classicalandmixedﬁniteelementsforstaticanddynamic ... · pdf file1136 e. carrera and m....

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2007; 70:1135–1181Published online 6 November 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1901

Classical and mixed finite elements for static and dynamicanalysis of piezoelectric plates

E. Carrera∗,†,‡ and M. Boscolo§

Department of Aeronautics and Aerospace Engineering Politecnico di Torino, Corso Duca degli Abruzzi,24, 10129 Torino, Italy

SUMMARY

This paper addresses the problem of multilayered plates with embedded piezoelectric layers by finiteelement method (FEM). Original ideas in previous papers (Int. J. Numer. Methods Eng. 2002; 195:191–231, 253–291) have been extended to the static and dynamic analysis of coupled electro-mechanicalproblems. Two variational statements, the Principle of Virtual Displacements (PVD) and the ReissnerMixed Variational Theorem (RMVT) are employed to derive classical and mixed finite element matrices,respectively. Transverse stress assumptions are made in the framework of RMVT and the resulting finiteelements describe a priori interlaminar continuous transverse shear and normal stresses. The unifiedformulation (UF) has been referred to in order to derive hierarchical finite elements (FEs) in term of a fewfundamental nuclei for a large variety of piezoelectric plate theories. Both modellings that preserve thenumber of variables independent from the number of layers (equivalent single layer models, ESLM) andlayer-wise models (LWM) in which the same variables are independent in each layer have been treated. Theexpansion order N assumed for displacement, transverse stress and electrical potential fields in the platethickness direction z as well as the number of the element nodes Nn have been taken as free parametersof the considered formulations. By varying N , Nn , variable treatment (LW or ESL) as well as variationalstatements (PVD and RMVT), a large number of FEs have been presented. Compliances and/or stiffnessare accumulated from layer to multilayered level according to the corresponding variable treatment.

The numerical evaluations and assessment for the presented plate elements have been provided. Thesuperiority of RMVT applications with respect to classical ones based on PVD has been confirmed forpiezolectric plates. The proposed RMVT elements, in fact, are able to give a quasi-three-dimensionaldescription of stress/strain mechanical and electrical fields in multilayered thick and thin piezolectricplates. Copyright q 2006 John Wiley & Sons, Ltd.

Received 1 March 2006; Revised 25 August 2006; Accepted 4 September 2006

KEY WORDS: finite element; multilayered plates; electro-mechanical problems; classical and mixedformulation; smart structures

∗Correspondence to: E. Carrera, Department of Aeronautics and Aerospace Engineering Politecnico di Torino, CorsoDuca degli Abruzzi, 24, 10129 Torino, Italy.

†E-mail: [email protected]‡Professor.§Currently at: Department of Power, Propulsion and Aerospace Engineering, Aerospace Vehicle Design Group, Schoolof Engineering, Cranfield University, Cranfield, MK43 0AL, U.K.

Contract/grant sponsor: STREP EU project CASSEM; contract/grant number: NMP-CT-2005-013517

Copyright q 2006 John Wiley & Sons, Ltd.

1136 E. CARRERA AND M. BOSCOLO

1. INTRODUCTION

Smart systems are the candidate for next generation structures of aerospace vehicles as well as forsome advanced products of automotive and ship industries. Piezoelectric materials are extensivelyused in that framework. These materials are characterized by the so-called ‘direct’ and ‘inverseeffect’: an applied mechanical stresses induces electrical potential and vice versa. Such an electro-mechanical coupling permits one to build up closed-loop control systems in which piezomaterialsplay the role of both actuators and sensors. An intelligent structure can be therefore built in which,for instance, deformations or vibrations are reduced by appropriate control laws.

A number of theoretical and practical problems arise in the applications, whose solution wouldplay a crucial role in the future development of smart structures. Among these problems thefollowing are herein mentioned: the low value of the saturation of the electric voltage can lead tosevere limitation for the magnitude of the forces that can be generated by a given actuator; thedifficulties to develop efficient technology for the integration of piezomaterials in both continuousand distributed forms; effective, efficient and robust real-time control may not always be possible;the realization of an efficient and intelligent network, fully integrated in the structures, whichmakes the use of embedded piezoelectric materials convenient with respect to other traditionalsensors and actuators must be completely demonstrated. An exhaustive discussion on what abovecan be read in the two articles by Chopra [1, 2] and related literature.

However, an appropriate use of piezoelectric materials, requires an accurate description ofelectrical and mechanical fields in the constitutive layers. Pioneering works on piezoelasticity arethose by Mindlin [3], Tiersten and Mindlin [4], ErNisse [5] as well as the book by Tiersten [6].The present paper focuses on the computational, finite elements (FEs), electro-mechanical two-dimensional modellings of smart structures embedding piezolayers.

Piezolectric plates appear as multilayered structures. Very often piezoelectric layers are embed-ded in laminated structures made by anisotropic composite materials. Accurate modelling of thesestructure requires appropriate description of mechanical and electrical variables in the thicknessplate direction. Zig-zag (ZZ) form of displacement fields, interlaminar continuity (IC) of transverseshear and normal stress components [7] and electrical displacements must be mandatory accountedfor in a reliable analysis. In [8] ZZ and IC were referred to as C0

z -requirements (see Section 4.1).The importance of appropriate modellings of piezolectric plates is clearly displayed by the largeamount of papers that have been published in the last two decades. Available FEs have beendeveloped accounting to various plate theories:

• classical plate analyses (classical lamination theory (CLT) and first-order shear deformationtheory (FSDT));

• refinements of classical theories (higher order theory, HOT);• ZZ theories (theories that describe ZZ effects, see the historical review in [7] in which ZZtheories were classified according to the fundamental contribution by Lekhnitskii [9], byAmbartsumian [10] and by Reissner [11]);

• layer-wise LW theories (in which the variables are defined in each layers, while the samevariables are used for the whole multilayer in the so-called equivalent single layer models,ESLM);

• formulation with displacement or stress unknowns and mixed ones.

A first FE formulation based on the early works [3–5] has been proposed by Allik andHughes [12]. Among the available review papers those by Saravanos and Heyliger [13],

Copyright q 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 70:1135–1181DOI: 10.1002/nme

ANALYSIS OF PIEZOELECTRIC PLATES 1137

Benjeddou [14], and Wang and Yang [15] are herein mentioned. A short review of some of thelatest contribution to FE analyses of piezolectric plates follows. Application of piezoelectric shellelements to vibration control has been provided by Lammering [16]. Layer-wise plate elementswith mechanical displacements and electric voltage as field variables were proposed by Heyligeret al. [17]; application were given to static problems with applied surface traction and specifiedsurface potentials. The extension to dynamic case was presented by Saravanos et al. [18]. A FEaccounting for a FSDT description of displacement and layer-wise form of the electric potentialhas more recently been developed by Sheik et al. [19]. The numerical, membrane and bendingbehaviour of FEs which are based on FSDT has been analysed by Auricchio et al. [20] in theframework of a suitable variational formulation. Third-order theory of HOT type has been appliedby Thornbuegh and Chattopadhyay [21] to derived FEs accounting for electro-mechanical cou-pling. Similar elements have been more recently considered by Shu [22]. Extension of third-orderZZ Ambartsumian multilayered theory to finite analysis of electromechanical problems has beenproposed by Oh and Cho [23]. Extension to piezolectric plate of numerically efficient plate/shellelements based on mixed interpolation of tensorial components (MITC) formulation has beenrecently provided by Kogl and Buchalem [24, 25].

Since almost one decade the first author [8, 26, 27] and co-workers have contributed to theapplication of Reissner Mixed Variational Theorem (RMVT) to multilayered made structures.Closed-form solution analyses [28–31] as well as FE applications [32, 33] have shown that RMVTconsists of a very suitable tool to provide quasi-three-dimensional description of stress and strainfields in anisotropic laminated structures. RMVT was employed in the framework of unifiedformulation (UF) which has been recently detailed in [34]. As main feature the UF permits one toformulated both ESLM and LW models in terms of a few fundamental nuclei whose form does notdepend on the order of expansion N (that have been used for the various variables) neither by thenumber of the node of element Nn . The Murakami ZZ Function (MZZF) was used to reproduceZZ form of displacement field in the ESLM case [35, 36]. Classical formulation based on principleof virtual displacements (PVD) was developed for comparison purpose.

A first application of RMVT to piezolectric plates was provided by the first author in [37] inwhich a MITC-type plate element was extended to nonlinear dynamic analysis of piezolectric,composite plate. UF formulation has been applied, in the PVD framework, to piezolectric platein [38]; the attention was restricted to analytical closed-form solutions. Extension of related FEsto the free vibration problems was given in [39]. RMVT closed-form solutions were presentedin [40], while extension to shell was provided in [41]. Attempts to extend RMVT to piezolec-tric plates were also made by Benjeddou and Andrianarison [42]; numerical results were notprovided in that last paper. A substantiation of the RMVT extension to piezoelectric continuouscan be read in the already mentioned paper [40]. RMVT has been also applied by Mota Soareset al. [43] to developed LW plate elements in the static case. Transverse component of electricdisplacement was also considered as an assumed variable. Attention was restricted to quadratic dis-tribution of displacements (mechanical and electrical) and transverse stress unknowns in the staticcases.

The present work extends the UF and RMVT to develop FEs for the static and dynamic analysisof piezoelectric plates. Previous authors’ findings are fully extended to RMVT and a number ofnew FEs are derived and systematically compared to classical ones which were formulated onthe basis of PVD applications. ESLM and LW variable descriptions analysis are compared tothree-dimensional available solutions. Up-to forth order expansions in the thickness plate/layershave been implemented.



The paper has been organized as in the following. Section 2 gives the necessary preliminary.Section 3 quotes an extended derivation of PVD and RMVT form suitable for piezolectric continua.The variationally consistent constitutive equations have been derived in the same section. The UFhas been described in Section 4 while Section 5 introduces the FE models. Fundamental nucleirelated to the various matrices are derived in Section 6. Numerical results are discussed in Section 7to which concluding remarks follow. Explicit form of the fundamental nuclei related to PVD andRMVT are quoted in Appendix A and B, respectively.

2. PRELIMINARY

2.1. Geometry

The geometry and co-ordinate system of the laminated plate of Nl layers, including the piezolayers,have been shown in Figure 1. The integer k, which is extensively used as both subscripts orsuperscripts, denotes the layer number that starts from the plate bottom. x and y are the platemiddle surface �k co-ordinates. �0 and � will be also used to denote the reference surface. �k isthe layer boundary on �k . z and zk are the plate and layer thickness co-ordinates; h and hk denoteplate and layer thickness, respectively. �k = 2zk/hk is the non-dimensioned local plate co-ordinate;Ak will denote the k-layer thickness domain. Symbols not affected by k subscript/superscriptsrefer to the whole plate.

A multilayered plate is obtained stacking layer over layer until the desired thickness and stiffnessare reached. Composite and piezoelectric layers are considered. Each composite layer is placedwith a different orientation. A first reference system coincides to the material reference system;it will be denoted by the axes 1, 2, 3; the axis 1 is parallel to the fibre direction, the 2-axis istransverse to the fibre and it lies in the plane of the lamina while the 3-axis is perpendicular tothe lamina (see Figure 1(a)). The second reference system coincides to the plate system x, y, z;x- and y-axis are in the mid-plane of the laminate and z-axis is perpendicular to the first ones(see Figure 1(b)). Material properties can be expressed in both reference systems by the use ofappropriate rotation matrices (see [32, 44]).

2.2. Constitutive relation

Material properties of a piezoelectric continua can be expressed in various form. The so-callede-form [45] is referred to. The related energy coincides to the electric Gibbs energy G2, which

23

1

(a)

x

yz

b

a

h

(b)

Figure 1. Geometry and notation for multilayered plates: (a) material reference system;and (b) laminate reference system.



takes the following form:

G2 = 12e

TCEe− ETee− 12E

T��E (1)

Where e is the strain vector, E is the electric field vector, CE is the stiffness matrix calculated atE-constant, e is the piezoelectric matrix which couples electrical and mechanical fields and �� isthe permittivity matrix calculated at e-constant. The superscript T denotes the transposed matrices.The coefficient of the introduced matrices are obtained by the electric Gibbs energy according tothe following:

ε�mn =− �2G2

�Em�En, eni j = − �2G2

�En��i j, CE

i jkl =�2G2

��kl�i j(2)

The stress vector r and the electric displacement vector D can be obtained from Equation (1), asfollows:

r= �G2

�e, D= −�G2

�E(3)

The e-form of the constitutive equations is therefore obtained

rk =Ckek − ekTEk

Dk = ekek + �kEk(4)

The explicit forms, in the material reference systems 1–2–3, are:

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�1

�2

�3

�4

�5

�6

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

k

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

C11 C12 C13 0 0 C16

C12 C22 C23 0 0 C26

C13 C23 C33 0 0 C36

0 0 0 C44 C45 0

0 0 0 C45 C55 0

C16 C26 C36 0 0 C66

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

k ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�1

�2

�3

�4

�5

�6

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

k

−

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 e31

0 0 e32

0 0 e33

e14 e24 0

e15 e25 0

0 0 e36

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

k

⎡⎢⎢⎣E1

E2

E3

⎤⎥⎥⎦k

(5)

⎡⎢⎢⎣D1

D2

D3

⎤⎥⎥⎦k

=

⎡⎢⎢⎣

0 0 0 e14 e15 0

0 0 0 e24 e25 0

e31 e32 e33 0 0 e36

⎤⎥⎥⎦k

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�1

�2

�3

�4

�5

�6

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

k

+

⎡⎢⎢⎣

ε11 ε12 0

ε12 ε22 0

0 0 ε33

⎤⎥⎥⎦k ⎡⎢⎢⎣E1

E2

E3

⎤⎥⎥⎦k

(6)



For convenience, stresses and strains are split into in-plane and out-of-plane components:

rkTp = [�1 �2 �6]k, rkTn = [�5 �4 �3]k (7)

ekTp = [�1 �2 �6]k, ekTn = [�5 �4 �3]k (8)

As a consequence the constitutive equations can be written

rkp =Ckppe

kp + Ck

pnekn − ek T

p Ek

rkn =CkTpne

kp + Ck

nnekn − ekTn Ek

Dk = ekpekp + ekne

kn + �kEk

(9)

The explicit form of the introduced matrices follows:

Ckpp =

⎡⎢⎢⎣C11 C12 C16

C12 C22 C26

C16 C26 C66

⎤⎥⎥⎦k

, Ckpn =

⎡⎢⎢⎣0 0 C13

0 0 C23

0 0 C36

⎤⎥⎥⎦k

, Cknn =

⎡⎢⎢⎣C55 C45 0

C45 C44 0

0 0 C33

⎤⎥⎥⎦k

(10)

ekp =

⎡⎢⎢⎣

0 0 0

0 0 0

e31 e32 e33

⎤⎥⎥⎦k

, ekn =

⎡⎢⎢⎣e15 e14 0

e25 e24 0

0 0 e36

⎤⎥⎥⎦k

, �k =

⎡⎢⎢⎣

ε11 ε12 0

ε12 ε22 0

0 0 ε33

⎤⎥⎥⎦k

(11)

EkT = [E1 E2 E3]k, DkT = [D1 D2 D3]k (12)

2.3. Geometric relations

Strain–displacement relations in the linear case are:

ekpG =Dpuk, eknG = (Dnp + Dnz)uk (13)

Where uk is the vector of the displacement components:

ukT =[ux uy uz]k (14)

The explicit forms of the differential matrices are:

Dp =

⎡⎢⎢⎣

�x 0 0

0 �y 0

�y �x 0

⎤⎥⎥⎦ , Dnp =

⎡⎢⎢⎣0 0 �x

0 0 �y

0 0 0

⎤⎥⎥⎦ , Dnz =

⎡⎢⎢⎣

�z 0 0

0 �z 0

0 0 �z

⎤⎥⎥⎦ (15)

The electric field E is related to the electric potential by the following gradient relation:

EkT =[−�x −�y −�z]�k (16)



Being the electric potential � a scalar, one has

Ek = (Dep + Dez)�k (17)

where

DTep =[−�x − �y 0], DT

ez =[0 0 − �z] (18)

3. VARIATIONAL STATEMENTS FOR ELECTRO-MECHANICAL PROBLEMS

A brief introduction to the variational tools that are used to derive FE matrices is given in thissection. PVD and RMVT are treated in two different subsections. Emphasis is given to theirextension to electro-mechanical problems.

3.1. Principle of virtual displacement (PVD)

For a better understanding of the extension of RMVT to piezoelectric materials it is convenient toderive first the PVD from Hamilton’s principle:

�∫ t

t0(K − �) dt = 0 ⇒ �

∫ t

t0K dt − �

∫ t

t0� dt = 0 (19)

Where K is the kinetic energy and � is the potential energy; � denotes variations; t denotes time,t0 and t1 are the initial and generic instant. The kinetic energy variation can be treated as follows:

�∫ t

t0K dt = �

∫ t

t0dt

∫V

(1

2�ui ui

)dV =

∫ t

t0

∫V

�ui�ui dV dt

=∫V

�ui�ui dV

∣∣∣∣t1

t0

−∫ t1

t0

∫V

�ui�ui dV dt (20)

V is the plate volume, � is the mass/volume ratio and dot denotes differentiation with respect totime. �u is equal zero in t = t0 and t = t1, so that

�∫ t

t0K dt =−

∫ t1

t0

∫V

�ui�ui dV dt (21)

It follows:

�∫ t

t0K dt = −

∫ t

t0�L in dt (22)

in which �L in denotes the variation of the work done by inertial forces.The variation of potential energy for a piezoelectric continuum is written as algebraic sum of

the variation of electric Gibbs energy and the variation of the work made by applied mechanical



and electrical loadings:

�∫ t

t0� dt = �

∫ t

t0

[∫VG2(�i j , E) dV −

∫A(t j u j − Q�)dA

]dt

= �∫ t

t0

∫VG2(�i j , E) dV dt −

∫ t

t0�Le dt (23)

Where A is the plate area; t j is the mechanical loading in the j-direction; Q is the charge densityon the plate surface and � is the electric potential. Upon introducing Equations (23) and (22) inEquation (19) follows:

�∫ t

t0

∫VG2(�i j , E) dV dt =

∫ t

t0�Le dt −

∫ t

t0�L in dt (24)

Differentiating the Gibbs free energy takes the following form:∫ t

t0

∫V

(�G2

��i j��i j + �G2

�Ei�Ei

)dV dt =

∫ t

t0�Le dt −

∫ t

t0�L in dt (25)

Upon substitution of Equation (3), the PVD for piezoelectric continua is obtained:∫ t

t0

∫V(�i j�εi j − Di�Ei ) dV dt =

∫ t

t0�Le dt −

∫ t

t0�L in dt (26)

By eliminating the time integral and upon introduction of the notation used for the multilayeredcase one has

Nl∑k=1

∫Ak

∫�k

[�ekTpGrkCp + �ekTnGrkCn − �EkT

G DkC] d�k dz = �Le − �L in (27)

The subscripts ‘G’, and ‘C’ mean that quantities have to be calculated from the geometric relationsand from the constitutive relations, respectively. This equation will be used to obtain the FEmatrices of the FE piezomechanical problem. To be noticed that a different choice of G energywould lead to a different choice for the field variables.

3.2. Reissner Mixed Variation Theorem (RMVT)

A consistent extention of the Reissner mixed variation theorem to piezoelectric continuum is givenin this paragraph.

Let us consider the first term of Equation (24)

�∫ t

t0

∫VG2(�i j , Ei ) = �

∫ t

t0

∫VG2(�pG, �nG, EG) (28)

Since the transverse stresses are unknown variables [11] the Lagrange multiplier �nM should beintroduced:

�∫ t

t0

∫VG2(epG, enG,EG) dV dt = �

∫ t

t0

∫V[G2(epG, enG,EG) + rTnM (enG − enC)] dV dt (29)



The subscript M denotes that the corresponding variables are assumed from a given ‘Model’; inother words these variables will be obtained a priori, that is without any post-processing techniques.Upon differentiation one has∫ t

t0

∫V

[�eTpG

�G2

�epG+ �eTnG

�G2

�enG+ �ET

G�G2

�EG+ �rTnM (enG − enC)

]dV dt (30)

Taking Equation (3) into account

�∫ t

t0

∫VG2 dV dt =

∫ t

t0

∫V[�eTpGrpC + �eTnGrnM − �ET

GDC + �rTnM (enG − enC)] dV dt (31)

Upon substitution of Equation (31) in Equation (24), the extension of RMVT to piezolectriccontinuum is obtained∫ t

t0

∫V[�eTpGrpC + �eTnGrnM − �ET

GDC + �rTnM (enG − enC)] dV dt

= +∫ t

t0�Le dt −

∫ t

t0�L in dt (32)

By introducing the multilayered notations

Nl∑k=1

∫Ak

∫�k

[�ekTpGrkpC + �ekTnGrknM − �EkT

G DkC + �rkTnM (eKnG − eKnC)] d�k dz = �Le − �L in (33)

3.3. Constitutive relation for RMVT applications

Equation (33) clearly states that RMVT application requires the use of appropriate constitutiveequations for rpC, enC and DC. A derivation of these, by referring to various techniques has beenprovided in [40]. Anyway these are herein obtained by direct calculation of rpC, enC and DC fromEquation (9). The physical constitutive equations are for convenience re-written, by introducing anotation which is consistent for RMVT purposes:

rkpC =Ckppe

kpG + Ck

pneknC − ekTp Ek

G

rknM =CkTpne

kpG + Ck

nneknC − ekTn Ek

G

DkC = ekpe

kpG + ekne

knC + �kEk

G

(34)

First eknC is taken out from the second equation and introduced in the first and in the third equation:

rkpC = (Ckpp − Ck

pnCk−1nn Ck

np)ekpG + (Ck

pnCk−1nn )rknM + (Ck

pnCk−1nn ekTn − ekTp )Ek

G

eknC = (−Ck−1nn Ck

np)ekpG + (Ck−1

nn )rknM + (Ck−1nn ekTn )Ek

G

DkC = (ekp − eknC

k−1nn Ck

np)ekpG + (eknC

k−1nn )rknM + (�k + eknC

k−1nn ekTn )Ek

G

(35)



This equation can be rewritten by defining a new set of matrices that are depicted as C

rpC = CppepG + CpnrnM + CseEG

enC = CnpepG + CnnrnM + CdeEG

DC = CedepG + CesrnM + CeeEG

(36)

Where

Ckpp =Ck

pp − CkpnC

k−1nn Ck

np, Ckpn =Ck

pnCk−1nn

Ckse =Ck

pnCk−1nn ekTn − ekTp = Ck

pnekTn − ekTp , Ck

np = −Ck−1nn Ck

np

Cknn =Ck−1

nn , Ckde =Ck−1

nn ekTn = Cknne

kTn

Cked = ekp − eknC

k−1nn Ck

np = ekp + eknCknp, Ck

es = eknCk−1nn = eknC

knn

Ckee = �k + eknC

k−1nn ekTn = �k + eknC

knne

kTn

(37)

To be noticed that

Cknn = CkT

nn , Ckpp = CkT

pp, Ckee = CkT

ee

Ckpn = −CkT

np, Ckde = CkT

es , Ckse = −CkT

ed

(38)

and

Ckpn =CkT

np and Ck−1nn =Ck−1

nnT

(39)

The constitutive equations in Equation (36) represent the relation to be used for the applicationsof RMVT to piezolectric continua. The explicit forms of the introduced matrices are for sake ofcompleteness written as follows:

Ckpp =

⎡⎢⎢⎢⎣C11 C12 C16

C12 C22 C26

C16 C26 C66

⎤⎥⎥⎥⎦ (40)

C11 =C11 − C13C13

C33, C22 =C22 − C23C23

C33, C66 =C66 − C36C36

C33

C12 =C12 − C13C23

C33, C16 =C16 − C13C36

C33, C26 =C26 − C36C23

C33

(41)

Ckpn =

⎡⎢⎢⎢⎣0 0 C13

0 0 C23

0 0 C36

⎤⎥⎥⎥⎦ (42)



C13 = C13

C33, C23 = C23

C33, C36 = C36

C33(43)

Ckse =

⎡⎢⎢⎢⎣0 0 Cse13

0 0 Cse23

0 0 Cse33

⎤⎥⎥⎥⎦ (44)

Cse13 = C13e33 − e31, Cse23 = C23e33 − e32, Cse33 = C36e33 − e36 (45)

Cknp =

⎡⎢⎢⎣

0 0 0

0 0 0

−C13 −C23 −C36

⎤⎥⎥⎦ (46)

C13 = C13

C33, C23 = C23

C33, C36 = C36

C33(47)

Cknn =

⎡⎢⎢⎢⎣C55 C45 0

C45 C44 0

0 0 C33

⎤⎥⎥⎥⎦ (48)

C55 = C44

−C245 + C44C55

, C44 = C55

−C245 + C44C55

C33 = 1

C33, C45 = C45

C245 − C44C55

(49)

Ckde =

⎡⎢⎢⎢⎣Cde21 Cde22 0

Cde11 Cde12 0

0 0 Cde33

⎤⎥⎥⎥⎦ (50)

Cde21 = C45e14 + C55e15, Cde22 = C45e24 + C55e25, Cde11 = C44e14 + C45e15

Cde12 = C44e24 + C45e25, Cde33 = C33e33(51)

Cked =

⎡⎢⎢⎣

0 0 0

0 0 0

Ced31 Ced32 Ced33

⎤⎥⎥⎦ (52)



Ced31 = e31 − C13e33, Ced32 = e32 − C23e33, Ced33 = e36 − C36e33 (53)

Ckes =

⎡⎢⎢⎢⎣Ces12 Ces11 0

Ces22 Ces21 0

0 0 Ces33

⎤⎥⎥⎥⎦ (54)

Ces12 = C45e14 + C55e15, Ces22 = C45e24 + C55e25, Ces11 = C44e14 + C45e15

Ces21 = C44e24 + C45e25, Ces33 = C33e33(55)

Ckee =

⎡⎢⎢⎢⎣Cee11 Cee12 0

Cee21 Cee22 0

0 0 Cee33

⎤⎥⎥⎥⎦ (56)

Cee11 = E11 + C44e214 + 2C45e14e15 + C55e

215

Cee12 = E12 + C44e14e24 + C45e15e24 + C45e14e25 + C55e15e25

Cee21 = E12 + C44e14e24 + C45e15e24 + C45e14e25 + C55e15e25

Cee22 = E22 + C44e24e24 + 2C45e24e25 + C55e225

Cee33 = C33e233 + E33

(57)

4. UNIFIED FORMULATION FOR PLATE THEORIES

This section describes the plate theories on which basis the finite plate element matrices have beendeveloped. A more detailed description has been provided in previous works already discussed inthe Introduction, see for instance [27, 32, 34].4.1. C0

z requirements for layered structures

Attention is restricted to the so-called axiomatic theories which postulate a certain field in thethickness plate/layer direction. Due to not continuous distribution of electro-mechanical propertiesalong the thickness, multilayered plate requires special attention. It is, in fact, well known thatclassical theories that were originally developed for one-layer metallic structures cannot be used ifaccurate analyses of multilayered plates is required. Transverse shear and normal stresses must be,in fact, continuous at the interfaces, see Figure 2. The continuity of the normal stresses requires thatstrains must be discontinuous since material properties change from layer to layer; as a consequencethe derivatives of the displacements must be discontinuous at each interface (ZZ effect). However,in-plane stresses could be discontinuous at the interface. The normal electric displacement mustbe continuous at the interface too, if free boundary conditions are postulated. The set of this ruleshas been summarized in [8] as C0

z -requirements; mechanical and electrical displacements as well



Figure 2. Equilibrium at the interface between two layers.

as transverse stresses (not their derivatives) must be C0 continuous function in the z-thicknessdirection. An example of stresses, displacements and electric displacements is shown in Figure 3.

4.2. Unified formulation

The unified formulation consists of a technique which permits to handle in a unified manner a largevariety of plate modellings. This is made by expressing governing equations and/or FE matricesin term of a few fundamental 3 × 3 nuclei which do not formally depend on: expansion N usedto in the z-direction; number of the node Nn of the element; variables description (LW or ESL).

4.2.1. Formulation based on PVD. Unknown variables are in this case the displacements and theelectrical potential. The following ‘axiomatic’ expansion is used for displacements:

uk(x, y, z) = Fb(z)ukb(x, y) + Fr (z)ukr (x, y) + Ft(z)ukt (x, y), r = 2, . . . , N − 1 (58)

or in compact form:

uk(x, y, z) = F�uk� (59)



Figure 3. Example of displacements, stresses and electric displacements distribution along the z-co-ordinate.

F�(z) are function of the thickness co-ordinate z which can assume different forms. u� are two-dimensional unknowns. Similarly, the electric potential is assumed

�k(x, y, z) = Fb(z)�kb(x, y) + Fr (z)�

kr (x, y) + Ft(z)�

kt (x, y), r = 2, . . . , N − 1 (60)

or in compact form

�k(x, y, z) = F��k� (61)

Thickness functions could be the same of those already used for displacements. Of course this isnot mandatory and in same cases it could be not convenient.

4.2.2. Formulation based on RMVT. The transverse normal stresses should be added as fieldvariables in the RMVT applications with respect to those already introduced for the PVD case.The transverse normal stresses are

rkn(x, y, z) = Fb(z)rknb(x, y) + Fr (z)r

knr (x, y) + Ft(z)r

knt(x, y), r = 2, . . . , N − 1 (62)

or in compact form:

rkn(x, y, z) = F�rkn� (63)

Also in this case the thickness functions can be the same of those already used for displacementsand electric potential. See next sub-sections.



4.3. Equivalent single layer models

Assumptions above can be used at layer or at multilayer level: LW and ESL theories are thanobtained, respectively. Depending on the made choice for the polynomials F(z) a large variety ofplate modellings can be introduced. In the ESL case, the laminate is treated as a single layer. Thesubscript k is therefore omitted.

A first natural choice for thickness functions F(z) consists to refer to power of z polynomials

Fb = 1, Fr = z(r−1), Ft = z(N−1), r = 2, . . . , N − 1 (64)

That is a Taylor-type expansion is used and the variables related to Fb corresponds to the middleplane values; higher order terms correspond instead to higher order derivative calculated withcorrespondence to the reference surface of the plate.

To be noticed that this model is not able to reproduce the ZZ form for the displacement field inthe thickness direction. Murakami [46] introduced a ZZ function, able to describe a ZZ form forthe displacements. The MZZF can be, for instance, introduced instead of the Ft polynomial

Fb = 1, Ft = (−1)k�k, Fr = z(r−1), r = 2, . . . , N − 1 (65)

The exponent k changes the sign of the ZZ term in each layer. Such an artifice permits one toreproduce the discontinuity of the first derivative of the displacements in the z-direction.

The displacements are the only unknowns that could be modelled by ESL model with theintroduction of Murakami’s ZZ function. The electrical potential as well as transverse stressesrequire layer-wise description.

4.4. Layer-wise models LW

If detailed response of individual layers is required and if significant variations in gradients betweenlayers exist, layer-wise model must be referred to. Each layer is seen as an independent layer andthe continuity of the field variables at the interfaces is imposed as a constraint. The use of Taylor-type expansion in each layer is not convenient. Top–bottom continuity would require additionalconditions on the field variables. In order to avoid this drawback, an appropriate combination ofLegendre polynomials could be conveniently used as base function according to the following:

Ft = P0 + P12

, Fb = P0 − P12

, Fr = Pr − Pr−2, r = 2, . . . , N (66)

Where Pi (�k) is the i-order Legendre polynomial in the domain−1� �k � 1. The first five Legendrepolynomials are

P0 = 1, P1 = �k, P2 = 3�2 − 1

2, P3 = 5�3 − 3�

2, P4 = 35�4 − 30�2 + 3

8(67)

The functions we have chosen have the following properties:

�k ={1 : Ft = 1, Fb = 0, Fr = 0

−1 : Ft = 0, Fb = 1, Fr = 0(68)



L

E

D

M

1

2

3

4

Z C

TYPE OF FORMULATION

Classical based on PVD

Mixed based on RMVT

D

M

Linear

Parabolic

Cubic

Fourth-order

z-EXPANSIONFORORDER OF USED

OPTIONAL FIELDS USED FOR ESLM MODELS

Zig-Zag Effects is Accounted forZ

C Interlaminar Equilibria is fulfilled

ACRONYM

Equivalent-Single-LayerE

Layer-WiseLTYPE OF THEORY

Layer-Wise Theory based on Classical Displacement formulation with cubic displacement fields in the layer

EMZC2

LD3

EXAMPLES

Mixed Equivalent-Single-Layer with parabolic displacement fields (and cubic stress fields) accounting for Zig-zag Effectsand fulfilling interlaminar transverse stresses Continuity

Figure 4. Meanings of the introduced acronyms.

For instance, in case of displacement variables ukt and ukb are the top and bottom displacement ofthe kth layer; the continuity condition is therefore easily written as follows:

ukt =u(k+1)b , con k = 1, . . . , NL − 1 (69)

Electric potential and transverse stresses are in this work always described as LW variables.

4.5. Acronyms

Depending on the used variational statement (PVD or RMVT), variables description and order ofexpansion N a number of two-dimensional theories and related FEs can be derived. In order toidentify the various FEs appropriate acronyms are introduced. Figure 4 shows how the acronymsare built. Examples of displacement field related to few particular-case theories have been plottedin Figure 5: the first field can be ‘E’ or ‘L’ according to ESL or LW description, respectively;the second field can be ‘D’ or ‘M’ according to PVD or RMVT application, respectively; the lastfield can assume the numbers 1–4 according to the order of the adopted expansion in the thicknessdirection; a third ‘Z’ and fourth ‘C’ field (which are optional in the ESL case), denote the use ofMZZF and/or IC fulfilment, respectively.



(a) (b)

(c) (d)

Figure 5. Examples of displacement field related to some particular-case theories: (a) linearand cubic ESL analysis; (b) linear and cubic LW analysis; (c) linear case of Murakami zig

zag function; and (d) cubic case of Murakami zig zag function.

5. FINITE ELEMENT APROXIMATIONS

Finite element approximations into the plate reference surface domain are introduced by means ofiso parametric descriptions for the various field variables.

uk�(x, y) = Niqk�i , i = 1, 2, . . . , Nn (70)

�k�(x, y) = Ni g

k�i , i = 1, 2, . . . , Nn (71)

rkn�(x, y) = Ni fk�i , i = 1, 2, . . . , Nn (72)

Where Ni = Ni (x, y) are the shape functions, qk�i are the nodal unknown displacement vectors,gk�i is the nodal unknown electric potential and fk�i are the nodal unknown normal stress vectors.Substituting Equation (70) in Equation (59) we obtain

uk(x, y, z) = F�Niqk�i (73)

Substituting Equation (71) in Equation (61) we obtain:

�k = F�Ni gk�i (74)



Figure 6. Summary of the introduced approximations.

Substituting Equation (72) in Equation (63) we obtain:

rkn(x, y, z) = F�Ni f k�i , i = 1, 2, . . . , Nn (75)

Standard serendipity and Lagrangian shape functions related to 4 (Q4), 8 (Q8) and 9 (Q9) nodescould be considered. In Figure 6 approximations introduced are gathered and displayed along withsuperscripts which will be employed in the stiffness matrices.

6. DERIVATION OF FE MATRICES FOR ELECTRO-MECHANICAL PLATES

6.1. Classical formulations based on PVD

By starting from Equation (27), the fundamental nuclei of various FE matrices related to PVDapplications are obtained according to the following steps:

1. The constitutive relations Equation (9) are introduced in the PVD Equation (27).2. The geometric relations Equations (13) and (17) are used to express strain in terms of

displacements and electric field in terms of electric potential.3. The through-the-thickness assumptions discussed Equations (59) and (61) are introduced to

integrate in the thickness directions.4. FE shape functions Equations (70) and (71) are used to integrate the in-plane plate

co-ordinates.5. Matrix product are made and explicit form of fundamental nuclei of various FE matrices is

obtained.



All the necessary steps are herein for sake of brevity, omitted. However, the final form of thegoverning equations is written as in the following:

�qkT�i : Kk�si juu qks j + Kk�si j

ue gks j =Pku� − Mk�si j

uu qks j

�gkT�i : Kk�si jeu qks j + Kk�si j

ee gks j =Pke�

(76)

Where

Pku� =Kk�si j

up ps j (77)

Pke� = −Kk�si j

e f �s j (78)

where ps j is the nodal array of the applied mechanical loadings; �s j is the nodal array of theapplied electrical loadings.

Equations (76) states the FE governing matrices for PVD application to piezolectric plates. Thefollowing fundamental nuclei are involved:

Kk�si juu ,Kk�si j

ue ,Kk�si jeu ,Kk�si j

ee ,Kk�si je f ,Kk�si j

up ,Mk�si juu

Their explicit forms are:

Kk�si juu =

∫�k

[(DTpNi )(Ck

ppEuu�s (DpN j ) + Ck

pn Euu�s (DnpN j ) + Ck

pn Euu�s,z N j )

+ (DTnpNi )(CkT

pn Euu�s (DpN j ) + Ck

nn Euu�s (DnpN j ) + Ck

nn Euu�s,z N j )

+ Ni (CkTpn E

uu�,zs(DpN j ) + Ck

nn Euu�,zs(DnpN j ) + Ck

nn Euu�,zs,z N j )] d�k (79)

Kk�si jue = −

∫�k

[(DTpNi )ekTp (Eue

�s (DepN j ) + Eue�s,z (N j I

∗))

+ (DTnpNi )ekTn (Eue

�s (DepN j ) + Eue�s,z (N j I

∗))

+ NiekTn (Eue�,zs(DepN j ) + Eue

�,zs,z (N j I∗))] d�k (80)

Kk�si jeu = −

∫�k

[(DTepNi )(ekpE

eu�s (DpN j ) + ekn E

eu�s (DnpN j ) + ekn E

eu�s,z N j )

+ (Ni I∗)T(ekpE

eu�,zs(DpN j ) + ekn E

eu�,zs(DnpN j ) + ekn E

eu�,zs,z N j )] d�k (81)



Table I. Dimension of the fundamental nuclei.

Fundamental nuclei Dimension

Kk�si juu [3 × 3]

Kk�si ju� [3 × 3]

Kk�si jue [3 × 1]

Kk�si j�u [3 × 3]

Kk�si j�� [3 × 3]

Kk�si j�e [3 × 1]

Kk�si jeu [1 × 3]

Kk�si je� [1 × 3]

Kk�si jee [1 × 1]

Mk�si juu [3 × 3]

Kk�si jup [3 × 3]

Kk�si je f [1 × 1]

Kk�si jee = −

∫�k

[(DTepNi )�

k(Eee�s (DepN j ) + Eee

�s,z (N j I∗))

+ (Ni I∗)T�k(Eee�,zs(DepN j ) + Eee

�,zs,z (N j I∗))] d�k (82)

Kk�si jup =

∫�k

F1� (Ni N jmk

s )F1s d�k (83)

Kk�si je f =

∫�k

F1� (Ni N jn

ks )F

1s d�k (84)

Mk�si juu =

∫�k

�k(Ni I)Euu�s (N j I) d�k (85)

in which I denote the unit array while I∗T =[0, 0, −1]. mks is a diagonal matrix related to the three

components of applied pressure loading, see [32]. Superscript 1 denotes those values calculated atthe position z in which correspondence mechanical and/or electrical loadings are applied.

The following thickness integrals have been introduced:

(E��s , E�

�,zs, E��s,z , E

��,zs,z ) =

∫Ak

(F�� F

s , F�

�,z Fs , F�

� Fs,z , F

��,z F

s,z ) dz (86)

with � and that can assume the values u, e.Table I summarizes the dimension of the various fundamental nuclei (the dimension 3 is pre-

served for displacements while the dimension 1 is related to potential). The explicit forms of thefundamental nuclei has been given in Appendix A.



Equation (76) has been written for a given node couple i, j and a given k-layer. In order toobtain the governing equations for the whole plate, the fundamental nuclei should be opportunelyassembled expanding the indices k, i, j, �, s up to their extents. Such a description has been hereinomitted, however, it can be found in the already mentioned authors’ work.

6.2. Advanced mixed formulation based on RMVT

In addition to the electrical and displacement fields, RMVT permits one to assume an independenttransverse stress fields. According to what done for the PVD case, upon introduction of therelated constitutive relations Equation (36) as well as transverse stress assumption Equation (75)in Equation (33), one has

�qkT�i : Kk�si juu qks j + Kk�si j

u� fks j + Kk�si jue gks j =Pk

u� − Mk�si juu qks j

�fkT�i : Kk�si j�u qks j + Kk�si j

�� fks j + Kk�si j�e gks j = 0

�gkT�i : Kk�si jeu qks j + Kk�si j

e� fks j + Kk�si jee gks j =Pk

e�

(87)

Where

Pku� =Kk�si j

up ps j (88)

Pke� = −Kk�si j

e f �s j (89)

Equation (87) gives the governing FE matrix related to RMVT applications. The following funda-mental nuclei have been introduced:

Kk�si juu ,Kk�si j

u� ,Kk�si jue ,Kk�si j

�u ,Kk�si j�� ,Kk�si j

�e ,Kk�si jeu ,Kk�si j

e� ,Kk�si jee ,Kk�si j

e f ,Kk�si jup ,Mk�si j

uu

The explicit array forms are

Kk�si juu =

∫�k

[DTpNi Ck

ppEuu�s DpN j ] d�k (90)

Kk�si ju� =

∫�k

[DTpNi Ck

pn Eu��s N j + NiDT

npEu��s N j + Ni ITEu�

�,zs N j ] d�k (91)

Kk�si jue =

∫�k

[DTpNi Ck

seEue�s DepN j + DT

pNi CkseE

ue�s,zI

∗N j ] d�k (92)

Kk�si j�u =

∫�k

[Ni E�u�s DnpN j + Ni E

�u�s,zIN j + Ni Ck

npE�u�s DpN j ] d�k (93)



Kk�si j�� = −

∫�k

[Ni Cknn E

��s N j ] d�k (94)

Kk�si j�e = −

∫�k

[Ni CkdeE

�e�s DepN j + Ni Ck

deE�e�s,zI

∗N j ] d�k (95)

Kk�si jeu = −

∫�k

[Ni I∗TCked E

eu�,zsDpN j + NiDT

epCked E

eu�s DpN j ] d�k (96)

Kk�si je� = −

∫�k

[NiDTepC

kes E

e��s N j + Ni I∗TCk

es Ee��,zs N j ] d�k (97)

Kk�si jee = −

∫�k

[NiDTepC

keeE

ee�sDepN j + NiDT

epCkeeE

ee�s,zI

∗N j

+ Ni I∗TCkeeE

ee�,zsDepN j + Ni I∗TCk

eeEee�,zs,zI

∗N j ] d�k (98)

Kk�si jup =

∫�k

F1� (Ni N jmk

s )F1s d�k (99)

Kk�si je f =

∫�k

F1� (Ni N jn

ks )F

1s d�k (100)

Mk�si juu =

∫�k

�k(Ni I)Euu�s (N j I) d�k (101)

Table I summarizes the nuclei’s dimension. The explicit, scalar form of the fundamental nucleican be seen in Appendix B. As for the PVD case, assembly of matrices for the different layersand different nodes is required to get the complete element matrices.

7. NUMERICAL RESULTS

This section shows the performances of the developed FEs for vibration and static response (sensorand actuator configuration) of piezolectric plates. In order to compare the analysis to closed-formexact solutions, attention has been restricted to simply supported plates. Concerning convergencerates and comparison of various elements Q4, Q8 and Q9 (4, 8 and 9 nodes, respectively) theobtained results do not differ from that were already documented in [32]; these are thereforeomitted in this work. The reduced integration technique that was successfully applied in [33]have been preserved in the present work. All the quoted results refer to the use of selectiveintegration technique and a 6 × 6 mesh of Q9 element for which a convergent solution wasobtained. Further analyses and discussion on the convergence rate and locking mechanisms ofthe considered elements, for the pure mechanical problems, have recently been considered by



Table II. Mechanical and electrical material properties.

Properties PZT-4 Gr/Ep

E1 (GPa) 81.3 132.38E2 (GPa) 81.3 10.756E3 (GPa) 64.5 10.75612 0.329 0.2413 0.432 0.2423 0.432 0.49G23 (GPa) 25.6 3.606G13 (GPa) 25.6 5.6537G12 (GPa) 30.6 5.6537e15 (C/m2) 12.72 0e24 (C/m2) 12.72 0e31 (C/m2) −5.20 0e32 (C/m2) −5.20 0e33 (C/m2) 15.08 0ε11/ε0 1475 3.5ε22/ε0 1475 3.0ε33/ε0 1300 3.0� (kg/m3) 1 1

Figure 7. Geometry of a piezolectric plate.

D’Ottavio et al. [47]. The conclusions in [47] have been confirmed in the present work. For sakeof brevity, these conclusions are not documented herein.

7.1. Modal analysis of adaptive plates

The multilayered piezoelectric plate consists of five layers whose core is a symmetric cross-ply[0/90/0] of graphite–epoxy while the two external skins are constituted by piezoceramic material



Table III. Comparison of mixed and classical finite elements for the first three circular frequencies(rad/s) for a thick piezoelectric plate a/h = 4 and for a thin one a/h = 50.

4 50

ah �1 �2 �3 �1 �2 �3

Exact [48] 57 074.5 191 301 250 769 618.118 15 681.6 21 492.8

LD4 FEM 57 096.9 191 361 250 803 618.450 15 686.9 21 560.4(+0.03%) (+0.03%) (+0.01%) (+0.05%) (+0.03%) (+0.31%)

An 57 074.0 191 301 250 768 618.104 15 681.6 21 492.6(+0.00%) (+0.00%) (+0.00%) (+0.00%) (+0.00%) (+0.00%)

LD3 FEM 57 097.2 191 361 250 803 618.450 15 686.9 21 560.4(+0.04%) (+0.03%) (+0.01%) (+0.05%) (+0.03%) (+0.31%)

An 57 074.0 191 301 250 768 618.104 15 681.6 21 492.6(+0.00%) (+0.00%) (+0.00%) (+0.00%) (+0.00%) (+0.00%)

LM2 FEM 57 096.0 191 367 250 828 618.379 15 686.9 21 497.3(+0.04%) (+0.03%) (+0.02%) (+0.04%) (+0.03%) (+0.02%)

An 57 078.3 191 301 250 779 618.106 15 681.5 21 492.6(+0.01%) (+0.00%) (+0.00%) (+0.00%) (+0.00%) (+0.00%)

LD2 FEM 57 105.1 191 371 250 821 618.450 15 686.9 21 562.4(+0.05%) (+0.03%) (+0.02%) (+0.05%) (+0.03%) (+0.32%)

An 57 081.9 191 311 250 786 618.105 15 681.6 21 492.6(+0.01%) (+0.00%) (+0.00%) (+0.00%) (+0.00%) (+0.00%)

LM1 FEM 57 074.2 194 763 251 160 618.269 15 688.7 21 498.7(+0.00%) (+1.81%) (+0.16%) (+0.02%) (+0.05%) (+0.03%)

An 57 056.6 194 696 253 955 617.996 15 683.3 21 493.9(−0.03%) (+1.77%) (+1.29%) (−0.02%) (+0.01%) (+0.01%)

LD1 FEM 57 275.5 194 907 255 689 619.473 15 688.8 21 499.2(+0.35%) (+1.88%) (+1.96%) (+0.21%) (+0.04%) (+0.03%)

An 57 252.5 194 840 255 646 619.022 15 683.4 21 499.4(+0.31%) (+1.85%) (+1.94%) (+0.14%) (+0.01%) (+0.03%)

Results of layer-wise analyses. An denotes the analytical closed-form solution of the same problem.Note: FEM, finite element method.

PZT-4. The material properties are those in Table II, where ε0 is the dielectric constant and itassumes value ε0 = 8.854188× 10−12 F/m. The value of the material density � coincides to thatof the considered three-dimensional exact solutions [48]. The electric boundary conditions coincideto zero potential on the upper and lower surfaces, see Figure 7.

Advanced (RMVT) and classical (PVD) FEs have been compared. Tables III–V quote thefirst three frequencies; LW, ESL and, ESL with MZZF are compared, respectively. For comparisonpurpose, the three dimensional exact solutions in [48] as well as the analytical closed-form solutionsin [38, 40] have been reported. Excellent agreement is shown between FE and analytical solutions.That agreement confirms the effectiveness of the developed computational models. Excellent



Table IV. As in Table III: results related to equivalent single layer analysis.

4 50

ah �1 �2 �3 �1 �2 �3

Exact [48] 57 074.5 191 301 250 769 618.118 15 681.6 21 492.8

ED4 FEM 58 740.3 194 660 254 787 618.913 15 698.9 21 502.5(+2.91%) (+1.75%) (+1.60%) (+0.12%) (+0.11%) (+0.04%)

An 58 713.8 194 592 254 740 618.464 15 693.5 21 497.8(+2.87%) (+1.72%) (+1.58%) (+0.05%) (+0.07%) (+0.02%)

EM3 FEM 56 656.2 191 945 253 188 616.498 15 697.0 21 496.1(−0.73%) (+0.33%) (+0.96%) (−0.26%) (+0.10%) (+0.02%)

An 58 576.0 195 807 259 495 618.504 15 693.0 21 499.7(+2.63%) (+2.36%) (+3.48%) (+0.06%) (+0.07%) (+0.03%)

ED3 FEM 58 845.6 195 842 259 649 618.999 15 699.6 21 504.8(+3.10%) (+2.37%) (+3.54%) (+0.14%) (+0.11%) (+0.05%)

An 58 818.6 195 825 259 586 618.550 15 694.2 21 500.1(+3.05%) (+2.36%) (+3.51%) (+0.07%) (+0.07%) (+0.03%)

EM2 FEM 68 548.7 193 307 253 443 619.188 15 697.3 21 500.9(+20.1%) (+1.05%) (+1.07%) (+0.17%) (+0.10%) (+0.04%)

An 68 710.5 195 835 262 092 620.214 15 693.3 21 504.6(+20.4%) (+2.37%) (+4.52%) (+0.34%) (+0.07%) (+0.05%)

ED2 FEM 69 537.1 195 930 262 267 620.753 15 700.2 21 510.0(+21.8%) (+2.42%) (+4.58%) (+0.42%) (+0.11%) (+0.08%)

An 68 413.7 195 860 261 780 620.229 15 694.9 21 505.2(+19.8%) (+2.38%) (+4.39%) (+0.34%) (+0.08%) (+0.05%)

EM1 FEM 69 728.7 195 987 266 202 678.125 15 694.6 21 510.2(+22.2%) (+2.45%) (+6.15%) (+9.71%) (+0.08%) (+0.08%)

An 71 619.2 195 939 266 276 679.117 15 689.3 21 505.5(+25.5%) (+2.42%) (+6.18%) (+9.87%) (+0.05%) (+0.06%)

ED1 FEM 74 232.1 195 842 266 411 690.338 15 700.3 21 512.7(+30.0%) (+2.37%) (+6.23%) (+11.68%) (+0.11%) (+0.09%)

An 74 105.9 196 021 266 337 689.867 15 695.0 21 507.4(+29.8%) (+2.46%) (+6.20%) (+11.6%) (+0.08%) (+0.06%)

Note: FEM, finite element method.

agreement with three dimensional solution has been achieved as well. The exact solution can bereached by LW FEs with 1 higher order of expansion N , such as LM2 and LD4. RMVT FEs showbetter performance than PVD ones: RMVT FEs require lower value of N with respect to PVDones to provide a fixed accuracy. The advantage of using RMVT FEs is much more evident inthe case of ESL analyses: for instance, for a thick plate case a/h = 4 the ED1 element leads 30%error while the same error related to EM1 is 22%; the error of EM3 element is −0.73% whilethe one related to ED3 is 3.10%. The convenience of using the Murakami ZZ function [35, 36] is



Table V. As in Table III: results related to equivalent single layer analysisincluding Murakami zig-zag function.

4 50

ah �1 �2 �3 �1 �2 �3

Exact [48] 57 074.5 191 301 250 769 618.118 15 681.6 21 492.8

EDZ3 FEM 57 681.4 195 779 259 632 618.831 15 692.4 21 501.2(+1.06%) (+2.34%) (+3.53%) (+0.11%) (+0.06%) (+0.03%)

An 56 656.1 195 711 259 022 618.382 15 687.1 21 496.5(−0.73%) (+2.33%) (+3.29%) (+0.04%) (+0.03%) (+0.01%)

EMZ2 FEM 58 422.6 194 263 259 007 617.080 15 593.8 21 492.9(+2.36%) (+1.54%) (+3.29%) (−0.17%) (−0.56%) (+0.00%)

An 60 428.9 195 714 260 774 619.012 15 692.6 21 496.5(+5.88%) (+2.31%) (+3.99%) (+0.14%) (+0.07%) (+0.02%)

EDZ2 FEM 60 629.7 195 790 260 923 619.496 15 698.9 21 501.2(+6.22%) (+2.34%) (+4.04%) (+0.22%) (+0.11%) (+0.03%)

An 60 605.5 195 722 260 861 619.046 15 693.6 21 496.5(+6.18%) (+2.31%) (+4.02%) (+0.15%) (+0.07%) (+0.01%)

EMZ1 FEM 61 500.9 195 969 262 604 681.717 15 697.7 21 503.2(+7.76%) (+2.44%) (+4.72%) (+10.29%) (+0.10%) (+0.05%)

An 62 832.0 195 948 266 157 683.442 15 692.4 21 498.5(+10.09%) (+2.43%) (+6.14%) (+10.57%) (+0.07%) (+0.03%)

EDZ1 FEM 63 229.7 196 036 266 275 688.548 15 699.0 21 503.2(+10.7%) (+2.47%) (+6.18%) (+11.3%) (+0.11%) (+0.04%)

An 63 204.7 195 965 266 196 688.082 15 693.6 21 496.5(+10.7%) (+2.43%) (+6.15%) (+0.07%) (+0.07%) (+0.01%)


Figure 8. Piezolectric plate as a sensor.



Table VI. Sensor piezolectric plate. Comparison of mixedand classical elements to evaluate electric potential

�(a/2, b/2, 0) × 103: layer-wise results.

ah 2 4 10 100

Exact 3D [49] — 6.11 — —LM4 FEM 0.9364 6.118 44.78 4691LD4 FEM 0.9363 6.118 44.78 4668LM3 FEM 0.9361 6.118 44.79 4691LD3 FEM 0.9359 6.117 44.78 4614LM2 FEM 0.9268 6.111 44.78 4691LD2 FEM 0.9215 6.101 44.76 4672LM1 FEM 0.8763 6.044 44.68 4694LD1 FEM 0.8753 6.011 44.39 4651

An 0.8597 6.030 44.18 4552


Table VII. As in Table VI: equivalent single layer results.

ah 2 4 10 100

Exact 3D [49] — 6.11 — —EMZ3 FEM 1.102 6.332 44.73 4681EDZ3 FEM 1.111 6.346 45.01 4676EMZ2 FEM 1.130 6.433 45.14 4689EDZ2 FEM 1.149 6.462 45.15 4676EMZ1 FEM 1.195 6.108 41.52 4287EDZ1 FEM 1.149 6.136 41.43 4261


confirmed. The results merge in thin plate case a/h = 50. This fact confirms the effectiveness ofthe used technique [33] to contrast locking mechanisms.

7.2. Static analysis of a sensor plate

Multilayered plates constituting four layers have been considered. The two inner layers coincideto a cross-ply [0/90] carbon fibres and two external skins are made by piezoceramic materialPZT-4. The material properties are those in Table II. Three-dimensional solution were provided byHeyliger [49] The two composite layers have thickness hl = 0.4h while hl = 0.1h for the two skins.The unit value is assigned to the plate thickness. A bi-sinusoidal distribution of transverse pressurewith amplitude pz = 1 is applied to the top surface. The same electric boundary condition ofprevious section have been considered. Figure 8 shows the problem configuration. Tables VI–VIIIcompare the results of various elements to compute the electric potential. The same comparisonis made in Tables IX–XI for the transverse displacements uz . Four value of thickness ratio havebeen considered: a/h = 2, 4, 10, 100.



Table VIII. As in Table VI: equivalent single layer resultsincluding Murakami zig-zag function.

ah 2 4 10 100

Exact 3D [49] — 6.11 — —EM4 FEM 0.9607 6.144 44.73 4681ED4 FEM 0.9525 6.135 44.70 4667EM3 FEM 0.7170 5.822 44.31 4672ED3 FEM 0.7281 5.838 44.31 4659EM2 FEM 0.9870 6.233 44.79 4674ED2 FEM 1.013 6.273 44.81 4659EM1 FEM 0.7630 3.213 20.14 2047ED1 FEM 0.7521 2.664 15.15 1502


Table IX. Sensor piezolectric plate. Comparison of mixedand classical elements to evaluate transverse displacements

uz(a/2, b/2, 0) × 1011: layer-wise results.

ah 2 4 10 100

Exact 3D [49] — 30.03 — —LM4 FEM 4.947 30.27 587.1 4 717 200LD4 FEM 4.909 30.03 582.2 4 675 500LM3 FEM 4.952 30.27 587.1 4 717 200LD3 FEM 4.909 30.03 582.2 4 675 600LM2 FEM 4.928 30.23 587.0 4 717 200LD2 FEM 4.894 29.98 581.9 4 647 500LM1 FEM 4.761 30.16 587.8 4 719 600LD1 FEM 4.807 29.85 579.3 4 675 400

An 4.808 29.85 579.3 4 647 300


Table X. As in Table IX: equivalent single layer results.

ah 2 4 10 100

Exact 3D [49] — 30.03 — —EM4 FEM 4.564 28.97 579.4 4 715 900ED4 FEM 4.498 28.58 573.5 4 674 100EM3 FEM 4.713 28.96 578.2 4 715 600ED3 FEM 4.621 28.50 571.9 4 673 700EM2 FEM 3.039 21.54 530.5 4 710 900ED2 FEM 2.901 20.91 523.1 4 668 800EM1 FEM 3.145 19.96 446.0 3 830 200ED1 FEM 2.829 18.45 423.3 3 669 100

An 2.898 18.68 425.1 3 676 300




Table XI. As in Table IX: equivalent single layer resultsincluding Murakami zig-zag function.

ah 2 4 10 100

Exact 3D [49] — 30.03 — —EMZ3 FEM 4.487 28.91 579.3 4 716 000EDZ3 FEM 4.554 28.69 574.1 4 674 337EMZ2 FEM 2.881 21.26 529.5 4 711 188EDZ2 FEM 2.795 20.84 523.4 4 669 249EMZ1 FEM 3.028 20.78 506.0 4 481 929EDZ1 FEM 2.885 20.13 498.2 4 435 300

An 2.991 20.92 522.0 4 663 700


-0.4

-0.2

0

0.2

0.4

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

z

σzz

LD2

(a)

-0.4

-0.2

0

0.2

0.4

0 0.2 0.4 0.6 0.8 1 1.2

z

σzz

LM2

(b)

Figure 9. Piezolectric sensor plate a/h = 4. Transverse normal stress distribution �zz(a/2, b/2) vs z. Comparison between LD2 and LM2 elements: (a) �zz(a/2, b/2) for LD2

elements; and (b) �zz(a/2, b/2) for LM2 elements.

In order to give a complete assessment of various FEs different variables should be consid-ered as well as the distribution of these variables in the thickness direction. That is done inFigures 9–14. Transverse normal stress related to LM2 and LD2 are considered in Figure 9. Thefact that classical LD2 plate element leads to discontinuous transverse normal stress is to be no-ticed. Selected FEs are compared in Figure 10 to evaluate transverse displacement uz and in-planenormal stress �xx . Larger discrepancies among theories are experienced in the piezoelectric layers:this is due to the strong discontinuity of electro-mechanical properties between PZT-4 and Gr/Eplayers. Better evaluations are obtained for uz . More exhaustive comparison of various FEs toevaluate in-plane displacement, in-plane stress, transverse normal stress and electric potential isgiven in Figures 11–14, respectively. Table XII compares the results of different mixed elementsto compute the normal stress and respect the boundary condition. The following comments can bemade. LW solutions can provide quasi-three-dimensional description of displacement and stressfields as well as electrical variables in each layers. Larger discrepancies among the various theories



-0.4

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0

0.2

0.4

2.7e-10 2.75e-10 2.8e-10 2.85e-10 2.9e-10 2.95e-10 3e-10 3.05e-10 3.1e-10 3.15e-10 3.2e-10

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0

0.2

0.4

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0

0.2

0.4

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0

0.2

0.4

ED4LD4

EDZ3

EM4LM4

EMZ3

LD4ED4

EDZ3

LM4EM4

EMZ3

(a)-20 -15 -10 -5 0 5

-8 -6 -4 -2 0 2 4 6 8

10 15 20

2.7e-10 2.75e-10 2.8e-10 2.85e-10 2.9e-10 2.95e-10 3e-10 3.05e-10 3.1e-10 3.15e-10 3.2e-10 3.25e-10

(c) (d)

(b)

Figure 10. Sensor plate a/h = 4. Comparison of transverse displacement uz(a/2, b/2)and in-plane stress �xx (a/2, b/2) for significant, finite elements: (a) uz(a/2, b/2);

(b) �xx (a/2, b/2); (c) uz(a/2, b/2); and (d) �xx (a/2, b/2).

Table XII. Sensor piezolectric plate. Comparison of mixedelements to evaluate transverse normal stress �zz(a/2, b/2, h/2).

Selected layer-wise results.

ah 2 4 10 100

Exact 3D [49] 1.000 1.000 1.000 1.000LM4 FEM 0.997 1.003 1.003 1.227LM3 FEM 0.999 1.004 1.018 1.351LM2 FEM 1.019 1.011 1.009 1.253LM1 FEM 1.065 1.022 1.023 1.419


must be registered for the in-plane stresses. The advantage of using MZZF are confirmed: EMZC3,EMZC1 (e.g. EDZ3, EDZ1) results are much better than corresponding EM4 and EM1 (e.g. ED4,ED1) analyses. FEs based on RMVT lead to better description than the corresponding ones whichare formulated on the basis of PVD.



-0.4

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0

0.2

0.4

-6e-11 -4e-11 -2e-11 0 2e-11 4e-11 6e-11 8e-11

LM4LM1

3D

(a)

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0

0.2

0.4

-6e-11 -4e-11 -2e-11 0 2e-11 4e-11 6e-11 8e-11

LD4LD1

3D

(b)

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0

0.2

0.4

-6e-11 -4e-11 -2e-11 0 2e-11 4e-11 6e-11 8e-11

ED4ED1

3D

(d)

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0

0.2

0.4

-6e-11 -4e-11 -2e-11 0 2e-11 4e-11 6e-11 8e-11

EDZ3EDZ1

3D

(f)

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0

0.2

0.4

-6e-11 -4e-11 -2e-11 0 2e-11 4e-11 6e-11 8e-11

EMZ3EMZ1

3D

(e)

-0.4

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0

0.2

0.4

-6e-11 -4e-11 -2e-11 0 2e-11 4e-11 6e-11 8e-11

EM4EM1

3D

(c)

Figure 11. Sensor plate a/h = 4. Comparison of in-plane displacement uy(a/2, 0) vs z for various classicaland mixed finite elements. Three-dimensional exact solution is taken by Heyliger and Saravanos [48]:(a) LW mixed elements; (b) LW classical elements; (c) ESL mixed elements; (d) ESL classical elements;

(e) ESL mixed with MZZF; and (f) ESL classical elements with MZZF.

To complete the pictures of the implemented FEs, Table XIII compares the computational effortsrequired by a given element with respect to the fastest one which coincide to ED1 (which has6 d.o.f in each node). By taking the previous evaluation into account the following could be



-8 -6 -4 -2 0 2 4 6 8

LM4LM1

3D

LD4LD1

3D

ED4ED1

3D

EDZ3EDZ1

3D

EMZ3EMZ1

3D

EM4EM1

3D

(a)

(c)

(e)

(b)

(d)

(f)

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0

0.2

0.4

-8 -6 -4 -2 0 2 4 6 8

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0

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0.4

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0

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0

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0.4

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0

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0.4

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0

0.2

0.4

-8 -6 -4 -2 0 2 4 6 8 10 -20 -15 -10 -5 0 5 10 15 20

-8 -6 -4 -2 0 2 4 6 8 -10 -8 -6 -4 -2 0 2 4 6 8 10

Figure 12. Sensor plate a/h = 4. Comparison of in-plane stress �xx (a/2, b/2) vs z for various classical andmixed finite elements. Three-dimensional exact solution is taken by Heyliger and Saravanos [48]:(a) LW mixed elements; (b) LW classical elements; (c) ESL mixed elements; (d) ESL classical elements;

(e) ESL mixed elements with MZZF; and (f) ESL classical elements with MZZF.

concluded: if a very accurate evaluation of local distribution of variable is required the use of LWmodel, even though expensive, is required; EMZC3 analysis appears the most efficient element interm of obtained accuracy and computational efforts required.



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0

0.2

0.4

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

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0

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0

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0

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0

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0.4

LM4LM1

3D

LD4LD1

3D

EM4EM1

3D

ED4ED1

3D

EMZ3EMZ1

3D

EDZ3EDZ1

3D

(a) (b)

(c) (d)

(e) (f)

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

-4 -2 0 2 4 6 8 10 -25 -20 -15 -10 -5 0 5 10 15 20 25

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 -8 -6 -4 -2 0 2 4 6 8

Figure 13. Sensor plate a/h = 4. Comparison of transverse normal stress �zz(a/2, b/2)vs z for significant, classical and mixed finite elements. Three-dimensional exact solution istaken by Heyliger and Saravanos [48]: (a) LW mixed elements; (b) LW classical elements;(c) ESL mixed elements; (d) ESL classical elements; (e) ESL mixed elements with MZZF;

and (f) ESL classical elements with MZZF.

7.3. Static analysis of an actuator plate

The piezolectric plate used for the sensor case is reconsidered. The mechanical load is removedand a bi-sinusoidal distribution of electric potential with amplitude �z = 1 is applied as actuator



-0.4

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0

0.2

0.4

-0.001 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008

LM4LM1

3D

(a) (b)

(c) (d)

(e) (f)

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0

0.2

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-0.001 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008

LD4LD1

3D

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0

0.2

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-0.001 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008

EM4EM1

3D

-0.4

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0

0.2

0.4

-0.001 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008

π

ED4ED1

3D

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0

0.2

0.4

-0.001 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009

EMZ3EMZ1

3D

-0.4

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0

0.2

0.4

-0.001 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

EDZ3EDZ1

3D

Figure 14. Sensor plate a/h = 4. Comparison of electric potential �(a/2, b/2) vs z for various classical andmixed finite elements. Three-dimensional exact solution is taken by Heyliger and Saravanos [48]:(a) LW mixed elements; (b) LW classical elements; (c) ESL mixed elements; (d) ESL classical elements;

(e) ESL mixed elements with MZZF; and (f) ESL classical elements with MZZF.

on the upper surface, see Figure 15. For sake of conciseness, results are restricted to the plate withthickness ratio a/h = 4. Table XIV shows the obtained results. Though the thickness distributionof various variable has been plotted in Figures 16 and 17, which compares various FEs to evaluateelectrical potential and in-plane displacements, respectively. The conclusion already drawn for the



Figure 15. Geometry and boundary condition for the actuator problem.

Table XIII. Comparison of relative computation timesfor the various elements.

Mixed elements Classical elements

Element Timex/TimeED1 Element Timex/TimeED1

LM1 13.92 LD1 3.23LM2 72.96 LD2 15.18LM3 201.99 LD3 41.45LM4 489.77 LD4 88.90EM1 7.18 ED1 1.00EM2 30.07 ED2 2.92EM3 81.32 ED3 6.57EM4 166.50 ED4 12.48EMZ1 29.86 EDZ1 2.92EMZ2 79.76 EDZ2 6.60EMZ3 160.28 EDZ3 12.83

Table XIV. Results for the actuator piezoelectric plate. Comparison of various mechanical and electricalvariables; + denote values corresponding to z-positive.

FE type �(a/2, b/2, 0) × 101 v(a/2, 0,−h/2) × 1012 �yz(a/2, 0, 0+) × 102 �xy(0, 0, 0+) × 102

Exact 3D [49] 4.476 −2.8625 −2.3866 1.2286

LM4 4.480 (0.089%) −2.6590 (−7.109%) −2.0457 (−14.28%) 1.4770 (20.21%)LM3 4.480 (0.089%) −2.6600 (−7.074%) −2.4082 (0.905%) 1.4832 (20.72%)LM2 4.480 (0.089%) −2.6559 (−7.217%) −0.9942 (−58.34%) 1.3512 (9.979%)LM1 4.470 (−0.134%) −2.8652 (0.094%) 1.4902 (−162.4%) 1.4963 (21.78%)LD4 4.480 (0.089%) −2.6155 (−8.629%) −1.9452 (−18.49%) 1.5325 (24.73%)LD3 4.480 (0.089%) −2.6158 (−8.618%) −7.0506 (195.4%) 1.4554 (18.46%)LD2 4.481 (0.112%) −2.6082 (−8.884%) 3.5829 (−250.1%) 1.6332 (32.93%)LD1 4.469 (−0.156%) −3.0308 (5.879%) −0.9403 (−60.60%) 0.9240 (−24.70%)EM4 4.484 (0.179%) −1.9461 (−32.01%) 12.048 (−604.8%) 2.9091 (136.7%)EM1 4.461 (−0.335%) −3.5838 (25.19%) −40.630 (1602.%) −2.5329 (−306.1%)ED4 4.484 (0.179%) −1.2422 (−56.60%) 13.119 (−649.6%) 3.0417 (147.5%)ED1 4.462 (−0.313%) −4.3550 (52.14%) 56.882 (−2483.%) −3.0549 (−348.6%)EMZ3 4.484 (0.179%) −2.6814 (−6.327%) 11.168 (−567.9%) −0.4796 (−139.0%)EDZ3 4.484 (0.179%) −2.1441 (−25.09%) 9.2168 (−486.1%) −0.8003 (−165.1%)



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0

0.2

0.4

-0.2 0 0.2 0.4 0.6 0.8 1π

LM4LM1

3D

(a) (b)

(c) (d)

(e) (f)

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0

0.2

0.4

-0.2 0 0.2 0.4 0.6 0.8 1π

LD4LD1

3D

-0.4

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0

0.2

0.4

-0.2 0 0.2 0.4 0.6 0.8 1π

EM4EM1

3D

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0

0.2

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ED4ED1

3D

-0.4

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0

0.2

0.4

-0.2 0 0.2 0.4 0.6 0.8 1π

EMZ3EMZ1

3D

-0.4

-0.2

0

0.2

0.4

-0.2 0 0.2 0.4 0.6 0.8 1π

EDZ3EDZ1

3D

Figure 16. Actuator plate. Comparison of electric potential �(a/2, b/2) vs z for various classical andmixed finite elements. Three-dimensional exact solution is taken by Heyliger and Saravanos [48]: (a) LWmixed elements; (b) LW classical elements; (c) ESL mixed elements; (d) ESL classical elements; (e) ESL

mixed elements with MZZF; and (f) ESL classical elements with MZZF.

sensor case analysis is confirmed. However, large discrepancies among different theories shouldbe registered with respect to the sensor cases. It is concluded that FEs accuracy can be influencedby boundary conditions.



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0

0.2

0.4

-3.5e-11 -3e-11 -2.5e-11 -2e-11 -1.5e-11 -1e-11 -5e-12 0 5e-12 1e-11v

LM4LM1

3D

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0

0.2

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-3.5e-11 -3e-11 -2.5e-11 -2e-11 -1.5e-11 -1e-11 -5e-12 0 5e-12v

LD4LD1

3D

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0

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-3.5e-11 -3e-11 -2.5e-11 -2e-11 -1.5e-11 -1e-11 -5e-12 0 5e-12 1e-11v

EM4EM1

3D

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0

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-3.5e-11 -3e-11 -2.5e-11 -2e-11 -1.5e-11 -1e-11 -5e-12 0 5e-12 1e-11v

ED4ED1

3D

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0

0.2

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-3.5e-11 -3e-11 -2.5e-11 -2e-11 -1.5e-11 -1e-11 -5e-12 0 5e-12 1e-11v

EMZ3EMZ1

3D

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-3.5e-11 -3e-11 -2.5e-11 -2e-11 -1.5e-11 -1e-11 -5e-12 0 5e-12 1e-11v

EDZ3EDZ1

3D

(a) (b)

(c) (d)

(e) (f)

Figure 17. Actuator plate. Comparison of in-plane displacement uy(a/2, 0) vs z for various classical andmixed finite elements. Three-dimensional exact solution is taken by Heyliger and Saravanos [48]: (a) LWmixed elements; (b) LW classical elements; (c) ESL mixed elements; (d) ESL classical elements; (e) ESL

mixed elements with MZZF; and (f) ESL classical elements with MZZF.

8. CONCLUDING REMARKS

The paper has extended the UF and RMVT to develop FEs for the static and dynamic analysis ofpiezolectric plates. The following main conclusion have been outlined.



1. It has been confirmed that UF consists of a valuable tool to provide hierarchical analysis ofpiezolectric plates by FE method. The implemented FEs, in fact, can provide very accuratedescriptions of mechanical and electrical fields.

2. RMVT appears as a very valuable tools to introduce ‘a priori’ the requested continuityconditions for mechanical and electrical variables.

Future works should be directed to consider the analysis of piezolectric plate with localized patchesas sensors and or actuators for their application to control. Other plates lay up, as well as theeffect of additional boundary conditions and geometries should be examined. A further possibleextension should include the fact that the transverse components of electrical displacement hasassumed unknowns.

APPENDIX A: EXPLICIT EXPRESSION OF PVD FUNDAMENTAL NUCLEI

The explicit forms of the fundamental nuclei related to PVD applications are listed below.

• Kk�si juu :

Kk�si juu =

⎡⎢⎢⎣Kuu11 Kuu12 Kuu13

Kuu21 Kuu22 Kuu23

Kuu31 Kuu32 Kuu33

⎤⎥⎥⎦ (A1)

Its elements are:

Kuu11 =C55�Ni N j��k Euu�,zs,z + C11�Ni,x N j,x��k E

uu�s

+C16�Ni,x N j,y��k Euu�s + C16�Ni,y N j,x��k E

uu�s

+C66�Ni,y N j,y��k Euu�s

Kuu12 =C45�Ni N j��k Euu�,zs,z + C16�Ni,x N j,x��k E

uu�s

+C12�Ni,x N j,y��k Euu�s + C66�Ni,y N j,x��k E

uu�s

+C26�Ni,y N j,y��k Euu�s

Kuu13 =C13�Ni,x N j��k Euu�s,z + C36�Ni,y N j��k E

uu�s,z

+C55�Ni N j,x��k Euu�,zs + C45�Ni N j,y��k E

uu�,zs

Kuu21 =C45�Ni N j��k Euu�,zs,z + C12�Ni,y N j,x��k E

uu�s

+C26�Ni,y N j,y��k Euu�s + C16�Ni,x N j,x��k E

uu�s

+C66�Ni,x N j,y��k Euu�s (A2)




uu�s


uu�s

+C26�Ni,x N j,y��k Euu�s


uu�s,z


uu�,zs


uu�s,z


uu�,zs


uu�s,z


uu�,zs


uu�s


uu�s

+C45�Ni,x N j,y��k Euu�s

• Kk�si jue :

Kk�si jue =

⎡⎢⎣Kue11

Kue21

Kue31

⎤⎥⎦ (A3)

Its elements are:

Kue11 = e15�Ni N j,x��k Eue�,zs + e25�Ni N j,y��k E

ue�,zs

+ e31�Ni,x N j��k Eue�s,z + e36�Ni,y N j��k E

ue�s,z

Kue21 = e14�Ni N j,x��k Eue�,zs + e24�Ni N j,y��k E

ue�,zs

+ e36�Ni,x N j��k Eue�s,z + e32�Ni,y N j��k E

ue�s,z (A4)

Kue31 = e15�Ni,x N j,x��k Eue�s + e14�Ni,y N j,x��k E

ue�s

+ e25�Ni,x N j,y��k Eue�s + e24�Ni,y N j,y��k E

ue�s

+ e33�Ni N j��k Eue�,zs,z



• Kk�si jeu :

Kk�si jeu = [Keu11 Keu12 Keu13] (A5)

Its elements are:

Keu11 = e15�Ni,x N j��k Eeu�s,z + e25�Ni,y N j��k E

eu�s,z

+ e31�Ni N j,x��k Eeu�,zs + e36�Ni N j,y��k E

eu�,zs

Keu12 = e14�Ni,x N j��k Eeu�s,z + e24�Ni,y N j��k E

eu�s,z

+ e36�Ni N j,x��k Eeu�,zs + e32�Ni N j,y��k E

eu�,zs (A6)

Keu13 = e15�Ni,x N j,x��k Eeu�s + e14�Ni,x N j,y��k E

eu�s

+ e25�Ni,y N j,x��k Eeu�s + e24�Ni,y N j,y��k E

eu�s

+ e33�Ni N j��k Eeu�,zs,z

• Kk�si jee nuclei is a scalar quantity:

Kee11 = −ε11�Ni,x N j,x��k Eee�s − ε12�Ni,x N j,y��k E

ee�s

− ε21�Ni,y N j,x��k Eee�s − ε22�Ni,y N j,y��k E

ee�s

− ε33�Ni N j��k Eee�,zs,z (A7)

• Mk�si juu is a matrix:

Mk�si juu =

⎡⎢⎣Muu11 0 0

0 Muu22 0

0 0 Muu33

⎤⎥⎦ (A8)

Muu11 = �k�Ni N j��k Euu�s



(A9)

• Kk�si jup :

Kk�si jup =

⎡⎢⎣Kup11 0 0

0 Kup22 0

0 0 Kup33

⎤⎥⎦ (A10)



Its elements are:

Kup11 =mxks�Ni N j��k E

1�s

Kup21 =myks�Ni N j��k E

1�s

Kup31 =mzks�Ni N j��k E

1�s

(A11)

• Kk�si ju f nuclei is a scalar quantity:

Kef 11 = nks�Ni N j��k E1�s (A12)

Where the symbol � . . . ��k denote a integration on the element domain �k .It must be noted that:

Kk�si jeu =Kk�si j

ueT

(A13)

APPENDIX B: EXPLICIT EXPRESSION OF RMVT FUNDAMENTAL NUCLEI

The fundamental nuclei related to RMVT applications are listed below.

• Kk�si juu :

Kk�si juu =

⎡⎢⎢⎢⎣Kuu11 Kuu12 Kuu13

Kuu21 Kuu22 Kuu23

Kuu31 Kuu32 Kuu33

⎤⎥⎥⎥⎦ (B1)

Its elements are:

Kuu11 = C11�Ni,x N j,x��k Euu�s + C16�Ni,y N j,x��k E

uu�s

+ C16�Ni,x N j,y��k Euu�s + C66�Ni,y N j,y��k E

uu�s


uu�s


uu�s

Kuu13 = 0


uu�s


uu�s (B2)




uu�s


uu�s

Kuu23 = 0

Kuu31 = 0

Kuu32 = 0

Kuu33 = 0

• Kk�si ju� :

Kk�si ju� =

⎡⎢⎣Ku�11 Ku�12 Ku�13

Ku�21 Ku�22 Ku�23

Ku�31 Ku�32 Ku�33

⎤⎥⎦ (B3)

Its elements are:

Ku�11 = �Ni N j��k Eu��,zs

Ku�12 = 0

Ku�13 = C13�Ni,x N j��k Eu��s + C36�Ni,y N j��k E

u��s

Ku�21 = 0


Ku�23 = C36�Ni,x N j��k Eu��s + C23�Ni,y N j��k E

u��s

Ku�31 = �Ni,x N j��k Eu��s

Ku�32 = �Ni,y N j��k Eu��s


(B4)

• Kk�si jue nuclei:

Kk�si jue =

⎡⎢⎢⎢⎣Kue11

Kue21

Kue31

⎤⎥⎥⎥⎦ (B5)



Its elements are:

Kue11 = −Cse13�Ni,x N j��k Eue�s,z − Cse33�Ni,y N j��k E

ue�s,z

Kue21 = −Cse23�Ni,y N j��k Eue�s,z − Cse33�Ni,x N j��k E

ue�s,z

Kue31 = 0

(B6)

• Kk�si j�u :

Kk�si j�u =

⎡⎢⎢⎣K�u11 K�u12 K�u13

K�u21 K�u22 K�u23

K�u31 K�u32 K�u33

⎤⎥⎥⎦ (B7)

Its elements are:

K�u11 = �Ni N j��k E�u�s,z

K�u12 = 0

K�u13 = �Ni N j,x��k E�u�s

K�u21 = 0


K�u23 = �Ni N j,y��k E�u�s

K�u31 = + C13�Ni N j,x��k E�u�s + C36�Ni N j,y��k E

�u�s

K�u32 = + C36�Ni N j,x��k E�u�s + C23�Ni N j,y��k E

�u�s


(B8)

• Kk�si j�� :

Kk�si j�� =

⎡⎢⎢⎣K��11 K��12 K��13

K��21 K��22 K��23

K��31 K��32 K��33

⎤⎥⎥⎦ (B9)

Its elements are:

K��11 = − C55�Ni N j��k E��s

K��12 = − C45�Ni N j��k E��s

K��13 = 0



K��21 = − C45�Ni N j��k E��s

K��22 = − C44�Ni N j��k E��s

K��23 = 0

K��31 = 0

K��32 = 0

K��33 = − C33�Ni N j��k E��s (B10)

• Kk�si j�e :

Kk�si j�e =

⎡⎢⎢⎣K�e11

K�e21

K�e31

⎤⎥⎥⎦ (B11)

Its elements are:

K�e11 = + Cde22�Ni N j,y��k E�e�s + Cde21�Ni N j,x��k E

�e�s

K�e21 = + Cde11�Ni N j,x��k E�e�s + Cde12�Ni N j,y��k E

�e�s

K�e31 = + Cde33�Ni N j��k E�e�s,z

(B12)

• Kk�si jeu :

Kk�si jeu =[Keu11 Keu12 Keu13] (B13)

Its elements are:

Keu11 = Ced31�Ni N j,x��k Eeu�,zs + Ced33�Ni N j,y��k E

eu�,zs

Keu12 = Ced32�Ni N j,y��k Eeu�,zs + Ced33�Ni N j,x��k E

eu�,zs

Keu13 = 0

(B14)

• Kk�si je� :

Kk�si je� =[Ke�11 Ke�12 Ke�13] (B15)

Its elements are:

Ke�11 = Ces12�Ni,x N j��k Ee��s + Ces22�Ni,y N j��k E

e��s

Ke�12 = Ces11�Ni,x N j��k Ee��s + Ces21�Ni,y N j��k E

e��s

Ke�13 = Ces33�Ni N j��k Ee��,zs

(B16)



• Kk�si jee :

Kee11 = − Cee11�Ni,x N j,x��k Eee�s − Cee12�Ni,x N j,y��k E

ee�s

− Cee21�Ni,y N j,x��k Eee�s − Cee22�Ni,y N j,y��k E

ee�s

− Cee33�Ni N j��k Eee�,zs,z (B17)

• Mk�si juu :

Mk�si juu =

⎡⎢⎢⎣Muu11 0 0

0 Muu22 0

0 0 Muu33

⎤⎥⎥⎦ (B18)




(B19)

• Kk�si jup :

Kk�si jup =

⎡⎢⎢⎣Kup11 0 0

0 Kup22 0

0 0 Kup33

⎤⎥⎥⎦ (B20)

Its elements are:

Kup11 =mxks�Ni N j��k E

1�s

Kup21 =myks�Ni N j��k E

1�s

Kup31 =mzks�Ni N j��k E

1�s

(B21)

• Kk�si je f nuclei is a scalar quantity:

Kef 11 = nks�Ni N j��k E1�s (B22)

ACKNOWLEDGEMENTS

This work has been carried out in the framework of STREP EU project CASSEM under contractNMP-CT-2005-013517.

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