symplectic model for piezoelectric wedges and its...

Philosophical Magazine,Vol. 87, No. 2, 11 January 2007, 225–251

Symplectic model for piezoelectric wedges and its applicationin analysis of electroelastic singularities

J.-S. WANGy and Q.-H. QIN*z

yDepartment of Mechanics, School of Mechanical Engineering,Tianjin University, Tianjin 300072, China

zDepartment of Engineering, Australian National University,Canberra, ACT 0200, Australia

(Received 10 April 2006; in final form 3 August 2006)

In this paper, a symplectic model, based on the Hamiltonian system, is developedfor analyzing singularities near the apex of a multi-dissimilar piezoelectricwedge under antiplane deformation. The derivation is based on a modifiedHellinger–Reissner generalized variational principle or a differential equationapproach. The study indicates that the order of singularity depends directly on thenon-zero eigenvalue of the proposed Hamiltonian operator. Using the coordinatetransformation technique and continuity conditions on the interface between twodissimilar materials, the orders of singularity for multi-dissimilar piezoelectricand piezoelectric–elastic composite wedges are determined. Numerical examplesare considered to show potential applications and validity of the proposedmethod. It is found that the order of singularity also depends on the piezoelectricconstant, in addition to the geometry and shear modulus.

1. Introduction

Piezoelectric materials are commonly used as sensors and actuators in adaptivestructures owing to their electro-mechanical coupling effect [1]. In practicalapplications, piezoelectric elements are often bonded to other composite materials.In particular, they are often used in bimaterial structures or wedge structures [2–4],which may exhibit singular electro-mechanical fields on the interface of bimaterialsor near the apex, induced by geometric and material discontinuities under strenuousmechanical and electrical loads [4–7]. The corresponding singular stresses andelectric fields may even become infinite, leading to dielectric breakdown, debondingand fracture.

Singularity analysis of stress fields at the apex of elastic wedges has attractedgreat interest over the past decades [2–6]. In 1952, Williams [6] studied the stresssingularities of an elastic wedge based on plane stress theory. Since then, severalmathematical approaches have been presented to determine the order of stresssingularity near the apex of elastic wedges, such as the eigenfunction expansion,Mellin transform and complex potential functions (see [7] for details).

*Corresponding author. Email: [email protected]

Philosophical Magazine

ISSN 1478–6435 print/ISSN 1478–6443 online 2007 Taylor & Francis

http://www.tandf.co.uk/journals

DOI: 10.1080/14786430600941579

For singularity analysis of piezoelectric wedges within the framework ofEuclidean space, Xu and Rajapakse [8] studied the in-plane singular behaviourof electroelastic fields at the corner of piezoelectric wedges and junctions usingextended Lekhnitskii’s complex potential functions and Williams’ eigenfunctionexpansion. Using a similar procedure, Chue and Chen [9] analyzed stress singularitiesof a piezoelectric composite wedge under generalized plane deformation. Theysubsequently extended their results to the antiplane problem of a piezoelectricbonded wedge using the Mellin transform [10] and eigenfunction expansion [11].More recently, Chen [7] investigated the singularities of thermoelectroelastic fieldsnear the apex of a piezoelectric bonded wedge based on the generalized Lekhnitskiiformulation and Mellin transform.

From the above review, it can be seen that most of the previous studies have beendevoted to cases of one or two materials and there is very limited work involvingmulti-material junctions, especially piezoelectric composite wedges. With growthin the number of materials, singularity analysis of electromechanical fields maybecome increasingly complex and difficult within the framework of Euclidean space.To address this problem, Zhang and Zhong [12], in their study of stress singularitynear the crack in an elastic multi-material junction with arbitrary vertex angles,developed a novel symplectic approach based on the Hamiltonian system rather thanthe conventional Lagrangian system in Euclidean space. The Hamiltonian systemused here was introduced to elasticity theory by Zhong [13] in 1995 and is basedon the theory of analogy between computational structural mechanics and optimalcontrol, later detailed in Yao and Zhong’s book [14]. With the Hamiltonian system,the symplectic eigenfunction expansion and the separation of variables are used tosolve elastic problems within the symplectic geometry space rather than Euclideanspace [15]. This feature allows the analytical solution of some elastic problems to beobtained more easily and rationally than traditional methods, such as the semi-inverse method. Although the Hamiltonian system theory has been widely appliedto elastic mechanics problems, applications of this new approach to multi-fieldproblems, such as elastic stress fields and electric fields, have been rarely reported inthe literature. To the authors’ knowledge, there are only a few studies dealing withpiezoelectric and magnetoelectroelastic materials [16–18]. Bian [16] derived the basicequations of electro-mechanical coupling problems in the Hamiltonian systemfor a piezoelectric strip. Yao [17] qualitatively analyzed the symplectic solutionsystem and Saint-Venant principle for a magnetoelectroelastic strip underantiplane deformation. Wang and Qin [18] extended Bian’s results to the caseof plane deformation in a magnetoelectroelastic strip. However, none of these studiesprovides examples to show the practical application and validity of this new method.

The main object of this paper is to develop a symplectic formulation of theHamiltonian system for analyzing singular behaviour near the apex of a multi-material piezoelectric wedge under antiplane deformation. For simplicity, the polarcoordinate system is employed. After a variable transformation to the polarcoordinate system, the radial coordinate is analogous to the ‘time coordinate’,and then the hoop direction becomes the transverse direction. Displacement andelectric potential are treated as state vectors, while shear stress and electricdisplacement in the radial direction are treated as dual vectors; then, the governingdifferential equations can be converted into the Hamiltonian system by a differential

226 J.-S. Wang and Q.-H. Qin

equation approach or a modified variational principle approach. Equations forthe dual state vector mentioned above can, then, be obtained. By means of acoordinate transform as well as using the boundary conditions and continuityconditions at the interface, the singularity order for multi-dissimilar piezoelectricand piezoelectric–elastic composite wedges can be determined. Some numericalexamples are considered to demonstrate the potential applications and validity ofthe proposed model to singularity analysis of multi-field materials. The singularityresults for the case of piezoelectric wedges with one or two dissimilar materials areshown to be identical with those obtained by other methods [10]. The study alsoshows that the order of singularity depends strongly on the piezoelectric constant,the geometry and shear modulus of the wedge.

2. Basic equations and their Hamiltonian system

2.1. Basic equations

In the case of anti-plane shear deformation involving out-of-plane displacementw and in-plane electric fields only, the constitutive equations become [19]:

zrzDr

D

8>><>>:

9>>=>>; ¼

c44 0 0 e150 c44 e15 00 e15 "11 0e15 0 0 "11

2664

3775

zrzEr

E

8>><>>:

9>>=>>; ð1Þ

in which ij are the shear stresses, ij are shear strains, Di and Ei (i¼ r, , j¼ z)are electric displacements and electric fields, respectively, c44 is the elastic modulusat the constant electric field, e15 is the piezoelectric constant and "11 the dielectricpermittivity at constant strains.

The corresponding equilibrium equations and Maxwell’s equation are:

@ðrrzÞ

@rþ@z@

¼ 0,

@ðrDrÞ

@rþ@D

@¼ 0:

ð2Þ

The shear strains and electric fields are given by:

rz ¼@w

@r, z ¼

1

r

@w

@

Er ¼ @’

@r, E ¼

1

r

@’

@

ð3Þ

where w and ’ are the displacement and electric potential, respectively.

2.2. Hamiltonian system by differential equation approach

There are several ways to obtain the Hamiltonian system for an elastic or multi-fieldmaterial system. In this section, we show how to derive the Hamiltonian system usingdifferential equation methods, a typical approach among the existing methods.

Analysis of singular behaviour of composite wedges under antiplane deformation 227

To do this, let r, represent radial and transverse coordinates, respectively.

Then, introduce the dual vectors q and p:

q ¼w’

, p ¼

Sr

SDr

ð4Þ

which are required in the Hamiltonian system, where

Sr ¼ rrz, SDr ¼ rDr ð5Þ

Furthermore, to convert variables and equations from Euclidean space to symplectic

geometry space, introduce a generalized time variable such that

¼ lnðrÞ ð6Þ

Since is now a generalized time variable, the symbol ‘’ is used to represent the

differential with respect to .Making use of equations (2), (5) and (6) we have:

@Sr

@r¼

1

r

@Sr

@¼

1

r_Sr ¼

@z@

¼ 1

rc44@2w

@2þ e15

@2’

@2

ð7Þ

Equation (7) can be further written in the form of the first equation of equation (8),

and the expression of S _Dr can be obtained in a similar way as:

_Sr ¼ c44@2w

@2 e15

@2’

@2, S _Dr ¼ e5

@2w

@2þ "11

@2’

@2ð8Þ

Considering equations (1) and (6), the variables Sr and SDr can be expressed as:

Sr ¼ c44 _wþ e15 _’, SDr ¼ e15 _w "11 _’ ð9Þ

Solving equation (9) for _w and _’, yields:

_w ¼"111

Sr þe151

SDr, _’ ¼e151

Sr c441

SDr ð10Þ

in which

1 ¼ e215 þ c44"11 ð11Þ

The combination of equations (8) and (10) provides following matrix equation:

_w

_’

_Sr

S _Dr

8>>>>><>>>>>:

9>>>>>=>>>>>;

¼

0 0"111

e151

0 0e151

c441

c44@2

@2e15

@2

@20 0

e15@2

@2"11

@2

@20 0

2666666666664

3777777777775

w

’

Sr

SDr

8>>>>><>>>>>:

9>>>>>=>>>>>;

ð12Þ


Furthermore, using the notation

v ¼qp

ð13Þ

Equation (12) can be simplified as:

_v ¼ Hv ð14Þ

in which

H ¼

0 0"111

e151

0 0e151

c441

c44@2

@2e15

@2

@20 0

e15@2

@2"11

@2

@20 0

266666666664

377777777775

ð15Þ

Equation (14) is the Hamiltonian dual equation. To prove that H is a Hamiltonianoperator matrix, a rotational exchange operator matrix J is introduced as follows:

J ¼0 I2

I2 0

, J2 ¼

I2 00 I2

, J1 ¼ J ¼ JT ð16Þ

where I2 is the order identity matrix. With the notation J, it is easy to prove thatH satisfies the following relation (see Appendix A for details of the proof):

vT1 ,Hv2

¼ vT2 ,Hv1

ð17Þ

where

vT1 ,Hv2

¼

Z 2

1

vT1 JHv2 d ð18Þ

Then, according to theory of the symplectic geometry [13, 14], H is a Hamiltonianoperator matrix.

2.3. Hamiltonian system by variational principle approach

In section 2.1, we derived a Hamiltonian system using a differential equationapproach. The same Hamiltonian system for the antiplane problem of a piezoelectricwedge can also be obtained using a variational principle approach. To illustrate thisapproach, consider following constitutive equation:

z

rz

Er

E

8>>><>>>:

9>>>=>>>;

¼

s44 0 0 g15

0 s44 g15 0

0 g15 11 0

g15 0 0 11

26664

37775

z

rz

Dr

D

8>>><>>>:

9>>>=>>>;

ð19Þ


where the constants s44, g15 g15 and 11 are defined by the relations:

s44 ¼"111

, g15 ¼e151

, 11 ¼c441

ð20Þ

Based on the constitutive relation (19), the modified Hellinger–Reissner generalized

variational principle can be stated as follows:

Z r2

r1

Z 2

1

rz@w

@rþ z

1

r

@w

@þDr

@’

@rþD

1

r

@’

@

1

2

s44

2z þ

1

2s44

2rz

1

211D

2r

1

211D

2

g15Dz g15rzDr

rdrd ¼ 0 ð21Þ

Making use of the variable transformation equation (4), the variational equality (21)

can be further written as:

Z 2

1

Z 2

1

Sr@w

@þS

@w

@þSDr

@’

@þSD

@’

@

1

2

s44S

2r þ

1

2s44S

2

1

211SD

2r

1

211SD

2

þ g15SDSþ g15SDrSr

dd¼ 0 ð22Þ

where

1 ¼ ln r1, 2 ¼ ln r2, S ¼ rz, SD ¼ rD ð23Þ

Taking variation with respect to S and SD, equation (22) leads to:

S ¼1

211

@w

@þ g15

@’

@

, SD ¼

1

2g15

@w

@ s44

@’

@

ð24Þ

in which

2 ¼ g215 þ s4411 ð25Þ

Substituting equation (24) into equation (22) we can obtain the Hamiltonian mixed

energy variational principle as follows:

Z 2

1

Z 2

1

Sr@w

@þ SDr

@’

@1

2s44S

2r þ

1

211SD

2r g15SDrSr þ

1

2

1

211

@w

@

2(

1

2

1

2s44

@’

@

2

þ1

2g15

@w

@

@’

@

)dd ¼ 0 ð26Þ

Making use of equation (4), equation (26) can be further simplified as:

Z 2

1

Z 2

1

pT _q<ðq, pÞ

dd ¼ 0 ð27Þ

where <ðq, pÞ is the Hamiltonian function defined by:

<ðq, pÞ ¼ 1

2qTBqþ

1

2pTDp ð28Þ


in which

B ¼

112

@2

@2g152

@2

@2

g152

@2

@2s442

@2

@2

26664

37775, D ¼

s44 g15

g15 11

" #ð29Þ

In the derivation of equation (27), homogeneous boundary conditions (see

Appendix A) have been used.From equation (27) the following equations can be obtained:

_q ¼ Dp

_p ¼ Bqð30Þ

Using the definition of v in equation (13), equation (30) can be rewritten in the form:

_v ¼ Hv ð31Þ

in which

H ¼0 DB 0

ð32Þ

where H* is used to distinguish the matrix H in equation (14). Making use of

the relation (20), it can be found that H* in equation (32) is the same as H in

equation (15). Therefore, equations (31) and (14) are identical.

3. Basic eigenvalues and singularity of stress and electric fields

In section 2, the dual state vector equations (14) and (31) were derived using

the differential equation method and the variational principle approach, respectively.

In this section, applications of the proposed Hamiltonian model to analyzing

the eigenvalues of the Hamiltonian operator matrix, which are associated with the

singularity behaviour of a piezoelectric wedge, are discussed.Noting that equation (14) can be solved by the separation of the variable and

the symplectic eigenfunction expansion, one can assume v in the form:

vð, Þ ¼ jðÞwðÞ ð33Þ

in which j(), w() are two functions of , respectively.Substituting equation (33) into equation (14) yields the solution for j() as:

jðÞ ¼ e ð34Þ

and the eigenvalue equation:

Hw ¼ w ð35Þ


in which is an eigenvalue of the Hamiltonian operator matrix and w is given by:

w ¼qðÞ

pðÞ

ð36Þ

Thus, we have:

v ¼ ewðÞ ð37Þ

It should be mentioned that the eigenvalues of the Hamiltonian operator matrix

have following property [13]: if i is an eigenvalue of equation (35), then i is also

an eigenvalue of equation (35). With this property, all eigenvalues of H can be

subdivided into the following three groups:

(a) i, ReðiÞ40 or ImðiÞ40 ðif ReðiÞ ¼ 0Þ i ¼ 1, 2,(b) i ¼ i

(c) ¼ 0

Their corresponding eigenfunction-vectors are written as wþi, wi and w0. Following

the procedure in [13], it is easy to prove that wþi and wi are of adjoint symplectic

orthonormalization, that is,

hwTþi, J,wji ¼ ij, hwT

i, J,wþji ¼ ij,

hwTþi, J,wþji ¼ 0, hwT

i, J,wji ¼ 0

ð38Þ

in which

Ti , J, j

¼

Z 2

1

Ti J jd ð39Þ

Equation (39) implies that wi satisfies the homogeneous boundary conditions at

¼ 1, 2.

Proof: Considering two eigenfunction-vectors, wi, wj, we have from equation (35)

that

Hwi ¼ iwi, Hwj ¼ jwj ð40Þ

in which H satisfies the following relation:

HT ðJwiÞ ¼ iJwi ð41Þ

Multiplying wTj on both sides of equation (41) and integrating it across the transverse

section, we can obtain the following equation:

wTj ,H

T, Jwi

D E¼ wT

i , JH,wj

¼ i w

Tj , J,wi

D E¼ i w

Ti , J,wj

, ð42Þ

in which equation (16) has been used. Similarly, it is easy to show that the following

relation holds true:

wTi , JH,wj

¼ j w

Ti , J,wj

ð43Þ


Making use of equations (42) and (43), we have:

ði þ jÞ wTi , J,wj

¼ 0 ð44Þ

If iþj 6¼ 0, equation (44) is reduced to:

wTi , J,wj

¼ 0 ð45Þ

When iþj¼ 0, i.e., j¼i, ti and wi satisfy the following relation:

wTi J i 6¼ 0 ði ¼ 1, 2, . . . , nÞ ð46Þ

Performing the orthonormality operation, we obtain:

wTi Jwi ¼ 1, and=or wT

iJwi ¼ 1 ði ¼ 1, 2, . . . , nÞ ð47Þ

This indicates that equation (38) holds true. The adjoint symplectic ortho-normalization relation between eigenfunctions has thus been proved.

Since wþi and wi are of adjoint symplectic orthonormalization, the state vectorw can be expressed by the linear combination of the eigenfunction-vectors as follows:

w ¼X1i¼1

ðaiwi þ biwiÞ ð48Þ

where ti and wi are eigenfunction vectors corresponding to i and i and ai, biare coefficients to be determined.

From equations (4) and (37) we can obtain the following expressions:

w’

¼ rqðÞ ð49Þ

rzDr

¼ r1pðÞ ð50Þ

From equation (50) it can be found that the stresses and electric displacements nearthe apex of a wedge are proportional to r 1; therefore, the singularity order of thestresses and electric displacements is Re() 1.

It is obvious that the stresses and electric displacements are singular if the realpart of is less than 1, i.e. Re()51. For the potential energy to be bounded at theapex of the wedge, it is required that Re()40. So we focus our attention on theinterval:

0 < ReðÞ < 1 ð51Þ

Using the notation of the generalized stress and electric displacement intensityfactors Kand KD, equation (50) can be rewritten as:

rzðr, Þ ¼ Kr1f ðÞ

Drðr, Þ ¼ KDr1f DðÞ

ð52Þ


where * 1 is the orders of the stress and electric displacement singularity, andf () and f D() are the angular function

From equations (50) and (52) it is easy to see that:

wðÞ ¼ KfðÞ, ¼ ð53Þ

in which

K ¼K 00 KD

, and fðÞ ¼ f ðÞ, fDðÞ

T:

The remaining task is to find the angular function f() and the generalized stressand electric displacement intensity factor K (the definition of the generalized stressintensity factors can be found in [11]). Therefore, we need to find eigenvaluesof equation (35), which satisfy the condition equation (51). To this end, rewritingequation (35) in terms of its matrix components, we have

0"111

e151

0 e151

c441

c44d2

d2e15

d2

d2 0

e15d2

d2"11

d2

d20

266666666664

377777777775

w

’

Sr

SDr

8>>><>>>:

9>>>=>>>;

¼ 0 ð54Þ

The order of singularity in the elastic and electric fields is determined by setting thedeterminant of the 4 4 matrix in equation (54) to zero. This is equivalent to:

0"111

e151

0 e151

c441

c442 e15

2 0e15

2 "112 0

¼ 0 ð55Þ

where is the eigenvalue in direction.Equation (55) leads to the following equation:

ð2 þ 2Þ2¼ 0 ð56Þ

Thus, the solutions of are:

1, 2 ¼ i, 3, 4 ¼ i ð57Þ

With solution (57), the general expressions of the elastic and electric fields can beexpressed as:

w ¼ A1 cosðÞ þ B1 sinðÞ þ C1 cosðÞ þD1 sinðÞ,

’ ¼ A2 cosðÞ þ B2 sinðÞ þ C2 cosðÞ þD2 sinðÞ,

Sr ¼ A3 cosðÞ þ B3 sinðÞ þ C3 cosðÞ þD3 sinðÞ,

SDr ¼ A4 cosðÞ þ B4 sinðÞ þ C4 cosðÞ þD4 sinðÞ:

ð58Þ


where Ai, Bi, Ci, Di (i¼ 1 4) are unknown constants to be determined.Substituting equation (58) into equation (54) yields the following relationships

among the four unknown constants (the equations of these constants are listed in

Appendix B):

A3 ¼ ðc44A1 þ e15A2Þ, A4 ¼ ðe15A1 "11A2Þ,

B3 ¼ ðc44B1 þ e15B2Þ, B4 ¼ ðe15B1 "11B2Þ,

Di ¼ Ci ¼ 0 ði ¼ 1 4Þ:

ð59Þ

Then, equation (58) can be rewritten as:

w

’

( )¼

A1 B1

A2 B2

" #cosðÞ

sinðÞ

( ),

Sr

SDr

( )¼

c44 e15

e15 "11

" #A1

A2

( )cosðÞ þ

c44 e15

e15 "11

" #B1

B2

( )sinðÞ:

ð60Þ

From equations (1) and (60) we have:

SSD

¼

c44 e15e15 "11

A1 B1

A2 B2

sinðÞcosðÞ

ð61Þ

To obtain explicit expression of the four unknown constants, consider a piezoelectric

wedge as shown in figure 1. The conditions at the boundary edges are assumed to be

free of traction and electrically insulated:

zðr,Þ ¼ zðr, Þ ¼ Dðr,Þ ¼ Dðr, Þ ¼ 0 ð62Þ

x

o

y

r

θ

α

β

Figure 1. Piezoelectric wedge.


Substituting equations (60) and (61) into equation (62) yields:

c44 sinðÞ e15 sinðÞ c44 cosðÞ c44 cosðÞe15 sinðÞ "11 sinðÞ e15 cosðÞ "11 cosðÞc44 sinðÞ e15 sinðÞ c44 cosðÞ e15 cosðÞe15 sinðÞ "11 sinðÞ e15 cosðÞ "11 cosðÞ

2664

3775

A1

A2

B1

B2

8>><>>:

9>>=>>; ¼ 0 ð63Þ

The condition for the existence of non-zero solutions of fA1 A2 B1 B2 gT is the

determinant of the coefficients matrix being zero, which leads to the following

equation:

ðc44"11 þ e215Þ2 sin2ððþ ÞÞ ¼ 0 ð64Þ

If ¼ ¼, we have ¼ 1/2, and the order of singularity is 1/2, which is the

classical square-root singularity for a semi-infinite crack. This result also verifies the

validity of this method for the case of a semi-infinite crack.Consider now a piezoelectric half plane, i.e. ¼ ¼/2. It can be easily found

that no root of equation (64) can satisfy the condition 05Re()51. Therefore, there

is no singularity for the piezoelectric half plane under the homogeneous boundary

condition. The same results have also been obtained for a piezoelectric half plane

under anti-plane deformation [11]. We also note that the singularity disappears for

þ 180. For 1805þ 5360, the variation of the order of singularity with

þ is plotted in figure 2. It can be seen that for a homogeneous piezoelectric

wedge, the order of singularity depends on the value of þ only.

180 200 220 240 260 280 300 320 340 360−0.5

−0.45

−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

α+β

Re

(µ–1

)

Figure 2. Variation of order of singularity with þ for a piezoelectric wedge.


It should be mentioned here that for the sake of simplicity only one type of boundarycondition on the edges, given as equation (62), is considered. However, for othertypes of boundary conditions such as clamped (w¼ 0) and electrically open (¼ 0),this procedure is also applicable and the results can be obtained in a similar way.

4. Piezoelectric bimaterial wedge

For a piezoelectric bimaterial wedges as shown in figure 3, the boundary conditionsare as follows:

ð1Þz ðr,Þ ¼ ð2Þz ðr, Þ ¼ Dð1Þ ðr,Þ ¼ D

ð2Þ ðr, Þ ¼ 0 ð65Þ

If the bimaterials are rigidly bonded at the interface, the continuity conditionson the interface are:

ð1Þz ðr, 0Þ ¼ ð2Þz ðr, 0Þ, wð1Þðr, 0Þ ¼ wð2Þðr, 0Þ,

Dð1Þ ðr, 0Þ ¼ D

ð2Þ ðr, 0Þ, Eð1Þ

r ðr, 0Þ ¼ Eð2Þr ðr, 0Þ:

ð66Þ

in which superscripts (1) and (2) denote materials 1 and 2, respectively.As shown in figure 3, each material can be viewed as a homogeneous wedge.

Using the general solution (60) of a homogeneous piezoelectric wedge and thecontinuity condition at its interface, we can easily obtain the solution for a bimaterialwedge. Keeping this in mind, substituting the solutions (60) and (61) for eachmaterial into equation (66) yields the relationship between the unknown constants:

Bð2Þ1

Bð2Þ2

( )¼

a11 a12a21 a22

Bð1Þ1

Bð1Þ2

( )ð67Þ

x

o

y

Material 1

Material 2

r

θ

α

β

Interface

Figure 3. Piezoelectric bimaterial wedge.


in which

a11 ¼1

3"ð2Þ11 c

ð1Þ44 þ e

ð2Þ15 e

ð1Þ15

, a12 ¼

1

3"ð2Þ11 e

ð1Þ15 e

ð2Þ15 "

ð1Þ11

,

a21 ¼1

3eð2Þ15 c

ð1Þ44 c

ð2Þ44 e

ð1Þ15

, a22 ¼

1

3eð2Þ15 e

ð1Þ15 þ c

ð2Þ44 "

ð1Þ11

,

ð68Þ

where

3 ¼ cð2Þ44 "

ð2Þ11 þ e

ð2Þ215 ð69Þ

Using the relation (67) and substituting the solutions (60) and (61) into equation (65)

leads to:

cð1Þ44 sinðÞ eð1Þ15 sinðÞ cð1Þ44 cosðÞ eð1Þ15 cosðÞ

eð1Þ15 sinðÞ "ð1Þ11 sinðÞ e

ð1Þ15 cosðÞ "ð1Þ11 cosðÞ

cð2Þ44 sinðÞ eð2Þ15 sinðÞ cð2Þ44 a11þ eð2Þ15 a21

h icosðÞ cð2Þ44 a12þ eð2Þ15 a22

h icosðÞ

eð2Þ15 sinðÞ "ð1Þ11 sinðÞ e

ð2Þ15 a11 "

ð2Þ11 a21

h icosðÞ e

ð2Þ15 a12 "

ð2Þ11 a22

h icosðÞ

26666664

37777775

Að1Þ1

Að1Þ2

Bð1Þ1

Bð1Þ2

8>>>>><>>>>>:

9>>>>>=>>>>>;¼ 0

ð70Þ

The non-zero solution of equation (70) requires that:

cð1Þ44 "

ð1Þ11 þ e

ð1Þ215

sin2ðÞ cos2ðÞ þ c

ð2Þ44 "

ð1Þ11 þ e

ð2Þ215

sin2ðÞ cos2ðÞ

þ cð1Þ44 "

ð2Þ11 þ 2e

ð1Þ15 e

ð2Þ15 þ c

ð2Þ44 "

ð1Þ11

sinðÞ sinðÞ cosðÞ cosðÞ ¼ 0

ð71Þ

or written as:

sin2ððþ ÞÞ þ R1 sin2ðð ÞÞ ¼ R2 sinððþ ÞÞ sinðð ÞÞ ð72Þ

with:

R1 ¼A11 þ A22 A12 A21

A11 þ A22 þ A12 þ A21, R2 ¼

2ðA22 A11Þ

A11 þ A22 þ A12 þ A21

A11 ¼ eð1Þ15 e

ð1Þ15 þ c

ð1Þ44 "

ð1Þ11 , A12 ¼ e

ð1Þ15 e

ð2Þ15 þ "ð1Þ11 c

ð2Þ44

A21 ¼ eð2Þ15 e

ð1Þ15 þ "ð2Þ11 c

ð1Þ44 , A22 ¼ e

ð2Þ15 e

ð2Þ15 þ c

ð2Þ44 "

ð2Þ11

ð73Þ

It is found that equation (72) is exactly the same as those of Chue and Chen [10]

and Chen and Chue [11]. The solution of equation (72) is given in detail in [11] for

two cases of ¼ and 6¼ to show the singular electro-mechanical field. For an

interface crack, i.e. ¼ ¼, we have ¼1/2, which returns to a classical 1/2

singularity. For other values of and , equation (72) shows that the order of

singularity strongly depends on the geometry and material constants of the two

piezoelectric materials. In addition, the angular function and generalized stress

and electrical intensity factors Kand KD can also be obtained easily using this

method. It should be mentioned, however, that equation (72) derives from the

4 4 homogeneous equation (70), while in [11], it derives from 8 8


homogeneous equations. Compared to the conventional method [11], the proposedsymplectic model can solve the singularity problem more rationally. In particular,with the increase of number of materials, the conventional method will induce alarge number of complicated equation systems which may be difficult to be solvedtheoretically. In contrast, the model developed in this paper solves the problemabove with the matrix operations and does not suffer the problem induced by thenumber of materials. In addition, this model is particularly useful for analyzingproblems with local effects, such as the field singularity problem and the Saint-Venant decay problem.

5. Multi-material wedge

In sections 3 and 4, the theory of the Hamiltonian system was used to developsymplectic models of a piezoelectric wedge and, then, to determine the orders ofsingularity for both a homogeneous wedge and a bimaterial wedge. In the following,the results obtained are extended to the case of multi-material wedges, includingmulti-piezoelectric materials and combinations of piezoelectric–elastic materials.

5.1. Multi-piezoelectric material wedge

Consider a piezoelectric wedge consisting of multi-piezoelectric material elementsas shown in figure 4, which is similar to the multi-elastic material wedge in [12].Here, N is the number of material elements. The polar coordinate is again selected

C0

1C

…… Ci−1

Ci

CN−1

CN

O

Ω1

Ωi

ΩN

M1

Mi

MN

……

a1

ai

aN

Figure 4. Multi-material wedge.


for simplicity, and C0,C1, . . . ,CN is adopted to indicate the 0N sub-polar

coordinate systems. The domain i denotes the material element Mi, and i is the

angle of Mi.The continuity conditions on the bonded interface region are:

ðiÞz ðr, 0Þ ¼ ðiþ1Þz ðr, 0Þ, wðiÞðr, 0Þ ¼ wðiþ1Þðr, 0Þ,

DðiÞ ðr, 0Þ ¼ D

ðiþ1Þ ðr, 0Þ, EðiÞ

r ðr, 0Þ ¼ Eðiþ1Þr ðr, 0Þ

ð74Þ

in which the superscript i runs from 1 to N 1, which represents the associated

variable defined in the domain i. It should be mentioned that the fields in the two

adjacent regions i and iþ 1 in equation (74) are written in terms of the coordinate

system Ci with both of the two regions having ¼ 0 at the interface. This is for the

sake of simplicity.The boundary conditions of this problem are:

ð1Þz ðr,1Þ ¼ Dð1Þ ðr,1Þ ¼ ðNÞ

z ðr, NÞ ¼ DðNÞ

ðr, NÞ ¼ 0 ð75Þ

Note that solution (60) also applies for each single domain i. Thus, substitution

of the general solutions (60) for the domain i and iþ 1 into equation (74)

yields the relationship of the unknown constants for any two adjacent domains as

follows:

F iþ1i

¼ Rði, iþ 1Þ½ F i

i

ð76Þ

where

Rði, iþ 1Þ½ ¼

1 0 0 0

0 1 0 0

0 0 r11 r12

0 0 r21 r22

26664

37775 ð77Þ

Fiþ1i

¼ Aiþ1

1i Aiþ12i Biþ1

1i Biþ12i

Tð78Þ

and

r11 ¼1

r"ðiþ1Þ11 c

ðiÞ44 þ e

ðiþ1Þ15 e

ðiÞ15

, r12 ¼

1

r"ðiþ1Þ11 e

ðiÞ15 e

ðiþ1Þ15 "ðiÞ11

,

r21 ¼1

reðiþ1Þ15 c

ðiÞ44 c

ðiþ1Þ44 e

ðiÞ15

, r22 ¼

1

reðiþ1Þ15 e

ðiÞ15 þ c

ðiþ1Þ44 "ðiÞ11

,

ð79Þ

with

r ¼ cðiþ1Þ44 "ðiþ1Þ

11 þ eðiþ1Þ215 ð80Þ


In equation (76), the subscript i represents the unknown constants expressed in terms

of the coordinates Ci, and the superscripts i and iþ 1 mean the domains i, iþ 1,

respectively.In the following, the coordinate transformation is used to find the relationships

between the unknown constants in general solutions of each material domain i

in two coordinate systems Ci and. Ci 1 Assuming the equality

wii

’ii

¼

wii1

’ii1

ð81Þ

and using the relationships among the trigonometric functions, we obtain:

Fii

¼ Tði, i 1Þ½ Fi

i1

ð82Þ

where

Tði, i 1Þ½ ¼

cosðiÞ 0 sinðiÞ 0

0 cosðiÞ 0 sinðiÞ

sinðiÞ 0 cosðiÞ 0

0 sinðiÞ 0 cosðiÞ

26664

37775 ð83Þ

The combination of equations (76) and (82) yields the relationship:

FNN1

¼ TRN½ F1

1

ð84Þ

where

TRN½ ¼Y2

i¼N1

Rði, iþ 1Þ½ Tði, i 1Þ½

Rð1, 2Þ½ ð85Þ

It can be seen from equation (84) that solutions in any domain can be expressed by

four independent unknown constants defined in 1. Considering the boundary

conditions (75), we obtain:

M½ F11

¼ 0 ð86Þ

where [M] is a 4 4 matrix which has a similar form to that in equation (70) (we omit

its details here). The existence of a nontrivial solution for fF11g requires a zero

determinant of the coefficients matrix [M]:

det M½ ¼ 0 ð87Þ

Then, the solution for can be obtained by solving equation (87) and the order

of singularity is again Re() 1 by considering the condition in equation (51).It should be mentioned that equation (87) is highly nonlinear in terms of

the variable and, therefore, an analytical solution to is usually impossible

except for a few simple cases. In the following, our focus is placed on a numerical

solution only. For illustration, consider a three-material wedge in which materials

1 and 3 are assumed to be PZT-4, and material 2 is PZT-5 (see figure 5). The material


properties used are:

PZT-4 :c44 ¼ 25:6 109 N=m2, e15 ¼ 12:7C=N, "11 ¼ 6:46 109 F=m

PZT-5 :c44 ¼ 21:1 109 N=m2, e15 ¼ 12:3C=N, "11 ¼ 8:11 109 F=m

As an example, table 1 lists the orders of singularity for differet values of 1 and eð1Þ15

when 2¼3¼/3. It can be seen from table 1 that there exist a pair of complex ortwo real singularity orders for some values of 1 (e.g. two real singularity orders for1¼/3 and a pair of complex roots for 1¼) or for some values of e

ð1Þ15 . It is found

that, at 1¼ (2/3), the singularity order may be complex or real depending on thevalue of e

ð1Þ15 , which indicates that e

ð1Þ15 can affect the singularity orders to some extent.

In addition, the singularity order may become zero for some special values of 1and e

ð1Þ15 . This is useful when designing a multi-material wedge. It is interesting to

note from table 1 that the singularity order of a three-material wedge is not equalto zero for a half-plane when 1¼/3 and not equal to 1/2 for an interfacecrack when 1¼ 4/3, which is different from the results for a homogeneous

Table 1. Singularity orders of a piezoelectric wedge of three dissimilar materials for differentvalues of 1 and e

ð1Þ15 when 2¼3¼/3 and e15¼ 12.7C/N.

1 0.8 eð1Þ15 0.9 eð1Þ15 1.0 eð1Þ15 1.1 eð1Þ15 1.2 eð1Þ15=3 0.269 0.300 0.324 0.339 0.339

0.155 0.16 0.182 0.214 0.2602=3 0.783 0.034i 0.765 0.812 0 0.913 0.226i 0.959 0.296i 0 0 0.985 0.372i 0.956 0.398i4=3 0.079 0.086 0.090 0.094 0.095

0.043 0.047 0.052 0.06 0.0690.967 0.355i 0.985 0.415i 0.963 0.435i

Inter

face 1

Interface 2

Material 2

Material 3

x

αγ

θr

β

y

o

Figure 5. Piezoelectric wedge of three dissimilar materials.


or bimaterial wedge. This behaviour is identical with that for an anisotropicthree-material wedge [20].

It is evident from table 1 that the values of 1 and the piezoelectric constant eð1Þ15

have an important effect on the order of singularity. In conclusion, the order ofsingularity is a function of geometry and material constants for a multi-piezoelectricmaterial wedge.

For a three piezoelectric materials wedge, the variation of singularity order with1 has been plotted in figure 6 for 2¼3¼/2 and in figure 7 for 2¼/2, 2¼. Itcan be seen from figure 6 that the singularity order increases with an increase in thevalue of 1 and, at about 1¼ (2/3), multi-orders disappear and the complex orderappears. It is observed from figure 7 that there are two singularity orders for anygiven value of 1, and they also increase with an increase in 1.

To prove the validity of the proposed formulation, two cases are consideredas follows:

case 1 : 1 ¼ 2 ¼ , 3 ¼ 0

case 2 : 1 ¼ 3 ¼ , 2 ¼ 0

The singularity order for both cases is 1/2, which is identical with the result foran interface crack.

5.2. Mixed multi-piezoelectric–elastic wedge

The wedge considered in this section is obtained by replacing the piezoelectricmaterial i in figure 4 with an elastic composite, while the other materials are stillassumed to be of piezoelectric material.

Figure 6. Variations of the singularity orders ( 1) for a three piezoelectric materialswedge with the wedge angle 1 (2¼3¼/2, e

ð1Þ15 ¼ 1:2 12:7C=N).


The boundary conditions and the continuity conditions on the bonded interface

are the same as those for the multi-piezoelectric materials wedge, except that the last

two equations of the continuity conditions for the interfaces between i1 and i, i,

and iþ1 are replaced by the following equation:

Dði1Þ ðr, 0Þ ¼ 0, D

ðiþ1Þ ðr, 0Þ ¼ 0 ð88Þ

For an elastic composite wedge, the general solution can be written as:

w ¼ A1 cosðÞ þ B1 sinðÞ

Sr ¼ c44A1 cosðÞ þ c44B1 sinðÞ

S ¼ c44A1 sinðÞ þ c44B1 cosðÞ

ð89Þ

in which there are only two independent unknown constants A1 and B1, and

S¼ rz.From the continuity conditions (74) and (88) and the general solution (89),

we can obtain the following relationships:

Aði1Þ1ði1Þ ¼ A

ðiÞ1ði1Þ, B

ði1Þ1ði1Þ ¼

cðiÞ44

i1BðiÞ1ði1Þ, B

ði1Þ2ði1Þ ¼

cðiÞ44e

ði1Þ15

i1"i111

BðiÞ1ði1Þ,

Aðiþ1Þ1ðiÞ ¼ A

ðiÞ1ðiÞ, B

ðiþ1Þ1ðiÞ ¼

cðiÞ44

iþ1BðiÞ1ðiÞ, B

ðiþ1Þ2ðiÞ ¼

cðiÞ44e

ðiþ1Þ15

iþ1"iþ111

BðiÞ1ðiÞ

ð90Þ

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.55

−0.5

−0.45

−0.4

−0.35

−0.3

II

p2

p2 III

I a1

a2 =

a3 =

Re

(m−1

)

a1 (×p)

Figure 7. Variations of the singularity orders ( 1) for a three piezoelectricmaterials wedge with the wedge angle 1 (2¼/2, 2¼, e

ð1Þ15 ¼ 1:2 12:7C=N).


where

i1 ¼ cði1Þ44 þ

eði1Þ215

"ði1Þ11

, iþ1 ¼ cðiþ1Þ44 þ

eðiþ1Þ215

"ðiþ1Þ11

ð91Þ

Making use of equations (76) and (82), we can obtain following equations:

F11 ¼ Mi1½ Fi1

i1, FNN1

¼ Ni½ Fiþ1

i

ð92Þ

where

Mi1½ ¼ TRi1½ 1 Tði 1, i 2Þ½

1,

Ni½ ¼Yi

i¼N1

Rðiþ 1, iþ 2Þ½ Tðiþ 1, iÞ½ ð93Þ

where [Mi1] and [Ni] are 4 4 matrices.Using the coordinate transformation, the following equation can be obtained:

AðiÞ1ðiÞ

BðiÞ1ðiÞ

( )¼

cosðiÞ sinðiÞsinðiÞ cosðiÞ

A

ðiÞ1ði1Þ

BðiÞ1ði1Þ

( )ð94Þ

Considering equation (94) and from the second and fourth equation of the boundary

conditions (75), we can arrive at:

Aði1Þ2ði1Þ ¼ a1A

ðiÞ1ði1Þ þ b1B

ðiÞ1ði1Þ, A

ðNÞ

2ðN1Þ ¼ a2AðiÞ1ði1Þ þ b2A

ðiÞ2ði1Þ ð95Þ

where ai and bi (i¼ 1, 2) are listed in Appendix C.Combined use of equations (90), (92) and (95) leads to the following

relationships:

F11 ¼ Mi1½

1 0

a1 b1

0cðiÞ44

i1

0cðiÞ44e

ði1Þ15

i1"ði1Þ11

2666666664

3777777775

AðiÞ1ði1Þ

BðiÞ1ði1Þ

8<:

9=; ð96Þ

and

FNN1 ¼ Ni½

1 0

a2 b2

0cðiÞ44

iþ1

0cðiÞ44e

ðiþ1Þ15

iþ1"ðiþ1Þ11

2666666664

3777777775

cosðiÞ sinðiÞ

sinðiÞ cosðiÞ

A

ðiÞ1ði1Þ

BðiÞ1ði1Þ

8<:

9=; ð97Þ


Then, substituting equation (96) and (97) into the first and third equation ofequation (75) yields:

M½ A

ðiÞ1ði1Þ

BðiÞ1ði1Þ

( )¼ 0 ð98Þ

in which [M] is a 2 2 matrix. The value of can be determined by setting thedeterminant of [M] to be zero.

For illustration, consider a three-material wedge in which materials 1 and 3 arePZT-4 and PZT-5, respectively, and material 2 is a graphite/epoxy composite.

The properties of PZT-4 and PZT-5 can be found in section 5.1, and the materialconstant of the graphite/epoxy composite is c44¼ 3.61109 Pa.

Table 2 lists the orders of singularity of the three-material wedge for different 2and the shear modulus c

ð2Þ44 . It is again found from table 2 that the singularity order is

not equal to zero for the case of a half-plane and not equal to 1/2 for an interfacecrack. However, when 2¼, the singularity order vanishes. Similar to table 1, itcan also be seen that the order of singularity depends strongly on the value of 2 andthe shear modulus. Knowledge of this behaviour can be applied to determinean optimum angle and optimal material constants for avoiding fracture anddelamination induced by singularities of the electroelastic fields. Although only oneelastic composite element is considered here for simplicity, this procedure is alsosuitable for the case of a wedge containing two or more elastic composite elements.

The results shown in figure 8 are for a three-material wedge with varying thewedge angle of material II (elastic material) from 0 to /2. It is observed that thesingularity order is complex when 0 2 0.22 and then becomes real when240.22.

6. Conclusion

In this paper, a symplectic model based on the Hamiltonian system has beendeveloped for analyzing singular behaviour near the apex of a multi-dissimilarpiezoelectric wedge under antiplane deformation. Explicit solutions of elastic andelectric fields are obtained for the cases of composite wedge consisting of one, twoand multiple piezoelectric materials and mixed piezoelectric–elastic materials.Numerical examples for cases of a three-material wedge are considered and theresults show that the order of singularity of the electroelastic field near the apex of

Table 2. Singularity order of piezoelectric–elastic composite wedge containing threedissimilar materials for different values of 2 and c

ð2Þ44 (1¼3¼/3).

2 1.01cð2Þ44 2.0c

ð2Þ44 3.0c

ð2Þ44 4.0c

ð2Þ44 5.0c

ð2Þ44

=3 0.892 0.892 0 0 02=3 0.946 0 0 0 0 0 0 0 0 04=3 0.245 0.241 0.237 0.234 0.231


the wedge is significantly affected by the geometry of the wedge, the piezoelectricconstant and the shear modulus. The study also indicates that for a three-materialwedge the singularity order is not equal to zero for the case of a half-plane andnot equal to 1/2 for an interface crack, unlike in the case of a homogeneous wedge.In addition, this method can be easily extended to cases of in-plane deformation.This work is under way.

Acknowledgements

The authors are indebted to the two anonymous reviewers for their helpfulcomments and suggestions on an earlier version of this paper.

Appendix A: Proof of equation (17)

Making use of the definitions (13), (15), (16), and (18), we have

vT1 ,Hv2

¼

Z 2

1

vT1 JHv2d

¼

Z 2

1

w1 ’1 Sr1 SDr1

c44

@2

@2e15

@2

@20 0

e15@2

@2"11

@2

@20 0

0 0 "111

e151

0 0 e151

c441

266666666664

377777777775

w2

’2

Sr2

SDr2

8>>><>>>:

9>>>=>>>;d

Figure 8. Variations of the singularity orders ( 1) for a piezoelectric composite materialswedge with the wedge angle 2 (1¼, 3¼/2, c

ð2Þ44 ¼ 5 3:61 109 Pa).


¼

Z 2

1

w1 ’1 Sr1 SDr1

c44

@2w2

@2 e15

@2’2@2

e15@2w2

@2þ "11

@2’2@2

"111

Sr2 e151

SDr2

e151

Sr2 þc441

SDr2

266666666666664

377777777777775d

¼

Z 2

1

w2 ’2 Sr2 SDr2

c44

@2w1

@2 e15

@2’1@2

e15@2w1

@2þ "11

@2’1@2

"111

Sr1 e151

SDr1

e151

Sr1 þc441

SDr1

266666666666664

377777777777775dþC

¼

Z 2

1

w2 ’2 Sr2 SDr2

c44

@2

@2e15

@2

@20 0

e15@2

@2"11

@2

@20 0

0 0 "111

e151

0 0 e151

c441

266666666666664

377777777777775

w1

’1

Sr1

SDr1

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;dþC

¼ vT2 ,Hv1

þC ðA:1Þ

where

C ¼ c44 w1@w2

@ w2

@w1

@

21

e15 ’1@w2

@ w2

@’1@

21

e15 w1@’2@

’2@w1

@

21

þ "11 ’1@’2@

’2@w1

@

21

ðA:2Þ

C is an integration constant representing the boundary condition of a wedge. In

our analysis, C represents the traction-charge free conditions:

z ¼ D ¼ 0 ð¼ 1, 2Þ ðA:3Þ

If v1, v2 satisfy the corresponding boundary condition (A.3), then:

vT1 ,Hv2

¼ vT2 ,Hv1

ðA:4Þ


Therefore, according to theory of the symplectic geometry [13, 14], H is the

Hamiltonian matrix. It should be mentioned that equation (A.4) is valid not only for

traction-charge free but also for other boundary conditions such as ¼ 0, w¼ 0 or

the mixed boundary condition ¼ 0, z¼ 0, although the derivation above is based

on the homogeneous boundary condition (A.3). To this end, equation (A.2) is

rewritten in the form:

C ¼ w21 þ ’2D1 w12 ’1D2½ 12

ðA:5Þ

in which i ¼ ðzÞ iand Di ¼ ðDÞ

i. It can be seen from (A.5) that equation (A.4) is

valid for other boundary conditions as only two among w, , , D equal to zero.

Appendix B: Equation for determining constants Ai, Bi, Ci, and Di (i¼ 1 4)

0"111

e151

0 e151

c441

c442 e15

2 0e15

2 "112 0

2666664

3777775

A1

A2

A3

A4

8>><>>:

9>>=>>; ¼

00

2ðc44C1 þ e15C2Þ

2ðe15C1 "11C2Þ

2664

3775 ðB:1Þ

0"111

e151

0 e151

c441

c442 e15

2 0e15

2 "112 0

2666664

3777775

B1

B2

B3

B4

8>><>>:

9>>=>>; ¼

00

2ðc44D1 þ e15D2Þ

2ðe15D1 "11D2Þ

2664

3775 ðB:2Þ

0"111

e151

0 e151

c441

c442 e15

2 0e15

2 "112 0

2666664

3777775

C1

C2

C3

C4

8>><>>:

9>>=>>; ¼

0000

2664

3775 ðB:3Þ

0"111

e151

0 e151

c441

c442 e15

2 0e15

2 "112 0

2666664

3777775

D1

D2

D3

D4

8>><>>:

9>>=>>; ¼

0000

2664

3775 ðB:4Þ


Appendix C: Constants ai and bi

a1 b1

¼1

mm1 m2 m3

1 0

0cðiÞ44

i1

0cðiÞ44e

ði1Þ15

i1"ði1Þ11

2666664

3777775,

a2 b2

¼1

nn1 n2 n3

1 0

0cðiÞ44

iþ1

0cðiÞ44e

ðiþ1Þ15

iþ1"ðiþ1Þ11

2666664

3777775

cosðiÞ sinðiÞ

sinðiÞ cosðiÞ

ðC:1Þ

m1 ¼ m11eð1Þ15 sinð1Þ m21"

ð1Þ11 sinð1Þ m31e

ð1Þ15 cosð1Þ þm41"

ð1Þ11 cosð1Þ




ð1Þ11 cosð1Þ




ð1Þ11 cosð1Þ

n1 ¼ n11eð1Þ15 sinðNÞ n21"

ð1Þ11 sinðNÞ þ n31e

ð1Þ15 cosðNÞ n41"

ð1Þ11 cosðNÞ




ð1Þ11 cosðNÞ




ð1Þ11 cosðNÞ

ðC:2Þ

in which mij and nij (i, j¼ 1 4) are the elements of [Mi 1] and [Ni] respectively, and

m ¼ "ð1Þ11m22 eð1Þ15m12

h isinð1Þ þ "ð1Þ11m42 e

ð1Þ15m32

h icosð1Þ ðC:3Þ

n ¼ "ðnÞ11 n22 eðnÞ15 n12

h isinðNÞ þ "ðnÞ11 n42 e

ðnÞ15 n32

h icosðNÞ ðC:4Þ

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symplectic model for piezoelectric wedges and its...

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