Complex guided waves in functionally graded piezoelectric cylindrical structures with sectorial cross-section

Accepted Manuscript

Complex guided waves in functionally graded piezoelectric cylindrical structures with sectorial cross-section B. Zhang , J.G. Yu , X.M. Zhang , P.M Ming PII: DOI: Reference:

S0307-904X(18)30306-8 10.1016/j.apm.2018.06.053 APM 12352

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

23 January 2018 7 June 2018 26 June 2018

Please cite this article as: B. Zhang , J.G. Yu , X.M. Zhang , P.M Ming , Complex guided waves in functionally graded piezoelectric cylindrical structures with sectorial cross-section, Applied Mathematical Modelling (2018), doi: 10.1016/j.apm.2018.06.053

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Highlights The modified polynomial method is proposed.

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some evanescent modes have high velocity and low attenuation.

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Changing section size can govern wave characteristics.

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Complex guided waves in functionally graded piezoelectric cylindrical structures with sectorial cross-section B. Zhang1, J.G. Yu*,1, X.M. Zhang1, P.M Ming1 1

School of Mechanical and Power Engineering, Henan Polytechnic University, Jiaozuo 454003, P.R. China *

corresponding author, E-mail: [email protected]

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Abstract: For the purpose of the design and optimization of piezoelectric transducers, the modified double orthogonal polynomial series method is proposed to investigate guided waves in functionally graded piezoelectric(FGP) cylindrical structures with sectorial cross-section. The real, imaginary and complex solutions are obtained simultaneously without iterative process. The real solutions represent propagative waves;

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the imaginary and complex solutions are evanescent waves. The boundary conditions are incorporated into the constitutive equations by virtue of the Heaviside function. Subsequently, the amplitudes are expanded into the double orthogonal polynomial series, and the motion equations are converted into a matrix eigenvalue problem about complex wavenumber. Numerical comparison with available reference result confirms

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the validity of the present method. Dispersion curves and the Poynting vector distributions are illustrated. The influences of angular measure, radius-thickness ratio

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and graded index on dispersion curves are analyzed. Results show that there exist some evanescent guided wave modes that have higher velocities than that of the propagative

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wave modes and simultaneously have low attenuation at high frequencies. These results can be utilized to improve the performance of transducers.

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Keywords: Complex guided waves; Funtionally graded piezoelectric material; Double orthogonal polynomial; Sectorial cross-section; the Poynting vector

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1. Introduction An army of piezoelectric transducers are multilayered structures by bonding

piezoelectric element and the protection layer together in practical applications[1]. There are many disadvantages for this kind of structures, such as creep and disbonding at high temperatures at interfaces[1] and stress concentration[2]. Accordingly, to overcome these disadvantages and improve mechanical reliability, the design and fabrication of functionally graded piezoelectric (FGP) transducers are paid increasing attentions[3, 4]. The performance of piezoelectric devices has a close relationship with 2

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wave characteristics, so guided waves in FGP structures also have drawn a lot of attentions in recent years. Various methods and models have been used for solving wave propagation in FGP structures, such as the Peano-series method[5], the stiffness matrix method[6, 7], the reverberation-ray matrix formulation[8], the Wentzel–Kramers– Brillouin

(WKB)

technique[9-12],

the

Liouville-Green’s

(LG)

approximation

technique[13], the power series method[14,15] and the Legendre orthogonal polynomial

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series expansion approach[16,17].

The above mentioned research objects are relatively simple geometric structures, i.e. half-space structures, hollow cylinders and infinite plates. With the development of the computation technique, many scholars pour their attentions into waves in waveguides with sectorial cross-section. Kosmodamianskii et al.[18] investigated

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dispersion spectrums of an anisotropic waveguide with a sector-shaped channel of arbitrary angular measure whose surfaces are covered with flexible and inextensible membranes. Controlling wave characteristics of piezoceramic cylinders by geometric cross-section was detailed by Awrejcewicz[19]. Its results showed that the desired wave characteristics, such as the cut-off frequency and wave speed, could be obtained by

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changing geometry cross-section, which could be used to improve the performance of transducers. Waves propagating in axially polarized piezoelectric hollow cylinders with

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sectorial cross-section were studied by Puzyrev and Storozhev [20] when the boundary surfaces of sectorial cut are covered by non-extensible membranes. Zhou et al. [21]

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investigated the elastic waves in piezoelectric cylinders with sectorial cross-section, and discussed the influence of the variation of angular measure on dispersion spectrums. For

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the cylindrical structures with sectorial cross-sections, the arc length is defined as l,

l  (b+a)  / 2 , where a, b and β represent the inner, outer radius and angular measure,

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respectively. And d is the thickness in the r direction. In reference[18-21], l is much bigger than d, the circumferential displacement is dealt by a similar way with that of cylinders with whole circle cross-sections. Its circumferential displacement is not unknown with respect to the variable θ, so it can be regarded as a 1-D model. If there is not obvious difference between l and d, the displacements in both radial and circumferential directions should be completely unknown with respect to the variables r and θ. The mathematical model is a 2-D structure. So far, rare references about 2-D FGP cylindrical structures with sectorial 3

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cross-section are available. Accordingly, an analytical approach, the modified double orthogonal polynomial series method, is proposed to investigate propagative and evanescent waves in 2-D FGP cylindrical structures with sectorial cross-section. Dispersion curves, electric potential and the Poynting vector distributions are illustrated. The stress-free and electrically open-circuit boundaries are assumed in this paper.

A functionally graded

piezoelectric

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2.Mathematics and formulation cylindrical

structure

with

sectorial

cross-section in cylindrical coordinates (θ, z, r) is considered, as shown in Fig.1. The

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radius-thickness ratio is denoted as ,  b / (b  a) .

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Fig.1 Schematic diagram of the cylindrical structure with sectorial cross-section

The body forces and electric charges are assumed to be zero. So the wave motion

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equations are controlled by[22]:

(1)

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Trr 1 Tr Trz Trr  T  2u      2r r r  z r t Tr 1 T T z 2Tr  2u     2 r r  z r t 2 Trz 1 T z Tzz Trz u      2z r r  z r t Dr 1 D Dz Dr    0 r r  z r

where Tij , ui , Di are the components of stress, displacement and electric displacement, respectively; ρ represents the mass density. The displacement-strain relationship can be written as:   

z 

1  1 u r u u  u u 1 u u r ,   ,   zz  z ,  rr  r ,  r   r  r r 2  r  r r  z

1  ur uz  1  u u z   ,    ,  rz   2  z r  2  z r  4

(2)

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with  ij are the strains. The relationship between the electric field and electric potential components can be expressed as: E  

1  , Ez    , r  z

Er  

 , r

(3)

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where Ei and  represent electric field and electric potential components, respectively. Based on linear electro-elasticity theory, the following constitutive equations can be obtained.

Tij  Cijkl  kl I (r , )  ekij Ek I (r , ) , Di  eikl kl I ( r, ) ik Eik I ( r, ),

(4)

where Cijkl , ekij and ik are the elastic, piezoelectric and dielectric coefficients,

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respectively. I (r , ) is a rectangular window function, which can be denoted as:

 1, a  r  b and 0       r ,    (r ) ( )   elsewhere 0,

open-circuit

boundary

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By introducing the rectangular window function I (r , ) , conditions

are

automatically

(5) the traction-free and

incorporated,

namely,

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Trr  Tr  Trz  Dr  0 at r=a and r=b; and T  Tr  T z  D  0 at θ=0 and θ=β. The method using the Heaviside function to deal with boundary problems has been utilized

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by a host of scholars[22-24].

For the FGP cylindrical structures, material properties vary in the r direction.

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Therefore, material parameters are the function of r, which can be written as follows：

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L (l ) r l Cijkl (r )   Cijkl ( ), d l 0

L L r l) r l e jkl (r )   e(jkl ( ) ，  jk (r )  (jkl ) ( )l ， d d l 0 l 0 L

r d

 ( r )    ( l ) ( )l . l 0

(6)

For the harmonic waves propagating in the z direction, the mechanical

displacements and electric potential are assumed to be of the following form:

ur (r, , z, t )  exp(ikz  it )U (r, )

(7a)

u (r , , z, t )  exp(ikz  it )V (r , )

(7b)

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uz (r , , z, t )  exp(ikz  it )W (r , )

(7c)

(r, , z, t )  exp(ikz  it ) X (r , )

(7d)

where U (r , ) , V (r , ) and W (r , ) are the amplitudes in the r, θ, z directions, respectively. X (r , ) is the amplitude of electric potential.

k represents the wave

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number and ω is the angular frequency. Substituting Eqs.(2)-(7) into Eq.(1), the controlling differential equations in terms of displacements and electric potential can be obtained as follows:

(r / d )l {I (r , )[C33(l ) r 2U ,rr C55(l )U , C33(l ) (l  1)rU ,r (C44(l ) k 2 r 2  C11(l ) )U  C13(l )lU 

(C13(l )  C55(l ) )rV ,r (C11(l )  C55(l ) )V , C13(l )lV , (C23(l )  C44(l ) )ikr 2W , r (C23(l )  C12(l ) )ikrW

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C23(l )likrW  e24(l ) k 2 r 2 X  (e33(l )  e31(l ) )rX ,r e33(l )lrX ,r e33(l ) r 2 X ,rr e15(l ) X , ]

(8a)

 I (r , ),r (C r U ,r C rU  C rV , C ikr W  e r X ,r ) (l ) 2 33

(l ) 13

(l ) 13

(l ) 23

2

(l ) 2 33

 I (r , ), (C55(l )U , C55(l ) rV ,r C55(l )V  e15(l ) X , )}  (r / d )l  (l ) 2 r 2U (r / d )l {I (r ,  )[(C55(l )  C13(l ) )rU ,r (C55(l )  C11(l ) )U , C55(l )lU , C55(l ) r 2V ,rr C11(l )V , C55(l ) (l  1)rV ,r (C66(l ) k 2 r 2  C55(l ) )V  C55(l )lV  (C12(l )  C66(l ) )ikrW , e15(l ) (l  1) X ,

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(e15(l )  e31(l ) )rX ,r ]  I (r ,  ),r (C55(l ) rU , C55(l ) r 2V ,r C55(l ) rV  e15(l ) rX , )

(8b)

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 I (r , ), (C11(l )V , C12(l )ikrW  C11(l )U  C13(l ) rU ,r e31(l ) rX ,r )}  (r / d )l  (l ) 2 r 2V (r / d )l {I (r , )[(C44(l )  C23(l ) )ikr 2U ,r (C44(l )  C12(l ) )ikrU  C44(l )likrU  (C12(l )  C66(l ) )ikrV , C44(l ) r 2W ,rr C66(l )W , C44(l ) (l  1)rW ,r C22(l ) k 2 r 2W  e24(l ) (l  1)ikrX  (e24(l )  e32(l ) )ikr 2 X ,r ]

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 I (r , ),r (C44(l )ikr 2U  C44(l ) r 2W ,r e24(l )ikr 2 X )  I (r , ), (C66(l )ikrV  C66(l )W , )}

(8c)

 (r / d )l  (l ) 2 r 2W

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l) (r / d ) l{I ( r, )[ e 24(l )k 2r 2U e 31( l)lU (e 31( l) e 33( l) )rU , r e 33( l)lrU , r e 33( l)r 2U , rr e 15( U ,  e 15( V)l , 

e31(l )lV , (e31( l)  e15( l) )rV , r e32( l) (l 1)ikrW (e 32( l) e 24( l) )ikr 2W , r  (22l) k 2r 2X (33)l (l 1)rX , r

(8d)

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33(l ) r 2 X ,rr 11(l ) X , ]  I (r , ),r (e31(l ) rU  e33(l ) r 2U ,r e31(l ) rV , e32(l )ikr 2W 33(l ) r 2 X ,r )  I (r , ), (e15(l )U , e15(l )V  e15(l ) rV ,r 11(l ) X , )}  0

where subscript commas indicate partial derivative. To solve the coupled wave equations, U(r,θ), V(r,θ), W(r,θ) and X(r,θ)are expanded into the double Legendre orthogonal polynomial series:

U (r ,  ) 





m, j 0

V (r ,  ) 

p1m, j Qm (r )Q j ( ),





m, j 0

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pm2 , j Qm (r )Q j ( ),

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W (r ,  ) 





m, j 0



X (r ,  ) 

pm3 , j Qm (r )Q j ( ),

r

m, j 0

m, j

Qm (r )Q j ( ),

(9)

where pmi , j (i  1, 2,3) and rm , j are expansion coefficients and 2m  1  2 r  d  Pm  , d  d 

Qm (r ) 

Q j ( ) 

2 j  1  2   Pj    

 , 

(10)

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where Pm and Pj are the mth and jth Legendre polynomials, respectively. Theoretically, m and j run from 0 to ∞. However, in fact, the summations over the polynomials in Eq.(9) are convergent at some finite values m=M and j=J.

Multiplying Eqs.(8) by Qn (r )  Qp ( ) with n running from 0 to M and p running

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from 0 to J, respectively, then integrating over r from a to b and over θ from 0 to β, and utilizing the orthogonality of the Legendre polynomial, the following system can be obtained:

A12n, p ,m, j l n, p ,m, j A22 l n, p ,m, j A32 l n, p ,m, j A42 C12n, p ,m, j l n , p ,m , j C22 l n , p ,m , j C32 l n, p ,m, j C42

A13n, p ,m, j l n, p ,m, j A23 l n, p ,m, j A33 l n, p ,m, j A43 l

C13n , p ,m, j l n , p ,m , j C23 l n , p ,m , j C33 l n , p ,m, j C43

1 A14n, p ,m, j   pm, j   l B11n, p ,m, j l B12n, p ,m, j l B13n, p ,m, j  l n, p ,m, j   2  A24   pm, j   l B21n, p ,m, j l B22n, p ,m, j l B23n, p ,m, j  k l n, p ,m, j   3  l n, p ,m, j l n, p ,m, j l n, p ,m, j A34 B32 B33  pm, j   B31  l n, p ,m, j  4  l n, p ,m, j l n, p ,m, j l n, p ,m, j A44   pm, j   B41 B42 B43 1 l n , p ,m , j C14   pm, j   l M n, p ,m, j 0 0 l n , p ,m , j   2   l C24   pm, j   0 M n, p ,m, j 0  l n , p ,m , j   3  l C34 0 M n, p ,m, j  pm, j   0 l n , p ,m , j   4   C44   pm, j   0 0 0 l

B14n, p ,m, j   pm, j  l n, p ,m, j   2  B24   pm, j  l n, p ,m, j   3  B34  pm, j  l n, p ,m, j   4  B44   pm, j  (11) 1 0   pm, j   2  0   pm, j   3 , 0   pm, j   4 0   pm, j  1

l

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l

l

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 lC11n, p ,m, j  l n , p ,m , j C   l 21n, p ,m, j  C31  l n , p ,m , j  C41

l

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 l A11n, p ,m, j  l n, p ,m, j A k 2  l 21n, p ,m, j  A31  l n, p ,m, j  A41

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where l M n, p , j ,m and l An, p, j ,m ( ,   1, 2,3, 4) are the elements of the non-symmetric matrices which are given in the appendix.

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For concision, we define

 l A11n , p ,m, j  l n, p ,m, j A A   l 21n , p ,m, j  A31  l n, p ,m, j  A41

l

A12n , p ,m, j l n, p ,m, j A22 l n, p ,m, j A32 l n, p ,m, j A42

l

A13n, p ,m, j l n, p ,m, j A23 l n, p ,m, j A33 l n, p ,m, j A43

A14n, p ,m, j   l B11n , p ,m, j  l n, p ,m, j  l n, p ,m, j A24  , B   B21 l n, p ,m, j   l B31n , p ,m, j A34  l n, p ,m, j l n, p ,m, j  A44   B41 l

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l

B12n , p ,m, j l n, p ,m, j B22 l n, p ,m, j B32 l n, p ,m, j B42

l

B13n, p ,m, j l n, p ,m, j B23 l n, p ,m, j B33 l n, p ,m, j B43

B14n, p ,m, j  l n, p ,m, j  B24 , l n, p ,m, j  B34 l n, p ,m, j  B44  l

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 l C11n , p ,m, j  l n, p ,m, j C C   l 21n , p ,m, j  C31  l n, p ,m, j  C41

l

C12n , p ,m, j l n , p ,m, j C22 l n , p ,m, j C32 l n , p ,m, j C42

 l M n, p , j ,m C14n , p ,m, j   l n , p ,m, j  C24 , M   0 l n , p ,m, j   0 C34   l n , p ,m, j C44   0

l

C13n , p ,m, j l n , p ,m, j C23 l n , p ,m, j C33 l n , p ,m, j C43

l

0 0

M n, p , j ,m 0 0

l

M n, p , j ,m 0

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 p1m , j   2  p  p   m3 , j  ,  pm , j   pm4 , j   

0 l

and introduce a new column vector

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q1m , j   p1m , j   2   2  qm , j   pm , j  q  kp , i.e.,  3   k  3  . qm , j   pm , j  4 qm , j   pm4 , j      So Eq.(11) can be written as

kAq  Bq    C  M  p .

0  0 , 0  0

(12)

(13)

Then the following equation can be obtained

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A1  C  M  p   A1B  q  kq  0 .

(14)

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The following matrix can be obtained by combining Eq.(12) with Eq.(14)

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0  I 4 M 1 J 1   p    p   k  1     , A1B q   A  C  M    q 

(15)

where I is a 4*(M+1)*(J+1) unit matrix. Here, a new column vector is introduced:

CE

R  Rm1 , j

AC

  p1m, j

Rm2 , j pm2 , j

Rm3 , j pm3 , j

Rm4 , j

Rm5 , j

pm4 , j qm1 , j

Rm6 , j qm2 , j

Rm7 , j Rm8 , j 

T

qm3 , j qm4 , j  . T

So Eq.(15) can be expressed as

0  I 4 M 1 J 1    1  R  kR . A1B  A  C  M  

(16)

Eq. (16) forms an eigenvalue equation about the complex wavenumber to be solved. The eigenvalues, namely, the wave numbers k, are complex-valued in general, and the profiles of the mechanical displacement components and the electrical potential can be obtained according to the corresponding eigenvectors. 8

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3.Numerical results The equivalent parameters of the FGP sectorial structures are calculated by using the Voigt-type model, which can be denoted as:

P(r )  PV 1 1 (r )  PV 2 2 (r ),

(17)

where Pi and Vi (r ) represent the corresponding material parameters and the volume fraction of the ith material, respectively. V1(r)+ V2(r)=1.

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P(r )  P2   P1  P2 V1 (r ),

(18)

In the present paper, two kinds of FGP sectorial structures are taken into account. Structure 1: the inner surface is PZT-4, and the outer surface is BSN, which means V1(r) represents the volume fraction of the BSN; Structure 2: the outer surface is PZT-4, and

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the inner surface is BSN, which means V1(r) represents the volume fraction of the PZT-4. Their corresponding material parameters under the radial polarization are listed in Table1. For brevity, the undermentioned structures are all Structure 1 unless otherwise specified.

Table 1 Material parameters of two materials under the radial polarization

parameters

C13 7.4 5 e31

C22 13.9 24.7 e32

C23 7.4 5.2 e33

C33 11.5 13.5

C44 2.56 6.5

C55 2.56 6.6

C66 3.05 7.6

∈11

∈22

∈33



12.7 2.8

12.7 3.4

-5.2 -0.4

-5.2 -0.3

15.1 4.3

650 196

650 201

560 28

7.5 5.3

PT

PZT-4 BSN

C12 7.8 10.4 e24

M

PZT-4 BSN

C11 13.9 23.9 e15

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parameters

units: Cij(1010N/m2),∈ij (10-11F/m2), eij (C/m),  (103kg /m3 )

CE

3.1 Comparison with an available reference To the best of our knowledge, to date, rare reference results about waves in the 2-D

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FGP sectorial structures can be found. However, when the radius to thickness η is enough large, the model can be deem to be a rectangular cross-section bar. Accordingly, such a sectorial structure is calculated: a=(106-1)mm, b=106 mm，β=10-6rad, and make a comparison with a square steel bar[25]. Its height and width are both 1mm. Its material parameters used in the present paper are: E=210Gpa, ν=0.3，ρ=7800kg/m3. Fig.2 illustrates their corresponding dispersion curves, where the lines are the results of the present method, and the hollow dotted lines are results from the two dimensional Rayleigh–Ritz method [25]. cp and cs represent the value phase velocity and shear 9

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velocity, respectively. It can be easily seen that the results of the two method agree extremely well.

c p / cs

4 3 2

0

0

1

2

 a / cs

3

4

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1

Fig 2. Dispersion curves for the structure with square cross-section; hollow dotted lines: results from the two dimensional Rayleigh–Ritz method [25], lines: the results from the present method.

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In the present paper, with M or J varing, if the relative error for the same mode :

 =(c( M 1)*JorM*(J+1)  cM *J ) / cM *J 

1 , 1000

(19)

the results can be regarded as convergent. Their corresponding phase velocity values are

M

listed in table.2 with kd=1.01. It can be seen from table.2 that the first two modes are convergent when M=J=4, and the first three modes are convergent when M=5 and J=6.

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Accordingly, the present method is convergent. There are more and more convergent modes as M or J increases.

Table.1 the phase velocity for propagative modes at kd=1.01

AC

CE

PT

M, J 4,4 4,5 4,6 5,4 5,5 5,6 6,4 6,5

The first mode 1.17922 1.17865 1.17864 1.1791 1.17853 1.17852 1.17921 1.17863

The second mode 1.46747 1.46713 1.46713 1.46695 1.46666 1.46667 1.46708 1.46675

The third mode 2.54253 2.54117 2.53942 2.53872 2.53712 2.53546 2.5379 2.53634

unit: km/s

3.2 Dispersion curves In this section, three FGP sectorial structures with different gradient fields are taken into account, i.e., V1(r)=[(r-a)/d]n, n=1, 2, 3, respectively. The complete 3D dispersion curves for a linearly FGP sectorial structure (A) with η=2 and β=π/6 are illustrated in Fig.3. It can be seen that the real (propagative waves), imaginary (evanescent waves, which just exist near the edge of a body) and complex 10

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solutions (evanescent waves too) are all obtained. They are represented by blue, green and red lines, respectively. It is generally known that the real roots correspond to propagative modes, whose amplitudes exhibit sine or cosine with the propagating distance. The purely imaginary roots represent evanescent modes, whose amplitudes exhibit an exponential attenuation with the propagating distance. The complex roots are also evanescent modes, but their amplitude decays follow damped sinusoidal

Im[kd] 2 4

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distribution rather than an exponential decay. 0

6 20

propagative modes

15

evanescent modes

10

 103 5

0 0 1 Re[kd]

(a)

3

20

10

5

4

3 Im[kd] (b)

2

1

5

0

0 0.0

0.5

1.0

1.5 Re[kd] (c)

2.0

2.5

3.0

CE

6

PT

5

15 10

ED

 103

15

 103

M

20

0

2

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evanescent modes

Fig.3 The complete dispersion curves of the FGP sectorial structure (A) with η=2 and

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β=π/6.(a): the complete 3-D curves; (b): Im[kd] vs ω; (c): Re[kd] vs ω .

In order to understand the wave characteristics in detail, the behaviors of the

branches are studied. For the propagative waves, the first four modes have no cut-off frequency. For the fifth and higher modes, a purely real branch starts from a cut-off frequency. For the evanescent waves with pure imaginary roots, some branches start from a cut-off frequency and end at another cut-off frequency. They intersect with propagative wave modes at the cut-off frequencies, where the phase velocity is infinite and the group velocity is zero. Some other branches start from ω=0. For the evanescent waves with complex solutions, most of them start from the plane ω=0 and end at a real 11

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wavenumber point that is located at the lowest frequency of propagative wave mode. Some other branches interconnect the gaps between two adjacent pure imaginary branches or two adjacent pure real branches. There are also some branches starting from ω=0 and ending at an inflection point of an imaginary branch. Im[k]

fd 2.0

1.5

01 2

1.0 0.5 0.0

3 4

Vph=ω/Re(k) 10

propagative modes

5

0 0

1

2

3

0

1.5

2.0

4

1.0

fd

0.5

0.0

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Im[k]

5

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evanescent modes

Vph[km/s]

Vph[km/s]

10

Fig.4 The phase velocity dispersion curves of the FGP sectorial structure (B), blue lines: propagative waves; red lines: evanescent waves. The published results [26-32] have shown that some pseudo surface acoustic wave modes have higher velocities than the classical surface acoustic waves (SAW), and

M

simultaneously have low attenuation. So the corresponding piezoelectric transducers and acoustic devices based on PSAW could have higher resolution. Accordingly, we

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find some similar evanescent wave modes in FGP cylindrical structures with sectorial cross-section. The 3-D phase velocity dispersion curves for the FGP sectorial structure

PT

(B) (η=3 and β=π/6) are illustrated in Fig.4. Here, the phase velocity is calculated via the relation Vph=ω/Re(k). And the imaginary part Im(k) corresponds to the attenuation.

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We can note that there exist some evanescent wave modes that have higher velocities than that of the propagative modes and simultaneously have low attenuation at high

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frequencies. For instance, the second red curve has a very high phase velocity (its Vph is more than 9 km/s but the velocity the first four propagative modes all below 5km/s) and a low attenuation with fd=1.5-1.6 MHz*m. Fig.5 shows the displacement and electric potential for the evanescent mode at fd=1.565MHz*m, k=0.992 +0.139i and r=1mm, β=π/6. We can note that the attenuation of the displacement and electric potential are small, and the evanescent mode could propagate a relatively long distance. These results obtained can be used to improve the performance of transducers. 12

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6

10

4

7.5

2

5

ur 0 u

-2 -4

2.5 0 -2.5

-6

-5

-8

0

5

10

15

20

0

25

5

10

15

20

25

z

z 8 1

6

0.5

4



uz 2

0 -0.5

0

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-1

-2

-1.5

-4

-2 0

5

10

15

20

0

25

5

10

15

20

25

z

z

Fig.5 The displacement and electric potential for the evanescent mode with fd=1.565MHz*m and k=0.992 +0.139i. 10

6 4 2

0

0

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 103

8

1

2 Re[kd]

3

4

M

Fig.6 The dispersion curves for FGP sectorial structure with β=π/6; red lines: η=3; green

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lines: η=4; blue lines: η=5.

To change the size of the cross-section is assumed to be an effective way to govern wave characteristics, such as the values of cut-off frequencies and the shapes of wave

PT

displacements. Firstly, to investigate the influence of the variation of radius-thickness ratios, three FGP sectorial structures with β=π/6 are considered, their radius-thickness

CE

ratios are: (B) η=3, (C): η=4 and (D): η=5. Their corresponding dispersion curves are illustrated in Fig.6. We can note that the cut-off frequencies decrease with the increase

AC

of the radius-thickness ratio. For this kind of structures, the area of the cross-section increases with the increase of η. Therefore, we harbor the idea that the cut-off frequencies decrease as the area of the cross-section increases, i.e., the cut-off frequencies are negatively related to the area of the cross-section. Fig.7 shows the phase velocity dispersion curves for evanescent wave modes. Examining the figure, it can be seen that the influence of the radius-thickness ratio on evanescent wave modes are significant and intricate. For evanescent wave modes at η=3, we can find a range of fd (about 1.4-1.6MHz*m) where its attenuation is low, the dispersion is weak, and its 13

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phase velocity is very high. However, this characteristic does not exist at the structure with η=4. Consequently, the cylindrical structures with sectorial cross-section with

12

12

10

10

η =3 η =4

8

8

Vph=ω/Re(k)

6 4 η =3 η =4

2 0 0.0

0.5

1.0 fd[MHz-m] (a)

1.5

6 4 2 0

2.0

0

1

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Vph[km/s]

Vph[km/s]

β=π/6 and η=3 is relatively more suitable for the evanescent wave transducers.

2 Im[kd]

3

4

(b)

Fig.7 The phase velocity dispersion curves for evanescent wave modes.

Subsequently, the influence of the variation in angular measure is investigated.

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Three FGP sectorial structures both with η=3 are taken into account, whose angular measures are: (B) β=π/6, (E) β=π/4 and (F): β=π/8. Fig.8 shows the corresponding phase velocity dispersion curves. The cut-off frequencies of the fifth and higher modes decrease as the angular measure decreases. For this kind of structures, as the angular

M

measure increases, the area of the cross-section increases, and the cut-off frequencies decrease. Accordingly, the assumption that the cut-off frequencies are negatively related

ED

to the area of the cross-section is confirmed again. Fig.9 shows the phase velocity dispersion curves for evanescent wave modes. We can note that as the angular measure increases, the frequency band of evanescent wave becomes wider, and the maximum of

PT

the phase velocity increases. In summary, variations in geometric sizes of the sectorial cross-section, such as the angular measure and the radius-thickness ratio, have

CE

significant influence on the dispersion curves, which can be used to adjust the

10

8

 10

3

AC

performance of transducers.

6 4 2 0

0

1

2 kl

3

4

Fig.8 The dispersion curves for FGP sectorial structure with η=3; red lines: β=π/4; green lines: β=π/6; blue lines: β=π/8. 14

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14

14

Vph=ω/Re(k)

8

6

8 6

β =π /6

4

2

β =π /4

2

0.5

1.0

1.5 fl[MHz-m]

2.0

β =π /4

10

4

00.0

β =π /6

12

10

Vph[km/s]

Vph[km/s]

12

0

2.5

0

1

2

3

4

Im[kl]

(a)

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(b)

Fig.9 The phase velocity dispersion curves for evanescent wave modes. 5

Vph[km/s]

4 3 2 1

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Vph=ω/Re(k)

0 0.0

0.5

1.0 fd[MHz-m]

1.5

2.0

Fig.10 The phase velocity dispersion curves for FGP Structure 1 (η=3 and β=π/6) with different power exponents; red lines: n=1; green lines: n=2; blue lines: n=3.

8

6 4 2

CE

PT

0 0.0

12

10

Vph[km/s]

Vph=ω/Re(k)

ED

Vph[km/s]

10

M

12

0.5

1.0 fd[MHz-m] (a)

8 6 4

2 1.5

2.0

0

0

1

2 Im[kd] (b)

3

4

Fig.11 The phase velocity dispersion curves for evanescent wave modes; red lines: n=1; green lines: n=2; blue lines: n=3.

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At last, the influence of the gradient index is studied. The phase velocity dispersion

curves for FGP Structure 1 (η=3 and β=π/6) with different power exponents are illustrated in Fig.10. We can note that phase velocities decrease with the increase of n. This phenomenon lies in the fact that the increase of n results in the increase of PZT-4 volume according to Eq.(18). It is generally known that wave velocity is determined by material properties, and wave velocity of BSN is higher than that of PZT-4. Fig.11 shows the phase velocity dispersion curves for evanescent wave modes. We can note that the graded index also has very significant influence on the dispersion curves for 15

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evanescent wave modes. In the range of high velocity and weak dispersion, the velocity values increase as the power exponent decrease. 3.3 The stress and electric displacement distributions

4 3

Trr 0.2 0 1

0.4 0.2 θ

1.25 1.5 r

D 2

0.4

1 0 1 1.25

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0.4

0.2

θ

1.5

1.75

r

20

1.75

20

Fig.12 The stress and electric displacement distributions of the first mode for the FGP

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sectorial structure (A) at kd=3.01.

The stress and electric displacement distributions of the first propagative mode for the FGP sectorial structure (A) at kd=3.01 are calculated, as shown in Fig.12. It is quite clear that the component of stress Trr is 0 at r=1mm and r=2mm, and the component of electric displacement Dθ is 0 at β=0 and β=π/6. So the traction-free and electrically

M

open-circuit boundaries are well satisfied. 3.4 The Poynting vector distribution

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The calculation equations for the Poynting vectors in the sound field can be written as follows:

(19-a)

P  Re[0.5i (Tr  ur*  T  u*  T z  u*z +D   * )]

(19-b)

PT

Pr  Re[0.5i (Trr  ur*  Tr  u*  Trz  u*z +Dr   * )]

CE

Pz 

Re[0.5i (Trz  ur*

 T z  u  Tzz  u*z +Dz *

  )] *

(19-c)

where superscript * represents the complex conjugation.

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As we all know, the Poynting vectors represent the power flow density. The power

flow through the entire cross-section of the cylindrical structure with sectorial cross-section is written as follows: W =

b

a





0

Pdrd , i

(20)

where Pi (i=r, θ, z) represent the Poynting vectors in the r, θ, z directions, respectively. According to Eq.(19), the Poynting vectors and the power flow for the FGP sectorial structure (A) are calculated. Fig.13 shows the Poynting vectors for the 16

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evanescent mode with k=0.992 +0.139i. It can be seen that the Poynting vectors Pi are not zeros, and there is local spatial propagation in a long distance, as shown in Fig.5. However, their integrals over the whole cross-section in all three directions are zeros, i.e., W=0. There is no power flow through the entire cross section, so they are local vibration and just have local energy. The Poynting vector distributions for the first propagative wave mode are

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illustrated in Fig.14. It can be seen that the Poynting vector in the direction z is not zero, and the Poynting vectors in the direction r and θ are zeros. The reason lies in the fact that the wave propagation direction is z direction, and energy also propagates in this direction. Besides, it can be seen from Fig.14 that the Poynting vector distributions are

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symmetrical with respect to variable θ.

2 Pr 0 -2

2 P 0 -2 1

0.4

1

0.2

1.25

1.25

θ

1.75

1.5 r

20

1.75

θ

20

M

1.5 r

0.4 0.2

5 2.5 0 -2.5 1

ED

Pz

0.4 0.2 θ

1.25 1.5 r

1.75

PT

20

AC

CE

Fig.13 The Poynting vectors for the evanescent modes with k=0.992 +0.139i.

0.5

0.5 Pr 0

P

0.4

-0.5 1

0.2 θ

1.25 1.5 r

1.75

Pz

0

0.4

-0.5 1

0.2 θ

1.25 1.5 r

20

3 2 1 1

1.75

20

0.4 0.2 θ

1.25 1.5 r

1.75

20

Fig.14 The Poynting vectors of the first propagative mode for the FGP sectorial 17

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structure (A) at k=2.05 Subsequently, the case of big wave number is analyzed. When k=100.01, Fig.15 illustrates the Poynting vector distributions in the direction z for FGP cylindrical Structure 1 and 2. Their geomitry both are η=2 and β=π/6. The full Poynting vector distribution figure can not illustrate detailed distributions, so the partial enlarged drawings are made to more clearly illustrate the Poynting vector distributions. The

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enlarged drawings of the Poynting vectors seem to be discontinuous because of the exist of platforms. In fact, the Poynting vectors are continuous, and the platform is resulted from the truncation of larger values. We can also note that the Poytning vectors mainly distribute in the region with more PZT-4, that is, the inner side for Structure 1 and the outer side for Structure 2. In conclusion, we hold the view that the Poynting vectors

0.4

1

0.2

1.25 1.5

r (a)

θ

3 2 Pz 1 0 1

1.75

0.6 0.4

1.25

M

10 7.5 Pz 5 2.5 0

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mainly distribute near the side with low wave velocity.

r (a)

20

θ

0.2

1.5 1.75

20

4. Conclusion

ED

Fig.15 Poynting vectors of the first mode at k=100.01; (a): Structure 1; (b): Structure 2

PT

Based on the 3-D linear electro-elasticity theory, guided waves in FGP cylindrical structures with sectorial cross-section are investigated by using the modified double

CE

orthogonal polynomial series method, which can obtain simultaneously real, imaginary and complex solutions without iterative process. The boundary conditions are

AC

incorporated into constitutive equations by virtue of the Heaviside function. Dispersion curves, electric potential and the Poynting vector distributions are illustrated, and the influences of angular measure, radius-thickness ratio and graded index on dispersion curves are analyzed. According to the above results, the following conclusions can be drawn: (1) The validity of the present method is confirmed by numerical comparison with available reference results. (2) There exist some evanescent wave modes in the FGP sectorial structures. They 18

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have higher velocities than that of the propagative modes and simultaneously have low attenuation at some frequencies. (3) The wave characteristics, such as wave speed and cut-off frequency, can be adjusted by changing the size of the cross-section. The cut-off frequencies are negatively related to the area of the cross-section.

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(4) For the evanescent wave modes, their Poynting vectors are not zeros, but their power flows over the entire cross-section are zeros. They are local vibration and just have local energy, but sometimes they can propagate long distances. Acknowledgement

The authors gratefully acknowledge the support by the National Natural Science

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Foundation of China (No.U1504106), the Program for Innovative Research Team of Henan Polytechnic University (No. T2017-3) and the Fundamental Research Funds for the Universities of Henan Province (No.NSFRF140301) and the Program for Science and Technology Innovation Team in Universities of Henan Province (15IRTSTHN013).

M

Appendix

The explicit expressions for the element are A11n, p ,m, j =-

1 (l ) n , p , m, j l n , p , m, j l n , p , m, j = A23 = A24 =0, C44 u[n, p, l  2, m, j,0,0], l A12n, p,m, j = l A13n, p,m, j = l A21 l d

l

A14n, p ,m, j =-

1 (l ) 1 (l ) e u[n, p, l  2, m, j,0,0], l A22n, p ,m, j =- l C66 u[n, p, l  2, m, j,0,0], l 24 d d

l

n , p , m, j l n , p , m, j A31n, p,m, j = l A32n, p,m, j = l A34n, p,m, j = l A42 = A43 =0, l A33n, p ,m, j =-

l

A41n, p ,m, j =-

PT

CE

AC l

ED

l

1 (l ) C22 u[n, p, l  2, m, j,0,0], dl

1 (l ) 1 e u[n, p, l  2, m, j,0,0], l A44n, p ,m, j = l (22l ) u[n, p, l  2, m, j,0,0], l 24 d d

n , p ,m , j n , p ,m , j n , p ,m , j B11n, p,m, j = l B12n, p ,m, j  l B14n, p ,m, j = l B21  l B22  l B24  0,

1 (l ) {i  C(23l ) +C(44l )  u[n, p, l  2, m, j,1, 0]  i  C(23l ) -C12  u[n, p, l 1, m, j, 0, 0] dl iC(23l )lu[n, p, l  1, m, j, 0, 0]  iC(23l ) K r [n, p, l  2, m, j, 0, 0]},

l

B13n, p ,m, j =

l

B23n, p ,m, j =

1 (l ) (l ) {i  C12 +C(66l )  u[n, p, l  1, m, j,0,1]  iC12 K [n, p, l  1, m, j,0,0]}, l d 19

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1 {i  C(23l ) +C(44l )  u[n, p, l  2, m, j,1, 0]  i  C(23l ) +C(44l )  u[ n, p, l  1, m, j, 0, 0] dl iC(44l )lu[n, p, l  1, m, j, 0, 0]  iC(44l ) K r [n, p, l  2, m, j, 0, 0]},

l

B31n, p ,m, j =

l

B32n, p ,m, j =

l

n , p ,m, j n , p ,m , j n , p ,m , j n , p ,m , j B33n, p,m, j = l B41  l B42 = l B43  l B44  0,

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1 (l ) {i  C12 +C(66l )  u[n, p, l  1, m, j,0,1]  iC(66l ) K [n, p, l  1, m, j,0,0]}, l d

1 (l ) (l ) (l ) {i  e24 +e32 u[n, p, l  2, m, j,1, 0]  ie24 l  1 u[n, p, l  1, m, j, 0, 0]  l d (l ) ie24 K r [n, p, l  2, m, j, 0, 0]},

l

B34n, p ,m, j =

1 (l ) (l ) (l ) {i  e24 +e32 u[n, p, l  2, m, j,1, 0]  ie32 l  1 u[n, p, l  1, m, j, 0, 0]  l d ie32(l ) K r [n, p, l  2, m, j, 0, 0]}, B43n, p ,m, j =

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l

1 (l ) (l ) {C33 u[n, p, l  2, m, j, 2, 0]  C55 u[ n, p, l , m, j, 0, 2] l d (l ) C33  l  1 u[n, p, l  1, m, j,1, 0]  C11(l )u[n, p, l, m, j, 0, 0]  C13(l )lu[n, p, l, m, j, 0, 0]

l

C11n , p ,m, j =

M

(l ) (l ) (l ) C33 K r [n, p, l  2, m, j,1, 0]  C13 K r [n, p, l  1, m, j, 0, 0]  C55 K [n, p, l , m, j, 0,1]},

1 (l ) (l ) (l ) (l ) {(C33  C55 )u[n, p, l  1, m, j,1,1]  (C11  C55 )u[ n, p, l , m, j, 0,1] dl (l ) (l ) C33 lu[n, p, l , m, j, 0,1]  C13 K r [n, p, l  1, m, j, 0,1] C12n , p ,m, j =

ED

l

l

PT

(l ) (l ) C55 K [n, p, l  1, m, j,1, 0]  C55 K [n, p, l , m, j, 0, 0]}, n , p , m, j C13n, p,m, j = lC23n, p,m, j  l C31n, p,m, j = l C32n, p,m, j  l C34n, p,m, j  l C43  0,

1 {(e33(l )  e31(l ) )u[n, p, l  1, m, j,1, 0]  e33(l )lu[n, p, l  1, m, j,1, 0] l d (l ) e33 u[n, p, l  2, m, j, 2, 0]  e15(l )u[n, p, l , m, j, 0, 2]  e33(l ) K r [n, p, l  2, m, j,1, 0] C14n , p ,m, j =

CE

l

AC

e15(l ) K [n, p, l , m, j , 0,1]}, 1 (l ) (l ) (l ) (l ) {(C13  C55 )u[n, p, l  1, m, j,1,1]  (C11  C55 )u[ n, p, l , m, j, 0,1] l d (l ) (l ) C55 lu[n, p, l , m, j, 0,1]  C55 K r [n, p, l  1, m, j, 0,1]

l

C21n , p ,m, j =

(l ) (l ) C13 K [n, p, l  1, m, j,1, 0]  C11 K [n, p, l , m, j, 0, 0]},

20

ACCEPTED MANUSCRIPT

1 (l ) (l ) {C55 u[n, p, l  2, m, j, 2, 0]  C55  l  1 u[n, p, l, m, j, 0, 0] l d (l ) C55  l  1 u[n, p, l  1, m, j,1, 0]  C11(l )u[n, p, l, m, j, 0, 2]

l

C22n , p ,m, j =

(l ) (l ) (l ) C55 K r [n, p, l  2, m, j,1, 0]  C55 K r [n, p, l  1, m, j, 0, 0]  C11 K [ n, p, l , m, j, 0,1]},

1 (l ) {(e15(l )  e31 )u[n, p, l  1, m, j,1,1]  e15(l ) (l  1)u[n, p, l  1, m, j,1, 0] l d (l ) (l ) e15 K r [n, p, l  1, m, j , 0,1]  e31 K [n, p, l  1, m, j,1, 0]}, C24n , p ,m, j =

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l

1 (l ) {C44 u[n, p, l  2, m, j, 2, 0]  C(44l )  l  1 u[n, p, l  1, m, j,1, 0] l d (l ) C66 u[n, p, l , m, j, 0, 2]  C(44l ) K r [n, p, l  2, m, j,1, 0]  C (66l ) K [ n, p, l , m, j, 0,1]},

l

C33n , p ,m, j =

1 {(e33(l )  e31(l ) )u[n, p, l  1, m, j ,1, 0]  e31(l )lu[n, p, l , m, j , 0, 0]  e33(l )lu[n, p, l  1, m, j ,1, 0] l d (l ) e33 u[n, p, l  2, m, j , 2, 0]  e15(l )u[n, p, l , m, j , 0, 2]  e33(l ) K r [n, p, l  2, m, j ,1, 0] C41n , p ,m, j =

AN US

l

e31(l ) K r [n, p, l  1, m, j , 0, 0]  e15(l ) K [n, p, l , m, j , 0,1]},

1 (l ) (l ) {(e15(l )  e31 )u[n, p, l  1, m, j,1,1]  e15(l )u[n, p, l , m, j, 0,1]  e31 lu[ n, p, l , m, j, 0,1] dl (l ) e31 K r [n, p, l  1, m, j , 0,1]  e15(l ) K [n, p, l  1, m, j,1, 0]  e15(l ) K [n, p, l , m, j, 0, 0]}, C42n , p ,m, j =

M

l

1 (l ) (l ) {- 33 u[n, p, l  2, m, j, 2, 0] 33  l  1 u[n, p, l  1, m, j,1, 0] dl (l ) (l ) (l )  11 u[n, p, l , m, j, 0, 2] 33 K r [n, p, l  2, m, j,1, 0] 11 K [n, p, l , m, j, 0,1]}, C44n , p ,m, j =

l

M11n, p, j ,m =--1(l ) 2u[n, p, l  2, m, j,0,0],

CE

where

PT

ED

l

q  g Qm (r )  Q j ( )  drd , a 0 r g  q q b  I ( r ,  )  g Qm (r )  Q j ( ) s K r [n, p, s, m, j, g , q]=    Qn (r )  Qp ( ) r   drd , a 0 r r g  q q b  I ( r ,  )  g Qm (r )  Q j ( ) s K [n, p, s, m, j, g , q]=    Qn (r )  Qp ( ) r   drd . a 0  r g  q b



I (r ,  )  Qn (r )  Qp ( ) r s 

AC

u[n, p, s, m, j, g , q]= 

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