Dynamic electromechanical behavior of a triple-layer piezoelectric composite cylinder with imperfect interfaces

Applied Mathematical Modelling 35 (2011) 1765–1781

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Dynamic electromechanical behavior of a triple-layer piezoelectric composite cylinder with imperfect interfaces H.M. Wang ⇑ Department of Mechanics, School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, PR China

a r t i c l e

i n f o

Article history: Received 30 May 2009 Received in revised form 15 September 2010 Accepted 4 October 2010 Available online 13 October 2010 Keywords: Electromechanical behavior Composite cylinder Imperfect interface Piezoelectric layer

a b s t r a c t The dynamic electromechanical behavior of a triple-layer piezoelectric composite cylinder with imperfect interfaces is investigated. The composite cylinder is constructed by two elastic layers and an embedded piezoelectric layer. A linear spring model is adopted to describe the weakness of imperfect interface. The exact analysis is performed by the state space method and normal mode expansion method. The determining procedure for the eigenfunction and the proof of the orthogonal property of the eigenfunction is presented for an imperfectly bonded triple-layer piezoelectric composite cylinder. The obtained solution is valid for analyzing the dynamic electromechanical behavior of composite cylinder with arbitrary thickness for both elastic and piezoelectric layers. Numerical results show that the weakness of imperfect interface has significant effect on the transient electromechanical responses of piezoelectric composite cylinder. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction Due to the special electromechanical coupling effects, piezoelectric materials have been widely used in many industrial applications such as vibration control, ultrasonic motor, structural health monitoring, intelligent systems etc. With the increasing requirement in materials, more and more piezoelectric composites are explored and many new devices and systems have been designed and manufactured [1–3]. The advantage of composites is that they usually exhibit the best qualities of their constituents. In order to know the performance of the piezoelectric composites exactly, a deeper investigation for their mechanical and electric characteristics is always required. Many theoretical investigations have been reported in the analysis for piezoelectric composite structures with perfectly bonded interfaces. The topics include static analysis, free vibration, wave propagation and transient response. Heyliger [4] studied the static behavior of simply supported laminated piezoelectric cylinders. Wang and Zhong [5] analyzed the mechanical, electric and magnetic fields in a finitely long circular cylindrical shell of a piezoelectric/piezomagnetic composite under pressuring and temperature change. Chen and Shi [6] completed the exact static analysis of double-layered piezoelectric hollow cylinder under some coupled loadings. Kharouf and Heyliger [7] studied the axisymmetric free vibrations laminated piezoelectric cylinders. Hussein and Heyliger [8] investigated the three-dimensional vibrations of layered piezoelectric cylinders. Chen et al. [9] presented a perfectly bonded laminate model to study the 3D free vibration of a functionally graded piezoelectric hollow cylinder filled with compressible fluid. Wang et al. [10] studied the SH wave propagating around a long metallic cylinder covered with a piezoelectric layer. Du et al. [11] investigated the SH wave propagation in a cylindrically layered piezoelectric structures with initial stress. Li and Lee reported the fracture analysis of a cylindrical piezoelectric

⇑ Tel.: +86 571 8795 2396; fax: +86 571 8795 2570. E-mail address: [email protected] 0307-904X/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2010.10.008

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H.M. Wang / Applied Mathematical Modelling 35 (2011) 1765–1781

composite [12]. Wang et al. [13,14] further studied the transient responses in two-layered elasto-piezoelectric composite hollow cylinder and multilayered piezoelectric hollow cylinder, respectively. On the other hand, for composite structures, there often have inherent disadvantages, for instance, the presence of crack or defect or delamination in the interface during the fabrication process. Recently, the investigations considering the interlaminar bonding imperfections have been increasing interested by scientist and engineers. Huang and Rohhlin [15] investigated the interface waves propagating along an anisotropic imperfect interface between anisotropic solids. Cheng et al. [16] studied the static bending behavior of multilayered anisotropic plates with weakened interfaces. Berger et al. [17] analyzed dispersion curves of torsional waves in a biomaterial elastic cylinder with an imperfect interface. Chen et al. [18] and Chen and Lee [19] obtained the exact solutions of laminated orthotropic piezoelectric rectangular plates and angle-ply piezoelectric-laminated cylindrical panels with weak interfaces, respectively. Wang et al. [20] studied the scattering of antiplane shear wave by a piezoelectric circular cylinder with an imperfect interface. Li and Lee [21–23] carried out the fracture analysis on the interfacial imperfections in piezoelectric layered structures. There are rare publications can be cited in transient analysis of piezoelectric composite structures by taking into account the effects of imperfectly bonding interface. The motivation for this study comes from the application of piezoelectric cylindrical actuators and sensors with their inner and outer layers are the electrodes. In theoretical analysis, the electrodes can be treated as elastic layers and then such actuators and sensors can be modeled as triple-layer piezoelectric composite cylinders and the effect of the electrodes on the static response has been considered [24]. Due to the difficulties, up to now, no results have been reported for the dynamic analysis for a triple-layer piezoelectric composite cylinder with imperfection interfaces. In this investigation, a general linear spring-layer model is employ to describe the weakness of the imperfect interface between the piezoelectric layer and the electrode layer. The elastodynamic solution of a triple-layer piezoelectric composite cylinder with imperfection interfaces is obtained and the dynamic electromechnical characteristics are illustrated graphically. 2. Basic equations Fig. 1 shows the cross-section of an infinite long triple-layer piezoelectric composite hollow cylinder. The radius of each interface from the inner to the outer is denoted as ri (i = 0, 1, 2, 3). The middle layer is piezoelectric and the inner and outer layers are elastic. In the following statements and equations, the quantities with superscripts ‘‘1” and ‘‘3” denote those for elastic layers and the quantities with superscript ‘‘2” denote those for piezoelectric layer. In the polar coordinate system, for radial vibration, we have

urðiÞ ¼ urðiÞ ðr; tÞ ði ¼ 1; 2; 3Þ;

ð1Þ

Uð2Þ ¼ Uð2Þ ðr; tÞ; ðiÞ

where ur is the radial displacement component in both elastic and piezoelectric layers and U(2) denotes the electric potential in piezoelectric layer. If both elastic and piezoelectric layers characterize material orthotropy and the piezoelectric layer is polarized radially, then the constitutive relations are ðiÞ

ðiÞ

ur ðiÞ @ur þ c13 ; r @r ðiÞ ðiÞ ðiÞ ur ðiÞ @ur ¼ c13 þ c33 ði ¼ 1; 3Þ; r @r

ðiÞ rðiÞ hh ¼ c 11

r

ðiÞ rr

ð2aÞ

piezoelectric layer r3

elastic layer

r2

r

θ

r1

2(t)

r0

outer interface

1(t)

inner interface

Fig. 1. Cross-section of piezoelectric composite hollow cylinder.

H.M. Wang / Applied Mathematical Modelling 35 (2011) 1765–1781

1767

and ð2Þ

ð2Þ

ð2Þ ur ð2Þ @ur ð2Þ @ U þ c13 þ e31 ; r @r @r ð2Þ ð2Þ ð2Þ ð2Þ ur ð2Þ @ur ð2Þ @ U ¼ c13 þ c33 þ e33 ; r @r @r ð2Þ ð2Þ ð2Þ ð2Þ ur ð2Þ @ur ð2Þ @ U ¼ e31 þ e33  e33 ; r @r @r

ð2Þ rð2Þ hh ¼ c 11

rð2Þ rr Dð2Þ rr

ð2bÞ

ðiÞ

where rjj ði ¼ 1; 2; 3; j ¼ r; hÞ are the components of stress, and Dð2Þ rr is the radial electric displacement in piezoelectric layer. ðiÞ ð2Þ ð2Þ ð2Þ cjm ði ¼ 1; 2; 3; j ¼ 1; 3; m ¼ 1; 3Þ; e31 ; e33 and e33 are the elastic, piezoelectric and dielectric constants, respectively. The equation of motion is ðiÞ

ðiÞ

ðiÞ

ðiÞ

@ rrr rrr  rhh @ 2 ur ði ¼ 1; 2; 3Þ; þ ¼ qðiÞ @r r @t2

ð3Þ

where q(i) is the mass density of the ith layer. For piezoelectric layer, in the absence of free charge density, the charge equation of electrostatics is

1 @ h ð2Þ i rDrr ¼ 0: r @r

ð4Þ

Suppose the piezoelectric composite cylinder is subjected to dynamic pressure Q1(t) and Q2(t) at the inner and outer surfaces, respectively. U1(t) and U2(t) are the prescribed electric potential excitations applied on the internal and external surfaces of piezoelectric layer, respectively. Then the boundary conditions are expressed as

rrrð1Þ ðr0 ; tÞ ¼ Q 1 ðtÞ; rrrð3Þ ðr3 ; tÞ ¼ Q 2 ðtÞ; ð2Þ

U ðr1 ; tÞ ¼ U1 ðtÞ;

ð2Þ

U ðr 2 ; tÞ ¼ U2 ðtÞ:

ð5Þ ð6Þ

For piezoelectric composites under electromechanical loading, the interface might be mechanically and/or electrically imperfect. It is found that the former has more remarkable effect on the fracture behavior of the composite [21–23], that is to say, the former is more serious than the latter. In the current paper, only the more serious interfacial imperfection (i.e., mechanical imperfection) is considered. Here, the imperfect interface is modeled as linear spring layers as extensively employed in many publications [18–23]. Thus ð1Þ ð2Þ rð2Þ rð3Þ rr ðr 1 ; tÞ ¼ rrr ðr 1 ; tÞ; rr ðr 2 ; tÞ ¼ rrr ðr 2 ; tÞ; ð1Þ ðr 1 ; tÞ; urð2Þ ðr 1 ; tÞ  urð1Þ ðr 1 ; tÞ ¼ v1 rrr ð3Þ ð2Þ ð2Þ ur ðr 2 ; tÞ  ur ðr 2 ; tÞ ¼ v2 rrr ðr 2 ; tÞ;

ð7Þ

where v1 and v2 are the compliance constants of the inner and outer interfaces. Especially, for perfectly bonded interface, we have vi = 0 (i = 1, 2). Suppose the composite cylinder is at rest at the prior, then the initial conditions (t = 0) for each layer are expressed as

urðiÞ ðr; 0Þ ¼ 0;

u_ ðiÞ r ðr; 0Þ ¼ 0 ði ¼ 1; 2; 3Þ;

ð8Þ

where a dot over a quantity denotes partial derivative with respect to time t. 3. Governing equations for mechanical field The solution of Eq. (4) can be obtained as ð2Þ Drr ðr; tÞ ¼ yðtÞ=r;

ð9Þ

where y(t) is an unknown function with respect to time t. Then the third equation in Eq. (2b) can be rewritten as. ð2Þ

ð2Þ

ð2Þ

ð2Þ

@ Uð2Þ e31 ur e @ur 1 yðtÞ : ¼ ð2Þ þ 33  ð2Þ ð2Þ @r @r r e33 e33 e33 r

ð10Þ

The substitution of Eq. (10) into the first two equations in Eq. (2b) derives ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

~ð2Þ rð2Þ hh ¼ c 11

ur e31 yðtÞ ð2Þ @ur ; þ ~c13  ð2Þ r @r e33 r

ð2Þ ¼ ~c13

e33 yðtÞ ur ð2Þ @ur ; þ ~c33  ð2Þ r @r e33 r

r

ð2Þ rr

ð11Þ

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H.M. Wang / Applied Mathematical Modelling 35 (2011) 1765–1781

where ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ~c11 ¼ c11 þ e31 e31 =e33 ;

ð2Þ ð2Þ ð2Þ ð2Þ ~cð2Þ 13 ¼ c 13 þ e31 e33 =e33 ;

ð2Þ ð2Þ ð2Þ ð2Þ ~cð2Þ 33 ¼ c 33 þ e33 e33 =e33 :

ð12Þ

For the sake of simplicity, the following non-dimensional quantities and variables are introduced as ðiÞ

uðiÞ ¼ n¼

ur ; r3

r ; r3

/1 ¼

rðiÞ j ¼

s¼

U1 ; U0

rðiÞ jj ð2Þ c33

ð2Þ

cv t; r3

/2 ¼

/ð2Þ ¼

ðj ¼ r; hÞ;

ð2Þ

ej

¼

e3j

e0

;

q ðiÞ ¼

Uð2Þ ; U0

ð1Þ cjm ¼

cjm

ð2Þ cjm ¼

; ð2Þ

c33

ð2Þ Drr ; e0

g¼

y ; r3 e0

qðiÞ Q Q ; q1 ¼ ð2Þ1 ; q2 ¼ ð2Þ2 ; qð2Þ c33 c33

ð1Þ

U2 ; U0

Dð2Þ r ¼

ð2Þ ~cjm

; ð2Þ

c33

ð3Þ

ð3Þ cjm ¼

cjm

ð2Þ

c33

ðj; m ¼ 1; 3Þ;

ð2Þ

r0 r1 r2 r3 c  1 ¼ 33 ; n1 ¼ ; n2 ¼ ; n3 ¼ ¼ 1; v r3 r3 r3 r3 r3 vffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffi u ð2Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2Þ uc c33 ð2Þ ð2Þ ; e0 ¼ c33 e33 ; cv ¼ U0 ¼ r3 t 33 : ð2Þ ð2Þ n0 ¼

ð13Þ

ð2Þ

v1 ; v 2 ¼

c33 r3

v2 ;

q

e33

By virtue of Eq. (13), Eqs. (2a), (11) and (3) can be rewritten as ðiÞ ðiÞ ðiÞ RhðiÞ ¼ c11 u þ c13 ruðiÞ ;

ð14aÞ

ðiÞ ðiÞ ðiÞ RrðiÞ ¼ c13 u þ c33 ruðiÞ ði ¼ 1; 3Þ; ð2Þ ð2Þ ð2Þ þ c13 Rð2Þ ruð2Þ  e1ð2Þ gðsÞ; h ¼ c 11 u

ð14bÞ

ð2Þ ð2Þ ð2Þ þ c33 Rð2Þ ruð2Þ  e3ð2Þ gðsÞ; r ¼ c 13 u

 ðiÞ rRrðiÞ  RhðiÞ ¼ n2 q

@ 2 uðiÞ ði ¼ 1; 2; 3Þ; @ s2

ð15Þ

@ : @n

ð16Þ

where

RhðiÞ ¼ nrhðiÞ ;

RrðiÞ ¼ nrrðiÞ ;

r¼n

Subsequently, Eqs. (5), (7) and (8) can be rewritten as

Rrð1Þ ðn0 ; sÞ ¼ n0 q1 ðsÞ;

Rrð3Þ ð1; sÞ ¼ q2 ðsÞ;

ð1Þ Rð2Þ r ðn1 ; sÞ ¼ Rr ðn1 ; sÞ;

Rrð3Þ ðn2 ; sÞ ¼ Rð2Þ r ðn2 ; sÞ;

 1 Rrð1Þ ðn1 ; sÞ=n1 ; u ðn1 ; sÞ  u ðn1 ; sÞ ¼ v ð2Þ

ð1Þ

ð3Þ

ð2Þ

2R u ðn2 ; sÞ  u ðn2 ; sÞ ¼ v uðiÞ ðn; 0Þ ¼ 0;

ð17Þ

ð2Þ r ðn2 ;

ð18Þ

sÞ=n2 ;

u_ ðiÞ ðn; 0Þ ¼ 0 ði ¼ 1; 2; 3Þ;

ð19Þ

In Eq. (19) and hereafter, a dot over a quantity denotes its partial derivative with respect to the non-dimensional time s. 4. Solution for mechanical field The solution of elastodynamic equations can be obtained by the eigenfunction expansion method [25,26]. In solving the elastodynamic problems with inhomogeneous mechanical boundary conditions, one key step is the homogenization of the mechanical boundary conditions. Based on this homogenization procedure, then we can deduce an eigenequation, from which the natural frequencies can be determined. The homogenization of the mechanical boundary conditions can be realized by the superposition method. In this method, the complete solution is treated as the superposition of a quasi-static solution and a dynamic one [25,26]. This section will present a detailed procedure in solving the considered elastodynamic problem. 4.1. The homogenization for mechanical boundary conditions To homogenize the mechanical boundary conditions, the displacement and stresses are treated as ðiÞ

uðiÞ ¼ usðiÞ þ ud ;

ðiÞ RrðiÞ ¼ RrsðiÞ þ Rrd ;

ðiÞ ðiÞ RðiÞ h ¼ Rhs þ Rhd ði ¼ 1; 2; 3Þ;

ð20Þ

1769

H.M. Wang / Applied Mathematical Modelling 35 (2011) 1765–1781 ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

where us ; RðiÞ rs and Rhs are known as quasi-static solution and ud ; Rrd and Rhd are dynamic solution. The governing equations for quasi-static solution are given as ðiÞ ðiÞ ðiÞ ðiÞ RðiÞ hs ¼ c 11 us þ c 13 rus ;

ð21aÞ

ðiÞ ðiÞ ðiÞ ðiÞ RðiÞ rs ¼ c 13 us þ c 33 rus ði ¼ 1; 3Þ; ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ Rð2Þ hs ¼ c 11 us þ c 13 rus  e1 gðsÞ;

ð21bÞ

ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ Rð2Þ rs ¼ c 13 us þ c 33 rus  e3 gðsÞ; ðiÞ rRrsðiÞ  Rhs ¼ 0 ði ¼ 1; 2; 3Þ;

R

ð1Þ rs ðn0 ;

sÞ ¼ n0 q1 ðsÞ; R

ð22Þ

ð3Þ rs ð1;

ð1Þ Rð2Þ rs ðn1 ; sÞ ¼ Rrs ðn1 ; sÞ;

sÞ ¼ q2 ðsÞ;

ð23Þ

Rrsð3Þ ðn2 ; sÞ ¼ Rð2Þ rs ðn2 ; sÞ;

 ð1Þ usð2Þ ðn1 ; sÞ  uð1Þ s ðn1 ; sÞ ¼ v1 Rrs ðn1 ; sÞ=n1 ; usð3Þ ðn2 ;

sÞ 

uð2Þ s ðn2 ;

sÞ ¼ v 2 R

ð2Þ rs ðn2 ;

ð24Þ

sÞ=n2 ;

Substituting Eq. (20) into Eqs. (14a), (14b) and (15) as well as Eqs. (17)–(19) and utilizing Eqs. (21a), (21b), (22)–(24), the ðiÞ ðiÞ ðiÞ governing equations for dynamic solution ud ; Rrd and Rhd are then constructed as ðiÞ ðiÞ ðiÞ ðiÞ RðiÞ hd ¼ c 11 ud þ c 13 rud ;

ð25Þ

ðiÞ ðiÞ ðiÞ ðiÞ RðiÞ rd ¼ c 13 ud þ c 33 rud ði ¼ 1; 2; 3Þ; ðiÞ ðiÞ  ðiÞ n2 ðu € dðiÞ þ u € sðiÞ Þ ði ¼ 1; 2; 3Þ; rRrd  Rhd ¼ q

ð26Þ

Rð1Þ rd ðn0 ; sÞ ¼ 0;

ð27Þ

Rð3Þ rd ð1; sÞ ¼ 0;

ð1Þ Rð2Þ rd ðn1 ; sÞ ¼ Rrd ðn1 ; sÞ; ð2Þ

ð3Þ ð2Þ Rrd ðn2 ; sÞ ¼ Rrd ðn2 ; sÞ;

ð1Þ

ð1Þ

 1 Rrd ðn1 ; sÞ=n1 ; ud ðn1 ; sÞ  ud ðn1 ; sÞ ¼ v ð3Þ ud ðn2 ;

sÞ 

ð2Þ ud ðn2 ;

ð28Þ

sÞ ¼ v 2 Rrdð2Þ ðn2 ; sÞ=n2 ;

ðiÞ

ðiÞ u_ d ðn; 0Þ ¼ u_ ðiÞ s ðn; 0Þ ði ¼ 1; 2; 3Þ:

ud ðn; 0Þ ¼ usðiÞ ðn; 0Þ;

ð29Þ

Eq. (27) denotes that the inhomogeneous mechanical boundary conditions (17) have been transformed into homogeneous ones. 4.2. Quasi-static solution ðiÞ

State space method will be employed to develop the quasi-static solution. First, the expressions for rus ði ¼ 1; 2; 3Þ are obtained from the second equations in Eqs. (21a) and (21b) as





ðiÞ ðiÞ =cðiÞ rusðiÞ ¼ RðiÞ rs  c13 us 33 ði ¼ 1; 3Þ; 



ð30Þ

ð2Þ ð2Þ ð2Þ ð2Þ rusð2Þ ¼ Rrsð2Þ  c13 us þ e3 gðsÞ =c33 ;

Then substituting the first equations in Eqs. (21a) and (21b) into Eq. (22) and utilizing Eq. (30), we obtain the expression for ðiÞ ðiÞ rRðiÞ rs ði ¼ 1; 2; 3Þ. Finally, expressions for rus and rRrs ði ¼ 1; 2; 3Þ can be rewritten in a matrix form as

rX ðiÞ ðn; sÞ ¼ N ðiÞ X ðiÞ ðn; sÞ ði ¼ 1; 3Þ;

ð31aÞ

rX ð2Þ ðn; sÞ ¼ N ð2Þ X ð2Þ ðn; sÞ þ Lð2Þ gðsÞ;

ð31bÞ

where

( X ðiÞ ðn; sÞ ¼

ðiÞ

us ðn; sÞ

RrsðiÞ ðn; sÞ

"

) ;

N ðiÞ ¼

ðiÞ

ðiÞ

a11

a12

ðiÞ

a22

a21

ðiÞ

#

( ði ¼ 1; 2; 3Þ;

Lð2Þ ¼

ð2Þ

a13

ð2Þ

a23

) ð32Þ

;

in which ðiÞ

ðiÞ

ðiÞ

a11 ¼ c13 =c33 ;

ðiÞ

ðiÞ

a12 ¼ 1=c33 ;

ðiÞ

ðiÞ

ðiÞ

ðiÞ

a21 ¼ c11 þ c13 a11 ;

ðiÞ

ðiÞ

ðiÞ

a22 ¼ c13 a12 ði ¼ 1; 2; 3Þ;

ð2Þ ð2Þ ð2Þ a13 ¼ e3 =c33 ;

ð2Þ

ð2Þ ð2Þ

ð2Þ

a23 ¼ c13 a13  e1 : ð33Þ

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H.M. Wang / Applied Mathematical Modelling 35 (2011) 1765–1781

Eqs. (31a) and (31b) are just the state equations for solving the quasi-static solution. The solutions of Eqs. (31a) and (31b) are

X ðiÞ ðn; sÞ ¼ T ðiÞ ðnÞX ðiÞ ðni1 ; sÞ ðni1 6 n 6 ni ; i ¼ 1; 3Þ;

ð34aÞ

X ð2Þ ðn; sÞ ¼ T ð2Þ ðnÞ½X ð2Þ ðn1 ; sÞ þ Gð2Þ ðnÞgðsÞ ðn1 6 n 6 n2 Þ;

ð34bÞ

where ðiÞ

T ðiÞ ðnÞ ¼ ðn=ni1 ÞN ði ¼ 1; 2; 3Þ; Z n Gð2Þ ðnÞ ¼ f1 ½T ð2Þ ðfÞ1 Lð2Þ df:

ð35Þ

n1

Utilizing the introduced state vector X(i)(n, s), Eq. (28), the transfer relations at the weakly interfaces, can be rewritten as

X ð2Þ ðn1 ; sÞ ¼ P ð1Þ X ð1Þ ðn1 ; sÞ;

ð36Þ

X ð3Þ ðn2 ; sÞ ¼ P ð2Þ X ð3Þ ðn2 ; sÞ; where

P ð1Þ ¼



1

v 1 =n1

0

1

 ;

P ð2Þ ¼



1

v 2 =n2

0

1

 ð37Þ

:

With the aid of Eqs. (34a), (34b) and (36), we have

X ð3Þ ðn3 ; sÞ ¼ HX ð1Þ ðn0 ; sÞ þ M gðsÞ;

ð38Þ

where

H ¼ T ð3Þ ðn3 ÞP ð2Þ T ð2Þ ðn2 ÞP ð1Þ T ð1Þ ðn1 Þ;

ð39Þ

M ¼ T ð3Þ ðn3 ÞP ð2Þ T ð2Þ ðn2 ÞGð2Þ ðn2 Þ: Utilizing the boundary conditions (23), Eq. (38) can be written in specified form as

(

ð3Þ

us ð1; sÞ q2 ðsÞ

)

 ¼

H11

H12

H21

H22

(

ð1Þ

us ðn0 ; sÞ n 0 q 1 ð sÞ

)

 þ

M1 M2



gðsÞ:

ð40Þ

From the second equation in Eq. (40), we obtain

uð1Þ s ðn0 ; sÞ ¼ ½q2 ðsÞ  H 22 n0 q1 ðsÞ  M 2 gðsÞ=H 21 : Substituting Eq. (41) into Eqs. (34a), (34b) and utilizing Eq. (36),

uðiÞ s ðn;

sÞ ¼

ðiÞ f1 ðnÞq1 ð

sÞ þ

ðiÞ f2 ðnÞq2 ð

sÞ þ

The detailed procedure for determining

ð41Þ ðiÞ us ðn;

sÞ ði ¼ 1; 2; 3Þ is then determined completely as

ðiÞ f3 ðnÞ

ðiÞ fj ðnÞ

gðsÞ ði ¼ 1; 2; 3Þ;

ð42Þ

ði ¼ 1; 2; 3; j ¼ 1; 2; 3Þ is presented in Appendix A.

4.3. Dynamic solution Substituting Eq. (25) into Eq. (26) and utilizing Eq. (42), we obtain ðiÞ

@ 2 ud @n2

þ

" # ðiÞ 2 2 2 l2 ðiÞ 1 @ 2 uðiÞ 1 @ud d q1 ðsÞ d q2 ðsÞ d gðsÞ ðiÞ ðiÞ ðiÞ d ði ¼ 1; 2; 3Þ; þ f ðnÞ þ f ðnÞ þ f ðnÞ  2i ud ¼ 2 1 2 3 n @n ds2 ds2 ds2 c i @ s2 n

ð43Þ

where

li

vffiffiffiffiffiffiffi u ðiÞ uc ; ¼ t 11 ðiÞ c33

sffiffiffiffiffiffiffi cðiÞ 33 ci ¼ ði ¼ 1; 2; 3Þ: q ðiÞ

ð44Þ

The normal mode expansion method, also named as eigenfunction expansion method, will be introduced to solve the dyðiÞ namic solution ud ðn; sÞ. We assume ðiÞ

ud ðn; sÞ ¼

1 X

ðiÞ Rm ðxm ; nÞXm ðsÞ ði ¼ 1; 2; 3Þ;

ð45Þ

m¼1

where Xm(s) is an undetermined function. xm and RðiÞ m ðxm ; nÞ are eigenfrequency and eigenfunction, respectively. The deterðiÞ mination of xm and RðiÞ m ðxm ; nÞ and the orthogonal property of Rm ðxm ; nÞ for a triple-layer piezoelectric composite cylinder with imperfect interface are presented in Appendices B and C, respectively.

H.M. Wang / Applied Mathematical Modelling 35 (2011) 1765–1781

1771

The substitution of Eq. (45) into Eq. (43) leads to 1 X

ðiÞ Rm ðxm ; nÞ

m¼1

" 2 # 2 2 2 d Xm ðsÞ d q1 ðsÞ d q2 ðsÞ d gðsÞ ðiÞ ðiÞ ðiÞ 2 þ x X ð s Þ ¼ f1 ðnÞ  f2 ðnÞ  f3 ðnÞ : m m 2 2 2 ds ds ds ds2

ð46Þ

With the aid of the orthogonal property of RðiÞ m ðxm ; nÞ, see Appendix C, Eq. (46) can be simplified as 2

d Xm ðsÞ þ x2m Xm ðsÞ ¼ pm ðsÞ ðm ¼ 1; 2; . . . ; 1Þ; ds2

ð47Þ

€1 ðsÞ þ I2m q €2 ðsÞ þ I3m g € ðsÞ; pm ðsÞ ¼ I1m q 3 Z ni X 1 Ijm ¼  q ðiÞ ffjðiÞ ðfÞRmðiÞ ðxm ; fÞ df ðj ¼ 1; 2; 3Þ; J m i¼1 ni1

ð48Þ

where

In which Jm is presented in Eq. (C.19) in Appendix C. The solution of Eq. (47) is

Xm ðsÞ ¼ Xm ð0Þ cos xm s þ

X_ m ð0Þ

xm

sin xm s þ

1

Z s

xm

0

pm ð1Þ sin xm ðs  1Þ d1:

ð49Þ

€1 ðsÞ; q €2 ðsÞ and g € ðsÞ are involved in pm(s) as shown in the first equation in Eq. (48). By employing the integrationNotice that q by-parts formula and further employing the initial conditions in Eq. (29), Eq. (49) can be rewritten as

Xm ðsÞ ¼ X1m ðsÞ þ I3m gðsÞ  I3m xm

Z s

gð1Þ sin xm ðs  1Þ d1;

ð50Þ

0

where

X1m ðsÞ ¼ I1m q1 ðsÞ  I1m xm

Z s 0

q1 ð1Þ sin xm ðs  1Þ d1 þ I2m q2 ðsÞ  I2m xm

Z s 0

q2 ð1Þ sin xm ðs  1Þ d1:

ð51Þ

4.4. Determination for electric field Utilizing the introduced non-dimensional quantities and variables in Eq. (13), Eq. (10) can be rewritten as ð2Þ ð2Þ @/ð2Þ gðsÞ ð2Þ u ð2Þ @u : ¼ e1 þ e3  n @n n @n

ð52Þ

The substitution of Eqs. (42) and (45) into the first equation in Eq. (20) derives 1 X

uðiÞ ðn; sÞ ¼

ðiÞ

ðiÞ

ðiÞ

ðiÞ Rm ðxm ; nÞXm ðsÞ þ f1 ðnÞq1 ðsÞ þ f2 ðnÞq2 ðsÞ þ f3 ðnÞgðsÞ:

ð53Þ

m¼1

Integrating Eq. (52) over the spatial interval [n1, n2] and utilizing Eqs. (50) and (53), we obtain

wðsÞ ¼ A1 gðsÞ þ

1 X

A2m

m¼1

Z s

gð1Þ sin xm ðs  1Þ d1;

ð54Þ

0

where

wðsÞ ¼ /2 ðsÞ  /1 ðsÞ  B3 q1 ðsÞ  B4 q2 ðsÞ 

1 X

B2m X1m ðsÞ;

m¼1 1 X

A1 ¼ B1 þ ð2Þ

B1 ¼ e1 B2m ¼ B3 ¼

m¼1 Z n2

Z

ð2Þ e1 ð2Þ

B4 ¼ e1

Z

n2

n1 n2

n1

Z

n2

n1

ð2Þ

A2m ¼ xm B2m I3m ; ð2Þ

f1 f3 ðfÞ df þ e3

n1

ð2Þ e1

B2m I3m ;

h

i ð2Þ ð2Þ f3 ðn2 Þ  f3 ðn1 Þ  lnðn2 =n1 Þ; ð2Þ

ð2Þ f1 Rm ðxm ; fÞ df þ e3

h

i ð2Þ Rm ðxm ; n2 Þ  Rð2Þ m ðxm ; n1 Þ ;

ð2Þ

ð2Þ

h

i ð2Þ ð2Þ f1 ðn2 Þ  f1 ðn1 Þ ;

ð2Þ

ð2Þ

h

i ð2Þ ð2Þ f2 ðn2 Þ  f2 ðn1 Þ :

f1 f1 ðfÞ df þ e3 f1 f2 ðfÞ df þ e3

ð55Þ

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H.M. Wang / Applied Mathematical Modelling 35 (2011) 1765–1781

Eq. (54) is a Volterra integral equation of the second kind [27]. By the newly developed method [28], Eq. (54) can be solved efficiently and the displacement, electric potential and stresses can be determined completely.

Table 1 Elastic, piezoelectric and dielectric constants. Material constant

BaTiO3

Aluminum

c11 (GPa) c12 (GPa) c13 (GPa) c23 (GPa) c33 (GPa) e31 (C/m2) e32 (C/m2) e33(C/m2) e33(109 F/m) q(103 kg/m3)

150.0 66.0 66.0 66.0 146.0 4.35 4.35 17.5 15.04 5.7

102.0 50.0 50.0 50.0 102.0 – – – – 2.7

Table 2  1 and v 2. The first five eigenfrequencies for different values of v

v 1

v 2

x1

x2

x3

x4

x5

0.0 0.1 0.2 0.3 0.5

0.0 0.1 0.2 0.3 0.5

1.414521 1.410613 1.406798 1.403076 1.395907

6.323589 5.378478 4.735825 4.286333 3.702749

13.39736 9.236587 7.402621 6.365203 5.195086

21.26951 15.48659 13.81301 13.07004 12.39642

28.77129 24.67354 23.70794 23.31265 22.96733

1.0

χ1 = χ2 =0

(a)

χ1 = χ2 =0.2

0.5

σr

0.0

-0.5

-1.0

-1.5

0

2

1.0

4

τ

6

8

χ1 = χ2 =0

(b)

χ1 = χ2 =0.2

0.5

10

σr

0.0

-0.5

-1.0

-1.5

0

2

4

τ

6

8

10

Fig. 2. Transient responses of radial stress rr at the interfaces for Case A (a) at the internal interface n = 0.6; (b) at the external interface n = 0.9.

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H.M. Wang / Applied Mathematical Modelling 35 (2011) 1765–1781

5. Numerical results and remarks Transient responses of a triple-layer piezoelectric composite infinite hollow cylinder will be considered in this subsection. Suppose the composite cylinder is made of Aluminum/BaTiO3/Aluminum. The non-dimensional radius of each interface from the inner to outer are n0 = 0.5, n1 = 0.6, n2 = 0.9 and n3 = 1.0, respectively. The material constants are listed in Table 1 [7]. Table 2 gives the first five eigenfrequencies xm(m = 1, 2, . . . , 5) of a triple-layer piezoelectric composite cylinder for differ 1 and v  2 . The results show that the eigenfrequencies decrease with the increase of v  1 and v  2 . That is to say, ent values of v similar with that having been reported investigations for imperfectly bonded laminated piezoelectric rectangular plate [18], the defect of the interface will also lead to the decrease of the natural frequency for the piezoelectric composite cylinder. The 1 ¼ v  2 ¼ 0:5 is only 98.68% of that for v 1 ¼ v  2 ¼ 0 (perfectly bonded interfaces). lowest eigenfrequency for v

3.0

χ1 = χ2 =0 χ1 = χ2 =0.2

σθ

2.0

(a)

1.0

0.0

-1.0

0

2

4

τ

6

8

10

3.0

χ1 = χ2 =0

σθ

(b)

χ1 = χ2 =0.2

2.0

1.0

0.0

-1.0

0

2

4

τ

6

8

10

3.0

χ1 = χ2 =0

σθ

2.0

(c)

χ1 = χ2 =0.2

1.0

0.0

-1.0

0

2

4

τ

6

8

10

Fig. 3. Transient rsponses of hoop stress rh at the three positions in the piezoelectric layer for Case A (a) at the internal surface n = 0.6; (b) at the middle surface n = 0.75; (c) at the external surface n = 0.9.

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H.M. Wang / Applied Mathematical Modelling 35 (2011) 1765–1781

In the following, the transient responses of the piezoelectric composite cylinder under two kinds of excitation will be considered. Case A: mechanical excitation. That is, the composite cylinder is subjected to a sudden constant pressure at the inner surface and both the inner and outer surfaces of the piezoelectric layer are electrically shorted. The boundary conditions are then expressed as

q1 ðsÞ ¼ HðsÞ; /1 ðsÞ ¼ 0;

q2 ðsÞ ¼ 0;

ð56Þ

/2 ðsÞ ¼ 0:

where H() denotes the Heaviside step function. Fig. 2 depicts the transient responses of radial stress rr at the two interfaces of piezoelectric composite cylinders subjected to mechanical excitation (Case A). It is found that at each interface, the responses of radial stress for imperfectly 1 ¼ v  2 ¼ 0:2Þ have significant difference with those for perfectly bonded interface ðv 1 ¼ v  2 ¼ 0Þ. We also bonded interface ðv observed that, for a piezoelectric composite cylinder subjected to a sudden constant pressure at the inner surface, the curves of radial stress response at the perfectly bonded interfaces vary more dramatically than those at the imperfectly bonded interfaces. For Case A, the responses of hoop stress rh at the three points (n = 0.6, 0.75 and 0.9) in the piezoelectric layer are shown in Fig. 3. The differences between the responses for the cylinder with perfectly bonded interfaces and those with the imperfectly bonded interfaces are also clearly presented. It is founded that the responses of hoop stress is almost quasi-periodic and the maximum hoop stress is tensile. Case B: electric potential excitation. That is, the outer surface of the piezoelectric layer is subjected to a sudden constant electric potential excitation while the inner surface of the piezoelectric layer is electrically shorted and both the inner and outer surfaces of composite cylinder are traction free. Thus

q1 ðsÞ ¼ 0;

q2 ðsÞ ¼ 0;

/1 ðsÞ ¼ 0;

/2 ðsÞ ¼ HðsÞ:

ð57Þ

For the electric potential excited piezoelectric composite cylinder, the elastic and electric fields are illustrated in Figs. 4–6. It is noticed that the weakness of the interface has notable effect on the transient responses of the radial and hoop stresses when the piezoelectric composite cylinder is excited by the external imposed electric potential. Unlike the behavior of the 2.0

χ1 = χ2 =0 χ1 = χ2 =0.2

( )

σr

1.0

0.0

-1.0

0

2

2.0

4

τ

6

8

χ1 = χ2 =0

( )

χ1 = χ2 =0.2

1.0

10

σr

0.0

-1.0

-2.0

0

2

4

τ

6

8

10

Fig. 4. Transient responses of radial stress rr at the interfaces for Case B. (a) at the internal interface n = 0.6; (b) at the external interface n = 0.9.

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H.M. Wang / Applied Mathematical Modelling 35 (2011) 1765–1781

1.0

χ1 = χ2 =0 χ1 = χ2 =0.2

( )

σθ

0.0

-1.0

-2.0

0

2

1.0

4

τ

6

8

χ1 = χ2 =0 χ1 = χ2 =0.2

10

( )

σθ

0.0

-1.0

-2.0

0

2

4

τ

6

8

10

1.0

χ1 = χ2 =0

( )

χ1 = χ2 =0.2

σθ

0.0

-1.0

-2.0

0

2

4

τ

6

8

10

Fig. 5. Transient responses of hoop stress rh at the three positions in the piezoelectric layer for Case B (a) at the internal surface n = 0.6; (b) at the middle surface n = 0.75; (c) at the external surface n = 0.9.

disturbed elastic field, in the piezoelectric layer, the distributions of the electric potential at the times s = 1.0 and 3.0, shown in Fig. 6, illustrate that the weakness of the interface have little effect on the electric field. We also noticed that the distribution of the electric potential in the piezoelectric layer is a weak nonlinear curve.

6. Conclusions An exact elastodynamic solution is developed by means of the state space method and normal mode expansion method for a triple-layer piezoelectric composite cylinder with imperfect interfaces which are modeled by a linear spring layers. The detailed procedure for determining the eigenfrequency and eigenfunction as well as the orthogonal properties of the eigenfunction for the piezoelectric composite cylinder with mechanical imperfect interfaces is presented. It should also be pointed

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H.M. Wang / Applied Mathematical Modelling 35 (2011) 1765–1781

1.0

χ1 = χ2 =0

0.8

χ1 = χ2 =0.2

φ

0.6

( )

0.4 0.2 0.0

0.6

0.7

ξ

0.8

0.9

1.0

χ1 = χ2 =0

0.8

χ1 = χ2 =0.2

φ

0.6

( )

0.4 0.2 0.0

0.6

0.7

ξ

0.8

0.9

Fig. 6. Distributions of electric potential / at the different times for case B (a) at the time s = 1.0; (b) at the time s = 3.0.

out that besides mechanical imperfection, there might also be electrical imperfection. Because the effect of the electrical imperfection is less serious than that of the mechanical one [21–23], the present work only model the interface to be mechanically imperfect. Numerical results show that the transient electromechanical responses are sensitive to the weakness of the mechanically imperfect interface. The present solution is an important foundation in the inversion analysis engineering for evaluating the interfacial properties, such as nondestructive testing, structural health monitoring etc. Acknowledgments The work was supported by the National Natural Science Foundation of China (Nos. 10872179 and 10725210), the Zhejiang Provincial Natural Science Foundation of China (No. Y7080298) and the Zijin and Xinxing Plan of Zhejiang University. Appendix A With the aid of Eq. (36), Eqs. (34a) and (34b) can be written as

X ð1Þ ðn; sÞ ¼ T ð1Þ ðnÞX ð1Þ ðn0 ; sÞ ðn0 6 n 6 n1 Þ; ð2Þ

ð2Þ

ð1Þ

ð1Þ

ð2Þ

X ðn; sÞ ¼ T ðnÞ½P X ðn0 ; sÞ þ G ðnÞgðsÞ ðn1 6 n 6 n2 Þ; h i X ð3Þ ðn; sÞ ¼ T ð3Þ ðnÞ P ð2Þ X ð1Þ ðn0 ; sÞ þ Gð2Þ gðsÞ ðn2 6 n 6 n3 Þ;

ðA:1Þ ðA:2Þ ðA:3Þ

where

P ð1Þ ¼ P ð1Þ T ð1Þ ðn1 Þ; G

ð2Þ

ð2Þ

¼P T

ð2Þ

P ð2Þ ¼ P ð2Þ T ð2Þ ðn2 ÞP ð1Þ T ð1Þ ðn1 Þ; ð2Þ

ðn2 ÞG ðn2 Þ:

ðA:4Þ

n oT ð1Þ ðiÞ Noticing that X ð1Þ ðn0 ; sÞ ¼ us ðn0 ; sÞ; n0 q1 ðsÞ , then by utilizing Eq. (41), us ðn; sÞ ði ¼ 1; 2; 3Þ is then determined as preðiÞ

sented in Eq. (42) and fj ðnÞ ði ¼ 1; 2; 3; j ¼ 1; 2; 3Þ is presented here as

H.M. Wang / Applied Mathematical Modelling 35 (2011) 1765–1781

1777

  H22 ð1Þ ð1Þ ð1Þ f1 ðnÞ ¼ n0 T 12 ðnÞ  T 11 ðnÞ ; H21 1 M2 ð1Þ ð1Þ ð1Þ ð1Þ T ðnÞ; f3 ðnÞ ¼  T ðnÞ; f2 ðnÞ ¼ H21 11 H21 11

ðA:5Þ

     H22 ð1Þ H22 ð1Þ ð2Þ ð2Þ ð1Þ ð2Þ ð1Þ P11 þ T 12 ðnÞ P22  P21 ; f1 ðnÞ ¼ n0 T 11 ðnÞ P12  H21 H21 i 1 h ð2Þ ð2Þ ð1Þ ð2Þ ð1Þ T ðnÞP 11 þ T 12 ðnÞP21 ; f2 ðnÞ ¼ H21 11     M2 ð1Þ M 2 ð1Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ P11 þ T 12 ðnÞ G2 ðnÞ  P21 ; f3 ðnÞ ¼ T 11 ðnÞ G1 ðnÞ  H21 H21

ðA:6Þ

     H22 ð2Þ H22 ð2Þ ð3Þ ð3Þ ð2Þ ð3Þ ð2Þ P11 þ T 12 ðnÞ P22  P21 ; f1 ðnÞ ¼ n0 T 11 ðnÞ P12  H21 H21 h i 1 ð3Þ ð3Þ ð2Þ ð3Þ ð2Þ T ðnÞP 11 þ T 12 ðnÞP21 ; f2 ðnÞ ¼ H21 11     M 2 ð2Þ M2 ð2Þ ð3Þ ð3Þ ð2Þ ð2Þ ð2Þ f3 ðnÞ ¼ T 11 ðnÞ G1  P 11 þ T 12 ðnÞ G2  P21 : H21 H21

ðA:7Þ

Appendix B The detailed procedure for determining the eigenfrequency xm and eignfunction RðiÞ m ðxm ; nÞ for a triple-layer piezoelectric composite cylinder with imperfect interfaces is shown in this Appendix. Inspecting the differential form at the left-hand side of Eq. (43), we have the hint that RðiÞ m ðxm ; nÞ can be assumed as ðiÞ

ðiÞ Rm ðxm ; nÞ ¼ E1 J li ðiÞ





xm ðiÞ n þ E2 Y li n ði ¼ 1; 2; 3Þ; ci ci

xm

ðB:1Þ

ðiÞ

where E1 and E2 are undetermined constants, J li ðÞ and Y li ðÞ are Bessel functions of the first and second kinds of order li, and xm is eigenfrequency. From the first equation in Eq. (16) and the second equation in Eq. (20), we know

rrdðiÞ ¼ RðiÞ rd =n:

ðB:2Þ

Substituting Eq. (45) into the second equation in Eq. (25) and utilizing Eq. (B.2), we obtain

rrdðiÞ ðn; sÞ ¼

1 X

rðiÞ m ðxm ; nÞXm ðsÞ;

ðB:3Þ

m¼1

where

h

i

rmðiÞ ðxm ; nÞ ¼ @ðiÞ RðiÞ m ðxm ; nÞ ði ¼ 1; 2; 3Þ:

ðB:4Þ

The operator @ðiÞ ðÞ is ðiÞ @ðiÞ ðÞ ¼ c33

@ ðiÞ 1 þ c13 ði ¼ 1; 2; 3Þ: @n n

ðB:5Þ

Substituting Eq. (B.1) into Eq. (B.4), we obtain



ðiÞ rmðiÞ ðxm ; nÞ ¼ EðiÞ J li 1 @



 

 xm ðiÞ n þ E2 @ðiÞ Y li n ði ¼ 1; 2; 3Þ: ci ci

xm

ðB:6Þ

Setting n = ni1 (i = 1, 2, 3) in Eqs. (B.1) and (B.6), we obtain



xm ðiÞ ni1 þ E2 Y li ni1 ; ci ci 

 

 x x rmðiÞ ðxm ; ni1 Þ ¼ E1ðiÞ @ðiÞ Jli m ni1 þ E2ðiÞ @ðiÞ Y li m ni1 : ci ci ðiÞ

ðiÞ Rm ðxm ; ni1 Þ ¼ E1 Jli

ðiÞ



xm

ðiÞ

ðB:7Þ

ðiÞ

ðiÞ

Then E1 and E2 can be determined from Eq. (B.7). Substituting the obtained E1 and E2 into (B.1) and (B.5), the expressions ðiÞ for rm ðxm ; nÞ and RðiÞ m ðxm ; nÞ can be written in a matrix form as ðiÞ ðiÞ ðiÞ Zm ðxm ; nÞ ¼ S m ðxm ; nÞZ m ðxm ; ni1 Þ ðni1 6 n 6 ni ; i ¼ 1; 2; 3Þ;

ðB:8Þ

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H.M. Wang / Applied Mathematical Modelling 35 (2011) 1765–1781

where

( ðiÞ Zm ð

xm ; nÞ ¼

ðiÞ ðxm ; nÞ Rm

)

rmðiÞ ðxm ; nÞ

" ðiÞ Sm ð

xm ; nÞ ¼

;

ðiÞ

ðiÞ

ðiÞ

ðiÞ

Sm;11 ðxm ; nÞ Sm;12 ðxm ; nÞ Sm;21 ðxm ; nÞ Sm;22 ðxm ; nÞ

# ;

ðB:9Þ

in which

  

    

xm xm i i ðiÞ Sm;11 ðxm ; nÞ ¼ @ðiÞ Y li ni1 J li km n  @ðiÞ J li ni1 Y li km n =KðiÞ ; ci ci



 

 x x x m m m i ðiÞ ni1 Y li n  Y li ni1 J li km n =KðiÞ ; Sm;12 ðnÞ ¼ J li ci ci ci

  

  





x x xm xm m m ðiÞ ni1 @ðiÞ J li n  @ðiÞ J li ni1 @ðiÞ Y li n =KðiÞ ; Sm;21 ðnÞ ¼ @ðiÞ Y li ci ci ci ci









xm xm xm xm ðiÞ ni1 @ðiÞ Y li n  Y li ni1 @ðiÞ J li n =KðiÞ ; Sm;22 ðnÞ ¼ J li ci ci ci ci 







xm xm xm xm J li  @ðiÞ J li Y li ði ¼ 1; 2; 3Þ: KðiÞ ¼ @ðiÞ Y li n n n n ci i1 ci i1 ci i1 ci i1

ðB:10Þ

Setting n = ni (i = 1, 2, 3) in Eq. (B.8), we derive ðiÞ ðiÞ Zm ðxm ; ni Þ ¼ S ðiÞ m ðxm ; ni ÞZ m ðxm ; ni1 Þ ði ¼ 1; 2; 3Þ:

ðB:11Þ

With the aid of Eq. (B.2), Eq. (28) can be written in a matrix form as ð2Þ Zm ðxm ; n1 Þ ¼ Y

ð1Þ

Z ð1Þ m ðxm ; n1 Þ;

ð3Þ Zm ð

ð2Þ

Z ð2Þ m ðxm ; n2 Þ;

xm ; n2 Þ ¼ Y

ðB:12Þ

where

Y

ð1Þ

 ¼

1

v 1

0

1

 ;

Y

ð2Þ

 ¼

1

v 2

0

1

 :

ðB:13Þ

By means of Eqs. (B.11) and (B.12), the relationship between the state vector at the external surface and that at the internal surface is then derived as ð1Þ Z ð3Þ m ðxm ; 1Þ ¼ S m ðxm ÞZ m ðxm ; n0 Þ;

ðB:14Þ

where

S m ðxm Þ ¼ S ð3Þ m ðxm ; n3 ÞY

ð2Þ ð2Þ Sm ð

xm ; n2 ÞY

ð1Þ ð1Þ Sm ð

xm ; n1 Þ:

ðB:15Þ

With the aid of Eqs. (45) and (B.3), the following equations can be derived from Eq. (27) as

rmð1Þ ðxm ; n0 Þ ¼ 0; rð3Þ m ðxm ; 1Þ ¼ 0:

ðB:16Þ

The substitution of Eq. (B.16) into Eq. (B.14) derives

(

ð3Þ Rm ðxm ; 1Þ

"

) ¼

0

Sm;11 ðxm Þ Sm;12 ðxm Þ Sm;21 ðxm Þ Sm;22 ðxm Þ

#(

Rð1Þ m ðxm ; n0 Þ 0

) :

ðB:17Þ

The existence of nontrivial solution in Eq. (B.17) leads to

Sm;21 ðxm Þ ¼ 0:

ðB:18Þ

Eq. (B.18), a transcendental equation, is the eigenequation from which a series of positive real roots, named as eigenfrequency xm (m = 1, 2, . . . , 1), can be obtained. Utilizing Eqs. (B.11) and (B.12), the eigenfunction RðiÞ m ðxm ; nÞ, can be obtained from the Eq. (B.8) as ð1Þ e Rð1Þ m ðxm ; nÞ ¼ S m;11 ðxm ; nÞRm ðxm ; n0 Þ; ð1Þ

ð1Þ e Rð2Þ m ðxm ; nÞ ¼ S m;11 ðxm ; nÞRm ðxm ; n0 Þ; ð2Þ

Rð3Þ m ð

xm ; nÞ ¼

ð3Þ e S m;11 ð

x

ð1Þ m ; nÞRm ð

xm ; n0 Þ;

ðB:19Þ

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H.M. Wang / Applied Mathematical Modelling 35 (2011) 1765–1781

where ð1Þ e S ð1Þ m ðxm ; nÞ ¼ S m ðxm ; nÞ; e S ð2Þ ðxm ; nÞ ¼ S ð2Þ ðxm ; nÞY ð1Þ S ð1Þ ðxm ; n1 Þ; m

m

e S ð3Þ m ð

S ð3Þ m ð

xm ; nÞ ¼

ðB:20Þ

m

x

ð2Þ ð2Þ Sm ð m ; nÞY

xm ; n2 ÞY

ð1Þ ð1Þ Sm ð

xm ; n1 Þ:

It is noted that Rð1Þ m ðxm ; n0 Þ in Eq. (B.19) is a common coefficient in each layer. In the calculation, we can adopt ðiÞ ðiÞ Rð1Þ m ðxm ; n0 Þ ¼ 1. Thus Rm ðxm ; nÞ is determined completely. The proof for the orthogonal property of Rm ðxm ; nÞ is presented in Appendix C. Appendix C The proof of the orthogonal property of RðiÞ m ðxm ; nÞ is presented in this Appendix. Suppose xm and xj are two different eigenfrequencies. Then we have





ðiÞ ðiÞ r2  l2i Rm ðxm ; nÞ þ ðkim nÞ2 Rm ðxm ; nÞ ¼ 0;



ðC:1Þ

 r  l2i RjðiÞ ðxj ; nÞ þ ðkij nÞ2 RðiÞ j ðxj ; nÞ ¼ 0; 2

ðC:2Þ

where

kim ¼ xm =ci ; Eq.

kij ¼ xj =ci :

ðiÞ (C.1)Rj ð

xj ; nÞ–Eq.

h

i

(C.2)RðiÞ mð

ðC:3Þ

xm ; nÞ derives

ðiÞ ðiÞ n2 ðkim Þ2  ðkij Þ2 Rm ðxm ; nÞRj ðxj ; nÞ ¼ Rm ðxm ; nÞr2 Rj ðxj ; nÞ  Rj ðxj ; nÞr2 RðiÞ m ðxm ; nÞ: ðiÞ

ðiÞ

ðiÞ

ðC:4Þ

With the aid of Eq. (44), Eq. (C.4) can be rewritten as

 q ðiÞ  2 d d ðiÞ ðiÞ 2 nRðiÞ x  x rRjðiÞ ðxj ; nÞ  RðiÞ rRðiÞ m ðxm ; nÞRj ðxj ; nÞ ¼ Rm ðxm ; nÞ m ðxm ; nÞ; m j j ðxj ; nÞ ðiÞ dn

c33

ðC:5Þ

dn

Integrating Eq. (C.5) at the spatial interval [ni1, ni] derives



q ðiÞ x2m  x2j

Z

ni

ni1

h in i ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ nRm ðxm ; nÞRj ðxj ; nÞ dn ¼ c33 Rm ðxm ; nÞrRj ðxj ; nÞ  Rj ðxj ; nÞrRm ðxm ; nÞ

ni1

ðC:6Þ

Eq. (B.4) can be rewritten as ðiÞ ðiÞ ðiÞ ðiÞ c33 rRðiÞ m ðxm ; nÞ ¼ nrm ðxm ; nÞ  c13 Rm ðxm ; nÞ;

ðC:7Þ

ðiÞ ðiÞ ðiÞ ðiÞ c33 rRðiÞ j ðxj ; nÞ ¼ nrj ðxj ; nÞ  c 13 Rj ðxj ; nÞ:

The substitution of Eq. (C.7) into Eq. (C.6) derives



q ðiÞ x2m  x2j

Z

ni

ni1

h in i ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ nRm ðxm ; nÞRj ðxj ; nÞ dn ¼ n RðiÞ : m ðxm ; nÞrj ðxj ; nÞ  Rj ðxj ; nÞrm ðxm ; nÞ ni1

ðC:8Þ

Considering the homogeneous boundary conditions in Eq. (27), then we have

rð3Þ rð3Þ m ðxm ; 1Þ ¼ 0; j ðxj ; 1Þ ¼ 0; ð1Þ rm ðxm ; n0 Þ ¼ 0; rjð1Þ ðxj ; n0 Þ ¼ 0:

ðC:9Þ

Eq. (C.8) contains three equations when we set i = 1, 2 and 3, respectively. The summation of the three equations derives 3 X i¼1



q ðiÞ x2m  x2j

Z

ni

ni1

ðiÞ

ð3Þ

ð3Þ

ðiÞ ð3Þ nRm ðxm ; nÞRj ðxj ; nÞ dn ¼ Rð3Þ m ðxm ; 1Þrj ðxj ; 1Þ  Rj ðxj ; 1Þrm ðxm ; 1Þ

n h i h io ð3Þ ð3Þ ð2Þ ð2Þ ð2Þ ð3Þ ð2Þ þ n2   Rð3Þ m ðxm ; n2 Þrj ðxj ; n2 Þ  Rj ðxj ; n2 Þrm ðxm ; n2 Þ þ Rm ðxm ; n2 Þrj ðxj ; n2 Þ  Rj ðxj ; n2 Þrm ðxm ; n2 Þ n h i h io ð2Þ ð2Þ ð1Þ ð1Þ ð1Þ ð2Þ ð1Þ þ n1   Rð2Þ m ðxm ; n1 Þrj ðxj ; n1 Þ  Rj ðxj ; n1 Þrm ðxm ; n1 Þ þ Rm ðxm ; n1 Þrj ðxj ; n1 Þ  Rj ðxj ; n1 Þrm ðxm ; n1 Þ h i ð1Þ ð1Þ ð1Þ ð1Þ  Rm ðxm ; n0 Þrj ðxj ; n0 Þ þ Rj ðxj ; n0 Þrm ð xm ; n 0 Þ : ðC:10Þ

1780

H.M. Wang / Applied Mathematical Modelling 35 (2011) 1765–1781

The following equations can be derived from Eq. (28) as ð1Þ  ð1Þ Rð2Þ m ðxm ; n1 Þ ¼ Rm ðxm ; n1 Þ þ v1 rm ðxm ; n1 Þ;

ð2Þ ð1Þ m ð m ; n1 Þ ¼ m ð m ; n1 Þ; ð2Þ ð1Þ j ð j ; n1 Þ ¼ j ð j ; n1 Þ;

r x x ð3Þ ð2Þ ð2Þ ð3Þ ð2Þ  2 rm ðxm ; n2 Þ; rm ðxm ; n2 Þ ¼ rm ðxm ; n2 Þ; Rm ðxm ; n2 Þ ¼ Rm ðxm ; n2 Þ þ v ð3Þ ð2Þ ð2Þ ð3Þ ð2Þ  2 rj ðxj ; n2 Þ; rj ðxj ; n2 Þ ¼ rj ðxj ; n2 Þ: Rj ðxj ; n2 Þ ¼ Rj ðxj ; n2 Þ þ v ð2Þ Rj ð

ð1Þ Rj ð

xj ; n1 Þ ¼

ð1Þ j ð

xj ; n1 Þ þ v 1 r

r

xj ; n1 Þ; r

x

x

r

ðC:11Þ

With the aid of Eq. (C.11), we learn

h i h i ð3Þ ð3Þ ð2Þ ð2Þ ð2Þ ð3Þ ð2Þ  Rð3Þ m ðxm ; n2 Þrj ðxj ; n2 Þ  Rj ðxj ; n2 Þrm ðxm ; n2 Þ þ Rm ðxm ; n2 Þrj ðxj ; n2 Þ  Rj ðxj ; n2 Þrm ðxm ; n2 Þ h i h i ð2Þ ð2Þ ð2Þ  ð2Þ  ð2Þ ¼  Rð2Þ m ðxm ; n2 Þ þ v2 rm ðxm ; n2 Þ rj ðxj ; n2 Þ þ Rj ðxj ; n2 Þ þ v2 rj ðxj ; n2 Þ rm ðxm ; n2 Þ ð2Þ

ð2Þ

ð2Þ ðxm ; n2 Þrj ðxj ; n2 Þ  Rj ðxj ; n2 Þrð2Þ þ Rm m ðxm ; n2 Þ ¼ 0:

ðC:12Þ

and

h i h i ð2Þ ð2Þ ð1Þ ð1Þ ð1Þ ð2Þ ð1Þ  Rð2Þ m ðxm ; n1 Þrj ðxj ; n1 Þ  Rj ðxj ; n1 Þrm ðxm ; n1 Þ þ Rm ðxm ; n1 Þrj ðxj ; n1 Þ  Rj ðxj ; n1 Þrm ðxm ; n1 Þ h i h i ð1Þ ð1Þ ð1Þ  ð1Þ  ð1Þ ¼  Rð1Þ m ðxm ; n1 Þ þ v1 rm ðxm ; n1 Þ rj ðxj ; n1 Þ þ Rj ðxm ; n1 Þ þ v1 rj ðxj ; n1 Þ rm ðxm ; n1 Þ ð1Þ

ð1Þ

ð1Þ ðxm ; n1 Þrj ðxj ; n1 Þ  Rj ðxj ; n1 Þrð1Þ þ Rm m ðxm ; n1 Þ ¼ 0:

ðC:13Þ

With the aid of Eqs. (C.9), (C.12) and (C.13), Eq. (C.10) is then simplified as 3 X



q ðiÞ x2m  x2j

Z

ni

ni1

i¼1

ðiÞ

ðiÞ nRm ðxm ; nÞRj ðxj ; nÞ dn ¼ 0:

ðC:14Þ

With the pre-assumption xm – xj, then we have 3 X

q ðiÞ

Z

ni ni1

i¼1

ðiÞ

ðiÞ nRm ðxm ; nÞRj ðxj ; nÞ dn ¼ 0:

d For the case xm = xj, we multiply Eq. (C.1) by 2 dn

ðC:15Þ h

i RðiÞ m ðxm ; nÞ and then obtain

 2 i2 i d d ðiÞ d h ðiÞ d h ðiÞ n Rm Rm ðxm ; nÞ þ ðkim nÞ2 RðiÞ Rm ðxm ; nÞ ¼ 0: ðxm ; nÞ  l2i m ðxm ; nÞ dn dn dn dn

ðC:16Þ

Integrating Eq. (C.16) at the spatial interval [ni1, ni] and utilizing Eqs. (C.3) and (44), we have

Z

ni

 ðiÞ

q n

ni1

h

( ) 2 ðiÞ  i2  n i 2 ðiÞ h l c33 ðiÞ 1 c33 d ðiÞ  2 ðiÞ i  n  xm ; nÞ dn ¼ n R ðxm ; nÞ þ q R ðx ; nÞ  :  2 x2m dn m x2m m m

RðiÞ mð

i2

ðC:17Þ

ni1

Then the summation of Eq. (C.17) derives 3 Z X i¼1

ni

h

i2

q ðiÞ n RðiÞ dn ¼ J m ; m ðxm ; nÞ

ðC:18Þ

ni1

where

( ) 2 ðiÞ h ðiÞ  3 i2  n i X l2i c33 1 c33 d ðiÞ  ðiÞ ðiÞ 2  Jm ¼ n Rm ðxm ; nÞ þ q n  Rm ðxm ; nÞ  : 2 2  2 dn x x m m i¼1

ðC:19Þ

ni1

The orthogonal property of the eigenfunction RðiÞ m ðxm ; nÞ is synthesized as 3 Z X i¼1

ni

ni1

q ðiÞ nRmðiÞ ðxm ; nÞRjðiÞ ðxj ; nÞdn ¼ Jm dmj ;

ðC:20Þ

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