A nonlinear formulation of piezoelectric plates

Article

Journal of Intelligent Material Systemsand Structures0(0) 1–11� The Author(s) 2012Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1045389X12447292jim.sagepub.com

A nonlinear formulation ofpiezoelectric plates

Michele Pasquali and Paolo Gaudenzi

AbstractA nonlinear piezoelectric plate model capable of accurately expressing the direct and converse piezoelectric effects ispresented. The developed semi-intrinsic theory is meant to encompass large strains, displacements, and rotations thatcan occur in the plate as a consequence of a complete coupling between the mechanical and the electrical fields. Basedon the assumption that a linear dependence of the electric potential on the plate thickness is not adequate to representthe potential electric energy, a specific structure to the field of admissible displacement is taken into account. A non-linear regression technique is successively developed to detect the nonlinear dependence of the obtained solutions onthe through-the-thickness coordinate. Particular warping functions characterized by the use of an ad hoc polynomialexpansion adopted to express their dependence on the plate thickness direction are here considered to describe theshear and the extensional deformability of the plate transverse fibers. Linear constitutive relations for a transversely iso-tropic continuum are considered. The governing equations of motions for the model are finally obtained.

Keywordspiezoelectricity, nonlinear plate models, semi-intrinsic plate theories

Introduction

The use of piezoelectric materials as sensors or actua-tors (or both) for several applications has induced aconsiderable interest in the modeling of piezoelectricplates (Gaudenzi, 2009). Several attempts are alsobeing made to model the presence of piezoelectriclayers in the frame of laminated plates and shells, asrecently reported in a review article (Wang and Yang,2000). The exact solutions of laminated piezoelectricplates can be obtained analytically for a few cases ofideal material type, geometry, and boundary conditions(Heyliger and Brooks, 1996; Ray et al., 1992; Vel andBatra, 2001). Numerous two-dimensional (2D) platemodels have been constructed in order to simplify theanalysis of piezoelectric plates, which generally startwith some assumption about the through-the-thicknessdistribution of the three-dimensional (3D) field quanti-ties. Piezoelectric plate models can be classified asequivalent single-layer models (Crawley and Lazarus,1991; Detwiler et al., 1995; Dimitriadis et al., 1991; Haet al., 1992; Hong and Chopra, 1999; Ray et al., 1994;Wang and Rogers, 1991; DiCarlo et al, 2001) if theassumptions are applied to the entire structure andlayer-wise models (Icardi and Sciuva, 1996; Pai et al.,1993; Saravanos et al., 1997; Zhou et al., 2000) if theassumptions are applied to each layer. It is not unusual

to find mixed models in which single-layer theory ofmechanical field is combined with layer-wise approxi-mation of the electric potential (Cen et al., 2002;Kapuria and Achary, 2005, 2006; Sheikh et al., 2001).In many plate models, only geometric nonlinearities ala Von Karman are taken into account, while a quadra-tic distribution for the electric potential in the thicknessdirection is considered (Behjat and Khoshravan, 2012).In other studies (Liao and Yu, 2009), no distribution ofthe electric field in the through-the-thickness directionis assumed, and variational asymptotic method is usedto construct a generalized Reissner–Mindlin piezoelec-tric plate model.

In many cases, only uncoupled models are proposedin which the sensing and the actuation of the piezoelec-tric plates are treated in a separate fashion, with stronglimiting hypotheses on the physical behavior of thematerial and the structure. A piezoelectric plate model

Department of Mechanical and Aerospace Engineering, Sapienza

University of Rome, 00184 Rome, Italy

Corresponding author:

Michele Pasquali, Department of Mechanical and Aerospace Engineering,

Sapienza University of Rome, 00184 Rome, Italy.

Email: [email protected]

at Dip Teoria Dello Stato on July 2, 2012jim.sagepub.comDownloaded from

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has been presented in Vidoli and Batra (2000), wherethe kinematical constraints have been related only tothe global geometry of the plate.

The model of piezoelectric plates here proposed aimsat properly modeling the coupled piezoelectric behaviorin the frame of the plate geometry. For these reasons,electrical state variables have been included in themodel and the assumptions on the distributions of themechanical and electrical quantities are verified basedon the coherency between the geometry of the localmaterial response and the global geometry. InGaudenzi and Bathe (1995), a quadratic variation ofthe electric potential through the thickness is obtained,confirming that, as also reported in Rahmoune et al.(1998), the assumption of a linear variation of the elec-tric potential along the plate thickness would result inneglecting a substantial part of the induced potential.In this sense, the results presented in Roccella andGaudenzi (2005) are extended to the nonlinear case,performing a numerical study of the response of thesystem to changes of the model geometry and of theboundary conditions, in terms of through-the-thicknessvariations of the electric potential. The obtained resultsconstitute the starting point of the derived nonlinearplate model: in particular, they are used to properlytruncate the polynomial expansion adopted to repre-sent the electric potential in the plate, so as to guaran-tee complete coupling between the mechanical and theelectrical phenomena.

2D linear piezoelectric flexural problem

Consider a rectangular strip of piezoceramics occupyingthe region xj j<l, zj j<h of a 2D space, with global basis(e1, e3) as shown in Figure 1. The piezoelectric materialhas been polarized along the thickness, that is, alongthe z-direction. The governing equations are those char-acterizing the linear theory of 2D piezoelectricity forthe steady-state case, which are reported in Ikeda et al.(1990). Assuming the following boundary conditions

f=6V0, s33 = t13 = 0 at z=6h

D = 0, s11 =s0 +s1z, t13 = 0 at x=6l

�ð1Þ

where f is the electric potential, D the dielectric displa-cement and s11, s33, t13, the normal and the shearstresses of the Cauchy stress tensor, respectively, aclosed-form solution for the problem in terms of elec-tric potential f and displacements u1 and u3 in the x-and z-directions can be found (Gaudenzi and Bathe,1995) as follows

f=V0

z

h� d31s1

2ε33

h2 � z2� �

ð2Þ

u1 = s11 s0 �d31V0

s11h

� �x+ s11 1� d2

31

s11ε33

� �s1xz ð3Þ

u3 = s13 s0 �d33V0

s13h

� �z+ s13 1� d31d31

s13ε33

� �s1

z2

2

�s11 1� d231

s11ε33

� �s1

x2

2ð4Þ

These results can be considered to develop a linearpiezoelectric plate model capable of assuring a com-plete coupling between the mechanical and the electri-cal fields, as shown in Roccella and Gaudenzi (2005);adopting the classical hypotheses of linear constitutiverelations and ‘‘small’’ deformations from a stress-freeplacement (Roccella and Gaudenzi, 2005), the follow-ing structure for the linearized Green–Lagrange straintensor can be found

e= e0 + e1

z

2h

g = g0 + g1

z

2h+ g2

z2

4h2

E = E0 + E1

z

2hð5Þ

where e is the deformation in the plate thickness direc-tion, g is the vector of the out-of-plane shear strains,and E is the restriction of the Green–Lagrange straintensor to the subspace of the plate membrane.

Analyzing the obtained deformation field, it isapparent that the classical Kirchhoff–Love plate mod-els are not adequate to guarantee a complete electrome-chanical coupling for the structure, leading tosubstantial misrepresentation of the piezoelectricphenomena.

2D nonlinear piezoelectric flexural problem

The considerations made in Roccella and Gaudenzi(2005) can be extended to the nonlinear case in order toderive a proper representation of the electric potential,so as to develop a nonlinear piezoelectric plate modelcapable of fully expressing the electromechanical cou-pling effect. In so doing, a complete expression for the

l

h

z

x

σ11= σ0+ σ1 z Φ = V0

Φ = −V0

e1

e3

Figure 1. The 2D piezoelectric problem.

2 Journal of Intelligent Material Systems and Structures 0(0)



Green–Lagrange deformation tensor in the global basisis taken into account

E=1

2ruT +ru� �

+1

2ruruT ð6Þ

where the u vector represents the displacements u and win the x- and z-directions, respectively. The electric fieldis defined as

H= �rf ð7Þ

As done for the linear problem, linear constitutiveequations are considered (Roccella and Gaudenzi,2005)

B =CE� eTHD= eE+ εH

�ð8Þ

where B is the second Piola–Kirchhoff stress tensor, C isthe elastic compliance tensor, e is the electromechanicalcoupling tensor, ε is the tensor of dielectric constants,and D is the vector of electric displacements. All thesequantities are expressed in the global basis (e1, e3). Theequilibrium is imposed on the deformed configurationand, making use of the deformation gradient tensor F, itis rewritten in the reference configuration domain byintroducing the first Piola–Kirchhoff stress tensor T

F= rp½ �T

T =FB ð9Þ

where p is the position vector of a point within the sys-tem. The equilibrium equation for the mechanical partof the model can be written as

Tij, j + fi = 0 ð10Þ

where fi are the external forces per unit volume. Theequilibrium equation for the electrical part can be writ-ten asððð

~L

r �Dd ~L= 0!ðð

~S

D � ~nd~S =

ðð~S

D � d~S= 0 ð11Þ

where the ‘‘;‘‘ symbol indicates quantities referred tothe deformed configuration. The following relationshipbetween infinitesimal area portions holds

d~S= det Fð Þ F�1� �T

dS ð12Þ

The previous integral can then be written asðð~S

D � d~S=

ððS

D � det Fð Þ F�1� �T� �

dS

=

ððS

det Fð ÞF�1D � dS=

ððS

G � dS= 0 ð13Þ

The vector G of the electric displacement in the refer-ence configuration can be thus expressed as

G= det Fð ÞF�1D ð14Þ

so that the equilibrium equation for the electric part ofthe model is

Gi, i = 0 ð15Þ

The boundary conditions of the nonlinear piezoelec-tric problem are the same as those considered in the lin-ear case except for the linear distribution of s11

imposed at x = 6l.Due to the definition given in equation (6), the

strains in E, and consequently the stresses expressed inB, have a quadratic dependence on the displacementgradient ru: this means that given a polynomial expan-sion in z of the displacements, the strains and the stres-ses of the second Piola–Kirchhoff tensor show adependence on the z-coordinate of the form

a0 + a1z+ a2z2 + � � � + anzn ð16Þ

where n is even. Because of their definition (9), it isapparent that the stresses of the first Piola–Kirchhofftensor share a similar structure, preventing the compo-nent T11 to assume a linear dependence on z at x = 6l.This problem is overcome by imposing the displace-ment u1 provided by the closed-form solution of the lin-ear problem at x = 6l instead of the stress s11.

Numerical analysis of the 2D linear and nonlinearpiezoelectric models

The linear as well as the nonlinear 2D theoretical mod-els introduced in the previous sections are numericallysolved with the software COMSOL Multiphysics�. Aconvergence analysis for the displacements u1 and u3 aswell as for the electric potential f is performed; in par-ticular, the study of the punctual convergence and theconvergence in norm of the said quantities are carriedout. A number of 1600 square four-node finite elementsfor the nondimensional model (h* = l* = 1) in Figure2 is found to be a good compromise between the solu-tion accuracy and the computational cost for both thelinear and the nonlinear cases.

The nondimensional electric potential f� ¼ f=V0

for the linear and the nonlinear models is numericallyestimated at 100 points on the z-axis. The obtained dataare interpolated with a nonlinear regression method,increasing progressively the order of the polynomialfunction adopted in the interpolation process. The resi-dual sum of squares is estimated for each order of inter-polation. In Figures 3 and 4, the percent variation ofthe estimated sum of residual squares is shown for thelinear and the nonlinear 2D piezoelectric problems.

Pasquali and Gaudenzi 3



For the linear model (Figure 3), a decrement ofabout 100% in the value of the residual sum of squaresis observed considering a linear interpolating functioninstead of a constant-value function. A similar behaviorcan be noted when passing from a linear to a quadraticinterpolation. As expected, further increases in theorder of the adopted regression function do not implysignificant changes in the values of the sum of residualsquares. This indicates that the functional dependenceof the electric potential on the z-coordinate is of the sec-ond order, as shown in equation (2).

For the nonlinear model (Figure 4), significantdecrements in the value of the residual sum of squarescan be noted (except for the fifth-order case) until asixth-order interpolation function is adopted, eviden-cing a sixth-order functional dependence of f* on thez-coordinate.

An analysis of the dependence of the nondimen-sional electric potential f* on the aspect ratio h = h/land on the load multiplier l of the stress s11 imposed at

x = 6l is also carried out for the linear and the non-linear cases (in the reference configuration h = 0.5 and l

= 1). The variation of the nondimensional potential f*along z* = z/h for different values of h and l is shownin Figure 5 for the linear model. The same investigationsare conducted for the nonlinear model: the results areillustrated in Figure 6. It is apparent that an increase inthe value of h and l results in a more evident nonlinearcharacter of the electric potential on the z*-coordinatefor both the linear and the nonlinear models. Thiseffect is more important when an increase in the value ofh rather than in the value of l is considered. More quan-titative conclusions can be drawn when analyzing theplot in Figure 7, where the percent difference betweenthe estimated value of f* at the point (0,0) for the linearand the nonlinear models considering different values ofh and l, respectively, is reported. In this case, major dif-ferences between the linear and the nonlinear models areobservable for low values of the h parameter. In anycase, the said differences remain below 0.2% for varia-tions of h and below 0.5% for variations of l. The varia-tion of nondimensional electric potential f* along z*for the linear and the nonlinear cases (h = 0.5 and l =1) is reported in Figure 8. The differences between thetwo numerical solutions remain small not only in per-cent, as shown in Figures 6 and 7, but also in absolutevalue.

The results obtained from the analyses of the differ-ences between the estimated electric potential f for thelinear and the nonlinear models can thus be summar-ized as follows:

� The electric potential f shows a higher orderdependence on z in the nonlinear model rather

Figure 3. Residual sum of squares (D%) for f* in the linear 2Dpiezoelectric model.

Figure 4. Residual sum of squares (D%) for f* in the nonlineartwo-dimensional piezoelectric model.

Figure 2. The 2D nondimensional model of plate.




than in the linear model (sixth order rather thansecond).

� The differences in the electric potential valuesestimated in the linear and the nonlinear modelsremain small from both a percent and an abso-lute point of view.

These considerations lead to the assumption that,for the nonlinear problem, a representation of the elec-tric potential f with a second-order dependence on z,as it holds for the linear problem, is also adequate as ithas been noted that the higher order terms do not

contribute in a substantial way to the definition of f.This means that also for a nonlinear model of piezo-electric plate, a quadratic variation of the electricpotential f with the z-coordinate has to be consideredas a starting point to develop a model capable ofexpressing a complete coupling between electrical andmechanical phenomena.

A nonlinear model of piezoelectric plate

The considerations made in the previous section aboutthe dependence of the electric potential f on the z-

Figure 5. Nondimensional electric potential f* in the linear model for different values of h and l.

Figure 6. Nondimensional electric potential f* in the nonlinear model for different values of h and l.




coordinate in the 2D case when a flexural load is takeninto account are here used to develop a nonlinearmodel of plate capable of encompassing large strains,displacements and rotations that can occur in the plateas a consequence of a complete coupling between themechanical and the electrical fields (Lacarbonara andPasquali, 2011).

The kinematics of the model are rooted into thebasic idea of representing the field of displacements ofthe plate as constituted by two contributions. The firstone is due to the motion of the transverse fiber, whichremains normal to the middle surface of the plate

during the deformation process, undergoing to a rigidrotation about an axis which lays in the plane tangen-tial to deformed middle surface of the plate at the inter-section with the fiber. The other contribution isrepresented by three warping functions that describethe deformation of the fiber and whose dependence onthe plate thickness direction is expressed by a polyno-mial expansion, successively truncated to guarantee acomplete electromechanical coupling.

Starting from these assumptions, the balance equa-tions are subsequently derived from the virtual worktheorem.

Figure 7. Percent difference in the f* value estimated at (0,0) in the linear and the nonlinear models for different values of h and l.

Figure 8. Nondimensional electric potential f* for the linear (solid line) and the nonlinear models (dashed line).




Kinematics of the plate

The geometry of the undeformed (stress-free) config-uration is described by the position vector of the mate-rial points of transversal fibers collinear with unitvector b03, with b03 [ e3, where (e1, e2, e3) is the globalframe reference. We let

r0 = x1e1 + x2e2 ð17Þ

be the position vector of the material points of a baseplane O of the reference configuration (see Figure 9).The position vector of the material points of the fiberthrough r0(x1, x2) is then represented by

r = r0 + ze3 = x1e1 + x2e2 + ze3 ð18Þ

where z [ x3 denotes the position along the fiber and z2 I = [2h, + h]. The subscript 0 characterizes allquantities referred to the base plane O.

The basis (b01, b02, b03) is a local basis employed toidentify material fibers collinear with the coordinatelines in the base plane as well as those collinear withthe thickness direction. The reference configuration(Figure 9) of the plate is thus

L= r (x1, x2, z)= r0 + zb03, r0 = x1e1 + x2e2;

x1 2 0, a½ �, x2 2 0, b½ �, z 2 Ig ð19Þ

The boundary of the reference configuration isdenoted by ∂L = O 3 I and is assumed to be Lipschitz-continuous. Moreover, we set S= ∂L [ O|+ h U O|2h.

We let p0 and p be the position vectors in the actualconfiguration of the material points that occupy posi-tions r0 and r in the reference configuration as follows

p0 = r0 + u0 = x1 + u01ð Þe1 + x2 + u02ð Þe2 + u03e3

ð20Þ

p= p0 + zb3 +w1e1 +w2e2 +w3e3 ð21Þ

where u0(x1, x2) represents the displacement vector ofthe material point r0 and wi the functions expressing thedeformation of the transverse fibers. We can thus definethe displacement vector u as

u= p� r= u01 + zb13 +w1ð Þe1 + u02 + zb23 +w2ð Þe2

+ u03 + z b33 � 1ð Þ+w3ð Þe3 ð22Þ

We further denote by b3 the unit vector collinear tothe thickness-wise fiber before it undergoes to the defor-mations expressed by the warping functions w1, w2, andw3, and passing through r0 in O as follows

b3 =n013n02

n013n02j j ð23Þ

where n01 and n01 are the stretch vectors of the baseplane at p0 given by

n01 =∂p0

∂x1

= 1+ u01, x1ð Þe1 + u02, x1

e2 + u03, x1e3 ð24Þ

n02 =∂p0

∂x2

= u01, x2e1 + 1+ u02, x2

ð Þe2 + u03, x2e3 ð25Þ

The current configuration of the plate is thusdescribed by

~L= p x1, x2, zð Þ= p0 + zb3 +w1 e1 +w2e2 +w3e3;fx1 2 0, a½ �, x2 2 0, b½ �, z 2 Ig ð26Þ

It is convenient to introduce the unit vectors b1 andb2, which, together with b3, make a suitable local basisfor the current configuration. They can be defined bythe following set of equations

b13b2 = b3; b1j j= 1; b2j j= 1;n01

n01j j � b2 =n02

n02j j � b1

ð27Þ

The relationship between the global frame (e1, e2, e3)and the local basis (b1, b2, b3) can be expressed by theorthogonal tensor R as follows

R= bij ei � ej, i= 1, 2, 3 ð28Þ

where the summation convention for repeated indices isadopted, with

b i =Re i, i= 1, 2, 3:

~y =RTy ð29Þ

where y is a generic vector and the ‘‘;‘‘ symbol indi-cates that it is expressed in the local reference frame.

The deformation gradient tensor F is defined as inequation (9) or using the dyadic product �

F=Fijei � ej, i, j= 1, 2, 3 ð30Þ

where

Fij =∂pi

∂xj

= nij ð31Þ

e3 e2

e1O

r0

p0b03 b02

b01

b1

b2b3

u0

Ω∂

Figure 9. A schematic of the plate in its referenceconfiguration.




For admissible deformations, it is required that detF . 0. In order to proceed with an easier descriptionof the constitutive behavior of the system, which isintroduced in the followings, it is useful to rewrite theF tensor in the local basis as

~F=Fij~ei � ~ej =FijRTe i � RTej

=FijRT~bi � RT~bj = ~Fij

~bi � ~bj ð32Þ

so that

~Fij =Fijbihbjk ð33Þ

The Green–Lagrange deformation tensor is thus

~E=1

2FTF� I�

ð34Þ

or, in componential form

~Eij =1

2FkhbkibhjFhkbhibkj � dij

� �ð35Þ

Electromechanical coupling

The previously introduced warping functions wi, whichdescribe the deformation of the transverse fibers, areexpressed by a polynomial expansion in the z-coordinate

w1 = zw11 + z2w21 + � � � + zlwl1

w2 = zw12 + z2w22 + � � � + zmwm2

w3 = zw13 + z2w23 + � � � + znwn3

8<: ð36Þ

Denoting with |z the order of dependence on the z-coordinate and considering equations (29), (30), and(35), we have

~n1jz = n1jz = max l,m, n½ �~n2jz = n2jz = max l,m, n½ �~n3jz = n3jz = max l � 1,m� 1, n� 1½ � ð37Þ

~e11jz = ~e22jz = ~e12jz = e11jz = e22jz = e12jz= max 2l, 2m, 2n½ �

~e13jz=~e23jz=e13jz=e23jz=max 2l�1,2m�1,2n�1½ �

ee33jz ¼ e33jz ¼ max½2l � 2; 2m� 2; 2n� 2� ð38Þ

It can be observed that, for given values of the para-meters l, m, and n, the order of the dependence of thequantities reported in equations (38) and (39) is deter-mined by the maximum parameter between l, m, and n.In other words, without losing in generality, we canconsider l = m = n.

Taking into account the analysis carried out in sec-tion ‘‘2D nonlinear piezoelectric flexural problem,’’ thefollowing expression for the electric potential isconsidered

f x1, x2, zð Þ=f0 x1, x2ð Þ+ zf1 x1, x2ð Þ+ z2f2 x1, x2ð Þð39Þ

Moreover, being the electric potential defined apartfrom a constant, we can set

f r0 + he3ð Þ+f r0 � he3ð Þ= 0 ð40Þ

so as to obtain

f(x1, x2, z)= zDf+ z2 � h2� �

f2(x1, x2) ð41Þ

The electric field H presented in equation (7) isstraightforwardly obtained in the local basis as

~H = � RTH ð42Þ

The linear constitutive equations adopted in equa-tion (8) are now expressed in the local basis as

~B =C~E� eT ~H~D = e~E+ e ~H

�ð43Þ

The Green–Lagrange deformation tensor ~E and theelectric filed vector can be conveniently recasted as

~E=~e11 ~e12

~e21 ~e22

� �; g =

~e13

~e23

� �; e= ~e33

~H=~H1

~H2

� �; z = ~H3

8>><>>: ð44Þ

so that the constitutive equations can be rewritten as

~B= ~C~E+C13~Ie� e13z

s =C13tr~E+C33 � e33z

t =Gg + e15~H

~D= e15g � ε11~H

d = e13tr~E+ e33e+ ε33z

8>>>>>><>>>>>>:ð45Þ

Observing equation (44), we can deduce that the con-dition of complete coupling between the electrical andthe mechanical behavior of the model imposes the fol-lowing structure to the Green–Lagrange deformationtensor ~E

~E= ~E0 + z~E1

g = g0 + zg1 + z2g2

e= e0 + ze1

8<: ð46Þ

To meet these requirements on ~E, the polynomialexpansion in equation (36) has to be arrested at aproper value of l = m = n; in particular, the minimumexpansion order for which equation (45) is satisfied is l= m = n = 2. This choice implies that

~e11jz = ~e22jz = ~e12jz = e11jz = e22jz = e12jz = 4

~e13jz = ~e23jz = e13jz = e23jz = 3

~e33jz = e33jz = 2 ð47Þ




and, of course

u= u01 + zb13 + zw11 + z2w21

� �e1

+ u02 + zb23 + zw12 + z2w22

� �e2

+ u03 + z b33 � 1ð Þ+ zw13 + z2w23

� �e3 ð48Þ

or, more simply

u= u01 + zw11 + z2w21

� �e1

+ u02 + zw12 + z2w22

� �e2

+ u03 + zw13 + z2w23

� �e3 ð49Þ

To complete the description of the model, the firstPiola–Kirchhoff stress tensor is introduced as

~T= ~F~B= ~Tij~bi � ~bj i, j= 1, 2, 3 ð50Þ

and in the global basis (e1, e2, e3)

T= ~Tijbi � bj = ~TijRei � Rej = ~Tijbhibkjeh � ek = Thkeh � ek

ð51Þ

with

Tij = ~Thkbihbjk ð52Þ

For the derivation of the balance equations, it is use-ful to introduce the following quantities

Tsym =

T11T12 +T21

2T12 +T21

2T22

� �

Tsym1 =

T11T12 +T21

2

� �T

sym2 =

T12 +T21

2

T22

� �

tsym =T13 + T31

2T23 + T32

2

� �ð53Þ

Balance equations

The balance equations are derived from the virtualwork theorem. The calculus of the elastic potentialenergy in the reference configuration is represented byððð

L

T : dEdL=ðððL

T :1

2rdu+rduT� �

+rduTrdu

� �dL

or neglecting the higher order term rduTrdu

’ððð

L

T :1

2rdu+rduT� �

dL ð54Þ

We can thus write

ðððL

T : dEdL ’ððð

L

Tsym : rdudL ð55Þ

The virtual potential must have the form of equation(40) (see Nicotra and Podio-Guidugli, 1998; Voigt,1910). We further denote with X, s, and c, the externalloads per unit volume and surface acting on L and Sand the surface charge density on S.

For du = du01e1, we have

r �N1 +B1 = 0, on ON1 � n= S1, on ∂O

�ð56Þ

with

N1 =Ð

IT

sym1 dz; B1 =

ÐI

X1dz+ s1jh + s1j�h

S1 =Ð

Is1dz

�ð57Þ


r � N2 +B2 = 0, on ON2 � n= S2, on ∂O

�ð58Þ

with

N2 =Ð

IT

sym2 dz; B2 =

ÐI


S2 =Ð

Is2dz

�ð59Þ


r � N3 +B3 = 0, on ON3 � n= S3, on ∂O

�ð60Þ

with

N3 =Ð

Itsymdz; B3 =

ÐI


S3 =Ð

Is3dz

�ð61Þ

For du= zd�w11e1, we have

r � M1 + c1 � t1 = 0, onOM1 � n=m1, on ∂O

�ð62Þ

with

M1 =Ð

IzT

sym1 dz; c1 =

ÐI

zX1dz+ h s1jh � s1j�h

� �t1 =

ÐI

Tsym31 dz; m1 =

ÐI

zs1dz

�ð63Þ


r � M2 + c2 � t2 = 0, onOM2 � n=m2, on ∂O

�ð64Þ

with

M2 =Ð

IzT

sym2 dz; c2 =

ÐI


� �t2 =

ÐI

Tsym32 dz; m2 =

ÐI

zs2dz

�ð65Þ





r �M3 + c3 � t3 = 0, onOM3 � n=m3, on ∂O

�ð66Þ

with

M3 =Ð

Iztsymdz; c3 =

ÐI


� �t3 =

ÐI

Tsym33 dz; m3 =

ÐI

zs3dz

�ð67Þ

For du = z2dw21e1, we have

r � L1 + g1 � f1 = 0, onOL1 � n= l1, on ∂O

�ð68Þ

with

L1 =Ð

Iz2T

sym1 dz; g1 =

ÐI

z2X1dz+ h2 s1jh + s1j�h

� �f1 =

ÐI

2zTsym31 dz; l1 =

ÐI

z2s1dz

�ð69Þ


r � L2 + g2 � f2 = 0, on OL2 � n= l2, on ∂O

�ð70Þ

with

L2 =Ð

Iz2T

sym2 dz; g2 =

ÐI


� �f2 =

ÐI

2zTsym32 dz; l2 =

ÐI

z2s2dz

�ð71Þ


r � L3 + g3 � f3 = 0, onOL3 � n= l3, on ∂O

�ð72Þ

with

L3 =Ð

Iz2tsymdz; g3 =

ÐI


� �f3 =

ÐI

2zTsym33 dz; l3 =

ÐI

z2s3dz

�ð73Þ

For du = zDf, we have

a� 2hcjh = 0, onOv= 0, on ∂O

�ð74Þ

with

a=Ð

Iddz

v=Ð

Izcdz

�ð75Þ

For du = (z2 2 h2) df2, we have

r � V2 � h2V0ð Þ � q= 0, onOV2 � h2V0ð Þ � n= p2 � h2p0, on ∂O

�ð76Þ

with

V2 =Ð

Iz2Ddz; V0 =

ÐIDdz

q=Ð

I2zddz; p2 =

ÐI

z2cdz; p0 =Ð

Icdz

(ð77Þ

Conclusion

In this article, a nonlinear model of piezoelectric platehas been proposed. A comparison between a linear anda nonlinear 2D piezoelectric model, simulating the flex-ural behavior of a 3D model cross section, has beenillustrated. The scope of the study has been to derive anexpression for the electric potential which could consti-tute the starting point of a new nonlinear piezoelectricplate model capable of expressing a complete electro-mechanical coupling. To meet these requirements, apolynomial expansion on the through-the-thicknesscoordinate z of the warping functions describing thedeformations of the plate transverse fibers has beenintroduced, leading to a particular structure of thedeformation field. The principle of virtual work hasfinally been considered to derive the balance equationsfor the model. The proposed theory can constitute thestarting point for the development of a specific finiteelement plate model.

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A nonlinear formulation of piezoelectric plates

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