaller for the low order modes. But the deviations become large for the high order modes. For

CE

the same order modes, the deviations become large gradually when the wavenumber

AC

increases gradually.

In the present study, the acoustic radiation at the outer surface is not taken into

consideration. When the acoustic radiation at the free surface is considered, there is one part of energy will leak into the surrounding air and this will result in the attenuation of the guided wave. Usually, the attenuation effect is negligible when the layered PE /PM cylinder is placed in air. However, the attenuation effect should be considered when it is placed in liquid. 20

ACCEPTED MANUSCRIPT

Acknowledgments The work is supported by Fundamental Research Funds for the Central Universities

AC

CE

PT

ED

M

AN US

CR IP T

(FRF-BR-15-026A) and National Natural Science Foundation of China (No. 10972029).

21

ACCEPTED MANUSCRIPT

References [1]. J. Liu, D.N. Fang, W.Y. Wei, X.F. Zhao, Love waves in layered piezoelectric/piezomagnetic structures, J. Sound. Vib. 315(2008) 146-156. [2]. Y. Pang, Y.S. Wang, J.X. Liu, D.N. Fang, Reflection and refraction of plane waves at the interface between piezoelectric and piezomagnetic media, Int. J. Eng. Sci. 46(2008) 1098-1110.

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[3]. W.J. Feng, E. Pan, X. Wang, J. Jin, Rayleigh waves in magneto-electro-elastic half planes, ACTA Mech. 202(2009) 127-134.

[4]. G. Iadonisi, C.A. Perroni, G. Cantele, D. Ninno, Propagation of acoustic and electromagnetic waves in piezoelectric, piezomagnetic, and magnetoelectric materials with tetragonal and hexagonal

[5]. W.H.

Sun,

G.L.

Ju,

J.W.

Pan,

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symmetry, Phys. Rev. B 80(2009) 094103. Y.D.

Li,

Effects

of

the

imperfect

interface

and

piezoelectric/piezomagnetic stiffening on the SH wave in a multiferroic composite, Ultrasonics 51(2011) 831-838.

M

[6]. D. Piliposyan, Shear surface waves at the interface of two magneto-electro-elastic media. Multi. Model. Mater. Struct. 8(2012) 417-426.

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piezomagnetic phononic crystal with functionally graded interlayers, Ultrasonics 70 (2016) 158-171. [20]. S. Blanes, F. Casas, J. Ros, Improved high order integrators based on the magnus expansion, BIT

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Figure captions Fig.1.

The layered piezoelectric/piezomagnetic (PE/PM) concentric cylinders with functionally graded interlayer (FGI).

Fig.2.

The curves of the gradient profiles corresponding with different kinds of functionally graded interlayers.

Fig.3.

Effects of the electro-magnetic surface conditions on dispersion curves of first three modes.

Fig.4.

Effects of functionally graded interlayers on dispersion curves under OO 23

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Effects of the electro-magnetic surface conditions and the FGI on dispersion curves for the higher order modes.

Fig.6.

The radial variation of dimensionless mechanical displacement uz ( uz / rm ) , electric potential  ( (11 ) (e15 rm )) and magnetic potential  ( (11 ) (q15 rm )) for the linear profile F1 of interlayer and the electro-magnetic surface case

Fig.7.

Comparisons of the layer-wise homogenization method and the direct integration method under OO electro-magnetic surface case.

Fig.8.

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OO.

Comparison of the present results (the linear profile F1 of interlayer with the

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thickness h) with that in reference [22] (without interlayer).

PM

SH Waves

FGI

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PE

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Fig.1. The layered piezoelectric/piezomagnetic (PE/PM) concentric cylinders with

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functionally graded interlayer (FGI).

Fig.2. The curves of the gradient profiles corresponding with different kinds of 24

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functionally graded interlayers.

modes.

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Fig.3 Effects of the electro-magnetic surface conditions on dispersion curves of first three

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Fig.4 Effects of functionally graded interlayers on dispersion curves under OO

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electro-magnetic surface case.

curves for the higher order modes.

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Fig.5 Effects of the electro-magnetic surface conditions and the FGI on dispersion

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Fig.6 The radial variation of dimensionless mechanical displacement uz ( uz / rm ) ,

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electric potential  ( (11 ) (e15 rm )) and magnetic potential  ( (11 ) (q15 rm )) for the

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linear profile F1 of interlayer and the electro-magnetic surface case OO.

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Fig.7 Comparisons of the layer-wise homogenization method and the direct integration

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method under electro-magnetic surface case OO.

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Fig.8 Comparison of the present results (the linear profile F1 of interlayer with the

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thickness h) with that in reference [22] (without interlayer).

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