A Review on the Three-Dimensional Analytical Approaches … · A Review on the Three-Dimensional Analytical Approaches of Multilayered and Functionally Graded Piezoelectric Plates

Copyright c© 2008 Tech Science Press CMC, vol.8, no.2, pp.93-132, 2008

A Review on the Three-Dimensional Analytical Approaches of Multilayeredand Functionally Graded Piezoelectric Plates and Shells

Chih-Ping Wu1,2, Kuan-Hao Chiu2 and Yung-Ming Wang2

Abstract: The article is to present an overviewof various three-dimensional (3D) analytical ap-proaches for the analysis of multilayered andfunctionally graded (FG) piezoelectric plates andshells. The reported 3D approaches in the liter-ature are classified as four different approaches,namely, Pagano’s classical approach, the statespace approach, the series expansion approachand the asymptotic approach. Both the mixedformulation and displacement-based formulationfor the 3D analysis of multilayered piezoelectricplates are derived. The analytical process, basedon the 3D formulations, for the aforementionedapproaches is briefly interpreted. The present for-mulations of multilayered piezoelectric plates canalso be used for the analysis of FG piezoelectricplates, of which material properties are heteroge-neous through the thickness coordinate, by artifi-cially dividing the plate as NL-layered plates withconstant coefficients in an average sense for eachlayer. The present formulations can also be ex-tended to the ones of piezoelectric shells using theassociated shell coordinates. A comprehensivecomparison among the 3D results available in theliterature using various approaches is made. Forillustration, the through-thickness distributions ofvarious field variables for the simply-supported,multilayered and FG piezoelectric plates are pre-sented using the asymptotic approach and doublychecked with a newly-proposed meshless method.The literature dealing with the 3D analysis of mul-tilayered and FG piezoelectric plates is surveyedand included. This review article contains 191 ref-erences.

1 Corresponding author. Fax: +886-6-2370804, E-mail:[email protected]

2 Department of Civil Engineering, National Cheng KungUniversity, Taiwan, ROC

Keyword: 3D solution; FG material; Piezoelec-tric material; Smart material; Shells; Plates

1 Introduction

Three-dimensional (3D) analysis of plates andshells made of a variety of advanced materials(or so-called smart materials) has attracted the re-searchers’ attention for a long time. It is mainlydue to the fact that 3D solutions of the benchmarkproblems may serve as a standard for assessingvarious approximate two-dimensional (2D) theo-ries of plates and shells. These 3D solutions alsoprovide as a reference for making the appropri-ate kinetics or kinematics assumptions prior to de-velop the advanced theories and numerical mod-eling of plates and shells. Hence, 3D analysis isinherently of much importance not only for aca-demic interest but also for industrial applications.

The review literature on 2D theoretical method-ologies and numerical modeling of multilay-ered composite elastic as well as piezoelectricplates/shells is surveyed and is tabulated in Table1. It is shown that the number of review litera-ture on laminated composite elastic plates/shellsis much larger than that on multilayered piezo-electric plates/shells. After a close examinationin the literature, we found that the review litera-ture on 3D analysis of multilayered and function-ally graded (FG) elastic/piezoelectric structures isscarce. Hence, the paper aims to making a com-prehensive survey on the present topics to replen-ish this insufficiency.

In general there are four different approaches,namely Pagano’s classical approach, the statespace approach, the series expansion approachand the asymptotic approach, drawn from the lit-erature to determine the 3D solutions of multilay-ered and FG piezoelectric (or elastic) plates and

94 Copyright c© 2008 Tech Science Press CMC, vol.8, no.2, pp.93-132, 2008

Table 1: Partial list of references on reviews for the analysis of multilayered elastic and piezoelectric struc-tures

Structures Assessments Literature

Multilayered composite plates Mechanics Noor, 1992

Computational models Liu and Soldatos, 2003; Noor and Burden, 1990a;

Subha, Shashidharan, Savithri and Syam, 2007

Mixed theories Carrera, 2000a, 2003

Displacement-based theories Liu and Li, 1996;

Matsunaga, 2002;

Noor and Burton, 1989;

Kant and Swaminathan, 2000

Bending Carrera and Ciuffreda, 2005

Thermal Carrera, 2000b

Vibration Carrera, 2004

Multilayered composite shells Mechanics Kapania, 1989;

Noor, 1990

3D dynamic analysis Soldatos, 1994

Computational models Noor, and Burton, 1990b;

Noor, Burton and Peters, 1991

Static Chandrashekhara and Pavan, 1995

Nonlinear Reddy and Chandrashekhara, 1987

Thermal Noor and Burton, 1992

Buckling Jaunky and Knight, 1999

Multilayered piezoelectric plates Laminated plate theories Gopinathan, Varadan and Varadan, 2000

Computational models Tang, Noor and Xu, 1996

Thermal Tauchert, Ashida and Noda, 1999

Sensing, actuation and control Rao and Sunar, 1994; Chee et al., 1998

Multilayered piezoelectric shells Laminated shell theories Kapuria, sengupta and Dumir, 1998;

Carrera and Brischetto, 2007

Computational models Saravanos and Heyliger, 1999

Nonlinear Yu, 1995;

Carrera and Parisch, 1997

Others Nontraditional theories Ambartsumian, 2002

A Review on the Three-Dimensional Analytical Approaches 95

shells. The pioneers who initiated the applicationsof various approaches to the structural behavior ofplates and shells were mentioned in each comingparagraph. The existing articles on exact 3D anal-yses of multilayered and FG elastic/ piezoelectricstructures were selectively mentioned in this sec-tion and comprehensively surveyed and tabulatedin the coming sections, respectively.

The first approach is based on Pagano’s study(Pagano, 1969) where a 3D problem of laminatedcomposite plates has been reduced as a planestrain problem by considering an infinite platewith the identical configuration and loading con-ditions along one of in-plane coordinates which isso-called the cylindrical bending problem. Thecompatibility equation has been expressed as afourth-order ordinary differential equation withconstant coefficients for each individual layer andin terms of the thickness coordinate only by in-troducing an Airy stress function satisfying thestress equilibrium equations in the formulation.The 3D solution of laminated plates under cylin-drical bending has been obtained by imposing thetraction conditions on the bottom and top sur-faces and the continuity conditions at the inter-faces between adjacent layers. This approach hasalso been extended to the 3D structural behaviorsof rectangular laminated plates (Pagano, 1970;Srinivas and Rao, 1970). However, Pagano’s clas-sical approach fails to exactly analyze the shellproblems of which governing equations are a sys-tem of differential equations with variable coeffi-cients. A successive approximation (SA) methodhas been proposed by Soldatos and Hadjigeorgiou(1990) to overcome the aforementioned straits. Inthe SA method, the shell is artificially dividedinto certain layers of smaller thickness so thatone may reasonably approximate the system ofthickness-varying differential equations to a sys-tem of thickness-invariant differential equations.That makes Pagano’s classical approach feasiblefor the 3D analysis of shells. The SA methodhas been demonstrated to be practically an ex-act method in the sense that it can approximatethe exact solution of relevant problems to any de-sired accuracy. Using the Pagano’s classical ap-proach in combination with the SA method and

matching the interface displacement and stresscontinuity conditions, Bhimaraddi (1991, 1993)and Bhimaraddi and Chandrashekhara (1992) ob-tained the approximate 3D solutions for the staticand dynamic responses of simply-supported, dou-bly curved shallow shells.

The second approach is based on the state spacemethod which is also called the method of trans-fer (or propagator) matrix (Bufler, 1971) or themethod of initial functions (Vlasov, 1957). Inthe approach, the sets of state variable equationswith constant coefficients for plates and with vari-able coefficients for shells have been obtained bymeans of direct elimination. Using a modal ma-trix composed of the eigenvectors of a full coef-ficient matrix of the coupled system of state vari-able equations and performing a similarity trans-formation, one may transform the coupled sys-tem equations to certain uncoupled system equa-tions. Afterwards, the unknowns can be inde-pendently determined. Again, for shell problems,the SA method has been used to reduce the sys-tem of thickness-varying equations to a series ofthickness-invariant equations. By imposing theboundary conditions on the lateral surfaces of theplates or shells, one may determine the state vari-ables through the thickness coordinate using themethod of transfer matrix. A detailed descrip-tion of the method of state space and its applica-tions to the bending, vibration, buckling analysesof laminated composite plates and shells has beenmade (Soldatos and Hadjigeorgiou, 1990; Fan andYe, 1990a, b; Fan and Zhang, 1992; Spencer etal., 1993; Ye and Soldatos, 1994a, b, 1995; Pan,2003; Ye, 2003; Tarn and Wang, 2001, 2003,2004).

The third approach is based on the series expan-sion method, i.e., the method of Frobenius, wherethe primary variables have been constructed inthe forms of a power series in the thickness co-ordinate (Ren, 1987, 1989; Varadan and Bhasksr,1991; Huang and Tauchert, 1991, 1992; Huang,1995). Substituting these specific forms of pri-mary variables into the system differential equa-tions and equating the coefficients of variouspower terms to be zero, one may obtain an in-dicial equation and a recurrence relation. Again,


imposing both the continuity conditions at inter-faces between adjacent layers and traction condi-tions on the lateral surfaces, one may iterativelysolve for the considered 3D problems. Kapuria etal. (1997a, b) and Xu and Noor (1996) proposeda modified Frobenius method for the 3D analysisof multilayered piezoelectric shells where the pri-mary field variables have been constructed in theform of a product of an exponential function anda power series in the thickness coordinate. Themethod of modified Frobenius has been demon-strated to yield better convergence properties.

The fourth approach is based on the method ofperturbation (Gol’denvaizer, 1961, 1963; Cicala,1965; Rogers et al., 1992, 1995; Wang and Tarn,1994; Wu, et al., 1996a, 1997, 2002). In the ap-proach, a geometric perturbation parameter hasbeen introduced in the asymptotic formulation.The 3D problems can be expanded as a series of2D problems governed by the equations of classi-cal plate or shell theories (CPT or CST). The 3Dsolutions can then be determined in a hierarchicand systematic manner. Studies of the dynamicanalysis of nonhomogeneous plates by means ofthe method of perturbation were presented longago (Widera, 1970; Johnson and Widera, 1971).In their formulation a straightforward expansionusing a single time variable has been used and cer-tain assumptions regarding the material compli-ances have been made. No numerical results havebeen given to demonstrate the applicability oftheir theories due to a rather complicated formu-lation. Asymptotic analysis for dynamic responseof nonhomogeneous plates and shells is not justa matter of applying the standard perturbationmethod. This will lead to not only equations toocumbersome to be useful but also nonuniform ex-pansions containing secular terms. The methodof multiple scales was therefore introduced in the3D asymptotic formulations of laminated platesand shells to eliminate the secular terms raisedfrom the regular asymptotic expansions (Tarn andWang, 1994; Wu et al., 1996b, 1998). The pre-vious asymptotic approach was successfully ex-tended to various benchmark problems for thebending, thermoelastic, vibration, buckling, dy-namic instability and nonlinear analyses of lam-

inated composite elastic plates and shells (Tarnand Wang, 1995; Tarn, 1996, 1997; Wu and Chi,1999, 2004, 2005; Wu and Hung, 1999; Wu andLo, 2000; Wu and Chiu, 2001, 2002; Wu andChen, 2001).

The extension of aforementioned approaches to3D analysis of multilayered and FG piezoelectricstructures was completely collected and tabulatedin the following sections. Several representa-tive articles dealing with 3D numerical modelingof multilayered and FG piezoelectric (or elastic)structures were also mentioned. For illustrationthe mathematical equations, based on the mixedformulation and the displacement-based formula-tion, for the 3D analysis of multilayered piezo-electric plates were derived. The analytical pro-cess using various approaches was briefly inter-preted. Comparisons of the 3D results of elasticand electric field variables obtained from a vari-ety of approaches were also presented.

2 Basic equations of 3D Piezoelectricity

As shown in Fig. 1, we consider a simply-supported, multilayered piezoelectric plate ofwhich thickness is 2h. The in-plane dimensionsin x1 and x2 directions are L1 and L2.

The linear constitutive equations valid for the na-ture of symmetry class of the piezoelectric mate-rial are given by

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

σ1

σ2

σ3

τ23

τ13

τ12

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

=

⎡⎢⎢⎢⎢⎢⎢⎣

c11 c12 c13 0 0 0c12 c22 c23 0 0 0c13 c23 c33 0 0 00 0 0 c44 0 00 0 0 0 c55 00 0 0 0 0 c66

⎤⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

ε1

ε2

ε3

γ23

γ13

γ12

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

−

⎡⎢⎢⎢⎢⎢⎢⎣

0 0 e31

0 0 e32

0 0 e33

0 e24 0e15 0 00 0 0

⎤⎥⎥⎥⎥⎥⎥⎦⎧⎨⎩

E1

E2

E3

⎫⎬⎭ (1)


( )zx or3

( )xx or1

NL

0z 1z 2z

1−kz kz

1−NLz NLz

k

12

a

2x

A3x A

2L

hh 1x 1L

b

Section A-A Figure 1: (a) The geometry and coordinates of a piezoelectric plate; (b) The configuration of section A-Aand the dimensionless thickness coordinate.

⎧⎨⎩

D1

D2

D3

⎫⎬⎭=

⎡⎣ 0 0 0 0 e15 0

0 0 0 e24 0 0e31 e32 e33 0 0 0

⎤⎦⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

ε1

ε2

ε3

γ23

γ13

γ12

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

+

⎡⎣η11 0 0

0 η22 00 0 η33

⎤⎦⎧⎨⎩

E1

E2

E3

⎫⎬⎭ , (2)

where σ1, σ2, σ3, τ13, τ23, τ12 and ε1, ε2, ε3, γ13,γ23, γ12 denote the stress and strain components,respectively; (D1,D2,D3) and (E1,E2,E3) denotethe components of electric displacement and elec-tric field, respectively. ci j, ei j and ηi j are the elas-


tic coefficients, piezoelectric and dielectric per-meability coefficients, respectively, relative to thegeometrical axes of the plate. The material prop-erties are considered to be layerwise constants.

The kinematic equations in terms of the curvilin-ear coordinates x1, x2 and x3 are

ε1 = u1,1, ε2 = u2,2, ε3 = u3,3

γ13 = u1,3 +u3,1, γ23u2,3 +u3,2,

γ33 = u3,3 +u3,3 (3)

in which u1, u2 and u3 are the displacement com-ponents.

The stress equilibrium equations without bodyforces are given by

σ1,1 +τ12,2 +τ13,3 = 0, (4)

τ12,1 +σ2,2 +τ23,3 = 0, (5)

τ13,1 +τ23,2 +σ3,3 = 0. (6)

The equation of electrostatics is

D1,1 +D2,2 +D3,3 = 0. (7)

The relations between the electric field and elec-tric potential are

E1 = −Φ,1 , E2 = −Φ,2 , E3 = −Φ,3 , (8)

where Φ denotes the electric potential.

The boundary conditions of the problem are spec-ified as follows:

On the lateral surface the transverse load and theelectric potential (or the normal electric displace-ment) are prescribed and given by

[τ13 τ23 σ3 φ ] =[0 0 q± φ±]

on ζ =±h (for closed-circuit surface conditions),(9)

[τ13 τ23 σ3 D3] =[0 0 q±3 D

±3

]on z = ±h(for open-circuit surface conditions).

(10)

The edge boundary conditions of the shells areconsidered as fully simple supports with electri-cally grounded and are given by

sigma1 = u2 = u3 = φ = 0 at x1 = 0 and x1 = L1

(11a)

σ2 = u1 = u3 = φ = 0textatx2 = 0 and x2 = L2

(11b)

It is noted that Eqs. (1)–(11) represent twenty-two basic equations in twenty-two unknowns witha set of appropriate boundary conditions for the3D analysis of a simply-supported, multilayeredpiezoelectric plate.

3 Nondimensionalization

For making the calculation more efficient and pre-venting from the ill-conditioned of system matrix,we select a set of dimensionless variables to nor-malize the coordinates and the variables of elec-tric and elastic fields. The dimensionless variablesare given as

x = x1/L, y = x2/L, z = x3/h,

u = u1/h, v = u2/h, w = u3/L,

σx = Lσ1/(hQ), σy = Lσ2/(hQ),

τxy = Lτ12/(hQ), τxz = L2τ13/h2Q,

τyz = L2τ23/h2Q, Dx = hD1/(Le), ,

Dy = hD2/(Le), Dz = LD3/(he),

φ = LeΦ/(h2Q), σz = L3σ3/(h3Q) (12)

where L denotes a typical in-plane dimension ofthe plate and is taken to be L =

√L1L2 in the pa-

per; −1 ≤ z ≤ 1; e and Q stand for a referencepiezoelectric and elastic modulus; Q is taken asQ = (1/2h)

∫ h−h c33dx3.

4 Direct Elimination

4.1 The mixed formulation

In the mixed formulation, the elastic displace-ments (u, v, w), transverse stresses (τxz, τyz, σz),


the electric potential (φ ) and the normal elec-tric displacement (Dz) are regarded as the pri-mary variables. Other fourteen variables are thedependent variables and can be determined bythe primary variables. By means of introducingthe set of dimensionless coordinates and variables(Eq. (12)) and then performing the direct elimina-tion, we may rewrite the basic 3D piezoelectricityequations (Eqs. (1)–(8)) as a system of eight par-tially differential equations in terms of eight pri-mary variables as follows:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

u,zv,z

σz,zDz,zτxz,zτyz,zw,zφ ,z

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 d15 0 d17 d18

0 0 0 0 0 d26 d27 d28

0 0 0 0 d17 d27 0 00 0 0 0 d18 d28 0 d48

d51 d52 d53 d54 0 0 0 0d61 d62 d63 d64 0 0 0 0d53 d63 d73 d74 0 0 0 0d54 d64 d74 d84 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

uvσz

Dz

τxz

τyz

wφ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

, (13)

where

d15 = Qh2/c55L2, d17 = −∂x,

d18 = −Qe15h2∂x/c55eL2, d26 = Qh2/c44L2,

d27 = −∂y, d28 = −Qe24h2∂y/c44eL2,

d48 = (c−155 e2

15 +η11)(Qh2/e2L2)∂xx

+(c−144 e2

24 +η22)(Qh2/e2L2)∂yy,

d51 = −(Q11∂xx + Q66∂yy),

d52 = −(Q12 + Q66)∂xy, d53 = −a1(h2/L2)∂x,

d54 = −b1(e/Q)∂x; d61 = (Q21 + Q66)∂xy,

d62 =−(Q66∂xx +Q22∂yy), d63 =−a2(h2/L2)∂y,

d64 = −b2(e/Q)∂y, d73 = ηQ(h4/L4),

d74 = eeh2/L2; d84 = −ce20/Q,

c = c33/(e2

33 +η33c33),

e = e33/(e2

33 +η33c33),

η = η33/(e2

33 +η33c33),

Qi j = ci j −a jci3 −b je3i (i, j = 1,2,6),

Qi j = Qi j/Q,

ai = (e33e3i +η33ci3)/(e2

33 +η33c33),

bi = (e33ci3 −c33e3i)/(e2

33 +η33c33).

The dimensionless form of boundary conditionsof the problem are specified as follows:

On the lateral surface the transverse load and elec-tric potential (or the normal electric displacement)are given by

[τxz τyz σz φ ] =[0 0 (q±3 L3/h3Q) (Φ±

Le/h2Q)]

on z = ±1

(for closed-circuit surface conditions), (14)

[τxz τyz σz Dz] =[0 0 (q±3 L3/h3Q) (D

±z L/he)

]on z = ±1

(for open-circuit surface conditions). (15)

At the edges, the boundary conditions are givenby

σx = v = w = φ = 0 at x = L1/L; (16a)

σy = u = w = φ = 0 at y = 0 and y = L2/L. (16b)

The method of double Fourier series expansion isfirstly applied to transform the system of partialdifferential equations (Eq. (13)) into a system ofordinary differential equations. In order to satisfy


the edge boundary conditions, we express the pri-mary variables in the following form

(u,τxz) =∞

∑m=1

∞

∑n=1

(umn(z),τxzmn(z))cosmxsin ny, (17)

(v,τyz) =∞

∑m=1

∞

∑n=1

(vmn(z),τyzmn(z)) sinmxcos ny, (18)

(w,σz,φ ,Dz) =∞

∑m=1

∞

∑n=1

(wmn(z),σzmn(z),φmn(z),Dzmn(z))

sinmxsin ny, (19)

where m = mπL/L1 and n = nπL/L2; m and n arepositive integers.

For brevity, the symbols of summation are omit-ted in the following derivation. Substituting Eqs.(17)–(19) in the basic 3D equations (Eq. (13)), wemay yield the resulting equations as follows:

ddz

F(z) = DF(z), (20)

where F(z) is called the state vec-tor of the plate and given as F(z) ={umn vmn σzmn Dzmn τxzmn τyzmn wmn φmn}T ; Dis the relevant coefficient matrix which is a con-stant coefficient matrix for plates and a variablecoefficient matrix for shells.

Similarly, the dimensionless dependent variablesof in-plane stresses and in-plane electric displace-ments can be expressed in terms of the primaryvariables as follows:

(σx,σy) =∞

∑m=1

∞

∑n=1

(σxmn(z),σymn(z))sinmxsin ny, (21)

τxy =∞

∑m=1

∞

∑n=1

τxymn(z)cosmxcos ny, (22)

Dx =∞

∑m=1

∞

∑n=1

Dxmn(z)cosmxsin ny, (23)

Dy =∞

∑m=1

∞

∑n=1

Dymn(z) sinmxcos ny, (24)

where

⎧⎨⎩

σxmn

σymn

τxymn

⎫⎬⎭=

⎡⎣l11 l12

l21 l22

l31 l32

⎤⎦{umn

vmn

}+

⎡⎣l13

l23

0

⎤⎦σzmn +

⎡⎣l14

l24

0

⎤⎦Dzmn,

{Dxmn

Dymn

}=[

l41 00 l52

]{σxzmn

σyzmn

}+[

l43

l53

]φmn,

li j are the relevant coefficients and given in Ap-pendix A.

Eq. (20) represents a system of eight simultane-ously linear first-order ordinary differential equa-tions in terms of eight primary variables. Vari-ous approaches have been applied to determinethe primary variables in the elastic and electricfields in the literature. Once these primary vari-ables are determined, the dependent variables canthen be calculated using Eqs. (21)–(24).

4.2 The displacement-based formulation

In the displacement-based formulation, the elasticdisplacements (u, v, w) and the electric potential(φ ) are regarded as the primary variables. Simi-larly, by means of introducing the set of dimen-sionless coordinates and variables and then per-forming the double Foruier series expansion andthe direct elimination, we may rewrite the basic3D piezoelectricity equations (Eqs. (1)–(8)) as asystem of four ordinally differential equations interms of four primary variables as follows:

Pd2Gdz2

+QdGdz

+RG = 0, (25)

where

G ={

umn vmn wmn φmn}T

,

P =

⎡⎢⎢⎣

p11 0 0 00 p22 0 00 0 p33 p34

0 0 p34 p44

⎤⎥⎥⎦ ,


Q =

⎡⎢⎢⎣

0 0 q13 q14

0 0 q23 q24

q13 q23 0 0q14 q24 0 0

⎤⎥⎥⎦ ,

R =

⎡⎢⎢⎣

r11 r12 0 0r12 r22 0 00 0 r33 r34

0 0 r34 r44

⎤⎥⎥⎦ ,

p11 = c55/Q, p22 = c44/Q, p33 = c33L2/Qh2,

p34 = −e33/e, p44 =(η33Qh2)/

(e2L2) ,

q13 = m(c13 +c55)/Q,

q14 = m(e31 +e15)h2/(eL2) ,

q23 = n(c23 +c44)/Q,

q24 = n(e32 +e24)h2/(eL2) ,

r11 = −(m2c11 + n2c66

)h2/

(QL2) ,

r12 = −mn(c12 +c66)h2/(QL2) ,

r22 = −(m2c66 + n2c22)h2/(QL2) ,

r33 =(m2c55 + n2c44

)/Q,

r34 = (m2e15 + n2e24)h2/(eL2) ,

r44 = −(m2η11 + n2η22)Qh4/(e2L4) .

The dependent variables can be determined by theprevious primary variables using Eqs. (1)–(8),(17)–(19) and Eqs. (21)–(24), and they are givenby

⎧⎨⎩

σxmn

σymn

τxymn

⎫⎬⎭=

⎡⎣l11 l12

l21 l22

l31 l32

⎤⎦{umn

vmn

}+

⎡⎣l13

l23

0

⎤⎦ dwmn

dz+

⎡⎣l14

l24

0

⎤⎦ dφmn

dz,

(26)

{τxzmn

τyzmn

}=

[l41 00 l52

]{dumndz

dvmndz

}+[

l43

l53

]wmn +

[l44

l54

]φmn,

(27)

{Dxzmn

Dyzmn

}=

[l61 00 l72

]{dumndz

dvmndz

}+[

l63

l73

]wmn +

[l64

l74

]φmn,

(28)

{σzmn

Dzmn

}=[

l81 l82

l91 l92

]{umn

vmn

}+[

l83

l93

]dwmn

dz+[

l84

l94

]dφmn

dz,

(29)

where li j are the relevant coefficients and aregiven in Appendix B.

5 3D analytical approaches

5.1 The Pagano’s classical approach

Based on the Pagano’s classical approach, the pri-mary variables in the displacement-based formu-lation are assumed as

G(z) = Geλz. (30)

Substituting Eq. (30) into Eq. (25) leads to aneighth-order polynomial equation as

A1λ 8 +A2λ 6 +A3λ 4 +A4λ 2 +A5 = 0, (31)

where Ai (i = 1−5) are the relevant coefficientsof the polynomial.

The roots of Eq. (31) must be functions of the ma-terial properties and the laminate geometry. Theroots can therefore be real, imaginary or com-plex. By using the material properties of a typicalk-th layer, one may obtain the eigenvalues λ (k)

i(i = 1− 8) and their corresponding eigenvectors

G(k)i (i=1–8) from Eqs. (31) and (30). The gen-

eral solutions of primary unknowns of k-th layercan then be written by

G(k)(z) =8

∑i=1

α (k)i G

(k)i eλ (k)

i z

(k = 1,2, . . .,NL; zk−1 ≤ z ≤ zk), (32)

where α (k)i (i=1–8 and k=1–NL) are the undeter-

mined coefficients. For a NL-layered plate, totally


there are (8×NL) undetermined coefficients in Eq.(32).

It is noted that the eigenvalues in Eq. (32) canbe obtained in a certain pair of complex conju-gate as λ1,2 = Re(λ)± Im(λ) and their corre-sponding eigenvectors are G1,2 = Re(G)± Im(G)where λ is a certain complex eigenvalue and itscorresponding eigenvector is G. There is nothingwrong with those complex-valued solutions (i.e.,G1,2). However, we may prefer to replace themwith two linearly independent solutions involvingonly real-valued quantities and they are given by

G1 = eRe(λ)z(Re(G)cos(Im(λ)z)− Im(G) sin(Im(λ)z)

),

(33a)

G2 = eRe(λ)z(Re(G) sin(Im(λ)z)+ Im(G)cos(Im(λ)z)

).

(33b)

The problem also supply with (8×NL) conditionsfor determining α (k)

i in Eq. (32). There are fourboundary conditions at each lateral surface (i.e.,either Eqs. (14) or (15)) and eight interfacial con-tinuity conditions at interfaces between adjacentlayers such that

F(k)(z = zk) = F(k+1)(z = zk)(k = 1,2, . . ., (NL−1)), (34)

where zk denotes the dimensionless thickness co-ordinate measured from the mid-plane of the plateto the top surface of the kth-layer (see Fig. 1); F(k)

can be calculated from G(k).

The complementary solutions of primary vari-ables of each layer can be determined by impos-ing the applied lateral surface conditions and thecontinuity conditions at interfaces between adja-cent layers. Afterwards, the dependent variablescan be obtained using Eqs. (26)-(29).

The Pagano’s classical approach originally pro-posed for the 3D analyses of laminated compos-ite elastic plates was extendedly used for a vari-ety of 3D structural analyses of multilayered and

FG piezoelectric plates and shells. The staticbehaviors of simply-supported laminated piezo-electric cylinders, plates and strips were studiedby Heyliger (1994, 1997a, b) and Heyliger andBrooks (1996), respectively. The free vibrationand cylindrical bending vibration of multilayeredpiezoelectric plates were analyzed by Heyligerand Saravanos (1995) and Heyliger and Brooks(1995), respectively. A partial publication list ofrelevant studies using the Pagano’s classical ap-proach is classified and tabulated in Table 2.

5.2 The state space approach

On the basis of the mixed formulaion, one mayexpress the primary variables of the typical kth-layer as

ddz

F(z) = DkF(z) zk−1 ≤ z ≤ zk, (35)

where Dk is a 8x8 constant coefficient matrixfor the kth-layer; zk−1 denotes the dimension-less thickness coordinate measured from the mid-plane of the plate to the bottom surface of the kth-layer (see Fig. 1).

With a known state vector at the bottom surface ofthe kth-layer (Fk−1), the solution of equation (35)is

F(z) = eDk(z−zk−1)Fk−1

= MkeΛk(z−zk−1)M−1k Fk−1

, (36)

where Mk is the modal matrix of Dk consist-ing of eight independent eigenvectors; Fk−1 =F(z = zk−1); eΛkz is a 8x8 diagonal matrix and

given by eΛkz =

⎡⎢⎢⎢⎣

eλ1z 0 . . . 00 eλ2z . . . 0...

.... . .

...0 0 . . . eλ8z

⎤⎥⎥⎥⎦ in which

λ1,λ2, . . .,λ8 are the set of eigen values of Dk.

According to Eq. (36), the state vector within thekth-layer can be determined. The state vector ofthe top surface of the kth-layer is then determinedas follows.

Fk = RkFk−1, (37)

where Fk = F(zk); Rk = MkeΛkhkM−1k .


Table 2: Partial list of references on 3D analysis of single-layer homogeneous, multilayered and FG piezo-electric plates using Pagano’s classical approach

Problems Literature

Static: Ray, Rao and Samanta, 1992, 1993; Ray, Bhattacharya and Samanta, 1993;

Heyliger, 1994; Heyliger and Brooks, 1996; Heyliger, 1997a, b

Thermal: Ootao and Tanigawa, 2000a, b

Vibration: Heyliger and Brooks, 1995; Heyliger and Saravanos, 1995; Batra and Liang, 1997;

Ray, Bhattacharya and Samanta, 1998; Vel, Mewer and Batra, 2004; Cupial, 2005

Others: Kapuria, Dumir and Sengupta, 1999

By analogy, the state vectors between the top andbottom surfaces of the shell are linked by

FNL = RNLF(NL−1)

= RNLR(NL−1) . . .R1F0. (38)

By defining a symbol of consectutive multiplica-tion, we rewrite Eq. (38) in the form of

FNL =

(NL

∏k=1

Rk

)F0, (39)

whereNL∏

k=1Rk = RNLR(NL−1) . . .R2R1.

Equation (39) represents a set of simultaneous al-gebraic equations. After imposing the boundaryconditions prescribed on the lateral surfaces, wecan determine the other unknowns in state vec-tors of lateral surfaces of the plate. After then,the state vector through the thickness coordinateof the shell can be obtained by

F(z) = MkeΛk(z−zk−1)M−1k Fk−1

= MkeΛk(z−zk−1)M−1k

(k−1

∏i=1

Ri

)F0

. (40)

Equations (39)-(40) provide the 3D solutions of asystem of the state equations.

The state space approach in conjunction with themethod of transfer matrix has been used for a va-riety of 3D structural analyses of multilayered andFG piezoelectric plates and shells. Vel and Batra(2001a, b) presented exact solutions for the cylin-drical bending of laminated plates and for rectan-gular sandwich plates, respectively, with embed-ded piezoelectric shear actuators. Exact solutions

for the static, piezothermoelectric and vibrationanalyses of the simply-supported, FG piezoelec-tric plates were presented by Zhong and Shang(2003, 2005) and Zhong and Yu (2006), respec-tively. Chen et al. (2001a, b) studied the staticand free vibration problems of a fluid-filled piezo-ceramic hollow sphere. A partial publication listof relevant studies is classified and tabulated inTable 3.

5.3 The series expansion approach

As we previously mentioned, the systems ofdifferential equations derived from both thedisplacement-based formulation and the mixedformulation are the ones with constant coeffi-cients for plates and with variable coefficients forshells. Pagano’s classical approach fails to ex-actly analyze the system of thickness-varying dif-ferential equations. Hence, the series expansionmethod becomes a feasible analytical approachfor shell problems.

5.3.1 Frobenius series expansion

For the Frobenius series expansion approach(Huang and Tauchert, 1992), the primary vari-ables in the displacement-based formulation areexpressed in the following form

G(z) = zp∞

∑q=0,1,2

Gqzq. (41)

Substituting Eq. (41) into Eq. (25), and equatingthe coefficients of the smallest power to zero lead


Table 3: Partial list of references on 3D analysis of single-layer homogeneous, multilayered and FG piezo-electric structures using the state space approach

Problems Literature

Static (Plates): Brissaud, 1996; Lee and Jiang, 1996; Bisegna and Maceri, 1996;

Ding, Xu and Guo, 1999; Benjeddou and uDe , 2001; Vel and Batra, 2001a, b; Zhong and Shang,

2003; Chen, Cai, Ye and Wang, 2004;

Chen, Ying, Cai and Ye, 2004; Lu, Lee and Lu, 2005, 2006

(Shells): Chen, Ding and Xu, 2001a; Wu and Liu, 2007

Thermal (Plates): Xu, Noor and Tang, 1995; Tarn, 2002; Zhong and Shang, 2005

(Shells): Xu and Noor (1996)

Vibration (Plates): Yang, Batra and Liang, 1994; Batra, Liang and Yang, 1996;

Xu, Noor and Tang, 1997; Ding, Xu and Chen, 2000; Chen and Ding, 2002; uDe and Benjeddou,

2005; Zhong and Yu, 2006

(Shells): Chen and Wang, 2002

Buckling (Plates): Kapuria and Achary, 2004

Others (Plates): Kapuria and Achary, 2005; Sheng, Wang and Ye, 2007

(Shells): Chen, Ding and Xu, 2001b; Wang and Zhong, 2003; Chen, Bian and Ding, 2004

to an indicial equation of the differential equa-tions (Eq. (25) and a recurrence relation relating(Gq, (q = 1,2, . . .,∞)) to G0. The indicial equa-tion is used to determine the values of p where pi

(i = 1−8). Hence, the complementary solution ofEq. (25) for a typical kth-layer can then be writtenas

G(k)(z) =8

∑i=1

B(k)i zp(k)

i

∞

∑q=0,1,2

G(k)q zq. (42)

Again, the unknown coefficients B(k)i (i =

1,2, . . .,8; k = 1,2, . . .,NL)are determined fromthe generalized traction conditions on the lateralsurface conditions (Eqs. (14)–(15)) and continu-ity conditions at the interfaces between adjacentlayers (Eq. (34)).

5.3.2 Modified Frobenius series expansion

For the modified Frobenius series expansion ap-proach (Kapuria et al., 1997a, b; Xu and Noor,1996), the primary variables in the mixed formu-

lation are expressed in the following form

F(z) = epz∞

∑q=0,1,2

Fqzq. (43)

Substituting Eq. (43) into Eq. (20), and equatingthe coefficients of each power of z to zero lead toa characteristic equation solving for p where pi

(i = 1− 8). The complementary solution of Eq.(20) for a typical kth-layer is the sum of the eightsolutions for the eight values of pi (i = 1−8) andcan then be written as

F(k)(z) =8

∑i=1

C(k)i ep(k)

i z∞

∑q=0,1,2

Fqzq. (44)

Again, the unknown coefficients C(k)i (i =

1,2, . . .,8; k = 1,2, . . .,NL) are determined fromthe generalized traction conditions on the lateralsurface conditions and continuity conditions at theinterfaces between adjacent layers.

The modified Frobenius method has been used fora variety of 3D structural analyses of multilayeredand FG piezoelectric plates and shells. Kapuriaet al. (1997a) presented exact solutions for the


Table 4: Partial list of references on 3D analysis of single-layer homogeneous, multilayered and FG piezo-electric shells using the series expansion approach

Problems Literature

Static: Chen, Shen and Wang, 1996; Dumir, Dube and Kapuria, 1997;

Kapuria, Dumir and Sengupta, 1997a; Kapuria, Sengupta and Dumir, 1997a;

Heyliger, 1997a; Chen, Shen and Liang, 1999

Thermal: Dube, Kapuria and Dumir, 1996a, b; Kapuria, Sengupta and Dumir, 1997b;

Kapuria, Dumir and Sengupta, 1997b; Wu, Shen and Chen, 2003

Vibration: Chen and Shen, 1998; Hussein and Heyliger, 1998

Others: Chen, Ding and Liang, 2001

coupled electro-elastic analysis of a piezoelectriccylindrical shell under axisymmetric load. Ka-puria, Sengupta and Dumir (1997b) and Kapuria,Dumir and Sengupta (1997b) studied for the 3Dpiezothermoelastic analyses of simply-supported,multilayered piezoelectric cylindrical shells un-der axisymmetric and nonaxisymmetric thermo-electric loads, respectively. A partial publicationlist of relevant studies using the series expansionapproach is classified and tabulated in Table 4.

5.4 The asymptotic approach

On the basis of the asymptotic approach, a per-turbation parameter ∈ (∈2= h/L) is introduced inthe mixed formulation. The primary variables arethen asymptotically expanded in the powers ε2 asfollows (Nayfeh, 1981):

f (x,y, z,ε) = f (0)(x,y, z)+ε2 f (1)(x,y, z)

+ ε4 f (2)(x,y, z)+ . . .. (45)

Substituting Eq. (45) into the basic 3D equations(Eq. (20)), collecting coefficients of equal powersof ε and performing the successive integration tothe resulting equations of various order problemsthrough the thickness coordinate, we finally ob-tain the recursive sets of governing equations ofvarious order problems as follows:

5.4.1 Plates with open-circuit surface condi-tions

In the cases of plates with open-circuit surfaceconditions, the governing equations of various or-

der problems are (Wu and Syu, 2007)

K11uk +K12vk +K13wk +K14Dkz = f1k(1), (46)

K21uk +K22vk +K23wk +K24Dkz = f2k (1) , (47)

K31uk +K32vk +K33wk +K34Dkz

= f3k(1)−m f1k(1)−n f2k(1), (48)

K41uk +K42vk +K43wk +K44Dkz = f4k (1) , (49)

where Ki j (i, j=1-4) are the relevant differentialoperators; fik (k = 0,1,2, . . .) are the revelantfunctions and f10 = f20 = 0, f30 = q+

z −q−z , f40 =D

+z −D

−z .

The boundary conditions for various order prob-lems are specified as follows:

On the lateral surface the transverse loads andnormal electric displacement are given by[

τ (0)xz τ (0)

yz σ (0)z D(0)

z

]=[0 0 q±z D±

z

]on z = ±1; (50a)

[τk

xz τkyz σ (k)

z D(k)z

]= [0 0 0 0]

(k = 1,2,3, . . .) on z = ±1; (50b)

Along the edges, the boundary conditions aregiven by

σ (k)x = v(k) = w(k) = φ (k) = 0 (k = 1,2,3, . . .)

at x = 0 and y = L1/L. (51a)

σ (k)y = u(k) = w(k) = φ (k) = 0 (k = 1,2,3, . . .)

at y = 0 and y = L2/L. (51b)


5.4.2 Plates with closed-circuit surface condi-tions

In the cases of plates with closed-circuit surfaceconditions, the governing equations of various or-der problems are (Wu and Syu, 2007)

L11uk +L12vk +L13wk +L14φ k = g1k(1), (52)

L21uk +L22vk +L23wk +L24φ k = g2k (1) , (53)

L31uk +L32vk +L33wk +L34φ k

= g3k(1)−mg1k(1)−ng2k(1), (54)

L41uk +L42vk +L43wk +L44φ k = g4k (1) , (55)

where Li j (i, j=1–4) are the relevant differentialoperators; gik (k = 0,1,2, . . .) are the revelantfunctions and g10 = g20 = 0, g30 = q+

z −q−z , g40 =φ+−φ−

.

The boundary conditions for various order prob-lem are specified as follows:

On the lateral surface the transverse loads andelectric potential are given by

[τ (0)

xz τ (0)yz σ (0)

z φ (0)]

=[0 0 q±z φ±]on z = ±1; (56a)

[τ (k)

xz τ (k)yz σ (k)

z φ (k)]

=[0 0 q±z φ±]on z = ±1; (56b)

Along the edges, the boundary conditions aregiven by

σ (k)x = v(k) = w(k) = φ (k) = 0 (k = 1,2,3, . . .)

at x = 0 and x = L1/L, (57a)

σ (k)y = u(k) = w(k) = φ (k) = 0 (k = 1,2,3, . . .)

at y = 0 and y = L2/L. (57b)

By observation of the governing equations of vari-ous order problems, we found that the differentialoperators among the various order problems re-main identical and the nonhomogeneous terms ofhigher-order problems can be calculated from the

lower-order solutions. Hence, it is shown that thesolution process of the leading-order problem canbe repeatedly applied to the higher-order prob-lems. The asymptotic solutions can be determinedorder-by-order in a hierarchic and consistent man-ner.

Recently, the asymptotic approach has been usedfor the 3D static analyses of laminated piezoelec-tric plates (Cheng and Batra, 2000a, 2000b), oflaminated piezoelectric shells (Wu, Lo and Chao,2005; Wu and Syu, 2006; Wu, Syu and Lo, 2007).The approach was further applied to determine ex-act solutions for the static and cylindrical bendingvibration analyses of FG piezoelectric cylindricalshells (Wu and Syu, 2007; Wu and Tsai, 2008). Apartial publication list of relevant studies using theasymptotic approach is classified and tabulated inTable 5.

5.5 Supplementary remarks

5.5.1 Material properties

Without loss of the generality, the material prop-erties can be considered as heterogeneous varyingthrough the thickness coordinate in the aforemen-tioned formulations. The structural behavior ofsingle-layer homogeneous, multilayered and FGplates and shells can be studied by assuming ap-propriate material–property variations through thethickness coordinate. The 3D analysis of severaltypes of FG plates and shells studied in the litera-ture with specific material-property variations aregiven as follows:

Type 1–single-layer homogeneous plates andshells.

For a Type 1 pate or shell, the material propertiesare assumed as homogeneous, independent uponthe thickness coordinate, and are given by

mi j (z) = mi j, (58)

where mi j = ci j , ei j , ηi j, etc.

Type 2–multilayered plates and shells.

For a Type 2 plate or shell, the material propertiesare assumed to be layerwise Heaviside functions


Table 5: Partial list of references on 3D analysis of single-layer homogeneous, multilayered and FG piezo-electric structures using the asymptotic approach

Problems Literature

Static (Plates): Cheng, Lim and Kitipornchai, 1999, 2000; Cheng and Batra, 2000a;

Kalamkarov and Kolpakov, 2001

(Shells): Cheng, Reddy, 2002; Wu, Lo and Chao, 2005; Wu and Syu, 2006;

Wu and Syu, 2007; Wu, Syu and Lo, 2007

Thermal (Plates): Cheng and Batra, 2000b

Vibration (Plates): Cheng and Reddy, 2003

(Shells): Wu and Lo, 2006; Wu and Tsai, 2008

and are given by

mi j(z) =NL

∑k=1

m(k)i j [H(z− zk−1)−H(z− zk)] , (59)

where H(z) is the Heaviside function; zk−1 andzk denote the dimesionless thickness coordinatemeasured from the middle surface of the structureto the bottom and top surfaces of the kth-layer.

Type 3–exponent-law varied, functionally gradedplates and shells.

For a Type 3 plate or shell, the material propertiesare assumed to obey the identical exponent-lawvaried exponentially with the thickness coordinateand are given by

mi j = m(b)i j eκp[(z+1)/2], (60)

where the superscript b in the parentheses denotesthe bottom surface; κp is the material-propertygradient index which represents the degree of thematerial gradient along the thickness and can bedetermined by the values of the material proper-ties at the top and bottom surfaces, i.e.,

κp = lnm(t)

i j

m(b)i j

, (61)

where the superscript t in the parentheses denotesthe top surface. ln(z) denotes the natural loga-rithm function of z which is the inverse of the ex-ponential function, ez.

Type 4–power-law varied, functionally gradedplates and shells.

For a Type 4 plate or shell, the material proper-ties are assumed to obey the identical power-lawdistribution of the volume fractions of the con-stituents and are given by

mi j (z) = m(t)i j Γ(z)+m(b)

i j [1−Γ(z)] , (62)

where Γ(z) denotes the volume fraction and is de-

fined as Γ(z) =(

z+12

)kp . kp is the power-law ex-ponent which represents the degree of the mate-rial gradient along the thickness. As kp=0 andkp = ∞, the present FG plate (or shell) reduces toa homogeneous piezoelectric plate (or shell) withmaterial properties m(t)

i j and m(b)i j , respectively. As

kp=1, it represents that the material properties ofthe FG plate (or shell) linearly varied through thethickness coordinate.

The solution process for the aforementioned 3Dapproaches will become unfeasible (such as thePagano’s classical and state space approaches)or become inconvenient (such as the series ex-pansion approach), as a system of differentialequations with variable coefficients resulting fromthe initial curvature of shells or from the varia-tions of material properties is treated. A succes-sive approximation method, originally proposedby Soldatos and Hadjigeorgiou (1990), has beencommonly used to make the aforementioned ap-proaches feasible. In the SA method, the FG plate(or shell) is artificially divided into a NL-layeredof small thickness and of homogeneous materialproperties. For a typical kth-layer, the materialproperties m(k)

i j are determined in a thickness av-


erage sense and given as

m(k)i j =

1hk

∫ zk

zk−1

mi j(z)dz. (63)

The SA method has been demonstrated that it canapproximate the exact solutions of relevant prob-lems of FG piezoelectric shells to any desired ac-curacy (Wu and Liu, 2007).

Unlike the Pagano’s classical, state space and se-ries expansion approaches existing the previouslymentioned straits, the interlaminar field variablesin the asymptotic formulations are derived as adefinite integration through the thickness direc-tion and can be directly determined without us-ing the SA method. The asymptotic approach hasbeen applied for the static and dynamic analysesof FG piezoelectric shells (Wu and Syu, 2007; Wuand Tsai, 2008). The asymptotic solutions havebeen demonstrated to rapidly converge and to bein excellent agreement with the relevant 3D solu-tions available in the literature (Zhong and Shang,2003; Zhong and Yu, 2006).

5.5.2 Coupled magneto-electro-elastic effects

Recently, the state space and asymptotic ap-proaches have also been extendedly used forthe 3D coupled analysis of multilayered and FGmagneto-electro-elastic plates and shells. Panand Heyliger (2002, 2003) and Pan and Han(2005) presented the exact solution for the staticand free vibration analyses of multilayered rect-angular plate made of FG, anisotropic and lin-ear magneto-electro-elastic materials using thestate space approach. A Pseudo-Stroh formal-ism combined with the propagator matrix methodfor the 3D analyses was developed and appliedto two FG sandwich plates made of piezoelec-tric BaTiO3 and magnetostrictive CoFe2O4. Itis shown that the coupled magneto-electro-elasticeffects has remarkable influence on the static anddynamic behaviors of the FG plate. Chen etal. (2005) studied on the free vibration of non-homogeneous transversely isotropic magneto-electro-elastic plates using the state space ap-proach. Other relevant studies using the statespace approach can be found in the literature (Lee

and Chen, 2003; Wang et al., 2003; Guan and He,2006; Chen et al., 2007).

Using the asymptotic approach, Wu and his col-leagues presented the 3D solutions for the staticand free vibration analyses of doubly curved,FG magneto-electro-elastic shells (Wu and Tsai,2007, 2008; Tsai et al., 2008; Tsai and Wu, 2008).The presented formulations can be reduced tothose of FG magneto-electro-elastic cylindricalshells and plates by letting the corresponding cur-vature radius zero. Parametric studies for both thecoupling magneto-electro-elastic effect and theinfluence of the gradient index of material prop-erties on the structural behavior of various shellshave been presented.

5.5.3 Approximate 3D numerical modeling andmethodologies

Several approximate 3D numerical modeling andmethodologies have been proposed for the anal-ysis of multilayered and FG elastic, piezoelec-tric and magneto-electro-elastic plates/shells witha variety of edge boundary conditions. They areselectively reported as follows.

Ramirez and his colleagues proposed a discretelayer approach in combination with the Ritzmethod (or a finite element method) for the ap-proximate 3D static and free vibration analy-ses of multilayered and FG elastic, piezoelectricand magneto-electro-elastic plates (Heyliger etal., 1994; Heyliger and Ramirez, 2000; Ramirezet al., 2006a, b). The discrete layer scheme hasbeen demonstrated to be not limited to specificboundary conditions and gradation functions.

Shu et al. (2003) recently proposed a local RBF-based differential quadrature (DQ) method. In themethod, the conventional DQ method is combinedwith the radial basis functions as the trial func-tions in the DQ scheme. The local RBF-basedDQ method has been successfully applied to studythe incompressible flows in the steady and un-steady regions (Shu et al., 2005), to solve 2Dincompressible Navier-Stokes equations (Shu etal., 2003) and to solve 3D incompressible viscousflows with curved boundary (Shan et al., 2008).Liew et al. (2005) and Zhang et al. (2006) de-veloped the 3D differential quadrature formula-


tions for the coupled thermo-electro-elastic andpiezoelectric analyses of multilayered plates, re-spectively. In their analyses, the approximate 3Dsemi-analytical formulations have been demon-strated to be in good agreement with the avail-able 3D solutions in the literature. In addition, theapproximate 3D solutions of the plates with vari-ous boundary conditions were also presented. Inconjunction with the methods of perturbation anddifferential quadrature, Wu and Wu (2000) andWu and Tsai (2004) presented the asymptotic DQsolutions for the free vibration analysis of lami-nated conical shells and for the bending analysisof functionally graded annular spherical shells, re-spectively.

Cheung and Jiang (2001) and Akhras and Li(2007) studied the 3D static analysis and 3Dstatic, vibration and stability analyses of piezo-electric composite plates, respectively, using a fi-nite layer method. In the finite layer method,the field variables are considered to be separa-ble and expanded as the Fourier functions in thein-plane coordinates and as the polynomials inthe thickness coordinate. Thus, the 3D analysisis transformed into a series of one-dimensionalanalysis. It is demonstrated that the approxi-mate 3D solution is accurate and the proposedmethod is efficient for computation. On the ba-sis of the Hellinger–Reissner (H–R) principle, Wuet al. (2001) proposed an asymptotic finite stripmethod (FSM) for the analysis of doubly curvedlaminated shells. In their analysis, recursive setsof element characteristic equations of first-orderdeformation theory have been obtained. It isdemonstrated that the asymptotic FSM convergesrapidly not only in mesh refinement and but alsoin higher-order modifications.

Sladek et al. (2006, 2007) proposed a meshless lo-cal Petrov-Galerkin method for the coupled anal-ysis of thermo-piezoelectricity and plane piezo-electricity. Hesthaven and Warburton (2004) pro-posed a high-order accurate method for the time-domain solution of 3D electromagnetics. Wu etal. (2008) proposed a differential reproducingkernel particle (DRKP) method for the analysisof multilayered elastic and piezoelectric plates.In their analysis, the Euler-Lagrange equations of

3D piezoelectricity and possible boundary con-ditions are derived using the stationary principleof H–R energy functional of the plate. A pointcollocation method, based on the DRKP approxi-mants, has been formulated for the approximate3D analysis of multilayered piezoelectric platesunder electro-mechanical loads.

6 Illustrative Examples

6.1 Multilayered hybrid piezoelectric and elas-tic strips

A simply-supported, two-layer laminate, com-posed of [PZT-4/ceramics] with equal thicknesslayers and with open-circuit surface conditions,has been studied using the Pagano’s classicalapproach (Heyliger and Brooks, 1996) and theasymptotic approach (Wu, Syu and Lo, 2007).The dimensions of length and total thickness ofthe strip are L2=0.1m and 2h=0.01m. The mate-rial properties of piezoelectric and ceramics layersare given in Table 6. The cylindrical bending typeof either mechanical load or electric potential (i.e.,q+

3 (x2) = sin(πx2/L2) or Φ+(x2) = sin(πx2/L2))is applied on the top surface of the strip. The 3Dsolutions of elastic and electric field componentsat the middle surface of each layer and at inter-faces between layers, obtained by the Pagano’sclassical approach and the asymptotic approachare quoted and given in Tables 7–8. It is shownthat the 3D solutions obtained from these two dif-ferent approaches are consistent.

6.2 Multilayered hybrid piezoelectric and elas-tic plates

The coupled analysis of simply-supported, mul-tilayered hybrid piezoelectric and elastic plates,i.e., [PZT-4/0◦/90◦/90◦/0◦/PZT-4] laminates, withclosed-circuit and open-circuit surface conditionsare studied using the asymptotic approach (Wu etal., 2005) and a newly-proposed meshless DRKPmethod (Wu et al., 2008) for double checking.The plates are composed of a [0◦/90◦/90◦/0◦] lam-inated composite elastic plate bounded with thepiezoelectric layers (PZT-4) on the outer surfacesof the laminate. The applied electro-mechanicalloads on lateral surfaces are given as follows:


Table 6: Elastic, piezoelectric and dielectric properties of composite and piezoelectric materials

Moduli Ceramics PZT-4 Fiber-reinforced

composite

PVDF

11c (Gpa) 138.28 139.021 134.855 3.0

22c 138.28 139.021 14.352 3.0

33c 128.07 115.449 14.352 3.0

12c 32.359 77.848 5.156 1.5

13c 27.821 74.328 7.133 1.5

23c 27.821 74.328 5.156 1.5

44c 53.5 25.6 3.606 0.75

55c 53.5 25.6 5.654 0.75

66c 53.0 30.6 5.654 0.75

24e ( )2/ mC 2.96 12.72 0.000 0.0

15e 2.96 12.72 0.000 0.0

31e 0.8 -5.2 0.000 -0.15e-02

32e 0.8 -5.2 0.000 0.285e-01

33e 6.88 15.08 0.000 -0.15e-01

11η ( )mF / 1.7885e-09 1.306e-08 0.3099e-10 0.1062e-09

22η 1.7885e-09 1.306e-08 0.2656e-10 0.1062e-09

33η 1.60257e-09 1.151e-08 0.2656e-10 0.1062e-09

For the closed-circuit surface conditions,

Case 1:

q+3 (x1,x2) = q0 sin(πx1/L1) sin(πx2/L2),

q−3 (x1,x2) = Φ+(x1,x2) = Φ−(x1,x2) = 0,(64)

Case 2:

Φ+(x1,x2) = Φ0 sin(πx1/L1) sin(πx2/L2),

Φ−(x1,x2) = q+3 (x1,x2) = q−3 (x1,x2) = 0;

(65)

For the open-circuit surface conditions,

Case 3:

q+3 (x1,x2) = q0 sin(πx1/L1) sin(πx2/L2),

q−3 (x1,x2) = D+3 (x1,x2) = D

−3 (x1,x2) = 0,

(66)

Case 4:

D+3 (x1,x2) = D0 sin(πx1/L1) sin(πx2/L2),

D−3 (x1,x2) = q+

3 (x1,x2) = q−3 (x1,x2) = 0(67)

The material properties of PZT-4 and compositelayers are given in Table 6 (Heyliger, 1994). Thedimensions of length and total thickness of the


Table 7: The elastic and electric field variables in a two-layer hybrid piezoelectric plate under the cylindricalbending type of mechanical load

3x Approaches 1332 10x),0( xu 10

33 10x),2

( xL

u 43 10x),

2( xLΦ ),

2( 32 xLσ ),0( 323 xτ ),

2( 33 xLσ 10

33 10x),2

( xL

D

h

Asymptotic approach

(Wu et al., 2007)

-170.415

1.05613

0.00000

57.8904

0.00000

1.000000

-2.21653

Pagano’s classical approach (Heyliger and Brooks, 1996))

-170.406

1.05609

0.00000

57.8914

0.00000

1.000000

-2.21625

0.5h

Asymptotic approach

(Wu et al., 2007)

-88.8861

1.06109

10.5729

30.1903

3.45473

0.850098

-2.68015


-88.8804

1.06105

10.5763

30.1904

3.45477

0.850095

-2.67988

0

Asymptotic approach

(Wu et al., 2007) -9.13402

1.06295

14.0662

2.93275

(3.77196) 4.75387

0.513739

-3.73427


-9.13120

1.06291

14.0706

2.93185 (3.77076)

4.75387

0.513734

-3.73402

-0.5h

Asymptotic approach

(Wu et al., 2007)

72.3128

1.06267

8.14370

-30.2007

3.71709

0.163625

-3.87361

Pagano’s classical approach (Heyliger and Brooks, 1996)

72.3126

1.06263

8.14620

-30.2007

3.71705

0.163623

-3.87336

-h

Asymptotic approach

(Wu et al., 2007)

154.769

1.06116

0.00000

-64.5538

0.00000

0.00000

-3.94362


154.765

1.06112

0.00000

-64.5526

0.00000

0.00000

-3.94337

Table 8: The elastic and electric field variables in a two-layer hybrid piezoelectric plate under the cylindricalbending type of electric potential

3x Approaches 1132 10x),0( xu 10

33 10x),2

( xL

u ),2

( 3xLΦ ),2

( 32 xLσ ),0( 323 xτ 233 10x),

2( xL 7

33 10x),2

( xL

D

h

Asymptotic approach

(Wu et al., 2007)

-17.2285

2.21653

1.00000

98.0664

0.00000

1.000000

-4.38171


-17.2277

2.21625

1.00000

98.0706

0.00000

1.000000

-4.38016

0.5h

Asymptotic approach

(Wu et al., 2007)

-11.6682

2.37365

0.935608

-38.9881

2.31023

-16.1157

-3.91986


-11.6676

2.37336

0.935611

-38.9872

2.31044

-16.1166

-3.91847

0

Asymptotic approach

(Wu et al., 2007) -6.21174

2.49838

0.874731

-175.591 (135.720)

-6.11243

-8.20466

-3.48676


-6.21137

2.49809

0.874736

-175.593 (135.710)

-6.11227

-8.20718

-3.48550

-0.5h

Asymptotic approach

(Wu et al., 2007)

-3.85896

2.74246

0.436357

38.9437

0.743825

7.90414

-3.45511


-3.85882

2.74217

0.436359

38.9426

0.743546

7.90257

-3.45386

-h

Asymptotic approach

(Wu et al., 2007)

-1.52083

2.98145

0.00000

-58.0171

0.00000

0.00000

-3.44464


-1.52091

2.98116

0.00000

-58.0090

0.00000

0.00000

-3.44340


plate are L1 = L2 = L and S = L/2h = 4,10,20.The thickness ratio of each layer is PZT-4 layer:0◦-layer: 90◦-layer: 90◦-layer: 0◦-layer: PZT-4layer =0.2h: 0.4h: 0.4h: 0.4h: 0.4h: 0.2h. A set ofnormalized elastic and electric variables are givenas follows:

For the loading conditions of Cases 1 and 3,

(u,w) = (u1,u3)c∗/q0 (2h) ,

(τxz,σ z) = (τ13,σ3)/q0,

φ = Φe∗/q0 (2h) ,

Dz = D3c∗/(q0e∗);

(68)

For the loading conditions of Case 2,

(u,w) = (u1,u3)c∗/(Φ0e∗),(τxz,σ z) = (τ13,σ3) (2h)/(Φ0e∗),

φ = Φ/Φ0,

Dz = D3c∗ (2h)/Φ0 (e∗)2 ;

(69)

For the loading conditions of Case 4,

(u,w) = (u1,u3)e∗/(2hD0),(τxz,σ z) = (τ13,σ3)e∗/(D0c∗),

φ = Φ (e∗)2 /(2hD0c∗),Dz = D3/D0;

(70)

where c∗ = 1010N/m2, e∗ = 10C/m2, q0 = 1N/m2,φ0 = 1V, D0 = 1C/m2.

Figures 2–3 and 4–5 present the through-thickness distributions of various elastic and elec-tric variables of the [PZT-4/0◦/90◦/90◦/0◦/PZT-4] laminated plates under loading conditions ofCases 1-2 and 3-4, respectively. It is shown thatthe transverse shear stresses produced in the platedecrease as the plates become thicker for the ap-plied mechanical load cases (Cases 1 and 3); con-trarily, they increase as the plates become thickerfor the applied electric load cases (Cases 2 and4). The maximum transverse shear stresses oc-cur in the composite material layer for the ap-plied mechanical load cases; they, however, oc-cur at interfaces between elastic and piezoelec-tric layers for the applied electric load cases. Thedistributions of the elastic displacements throughthe thickness coordinate are merely linear func-tions for the applied mechanical load cases; they,

however, reveal approximately layerwise linear orhigher-order polynomial functions for the appliedelectric load cases. It is observed that the through-thickness distributions of elastic and electric vari-ables reveal large difference between the appliedmechanical load cases and the applied electricload cases.

6.3 Single-layer piezoelectric shells

The coupled cylindrical bending analysis ofsimply-supported, homogeneous piezoelectriccylindrical shells with closed-circuit surface con-ditions has been studied using the state space ap-proach (Wu and Liu, 2007), the series expansionapproach (Dumir et al., 1997) and the asymptoticapproach (Wu and Syu, 2007). The configurationand coordinates (x,θ , r) of the cylindrical shell aregiven in Fig. 6 where aθ = Rθ θα . Rθ and aθdenote the curvature radius to the middle surfaceand the curvilinear dimension in circumferentialdirection, respectively. 2h and θα denote the to-tal thickness and the angle between two edgesof the shell, respectively. The shells are con-sidered to be composed of polyvinyledence fluo-ride (PVDF) polarized along the radial direction.The elastic, piezoelectric and dielectric proper-ties of PVDF material are given in Table 6. Thecylindrical bending types of applied mechanicalload and applied electric potential (i.e., q+

3 (θ ) =q0 sin(πθ/θα) and Φ+(θ ) = φ0 sin(πθ/θα)) areapplied on the top surface of the shells. The di-mensionless variables are denoted as the sameforms of those in the Reference (Dumir et al.,1997) and given as follows:

For the cases of applied mechanical load,

(uθ ,ur) =100Yr

2hS4R |q0|

(uθ ,ur) ,

(σ x,σθ ,σ r,τθr)

=(σx/S2

R,σθ /S2R,σr,τθr/SR

)/ |q0| ,(

Dθ ,Dr)

= (Dθ ,Dr)/ |d1|SR |q0| ,φ = |d1|YrΦ/2hS2

R |q0| , (71)

and S2R = Rθ/2h, Yr = 2.0GPa, d1 =

−30x10−12CN−1;


a b

-80 -40 0 40 80u ( 0 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

S = 4S = 10S = 20

-800 -400 0 400 800

w ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

S = 4S = 10S = 20

c d

0 1 2 3 4 5 τ xz ( 0 , Lb / 2 , z )

-1

-0.5

0

0.5

1

zS = 4S = 10S = 20

0 0.2 0.4 0.6 0.8 1 σz ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

S = 4S = 10S = 20

e f

0 0.4 0.8 1.2 1.6 2 φ ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

zS = 4S = 10S = 20

-0.02 -0.01 0 0.01 0.02Dz ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

S = 4S = 10S = 20

Figure 2: The through-thickness distributions of various field variables in a laminated [PZT-4/0◦/90◦/90◦/0◦/PZT-4] plate with closed-circuit surface conditions (Case 1).


a b

-0.04 -0.03 -0.02 -0.01 0 0.01u ( 0 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

S = 4S = 10S = 20

-0.016 -0.015 -0.014 -0.013 -0.012w ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

S = 4S = 10S = 20

c d

-0.004 0 0.004 0.008 τ xz ( 0 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

S = 4S = 10S = 20

-0.002 -0.001 0 0.001 σz ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

S = 4S = 10S = 20

e f

0 0.4 0.8 1.2 φ ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

S = 4S = 10S = 20

-0.3 -0.2 -0.1 0 0.1Dz ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

S = 4S = 10S = 20



a b

-80 -40 0 40 80u ( 0 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

S = 4S = 10S = 20

-800 -400 0 400 800w ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

S = 4S = 10S = 20

c d

0 1 2 3 4 5 τ xz ( 0 , Lb / 2 , z )

-1

-0.5

0

0.5

1

zS = 4S = 10S = 20

0 0.2 0.4 0.6 0.8 1 σz ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

S = 4S = 10S = 20

e f

0 1 2 3 φ ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z S = 4S = 10S = 20

-0.004 -0.002 0 0.002 0.004Dz ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

S = 4S = 10S = 20



a b

-0.2 0 0.2 0.4 0.6u ( 0 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

S = 4S = 10S = 20

-1.6 -0.8 0 0.8 1.6w ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

S = 4S = 10S = 20

c d

-0.03 -0.02 -0.01 0 0.01 0.02 τ xz ( 0 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

S = 4S = 10S = 20

-0.002 0 0.002 0.004 0.006 0.008 σz ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

S = 4S = 10S = 20

e f

-100 -75 -50 -25 0 25 φ ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

S = 4S = 10S = 20

-0.25 0 0.25 0.5 0.75 1Dz ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

S = 4S = 10S = 20



θRαθ

h

h

θ

r

Figure 6: The geometry and coordinates of apiezoelectric strip.

For the cases of applied electric potential,

(uθ ,ur) =100

|d1|SR |φ0| (uθ ,ur) ,

(σ x,σθ ,σ r,τθr)

=(σx,S2

Rσθ ,S3Rσr ,S3

Rτθr)

2h/Yr |d1| |φ0| ,(Dθ ,Dr

)= (SRDθ ,Dr)2h/ |d1|2 Yr |φ0| ,

φ = Φ/ |φ0| . (72)

Tables 9–10 show the 3D solutions of elastic andelectric field variables at crucial positions in thecylindrical shells under the applied mechanicalload and the applied electric potential, respec-tively. The geometric parameters are taken asRθ/2h= 4, 10, 100 and θα = π/3. Again itis shown that the 3D solutions of the piezoelec-tric cylindrical shells obtained using the the statespace approach, the series expansion approachand the asymptotic approach are consistent.

6.4 Functionally graded piezoelectric plates

The coupled analysis of simply-supported, FGpiezoelectric plates with closed-circuit and open-circuit surface conditions are studied by theasymptotic approach (Wu and Syu, 2007) and ameshless DRKP method (Wu et al., 2008) fordouble checking. Firures 7–10 show the through-thickness distributions of mechanical and electricvariables for the moderately shells (R/2h=10) un-der the loading conditions of Cases 1-4, respec-tively. The material properties are assumed to

obey the identical exponent-law varied exponen-tially with the thickness coordinate and are givenas Eq. (60). The material properties of PZT-4 are used as the reference material properties(Table 6) and placed on the bottom surface (i.e.,c(b)

i j ,e(b)i j ,η(b)

i j ). According to Eq. (61), the ratio ofmaterial properties between top surface and bot-tom surface is

c(t)i j

c(b)i j

=e(t)

i j

e(b)i j

=η(t)

i j

η(b)i j

= eκp , (73)

where κp are considered to be -3.0, -1.5, 0.0, 1.5,3.0 so that the ratio of material properties betweentop surface and bottom surface is approximatelyfrom 0.05 to 20. In addition, the present resultsfor FG piezoelectric plates with a particular valueof κp = 0 may reduce to the results of single-layerhomogeneous piezoelectric plates.

The electric and elastic field variables are nor-malized as the identical forms of Eqs. (68)–(70). The influence of material-property gradi-ent index on the mechanical and electric vari-ables is studied. Figures 7(a)-(d) and 9(a)-(d)show that the through-thickness distributions ofmechanical variables in the FG plates with closed-circuit and open-circuit surface conditions arealmost identical as the mechanical load is ap-plied. Figures 8(c)–(d) and 10(c)–(d) show thatthe through-thickness distributions of transversestresses change dramatically as the index κp be-comes a positive value for the loading conditionsof Case 2; conversely, they change dramaticallyas the index κp becomes a negative value for theloading condition of Case 4. The distributionsof transverse stresses across the thickness coor-dinate in the loading conditions of Cases 2 and 4are higher-degree polynomials and are back andforth among the positive and negative values. Itis also shown from Figs. 7(e)–(f) that the distri-butions of normal electric displacement throughthe thickness coordinate are approximately lin-ear functions and parabolic functions for elec-tric potential in the loading condition of Case 1.The through-thicknessdistributions of electric po-tential and normal electric displacement in Figs.8(e)–(f) and Figs 10(e)–(f) are shown to be differ-ent patterns between homogeneous piezoelectric


a b

-40 -30 -20 -10 0 10 20u ( 0 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

κp = 3κp = 1.5κp = 0κp = -1.5κp = -3

0 50 100 150 200 250w ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

κp = 3κp = 1.5κp = 0κp = -1.5κp = -3

c d

0 1 2 3 τ xz ( 0 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

κp = 3κp = 1.5κp = 0κp = -1.5κp = -3

0 0.2 0.4 0.6 0.8 1 σz ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

κp = 3κp = 1.5κp = 0κp = -1.5κp = -3

e f

0 0.6 1.2 1.8 2.4 φ ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

κp = 3κp = 1.5κp = 0κp = -1.5κp = -3

-4 -2 0 2Dz ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

κp = 3κp = 1.5κp = 0κp = -1.5κp = -3

Figure 7: The through-thickness distributions of various field variables in a functionally graded piezoelectricplate with closed-circuit surface conditions (Case 1).


a b

-0.6 -0.4 -0.2 0 0.2u ( 0 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

κp = 3κp = 1.5κp = 0κp = -1.5κp = -3

-6 -4 -2 0 2

w ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

κp = 3

κp = 1.5κp = 0κp = -1.5

κp = -3

c d

-0.15 -0.1 -0.05 0 0.05 0.1 τ xz ( 0 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

κp = 3κp = 1.5κp = 0κp = -1.5κp = -3

0 0.01 0.02 0.03 σz ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

κp = 3κp = 1.5κp = 0κp = -1.5κp = -3

e f

0 0.2 0.4 0.6 0.8 1 φ ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

κp = 3κp = 1.5κp = 0κp = -1.5κp = -3

-8 -6 -4 -2 0Dz ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

κp = 3κp = 1.5κp = 0κp = -1.5κp = -3



a b

-30 -20 -10 0 10 20u ( 0 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

κp = 3κp = 1.5κp = 0κp = -1.5κp = -3

-100 -50 0 50 100 150w ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

κp = 3κp = 1.5κp = 0κp = -1.5κp = -3

c d

0 1 2 3 τ xz ( 0 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

κp = 3κp = 1.5κp = 0κp = -1.5κp = -3

0 0.2 0.4 0.6 0.8 1 σz ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

κp = 3κp = 1.5κp = 0κp = -1.5κp = -3

e f

-2 0 2 4 6 8 φ ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

κp = 3κp = 1.5κp = 0κp = -1.5κp = -3

-0.3 -0.2 -0.1 0 0.1 0.2Dz ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

κp = 3κp = 1.5κp = 0κp = -1.5κp = -3



a b

-0.5 0 0.5 1 1.5 2u ( 0 , Lb / 2 , z )

-1

-0.5

0

0.5

1

zκp = 3κp = 1.5κp = 0κp = -1.5κp = -3

-6 -4.5 -3 -1.5 0 1.5w ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

κp = 3κp = 1.5κp = 0κp = -1.5κp = -3

c d

-0.04 -0.02 0 0.02 0.04 τ xz ( 0 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

κp = 3κp = 1.5κp = 0κp = -1.5κp = -3

-0.008 -0.006 -0.004 -0.002 0 0.002 σz ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

κp = 3κp = 1.5κp = 0κp = -1.5κp = -3

e f

-16 -12 -8 -4 0 φ ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

zκp = 3κp = 1.5κp = 0κp = -1.5κp = -3

0 0.2 0.4 0.6 0.8 1Dz ( La / 2 , Lb / 2 , z )

-1

-0.5

0

0.5

1

z

κp = 3κp = 1.5κp = 0κp = -1.5κp = -3

Figure 10: The through-thickness distributions of various field variables in a functionally graded piezoelec-tric plate with closed-circuit surface conditions (Case 4).


Table 9: Mechanical and electric components at the crucial positions in single-layer piezoelectric cylindricalshells under the cylindrical bending type of mechanical load

hR

2θ

Approaches ( )hu +,0θ 0,2αθ

ru +hx ,2αθσ +h,

2α

θθσ 0,

2αθσ r

( )0,0rθτ0,

2103 αθφ +hDr ,

210 αθ

4

10

100

The state space approach (Wu and Liu, 2007)

The series expansion approach (Dumir et al., 1997)

The asymptotic approach (Wu and Syu, 2007)







-0.8806

-0.8806

-0.8806

-3.5723

-3.572

-3.5723

-5.2969

-5.297

-5.2969

-21.1031

-21.10

-21.1029

-17.6845

-17.68

-17.6844

-16.5535

-16.55

-16.5535

-0.2747

-0.275

-0.2750

-0.2546

-0.2548

-0.2548

-0.2510

-0.2510

-0.2510

-0.7279

-0.7597

-0.7597

-0.7460

-0.7514

-0.7514

-0.7499

-0.7500

-0.7500

0.2170

0.217

0.2170

1.4195

1.420

1.4195

18.3389

18.34

18.3389

-0.6238

-0.6238

-0.6238

-0.5893

-0.5893

-0.5893

-0.5653

-0.5653

-0.5653

2.4428

2.443

2.4428

2.5600

2.560

2.5600

2.5035

2.504

2.5035

0.0214

0.0213

0.0213

-0.1574

-0.1575

-0.1575

-0.2040

-0.2041

-0.2041

plates (κp=0) and FG piezoelectric plates in thecases of applied electric loads (Cases 2 and 4). Byobservation through Figs. 7–10, we found that thedistributions of mechanical and electric variablesthrough the thickness coordinate in FG piezoelec-tric plates reveal different patterns from homoge-neous piezoelectric plates. Hence, it is suggestedthat an advanced 2D theory may be necessary tobe developed for the analysis of FG piezoelectricplates, especially when the plates are subjected toelectric loads.

7 Conclusions

Various 3D coupled analyses of multilayeredand functionally graded piezoelectric plates/shellsin the literature are surveyed. The theoreticalmethodologies for those analyses are classifiedas four different approaches and are briefly inter-preted. For comparison purposes, several avail-able 3D results obtained form these four ap-proaches are quoted and presented. It is pleasedto see that the 3D solutions obtained from dif-

ferent approaches are demonstrated to be consis-tent. The existing 3D solutions may serve as astandard for assessing a variety of 2D theories ofpiezoelectric plates. The through-thickness dis-tributions of various field variables in the multi-layered and FG piezoelectric plates under electro-mechanical loads are presented using an asymp-totic approach and doubly checked using a mesh-less DRKP method. These distributions mayserve as a reference for making the appropriate ki-netic or kinematics assumptions in advance, as anadvanced 2D theory is to be developed. Totallythere are 191 references included in the presentreview article. The authors hope the present re-view article may provide a comprehensive aspectfor the theoretical methodologies of 3D analysisof plates/shells in the existing literature. However,this survey is inevitable to miss some scientistswho might make significant contributions on thepresent subject since the authors’ academic infor-mation might be lacking. The authors hope to ex-press their apologies at not being able to includeall the relevant publishing papers on the present


Table 10: Mechanical and electric components at the crucial positions in single-layer piezoelectric cylindri-cal shells under the cylindrical bending type of electric potential

hR

2θ

Approaches ( )hu +,0θ 0,2αθ

ru +hx ,2αθσ −hx ,

2αθσ −h,

2α

θθσ −

2,0 h

rθτ 0,2αθφ +hDr ,

2αθ

4

10

100




The state space approach

(Wu and Liu, 2007) The series expansion approach

(Dumir et al., 1997) The asymptotic approach

(Wu and Syu, 2007)




-28.7436

-28.74

-28.7436

-26.4865

-26.49

-26.4863

-25.9902

-25.99

-25.9899

11.8943

11.90

11.8952

6.3387

6.340

6.3397

2.5221

2.523

2.5231

-0.1311

-0.1307

-0.1307

-0.1026

-0.1024

-0.1024

-0.0996

-0.0996

-0.0996

-0.1363

-0.1359

-0.1359

-0.1087

-0.1086

-0.1086

-0.1005

-0.1005

-0.1005

-1.5552

-1.536

-1.5361

-1.5189

-1.410

-1.4705

-1.8661

-1.411

-1.4108

0.3924

0.3925

0.3925

0.3982

0.3984

0.3984

0.3928

0.3953

0.3953

-0.4978

-0.4978

-0.4978

-0.5070

-0.5070

-0.5070

-0.5012

-0.5012

-0.5012

62.7680

62.77

62.7679

58.9644

58.96

58.9643

59.9193

59.92

59.9193

subject.

Acknowledgement: This work is supportedby the National Science Council of Republic ofChina through Grant NSC 96-2221-E-006-265.

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Appendix A

The relevant coefficients of li j in Eqs. (21)-(24)are given by

l11 = −mQ11, l12 = −nQ12, l13 = a1h2/L2,

l14 = b1e/Q, l21 = −mQ21, l22 = −nQ22,

l23 = a2h2/L2, l24 = b2e/Q, l31 = nQ66,

l31 = mQ66, l41 = e15Qh2/(ec55L2),

l43 = −m(c−155 e2

15 +η11)Qh2/(e2L2),

l52 = e24Qh2/(ec44L2),

l53 = −n(c−144 e2

24 +η22)Qh2/(e2L2).

Appendix B

The relevant coefficients of li j in Eqs. (26)-(29)are given by

l11 = −mc11, l12 = −nc12, l13 = c13L2/h2,

l14 = e31/e, l21 = −mc12, l22 = −nc22,

l23 = c23L2/h2, l24 = e32/e, l31 = nc66,

l32 = mc66, l41 = c55L2/h2, l43 = mc55L2/h2,

l44 = me15/e, l52 = c44L2/h2, l53 = nc44L2/h2,

l54 = ne24/e, l61 = e15h/(eL),

l63 = me15h/(eL), l64 = −mη11h3Q/(e2L3),

l72 = e24h/(eL), l73 = ne24h/(eL),

l74 = −nη11h3Q/(e2L3),

l81 = −mc13L2/h2, l82 = −nc23L2/h2,

l83 = c33L4/h4, l84 = e33L2/(eh2),

l91 = −me31/e, l92 = −ne32/e,

l93 = e33L2/(eh2), l94 = −η33Q/e2,

ci j = ci j/Q.

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