domain: phys spi

text:
Reactive transport in fractured porous media L. Jasinski a , J.-F. Thovert b , V.V. Mourzenko b , P.M. Adler a a UPMC-Sisyphe, tour 46, 4 Place Jussieu, 75252 Paris Cedex 05 b Institut SP2MI,BP 30179,86960 Futuroscope Chasseneuil Cedex Key words: Porous medium, Fracture network, Reactive transport Reactive flow through geological formations can result from natural processes, such as chemical dissolution/precipitation in diagenesis or karstification, or be due to human intervention, provoked as in acidization for increasing a well productivity, or accidental in CO2 sequestration. The description of the injection of a reacting fluid from a well, the quantitative analysis and prediction of the improvements or damages caused to the well itself and to its immediate surroundings are a subject of high interest

The simultnaeous account of the flows through the matrix rock and through the fractures that it contains is a challenging problem, due to the wide range of associated space and time scales. It is addressed here by direct numerical simulation, in a discrete fracture model. The coupled system of equations to be solved includes the descriptions of the fluid flow, of the solute transport by convection and diffusion/dispersion and of the chemical reactions, on the Darcy scale in the porous matrix and in the fracture network. In addition, constitutive equations are required for the evolution of these two structures, such as evolution laws for permeability and reactivity as functions of porosity.

An unstructured tetrahedral mesh of the fractures and of the porous matrix is built in the first place. Then, Darcy's equations are discretized and solved in a finite volume formulation. These two steps have been described in earlier works, which are summarized in [1]. Examples are shown in Fig. 1. The simulation of the evolution of the solute concentration is the most difficult point. A time implicit, non linear flux limiting scheme was implemented to avoid numerical diffusion. First-order dissolution/deposition reactions can take place, both in the matrix and the fractures, which modify their porosity and aperture. This in turn modifies the flow field which has to be continuously updated during the simulations. The effective properties such as the permeability, diffusivity and reactivity coefficients are modeled as functions of the local matrix porosity and fracture aperture.

Simple model situations have been considered for validation purposes or to illustrate some physical points. In particular, it is shown that even when the matrix permeability is small convective matrix/fracture exchanges can still widely exceed diffusive ones and strongly affect the solute transport and its residence time distribution. Finally, simulations of passive and reactive solute transport have been performed in large samples containing fracture networks, percolating or not. Various parameters have been systematically investigated, including the fracture transmissivity σ ′ , the flow regime characterized by Péclet numbers in the fractures and matrix (Pe m , Pe f ), and the Damköhler numbers of the reaction process in the matrix and fractures. The passive transport behavior and the effect of the gradual clogging of the fractures and/or matrix pore space in the case of precipitation are analyzed. Examples of concentration fields in the matrix and fractures are shown in Fig. 2 for a pulse injection of passive solute in a long rod in different regimes. Typical concentration profiles are shown in Fig. 3 for a continuous injection of reactive solute in such a rod. At early times (a), the solute front progresses but decreases in magnitude due to deposition. However, deposition does not yet cause significant changes in the medium properties and a nearly steady profile is reached. Due to strong convective exchanges between matrix and fractures, the solute carried by the fractures is transfered in the surrounding matrix and the two concentration profiles are only slightly shifted. Later (b,c), clogging becomes significant, the flow rate decreases and the concentration profiles recede. The recession is more pronounced in the matrix, since it is clogged along the fractures which prevents the solute to reach it.

[1] Adler P.M., J.-F. Thovert, V.V. Mourzenko, Fractured porous media, Oxford University Press, 2012.

Figure 1 :

 1 Figure 1: A fracture network meshed by unstructured triangles (a). Meshing of the matrix by unstructured tetrahedra (b). Triple control volume around a fracture node for the finite volume formulation (c).

Figure 2 :Figure 3 :

 23 Figure 2: Transport of a pulse of passive solute injected at x = 0 in a fractured porous matrix. Concentration at the rod walls in the matrix (left), in the fractures (middle), and in average over the cross section (right). The rows correspond to different regimes, Pem=1 and σ ′ =1 (top), Pem=100 and σ ′ =1 (middle), Pem=100 and σ ′ =10 (bottom). The concentration fields are captured at identical mean travel distances in all cases.

This abstract was prepared for the InterPore2013 conference held in Prague onMay 21-24, 2013.


Ayech Benjeddou

Roger Ohayon

M A Trindade
email: trindade@cnam.fr

Finite element modelling of hybrid active-passive vibration damping of multilayer piezoelectric sandwich beams-part II: System analysis

Keywords: nite element, multilayer sandwich beam, piezoelectric material, viscoelastic material, active-passive vibration damping

published or not. The documents may come

Finite element modelling of hybrid active-passive vibration damping of multilayer piezoelectric sandwich beams-part II: System analysis

INTRODUCTION

The model presented in Part 1 of this paper [START_REF] Trindade | Finite element modelling of hybrid active-passive vibration damping of multilayer piezoelectric sandwich beams. part 1: Formulation[END_REF], using a nite element modelling associated with the anelastic displacement elds (ADF) viscoelastic model and a linear state-space feedback control law, has led to the following control system: ẋ =( Â -BK g ) x + p; y = Ĉ x [START_REF] Trindade | Finite element modelling of hybrid active-passive vibration damping of multilayer piezoelectric sandwich beams. part 1: Formulation[END_REF] where Â; B and p are the system dynamics, input distribution and perturbation matrices, respectively, of the state-space modal reduced system. The measured variables y are written in terms of the reduced state vector x through the reduced matrix Ĉ. It should be noticed that the use of ADF viscoelastic model has led to a linear system with constant matrices but able to represent the frequency-dependence of the viscoelastic material. The problem of system dimension increase due to additional variables introduced by the ADF model was solved through a complex-based modal reduction of the state-space system. The feedback control gain matrix K g is computed using an iterative linear quadratic regulator (LQR) optimal control algorithm [START_REF] Yildirim | Free vibration analysis of symmetric cross-ply laminated composite beams with the help of the transfer matrix approach[END_REF] with weight matrices Q and R = I, being evaluated to respect maximum beam deection and control voltage applied to the actuators.

In this part of the paper, the two-node sandwich piezo-visco-elastic nite element model with layered faces is rst validated through comparisons with analytical, numerical and experimental results found in the literature. Then, the performance of the above hybrid active-passive control system is evaluated through the active control of a viscoelastically damped cantilever sandwich beam.

MODEL VALIDATION

The validation of the piezo-visco-elastic nite element model presented in Part 1 of this paper [START_REF] Trindade | Finite element modelling of hybrid active-passive vibration damping of multilayer piezoelectric sandwich beams. part 1: Formulation[END_REF] will be carried out through the analysis of three examples: a laminate elastic=adhesive= piezoelectric beam, to validate the piezoelectric stiness augmentation due to the induced potential;a4p l i e sl aminate composite beam, for the validation of the multilayer characteristic of the model, and a sandwich damped beam with bonded piezoelectric actuators, for a global validation of the piezo-visco-elastic model.

Piezoelectric layer bonded to a Timoshenko beam

As a rst example, let us consider a cantilever sandwich beam made of aluminium with thickness 15.24 mm, width 25.4 mm and length 152.4 mm completely covered by a piezoelectric layer 10 times thinner. The two layers are bonded by an adhesive layer of thickness 0.254 mm (Figure 1). Material properties are given in Table I.

The rst ve eigenfrequencies are evaluated for two situations: open-and closed-circuit on the piezoelectric layer. For closed-circuit condition, the voltage in the piezoelectric layer vanishes, therefore its stiness has only mechanical contributions. For open-circuit condition, the deformation of the sandwich beam induces a voltage in the piezoelectric layer, leading to an additional induced deformation of this layer. Notice that this could also be accounted for through a piezoelectric passive contribution to the layer stiness (see Reference [START_REF] Trindade | Finite element modelling of hybrid active-passive vibration damping of multilayer piezoelectric sandwich beams. part 1: Formulation[END_REF], Equation (31)). Present nite element results, using 15 elements, are compared to those given by Krommer and Irschik [START_REF] Krommer | On the inuence of the electric eld on free transverse vibrations of smart beams[END_REF] and shown in Table II. PT and AB state for the Timoshenko multilayer theoretical and Abaqus nite element results presented in Reference [START_REF] Krommer | On the inuence of the electric eld on free transverse vibrations of smart beams[END_REF]. The results found using the layerwise theory of Saravanos and Heyliger [START_REF] Saravanos | Coupled layerwise analysis of composite beams with embedded piezoelectric sensors and actuators[END_REF], which are able to reect a non-linear distribution of axial displacements and electric potentials through division of piezoelectric layers in multiple sublayers, have been considered as the reference results (SH).

Table II also shows the present results (FEM) error relative to the reference ones (SH). One may notice that present nite element results match well with previous theoretical and numerical results for both open-and closed-circuit conditions of piezoelectric layers.

Laminate composite beam

To validate the faces multilayer characteristic of the model, a modal analysis of laminate composite beams is performed and its results are compared to those presented in Reference [START_REF] Yildirim | Free vibration analysis of symmetric cross-ply laminated composite beams with the help of the transfer matrix approach[END_REF]. Clamped-free (CL), clamped-simply supported (CS) and clamped-clamped (CC) 4 plies symmetric laminate beams (0=90) s , made of three composite materials, namely Kevlar 49epoxy, AS4/3501-6 and T300/N5208, are considered. The material properties are presented in Table III. All plies have the same thickness, the sum of which is denoted as h = L=120, length L = 762 mm and width b = L=15.

It can be seen, in Table IV, that the rst three eigenfrequencies of the laminate composite beams, using 20 elements of the present nite element model (sandwich=classical laminate theory, SCLT), match well with those presented in Reference [START_REF] Yildirim | Free vibration analysis of symmetric cross-ply laminated composite beams with the help of the transfer matrix approach[END_REF], using rst-order shear deformation theory (FSDT). One may also see that the model accuracy increases for stier (T300=N5208) and less constrained (clamped-free) beams, since in these cases less shear is produced by deection.

It is well known that classical laminate theory leads only to rst approximations for crossply laminate composite beams. However, since most of the applications of the present sandwich beam model consists of sti laminate faces and less sti cores, it is assumed that most of the shear deformation will be performed by the core.

Sandwich damped beam with bonded actuators

Experimental validation was carried out through comparison with results presented by Wang and Wereley [START_REF] Wang | Frequency response of beams with passively constrained damping layers and piezoactuators[END_REF]. In this case, a sandwich cantilever beam, made of aluminium faces and ISD112 viscoelastic core, with a symmetrically attached pair of PZT5H piezoelectric patches is considered (Figure 2). Material and geometrical properties, adapted from Reference [START_REF] Wang | Frequency response of beams with passively constrained damping layers and piezoactuators[END_REF], are shown in Table V.

The 3M ISD112 viscoelastic material is modelled using three ADF, which parameters, evaluated from material master curves curve-tting and valid, at 27 are then evaluated for each viscoelastic core thickness. Our nite element results, using 20 elements, are then compared to numerical ones, carried out using the progressive wave method (PWM), which accounts for frequency-dependent viscoelastic properties, and experimental ones presented in Reference [START_REF] Wang | Frequency response of beams with passively constrained damping layers and piezoactuators[END_REF]. Present results, shown in Table VI, match quite well with both experimental and numerical ones. It should be noticed that, for the thinner viscoelastic core conguration, our results are acceptable except for the third bending eigenfrequency which presents a large error of 21.20 per cent. However, it can be seen that the PWM leads to a similar error (20.06 per cent). This suggests that this reference value (experimental) may be not well measured.

HYBRID DAMPING PERFORMANCE

In this section, the nite element model, validated in the previous section, is used for the control synthesis and analysis of a cantilever sandwich damped beam treated by bonded piezoelectric actuators similar to that studied in the validation section (Figure 2), material and geometric properties of which are shown in Table VII. The control strategy presented in Part 1 [START_REF] Trindade | Finite element modelling of hybrid active-passive vibration damping of multilayer piezoelectric sandwich beams. part 1: Formulation[END_REF] is considered, using the LQR iterative algorithm and supposing that the full state is measured. The weight matrix Q is considered to be diagonal and composed of the rst three eigenfrequencies so that Q = diag(! 2 1 ;! 2 2 ;! 2 3 ; 0; 0; 1; 1; 1; 0; 0) to maximize the damping of the energy of these rst modes. For the example treated in what follows, the output y is taken to be the cantilever sandwich beam tip deection and a perturbation transversal force applied at x = 100 mm is considered. Voltages applied to the piezoelectric actuators are such that the maximum electrical eld of 500 V=mm is not exceeded.

Furthermore, analysis of the variation, with the actuator length and damping core thickness, of passive damping, supplied by the viscoelastic damping layer, and corresponding damping gain relative to the passive one, developed by the piezoelectric feedback control, is carried out. First, the upper and lower piezoelectric actuators lengths are set to vary in the range 20-70 mm. As it can be seen in Figure 4, the actuator length variation does not aect much the passive viscoelastic damping, although one can notice a little overall augmentation, since only a face stiening eect due to the piezoelectric layers is obtained. However, the active damping provided by the piezoelectric actuators increases greatly with their length (Figure 5). The rst and third eigenmodes damping are augmented up to 140 per cent with respect to the passive one, for the longest actuators. Nevertheless, the second modal damping does not ameliorate that much. Additionally, the viscoelastic core thickness is set to vary in the range 0.01-2 mm for a xed actuator length of 50 mm. As it can be seen in Figure 6, the damping ratio of rst mode is greater for increasing thickness, as was also shown numerically by van Nostrand and Inman [START_REF] Van Nostrand | Finite element model for active constrained layer damping[END_REF]. However, for the second and third modes, modal dampings are optimal for relatively thin cores (around 0:06 mm). Although increasing viscoelastic material thickness may produce more passive damping, it also reduces the transmissibility between each actuator and its opposite elastic face, as reported in Reference [START_REF] Van Nostrand | Finite element model for active constrained layer damping[END_REF]. This is the reason for the decrease in the hybrid damping performance for the rst mode with increasing core thickness, as seen in Figure 7. Nevertheless, for the second mode, the eect is the opposite, that is, for its low passive damping range, active damping compensates, but does not surpass, the global damping of the sandwich beam. It is also clear that, as the controller loses its control performance for the rst mode, it improves the second mode damping.

The previous results show that the active damping performance is improved for long actuators and thin cores, however in order to guarantee enough passive damping for the three modes, the viscoelastic core thickness is set to 0:06 mm. For these parameters, open-and closed-loop perturbation impulse responses of the sandwich beam tip deection (y= p) are shown, in the frequency and time domains, in Figures 8 and9. From the frequency response function (Figure 8), one may notice that the rst and third modes are greatly damped by the piezoelectric actuation, whereas the other modes damping remains almost constant. This is justied by the choice of the state ponderation, that is the choice of the elements of matrix Q, in the control design. Nevertheless, combination of passive and active damping not only allows to increase the damping of some selected modes but also guarantees some passive damping for uncontrolled modes. Furthermore, passive damping may prevent spillover eects.

The frequency response function of the sandwich beam, when excited by the actuators (y=u), is also presented in Figure 8 and shows that the third mode resonance is almost cancelled.

For some applications, it is also necessary to analyse the transient response of the sandwich beam. Using the time-domain ADF viscoelastic modelling, this analysis is direct since all statespace matrices are frequency independent. Figure 9 shows the beam tip deection impulse responses for open-and closed-loop situations. One may notice that output is quickly damped.

The actuator voltage needed for such performance, shown in Figure 10, is clearly limited by the input constraint of 250 V considered in the control algorithm.

CONCLUSIONS

The validation of the electromechanically coupled nite element model, presented in Part 1 of this paper [START_REF] Trindade | Finite element modelling of hybrid active-passive vibration damping of multilayer piezoelectric sandwich beams. part 1: Formulation[END_REF], for the analysis of active-passive damped multilayer sandwich beams, was carried out through the analysis of three examples: a laminate elastic=adhesive=piezoelectric beam, a laminate composite beam and a sandwich damped beam with bonded piezoelectric actuators.

Present nite element results for modal analysis of the laminate elastic=adhesive=piezoelectric beam, under open-and closed-circuit conditions, match well with nite element and analytical results found in the literature. Modal analysis of laminate composite beams was performed to evaluate the accuracy of the classical laminate theory used for the faces. Its results were then compared to numerical ones, using rst-order shear deformation theory, found in the literature and they matched well. Nevertheless, accounting for the shear strain in each piezoelectric layer was not considered but is being considered as future work. Furthermore, comparison of modal analysis results for a ve-layer damped beam with experimental results found in the literature showed that the present model provides a good representation of both laminate=sandwich behaviour and viscoelastic material properties frequency-dependence. The numerical results indicate that the present nite element model leads to a simple but eective representation of the behaviour of multilayers active-passive constructions. The latter example was then considered for the analysis of the hybrid damping performance under viscoelastic layer thickness and actuators length variations. It was found that hybrid active-passive damping allows to increase damping of the rst and third modes while preventing instability of uncontrolled ones. Analysis of viscoelastic material thickness variation showed that modal damping distribution can be optimized by proper choice of this parameter. In this particular case, thin cores yielded better active and passive overall performances.

Application to the analysis of other active-passive damping constructions such as active constrained layer damping (ACLD) [START_REF] Trindade | Modeling of frequency-dependent viscoelastic materials for active-passive vibration damping[END_REF] and active control with passive constrained layer damping (AC=PCLD) variations has been also conducted in order to assess the various hybrid active-passive congurations proposed in the literature. Also, comparison of the performance of several control algorithms has being carried out [START_REF] Trindade | Piezoelectric active vibration control of sandwich damped beams[END_REF] and analysis of the viscoelastic material temperature-dependence eect on the control performance was presented in Reference [START_REF] Trindade | Finite element analysis of frequency-and temperature-dependent hybrid active-passive vibration damping[END_REF].

Figure 1 .

 1 Figure 1. Timoshenko beam with a bonded piezoelectric layer (dimensions in mm).

Figure 2 .

 2 Figure 2. Cantilever sandwich beam with bonded actuators (dimensions in mm).

Figure 3 .

 3 Figure 3. First three natural bending modes of the ve-layer beam of Reference [5].

Figure 4 .

 4 Figure 4. Three rst bending modes passive damping vs. actuator length (hv =0:25 mm).

Figure 5 .

 5 Figure 5. Three rst bending modes active damping gain vs. actuator length (hv =0:25 mm).

Figure 6 .

 6 Figure 6. Three rst bending modes passive damping vs. viscoelastic layer thickness (a =50mm).

Figure 7 .

 7 Figure 7. Three rst bending modes active damping gain vs. viscoelastic thickness (a =50mm).

Figure 8 .

 8 Figure 8. Frequency response function of the sandwich damped beam tip deection (a =70mm;h v =0:06 mm).

Figure 9 .

 9 Figure 9. Impulse response of the sandwich damped beam tip deection (a =70mm;h v =0:06 mm).

Figure 10 .

 10 Figure 10. Actuator voltage needed to control impulse loaded sandwich beam (a =70mm;h v =0:06 mm).

Table I .

 I Material properties for the cantilever Timoshenko sandwich beam.

			Aluminium	Piezoelectric	Adhesive
	Elastic modulus (GPa), e * 11	68.97		68.97	6.90
	Shear modulus (GPa), c55	27.59		-	-
	Density (kg m -3 ),		2769		7600	1662
	Piezoelectric constant (C m -2 ), e * 31 Dielectric constant (F m -1 ), * 33	--1 :15 × 10 -8 -8:41	--
	Table II. First ve eigenfrequencies (Hz) of the cantilever Timoshenko sandwich beam.
		12345
	Closed circuit
	SH [3]	538.10	3199.00	7580.00	8350.00	15039.00
	PT [2]	538.60	3211.00	7580.30	8394.86	15139.53
	AB [2]	539.60	3200.40	7584.70	8326.70	14922.00
	FEM	538.43	3206.67	7584.04	8388.22	15143.26
	Error (%)	0.06	0.24	0.05	0.46	0.69
	Open circuit
	SH [3]	544.10	3232.00	7614.00	8428.00	15167.00
	PT [2]	544.10	3241.79	7611.03	8469.41	15262.68
	AB [2]	543.81	3223.30	7608.70	8378.90	15004.00
	FEM	543.34	3234.35	7613.47	8454.78	15252.56
	Error (%)	-0:14	0.07	-0:01	0.32	0.56

Table III .

 III Material properties for the laminate composite beam.

			E1	E2 = E3	G12 = G13	G23	12 = 13	23
	Kevlar 49-epoxy	76.0 GPa	5.56 GPa	2.30 GPa	1.60 GPa	0.34	0.7	1460 kg m -3
	AS4=3501-6		144.8 GPa	9.65 GPa	4.14 GPa	3.45 GPa	0.30	0.4	1389 kg m -3
	T300=N5208		181.0 GPa	10.3 GPa	7.17 GPa	3.40 GPa	0.28	0.5	1600 kg m -3
	Table IV. Comparison of rst three bending non-dimensional eigenfrequencies ( = !L 2 (=E1h 2 ) 1=2 )o fa4
			plies symmetric (0=90)s laminate composite beam (L/h = 120).
		Kevlar 49-epoxy		AS4=3501-6	T300=N5208
		CF	CS	CC	CF	CS	CC	CF	CS	CC
	1(FSDT)	0.954	4.171	6.037	0.954	4.170	6.035	0.953	4.168	6.035
	1(SCLT)	0.958	4.203	6.099	0.957	4.196	6.088	0.955	4.190	6.079
	2(FSDT)	5.957	13.45	16.52	5.955	13.44	16.52	5.954	13.45	16.53
	2(SCLT)	6.006	13.62	16.81	5.996	13.60	16.78	5.987	13.58	16.76
	3(FSDT)	16.59	27.84	32.10	16.59	27.84	32.09	16.59	27.87	32.15
	3(SCLT)	16.81	28.41	32.95	16.79	28.36	32.89	16.76	28.32	32.85

Table VI .

 VI Three rst bending eigenfrequencies of the ve-layer beam of Reference[START_REF] Wang | Frequency response of beams with passively constrained damping layers and piezoactuators[END_REF].

		Experimental [5]	PWM [5]	FEM
		Freq. (Hz)
	Mode		Freq. (Hz)	Error (%)	Freq. (Hz)	Error (%)
	0.051 mm viscoelastic core
	1	6.64	6.37	-4:07	6.90	3.92
	2	36.21	37.88	4.61	39.27	8.45
	3	85.23	102.33	20.06	103.30	21.20
	0.127 mm viscoelastic core
	1	6.84	6.69	-2:19	7.05	3.07
	2	35.62	37.40	5.00	36.62	2.81
	3	92.42	96.13	4.01	92.25	-0:00
	0.254 mm viscoelastic core
	1	7.22	7.00	-3:05	7.30	1.11
	2	36.79	36.76	-0:00	34.62	-5:90
	3	89.50	89.60	0.11	84.66	-5:41

Table VII .

 VII Material and geometrical properties of the sandwich damped beam with bonded piezoelectric actuators.

ACKNOWLEDGEMENTS

Support of the "D el egation G en erale pour l'Armement"-Advanced Materials Branch, under contract D.G.A.=D.S.P.= S.T.T.C.=MA. 97-2530, is gratefully acknowledged. The rst author acknowledges also the support of the Brazilian government (CAPES) through a doctoral scholarship award.


Daniel Ohayon

M Rahmoune

A Benjeddou

R 0hayon

D Osmont

New Thin Piezoelectric Plate Models

INTRODUCTION

P

IEZOELECTRIC materials find wide use in deformations and motion of structure detection, as well as in active structural control. They are either bonded or embedded in the structure in order to measure strains and displacements (sensing effect) or to provide localized strains through which the deformation of the structure can be controlled (actuation effect). Effective sensing and actuation of the resulting smart structure need that its electroelastic behavior be well mod eled.

The concept of smart structure was first validated analyti cally and experimentally on beam elements, and rapidly structural problems of piezoelectric plates arose in modeling flat piezoelectric sensors and actuators. These were mainly analyzed with approximate theories [START_REF] Destuynder | Theoretical, numerical and experimental discussion on the use of piezoelectric de vices for control-structure interaction[END_REF][START_REF] Tzou | Piezoelectric Shells[END_REF][START_REF] Drozdov | Intelligent Composite Struc tures: general theory and applications[END_REF], based on as sumed simplifying, a priori, hypotheses concerning direc tions of the electric field and displacement, and the represen tative form of the electric potential and even the constitutive behavior [START_REF] Drozdov | Intelligent Composite Struc tures: general theory and applications[END_REF]. Indeed, only transverse components of electric displacement and field are retained. The electric potential is generally considered linear through thickness, and sometimes constant. Besides, for 1 Author to whom correspondence should be addressed.

most actuation applications, the induced potential due to the direct piezoelectric effect is often supposed negligible com pared to the imposed potential [START_REF] Destuynder | Theoretical, numerical and experimental discussion on the use of piezoelectric de vices for control-structure interaction[END_REF][START_REF] Tzou | Piezoelectric Shells[END_REF]. Classical structural theories are also often adapted and the presence of piezoelectric elements is taken into account by a thermal analogy approach through the in troduction of prescribed strains [START_REF] Destuynder | Theoretical, numerical and experimental discussion on the use of piezoelectric de vices for control-structure interaction[END_REF]. But, in this way, the full coupled electroelastic behavior of smart structures is not completely modeled, since the mass and stiffness of the piezoelectrics are not considered. There fore, good representations of the piezoelectric effect have been achieved using either additional electronic circuits that take into account the induced potential, as in self-sensing ac tuation techniques [START_REF] Dosch | A self-sensing piezoelectric ac tuator for collocated control[END_REF] and shunted piezoelectrics [START_REF] Hagood | D amping of structural vibrations with piezoelectric materials and passive electrical networks[END_REF], or using through-thickness quadratic variation of the electric potential as proposed by [START_REF] Rogacheva | The Theory of Piezoelectric Shells and Plates[END_REF] using an asymp totic analysis.

By the beginning of this decade, several asymptotic theo ries were proposed to deal with piezoelectric plates. They differ in the scaling techniques adopted and their results de pend on the type of electric boundary conditions applied on upper and lower faces of the plate. Most common conditions, in technical applications, are either of Neumann (prescribed surface electric charges) or Dirichlet (imposed surface elec tric potentials) type. [START_REF] Maugin | An asymptotic theory of thin piezoelec tric plates[END_REF] used asymp-totic integration theory, in its variational form, that exploits the "zoom technique," to establish the first two orders of an asymptotic theory of thin piezoelectric plates in the frame work of electrostatics. At the first-order of approximation, a purely mechanical Love-Kirchhoff theory emerged, while the electric potential satisfies a two-dimensional Poisson Neumann problem, with an effective dielectric constant ac counting for electromechanical coupling. Nevertheless, these results do not agree with the thin-plate limit obtained by Bisegna and Maceri from exact three-dimensional solu tions (Bisegna and Maceri, 1996a), and with the consistent piezoelectric plate theory they derived (Bisegna and Maceri, 1996b). In particular, the deflection given by [START_REF] Maugin | An asymptotic theory of thin piezoelec tric plates[END_REF] depends on elastic constants and not on piezo electric and dielectric ones, contradicting results of Bisegna andMaceri (1996a, 1996b). The procedure used in Bisegna and Maceri (1996b) to derive field equations governing the piezoelectric problem is based on the initial functions method in conjunction with a re-scaling of the applied loads. A piezoelectric plate model was also derived by [START_REF] Rogacheva | The Theory of Piezoelectric Shells and Plates[END_REF] as a zero-curvatures shell model. The latter was based on a priori assumptions from an asymptotic analysis similar to that used by [START_REF] Maugin | An asymptotic theory of thin piezoelec tric plates[END_REF] for isolated piezo electric plates. Similar results were then obtained. However, [START_REF] Rogacheva | The Theory of Piezoelectric Shells and Plates[END_REF] has also considered tangential polariza tion and short-circuited plates.

The major difficulty introduced by Dirichlet-type electric boundary conditions, besides electromechanical coupling, is the non-homogeneity (in the mathematical sense). To deal with this problem in piezoelectric shells, Bemadou andHaenel (1995a, 1995b) simply used a variable change in or der to make the electric boundary conditions homogeneous. They take their inspiration from the classical handling of mechanical Dirichlet boundary conditions. [START_REF] Canon | Modeling of active materials systems including electronic devices using asymptotic methods[END_REF] and [START_REF] Lenczner | Modeling of a plate including a distribution of piezo electric transducers and an electronic network optimization of circuit pa rameters[END_REF] used Lagrange's mul tipliers to deal with electric Dirichlet conditions applied to multilayer elastic plates containing piezoelectric inclusions [START_REF] Canon | Modeling of active materials systems including electronic devices using asymptotic methods[END_REF] or including distributed piezo electric actuators and a distributed electric circuit [START_REF] Lenczner | Modeling of a plate including a distribution of piezo electric transducers and an electronic network optimization of circuit pa rameters[END_REF]. Their plate models were derived as the limit of the three-dimensional problem when the thickness vanishes in the framework of asymptotic methods. The electric potential and displacement were found to be independent of the trans verse coordinate. Besides, electromechanical couplings exist between in-plane components of electric field and stresses on one hand, and between the jump of the electric potential on the upper and lower faces and plane stresses, on the other hand. However, these results do not agree with those of Bisegna and Maceri (1996b) who found that the transverse component of the electric displacement is linear in the thick ness direction and that the electric potential has quadratic law variation with respect to thickness coordinate, even in the case of thin piezoelectric plates.

Recently, [START_REF] Saravanos | Layerwise me chanics and finite element for the dynamic analysis of piezoelectric com posite plates[END_REF] pro posed layer-wise (or discrete layer) theories using piece wise linear continuous approximations along the thickness direction for both displacement and electric potential fields. These theories can model both global and local electrome chanical responses of smart composite laminates. Finite ele ment formulations with added electric potential degrees of freedom were also developed for quasi-static and dynamic analyses of smart composite structures containing piezoelec tric layers. Numerical analyses indicate that electric fields exist in the piezoelectric layers even with closed-circuit con ditions (zero potential). Moreover, figures show that the in duced electric potential is parabolic inside piezoelectric lay ers and vanishes at their upper and lower skins.

It is the objective of this paper to present an asymptotic theory for piezoelectric plates in the presence of Dirichlet electric boundary conditions. These are taken into account by considering the electric field induced by the imposed elec tric potentials. That is, the electric potential is decomposed into a known component, completely defined from pre scribed electric potentials and an unknown component repre senting an induced electric potential. It will be shown that, for the bending problem, this induced potential is propor tional to bending strains and its contribution in the system be havior appears through a modification of the constitutive elastic equations and an additional electric force. Associate numerical handling is then made easier, since standard finite elements could be used provided that modified elastic con stants and electric force could be entered (Rahmoune et al., 1996;Rahmoune and Osmont, 1996). It is found here that the consistent hypothesis for transverse electric displacement and field for thin layers is "the electric displacement and field are respectively constant and linear through thickness," con trary to above asymptotic theories where it is generally as sumed or derived that "the electric displacement and field are respectively constant through thickness." Moreover, it is found that the mechanical problem is of a Kirchhoff-Love type.

In the following, the paper focuses on developing Kirchhoff-Love models for piezoelectric plates using an as ymptotic approach extending to the piezoelectric media the theory of [START_REF] Ciarlet | A junction of two-dimensional plate model[END_REF] devoted to elastic media. Numerical implementation was described in Rahmoune et al. (1996), Rahmoune and Osmont ( 1996), and detailed in [START_REF] Rahmoune | Smart piezoelectric plates: modeling and applica tion to health monitoring[END_REF]. Local three-dimensional equations and their associated variational formulation will first be recalled. Next, two-dimensional variational and local equations of short-circuited and insulated piezoelectric plates will be deduced as a limit when the aspect ratio of the plate vanishes. Applications are then made to homogeneous and orthotropic plates.

THREE DIMENSIONAL PROBLEM

In the three-dimensional space, of reference frame (o, x f , x � ' x n, let us consider a homogeneous piezoelectric body electric and dielectric constants, C !Jk1 , ef ij and Eij respec tively. QQE denotes itS regular boundary Which is decom posed into upper, lower and lateral surfaces, L+E, L-E and fE, respectively. The latter is clamped on rg and free on its com plementary part f, E. The middle SUrface Of the piezoelectric body is denoted ro and has a boundary aro = y, split into a clamped part y0 and a free complementary part y 1 . The do main can carry mechanical body forces rand surface forces g ±E on upper and lower faces L±E. Electrically, it could be ei ther short-circuited, through prescribed potentials V±E on L±E and surface charge qE on r E, or insulated, i.e., charge-free on an.£.

Local Equations

Local three-dimensional problems consist of finding me chanical displacement uE and electric potential 'PE satisfying the following equations:

• mechanical equilibrium and boundary conditions:

-aij.j = N in QE (1 ) crijn j = g f E on L±E, crijn j =O on f1 E, (2 ) UE l =O on rg
where crE is the symmetric Cauchy stress tensor, and nj are outward normal components,

• electric equilibrium and boundary conditions:

(3)

where DE is the electric displacement vector,

• linear piezaelectric constitutive equations:

(5) representing the converse (actuator) and direct (sensor) pi ezoelectric effects. Notice that these are expressed using the short-circuited elastic, piezoelectric and dielectric con stants. In Equation (5), £ and E are the symmetric linear strain tensor and electric field vector related to the me chanical displacement uE and electric potential <p\ respec tively, through, It is worthwhile to notice that two important regular prob-!ems could be distinguished according to the electric bound ary conditions considered. The first corresponds to the short circuited piezoelectric media, and the second to the electri cally insulated piezoelectric body, for which the potential is completely unknown, in particular on the boundary. The nat ural electric boundary condition is then, assumed to be, for the latter case, Dfn; = 0 on aQE (7)

Variational Formulation

The three-dimensional variational problem, associated to local equations (1-5), consists of finding the couple (uE, <p E ) in the admissible space satisfying the variational equation:

where V0(QE ) = {( v E, \j/e) E [ H1 (QE ) ] 3 x H1 (QE ), v E = O on r g, 'l'E = 0 on L±E}. This variational problem has a unique solution [START_REF] Rahmoune | Smart piezoelectric plates: modeling and applica tion to health monitoring[END_REF].

ASYMPTOTIC ANALYSIS

The asymptotic development technique (Ciarlet and

Destuynder, 1 979) is adapted here to get equivalent Love Kirchhoff piezoelectric plate models as limits of the above three-dimensional variational problem when £ tends to van ish.

In order to apply the asymptotic analysis, the domain QE is first made fixed; i.e., independent of£ through the following geometrical variable change:

Q = (t) x ] -1, + 1[ (9)
Then mechanical and electric parameters scaling are defined.

The former is classically treated, as detailed in Ciarlet and Destuynder (1979). However, the latter depends on electric boundary conditions.

Mechanical Displacement and Forces Scaling

Geometrical considerations lead to the following a priori estimates: uf E H 1 (QE) is a priori associated to u;(E) E H1(Q) such that Similar relations hold forvf E H 1 (Q e) associated, a priori, to v;(e) E H 1 (Q) . It is convenient for mathematical purposes to introduce the following scaling on body and surface forces defined as (11) where f and g are independent of e. Material constants are supposed independent of E, i.e.,

Cijkl(xE) = cijk1 ( x, , Xz , X3 ), ef;ij ( x E) = ek;ij ( x, , Xz , X3 ), ( 12
)
Electric Potential Scaling for the Insulated Plate

The plate is assumed insulated, i.e., Equation (7) holds. Thus, the variational problem (8) reduces to finding (uE, rpe) E V0 1 (Qe), so that,

(13) where V0 1 (QE) = {( v e, 'l' E) E [ H 1 (QE) ] 3 x H l (QE) , v e = 0 on r8 }.
The order of E in the unknown potential rpe is selected such that the variational problem ( 13) has an asymptotic solution. This is found to be in the form [START_REF] Rahmoune | Smart piezoelectric plates: modeling and applica tion to health monitoring[END_REF]:

Similar relations hold for 'l'E associated to 'lf(E). The plate is now assumed short-circuited, i.e., electric boundary conditions ( 4) hold. Prior to the direct application of the asymptotic development technique, non-homage-neous Dirichlet electric boundary conditions ( 4.1) should be made homogeneous. To this end, the electric potential rpE is split into a linear part rpOE, known through the given poten tials on L ±e, and an unknown part <I>\ so that, where ----+

x VU±£ E H 1 12 ( L± E)

{u+e + u-e u+E -u-£ } 2 2E 3 ,
Hence, the variational problem ( 8), reduces to finding (u e, <l>e) E V0(QE), such that:

(16)

The second integral in the left hand side (l.h.s.) of Equation ( 16) is the contribution of the unknown potential <l>E, shown here to represent the induced potential, often neglected in the literature. However, the last integral in the r.h.s. of Equation ( 16) is the equivalent electric force due to the imposed poten tial rp 0 E. The above variational formulation can be used either for actuator or sensor analysis. But, in the latter case, the last two integrals in Equation ( 16) should be dropped, according to Equation ( 7). This is more detailed in the next sub-section. Now, scaling orders of rpOE, <l>E and qE are chosen so that an asymptotic solution exists for the variational problem ( 16). These are found to be in the form [START_REF] Rahmoune | Smart piezoelectric plates: modeling and applica tion to health monitoring[END_REF], where q is independent of E. Using Equation (1 5), the total potential is then taken in the form, (18) Similar relations hold for 'l'E associated, a priori, to 'lf(E) which can be also decomposed into a known and an unknown part as in Equations ( 15), ( 17) and ( 18). Notice that Equa tions ( 14), ( 17) and ( 18) ensure that the corresponding elec-tric parameters appear in the zeroth order of the asymptotic solution.

SHORT-CIRCUITED PIEZOELECTRIC PLATE MODEL

This section aims to present the limit variational problem associated with a short-circuited piezoelectric plate when E tends to zero. Starting from the three-dimensional variational problem ( 16), in conjunction with mechanical [Equations ( 10)-( 12)] and electric [Equations (15,17,18)] parameters scaling, the asymptotic mechanical displacement field and electric potential are first deduced through an asymptotic analysis; then a limit two-dimensional variational problem is defined, together with its associated local equations.

Asymptotic Mechanical Displacement and Electric Potential

According to the mechanical displacement scaling Equa tion (10), the strains can be scaled as It can be shown [START_REF] Rahmoune | Smart piezoelectric plates: modeling and applica tion to health monitoring[END_REF] where ( Sa,S 3 )

E VKL x WKL with VKL= {S a E H 1 (ro), sa=Oon Y o L WKL = {S3 E H 1 (ro), S3 = a v S3 = 0 on Y o }.
From the electric potential scaling (18), the electric field scaling has the form

Ea(<fJE) = £ 3 Ea(<fJ(E)), £3( <p E) = E 2 £3( <p (E)) (22)
It can be shown [START_REF] Rahmoune | Smart piezoelectric plates: modeling and applica tion to health monitoring[END_REF] that when E tends to zero

1 -<j>(E) -+ <j>* 3 (23)
Consequently the electric potential tends to <p such that:

<p = <p o + <!> * ( 24
)
where and

VL = {<p o E H 1 (Q ), <p o = V ± on I ± } V Q = {w E H 1 (il), 'I' = O on I ± ,
It is worthwhile noticing that Equations ( 20) and ( 23) are weak convergences. It was shown that these are also strong convergences (see [START_REF] Rahmoune | Smart piezoelectric plates: modeling and applica tion to health monitoring[END_REF] for details). These results suggest the following comments:

• The asymptotic mechanical displacement field ( 21) is of the Kirchhoff-Love type (E;3 = 0). • The asymptotic electric potential (24) has a known part ip 0 entirely defined by the known potentials on the upper and lower faces, and an unknown part <j>* corresponding to the induced potential. <!> * is often neglected in the literature (Rahmoune et al., 1996). • Only transverse components of the electric field are of im portance, which is often retained in the literature, but finds here a mathematical justification [Equation ( 23)].

Asymptotic Electric Displacement and Stresses

In order to satisfy the electric equilibrium (3) and bound ary conditions (4.2), in-plane and transverse electric dis placement components are scaled separately: where Using Equations ( 19), ( 22), ( 25) and (27), the three dimensional variational problem (8) becomes: = J f;v ;dD. + J + g fv ;dI. ± + J eq'lf(E) dr (28) n L-r

where 'lf(E) = '1' 0 (e) + e-' x(e) .

Keeping in mind that K;/E) are bounded, then 1. Multiplying Equation (28) by E, setting v =Va and 'I'= 0,

and tending e to zero 2. Multiplying Equation (28) by e 2 , setting v = v 3 and 'I'= 0,

and tending E to zero, weak [shown to be also strong (Rahmoune, 1997)] convergence of cr;3(e) to cr; 3( 0) is ob tained with:

(29)

These results indicate that:

• Only the transverse component of the electric displace ment is of importance [Equation (26)], in-plane compo nents may be neglected as it is the case for the electric field; this is a classical hypothesis, but finds its justification here.

• Transverse stresses cr;3 may also be neglected compared to in-plane components [Equation ( 29)]. can be written in terms of bending strains as:

(37)

The constant of integration is determined through the condi tion f � l £3( tj> *)dx3 = 0, leading to the final expression of the unknown electric field,

Hence, the unknown electric potential can be simply defined as (39) since tj>* = 0 on L±. Unknown electric field (38) and potential (39) are interpreted as induced electric field and potential. These are often neglected in the literature (Rahmoune et al., 1996). Expression (38) of the induced electric field is now substi tuted in the constitutive equation ( 31 Substituting Equations ( 43) into (41), then in the varia tional problem (33), gives the following new variational property:

(45) Compared to a classical elastic variational problem, the above equation, indicates that the piezoelectric effect in duces a modification of the constitutive equations according to Equation ( 43) and an extra generalized electric force/mo ment vector represented by the last r.h.s. integral in Equation ( 45). Hence, the numerical implementation of Equation ( 45) would be very easy provided that Equations ( 42) and ( 44) could be entered into a classical structural software. The so-lution of the above variational problem gives the mechanical displacement !;. Now let's see how to derive electric quantities such as <I>* or £ 3 (<!>*). For this, let v = 0 in the variational problem (28), which becomes, (46) Taking account of Equation ( 26) and making E vanish, leads to: which implies that D 3 , 3( 0) = 0 in Q. Hence, using Equations ( 25) and (35), we get:

In summary, to get the solution of the electromechanical coupled problem for a smart short-circuited plate, the variational problem (45) is first solved in order to obtain the mechanical displacement, then the electric displacement, field and potential could be computed, a posteriori, by Equa tions (48), ( 38) and (39) respectively. Beside, the piezoelec tric effect induces a modification of the constitutive equa tions, as indicated in Equation (43) and an extra electrical force term as defined in Equation (42). Here it was found that the induced electric potential and field are proportional to bending strains, whereas the electric displacement is con stant through the thickness.

Local Problem

Integrating by parts the variational equation (33), gives the following local equations:

• membrane problem -o p Na p = P a mro N a p V p = 0 on y1 (49)
where Na p are given by Equations ( 41)-( 44) and V p are normal components of the outward normal unit vector to

YI•

• bending problem The present two-dimensional formulation shows that in plane electric displacement and field do not appear. Hence, only their transverse components and in-plane stresses are retained in this model. These are usual hypotheses in the lit erature, but find here a mathematical background.

It was found that, when the induced potential is not ne glected, electric displacement and field are constant and lin ear in the thickness direction, respectively. When the induced field is substituted in the constitutive equations, by its ex plicit expression in terms of the bending strains, the electro mechanical problem uncouples. Thus, the mechanical prob lem is solved first for mechanical displacements, then these are used to compute, a posteriori, the electric quantities (field, potential, charge, displacement).

If the induced electric potential is neglected, for consis tency reasons, the transverse electric field component is con stant in the thickness direction. Consequently, the transverse electric displacement becomes linear, i.e., D 3 (0) = e 3 ;0,_(y0;i,.x 3 a 0,_ 1; 3 ) + E 33 E 3( rp 0 ) , but does not ensure the electric equilib rium equation (51). Hence, assuming constant transverse electric field and displacement, as retained in many papers, is contradictory. The right electric assumptions for short circuited piezoelectric plate should be constant electric dis placement and linear electric field as explained above.

According to Equation (48), bending strains do not induce electric charge, only membrane strains can do so for homo geneous piezoelectrics. However, both strains can generate electric charges for the practical case of multilayer configu rations where piezoelectrics are either surface-mounted to or embedded in a host structure.

INSULATED PIEZOELEC TRIC PLATE MODEL

The objective of this section is to develop the limit variational problem associated with an insulated piezoelec tric plate when E tends to zero. Starting from local equations (1)-( 3) and ( 5)-( 7) and their corresponding three-dimen sional variational problem (13), in conjunction with mechan ical [Equations (10)-( 12)] and electric [Equation (14)] parameters scaling, asymptotic electric potential and dis placement are first deduced through an asymptotic analysis; then an equivalent two-dimensional variational model is for mulated together with its associated local equations.

Asymptotic Electric Displacement and Potential

Compared to the above short-circuited model, scaling of electric parameters are no longer those obtained previously, since the electric potential is now completely unknown. Its scaling is given by Equation ( 14), i.e., (52) Hence, in-plane and transverse components of the electric field can be scaled as, (53) Scaling of in-plane and transverse components of the elec tric displacement are now taken so as to satisfy the electric equilibrium equation ( 3) and boundary condition ( 7) :

(54) where and Using Equations (1 9), ( 27), ( 53) and (54), the three dimensional variational problem (13), written on the fixed domain, is now,

+ D3(£)£-1 £3(\j/(E))]d.Q = f �J;v;d.Q + Jr.± gfv;dI.± (55)
Contrary to the short-circuited plate model, it may be shown that the asymptotic electric potential is now constant in the plate thickness [START_REF] Rahmoune | Smart piezoelectric plates: modeling and applica tion to health monitoring[END_REF], i.e., when£ goes to zero, (56) Besides, in-plane components of the electric displacement converge weakly according to:

D0 (£) � Da (0) in L 2 (57)
when£ is made to vanish [START_REF] Rahmoune | Smart piezoelectric plates: modeling and applica tion to health monitoring[END_REF]. Here, in-plane electric displacement components are dominant; D3£3, the transverse electric work is negligible versus D0E0•

This equation indicates that the electromechanical prob lem cannot be uncoupled in general, and that the coupling is between in-plane components of stresses and the electric field. However for orthorhombic piezoelectric media like PZT, the electrical and mechanical problems are not coupled.

Local Problem

Integrating by parts the above two-dimensional varia tional problem leads to the following local equations:

• membrane problem -apN a p = Pa in co (61)

• bending problem in co (62)

• electric problem (63 )

Discussion of the Insulated Piezoelectric Model

The present theory shows that, in general, in-plane compo nents of the electric field couples with all in-plane strains, leading to a coupling between mechanical and electrical phe nomena.

The asymptotic electric potential was found to be inde pendent of x3 at the zeroth order in £. This potential can be identified to the "mean value" of the electric potential through the thickness of the plate.

Moreover for orthorhombic piezoelectric plates, the elec tric and mechanical problems uncouple due to the constitu tive laws. The electrical problem gives the "mean value" of the potential through the thickness of the plate. The mechani cal problem, using insulated elastic constants, gives the dis placements and as a consequence an additional electrical po tential induced by the deformation.

APPLICATIONS

Mechanical membrane-bending coupling and electro mechanical couplings depend on the symmetry and homoge neity of the piezoelectric plate. This section discusses these coupling phenomena for a homogeneous and orthorhombic piezoelectric media. In particular, such plates have no mem brane-bending coupling.

Homogeneous Piezoelectric Plates

Transversely homogeneous piezoelectric medium has uni form through thickness elastic, piezoelectric and dielectric constants, i.e., independent of coordinate x3. Hence, previous results can be simplified. In particular the variational prob lems are symmetric.

SHORT-CIRCUITED PIEZOELECTRIC PLATE

The induced electric field defined by Equation (38), re duces to, (64) Thus, the induced potential (39) is now simply, (65) Equations ( 64) and (65) show that both induced electric field and potential are proportional to bending strains. The former is linear in x3, whereas the latter is parabolic.

From these relations, generalized constitutive equations Since there is no membrane-bending coupling, two uncou pled mechanical problems can be defined:

• a membrane problem, for which:

(67)

• a bending problem, for which:

(68 )

The electric displacement is still defined by equation ( 48).

For constant prescribed electric field E3(rp 0 ), Equations (42) reduce to:

(69) Therefore, the piezoelectric coupling induces modifications of the classical plate model:

• a membrane constitutive law using short-circuited plane stress elastic constants

• a bending constitutive law using insulated plane stress elastic constants

• additional generalized electric in-plane forces N g p ( tp 0 ) proportional to the prescribed electric field E 3( tp 0 ) for the membrane problem

• no additional generalized electric moments M 2 p ( tp 0 ) rela tive to the prescribed electric field E 3( tp 0 ) but an additional potential proportional to bending So, to solve the problem for the short-circuited thin plate, one calculates the prescribed electric potential tp 0 , then the displacements s of the plate with modified constitutive laws, then the induced potential <I>*• If it is desired to solve the variational problem using finite element codes, one can have trouble introducing different elastic constants for membrane and bending. This can be overcome by replacing the piezo electric plate by an equivalent three layered plate having usual constants for plates.

INSULATED PIEZOELECTRIC PLATE

The two-dimensional electromechanical variational prob lem is that defined in Equation ( 60), for which generalized constitutive equations read now:

(70) (71)

Here, the electromechanical variational problem (60) does not uncouple. Moreover the coupling is between the in-plane components of membrane strain or stresses and plane com ponents of electric field or displacement. From Equations ( 69), ( 70) and (71) it is clear that bending strains could not be correlated to any electric quantity measured on the plate sur face.

Orthorhombic Homogeneous Piezoelectric Plates

Actually, commonly used piezoelectric materials are transverse isotropic (like PZT) or orthorhombic (like PVDF). That is the reason why we focus here on ortho rhombic piezoelectric materials for which simplifications occur. These materials have the following short-circuited elastic matrix, Piezoelectric material constants depend on the direction of the material axes and electric polarization. Here, the former is supposed to be parallel to the reference axes and the latter is supposed perpendicular to the plate. Modified material constants, taking account of the piezoelectric effect in a clas sic mechanical problem, are now presented.

SHORT-CIRCUITED PIEZOELECTRIC PLATE 1. For an extension problem, the plane stress elastic con stants are,

_ C C 33Ali C 33ap C aPAli -aPAli ---- C 3333
where Cij k l are the short-circuited elastic constants. which can be used to find the constant potential. This result is similar to that given in [START_REF] Maugin | An asymptotic theory of thin piezoelec tric plates[END_REF].

CONCLUSION

The present theory shows that, for thin homogeneous pi ezoelectric plates, a priori consistent mechanical hypotheses are:

1. Only in-plane stresses have to be retained (cr3 i = 0). 2. The transverse electric displacement is constant through the plate thickness. 3. Mechanical displacement field is of Kirchhoff-Love type;

i.e., in-plane and transverse components are, respectively, linear and constant through thickness. 4. The electric potential is the sum of two potentials <p e and 'Pi :

• The former 'Pe• which vary linearly through the thick ness of the plate, depends on the electrical boundary conditions. • The latter 'Pb which vary quadratically through the thickness of the plate, does not depend on the electrical boundary conditions. However, a priori, other electric assumptions depend on the electric boundary conditions. They are:

• for a short-circuited plate: The present theory also indicates that bending strains could not be correlated to any measurable or applied electric quantity on piezoelectric plate surfaces.

These results make it possible to develop a similar plate theory for laminates with some of the layers being piezoelec-tric. Moreover, classical finite element codes can be used to find approximations of multilayer plate problems with pi ezoelectric layers. In practice, it is possible to solve a me chanical problem for the displacements; then the potential is obtained explicitly from these displacements.

For plates continuously heterogeneous through thickness, the results are more complicated.

Electric

  Potential and Charge Scaling for the Short-Circuited Plate

  that, when E goes to zero,

  ) is defined in Equation (24). Since Kk1(E) and Fk(<j>(E)) are bounded, then when E tends to zero, electric displacement components converge weakly to (26) Classical scaling of the stresses is defined as(Ciarlet and Destuynder, 1979), O'�� = E20'a�(E) , 0'�3 = E 3 0'a3(E) , 0'�3 = E40'33(E)(27)

  Two-Dimensional Variational Problem Let's set v a (x1, x2, x3) = lla(Xi. x2)-x3aa113(xi. x2) , v3(xi. x2, x3) = 113(xi. x2) and 'I'= 0 in the variational problem (28) and e tends to zero, in order to obtain where and O'ap(O) is the limit of O'a�(E), given by, in which are modified elastic and piezoelectric constants due to cr;3 = 0. The 3 x 3 matrix (S3j3 k ) is the inverse of the 3 x 3 matrix (C3j3 k ) • Stress resultants Nap and moments Map are now intro duced: f+I Map = _ 1 X30'ap(O) dx3 (32) then included in Equation (30) to get the following two dimensional variational problem: (33) where To prove the importance of the unknown potential <I>*, we shall first show that the unknown electric field £3( <!>*) [Equa tion (31)] may be expressed in terms of bending strains. To this end, let us set v = 0 and 'l' o = 0 in Equation (28), then mul tiplying by e, and make e vanish to obtain f n [e 3 ; k/1Ck1(E,) + E 33 F3 (<l>*) ]F3 (X)dQ = 0 'r:fx E VQ (34) From Equations (27) and (29), K; 3 can be expressed in terms of K ap and F3, (35) These are included in Equation (34 ), which is integrated by parts in order to get (36) where E values are modified dielectric constants due to c;i3 = 0. From the above equation, the unknown electric field £3( tj>*)

  32) of the normal and moment stress re sultants, we get,(41) where N � P and M � P are related to the applied potential cp0:

  Nip and M ip include the effect of induced electric field.They are linked to membrane and bending strains through the following modified constitutive equations:N * -cm cmb* iJ io apap/i'-Y o'--aPo'-'6'-':> 3 aPli'-, ap/i'-, apo-. , a Pli'-are mem rane, mem ranebending and bending modified elastic plane stress constants. They are given by:C mb* f +l c* d ap/iA = -I ap/iA X3 (44)It's worthwhile noticing that modified constitutive equations (43) are not symmetric due to membrane-bending elastic constant because of the piezoelectric coupling.

  Two-Dimensional Variational Problem Let's set v0(£) = T)0(x1, x2)-x3a0113(x1, x2) , v3(£) = TJ3(x1, x2) and 'Jf(E) = 'Jf(x1, x2) in the variational problem (55) and tend£ to zero, in order to get, (58) where �a l30 J.. , !!J.. ;ap are � odified elastic and piezoelectric constants due to cr;3 = 0 andD3,3 = 0. They are given in the Appendix.Introducing the following electric charge : . a are modified dielectric constants given in the Ap pendix), together with stress resultants (32) and generalized forces (33) in the above three-dimensional variational prob lem, reduces it to the following two-dimensional one,

  (44) . Notice that only bending con stants are modified by the piezoelectric effect. In fact C��x. o constants are insulated plane stress elastic constants.

  C � PA ii are the insulated elastic constants.The only difference with the homogeneous case is a simplification of the plane stress elastic constants.INSULATED PIEZOELECTRIC PLATESince modified piezoelectric constants g A ; ap are nil here, there is no coupling between in-plane components of stresses and electric field. The mechanical problem and the electric problem are not coupled. One obtains a classical elastic plate model with plane stress elastic constants having for mem brane and bending the expressions: where= c ; _ C �3"o c � 3a P � aPAli aPAli C i 3333The electrical potential <p is then the solution of the follow ing problem: !!. <p = 0 in ro, a ip = 0 on 'Y = aro an (75)

  (a) Only transverse components of the electric field and displacement have to be retained. (b) The electric potential <p e is known explicitly from the prescribed potentials on the upper and lower faces. (c) The electric potential <p; is known explicitly from the flexural displacement and is called the induced poten tial. • for an insulated plate (a) Only in-plane components of the electric field and dis placement have to be retained. (b) The electric potential 'Pe is constant through the thick ness and is only coupled to the membrane displace ment; moreover, for an orthorhombic plate, this poten tial does not depend on the displacement. (c) The electric potential 'Pi is known explicitly from the flexural displacement and is called the induced poten tial.

QE, of thickness 2£ and characterized by its elastic, piezo-

ACKNOWLEDGEMENT

The first author gratefully acknowledges the DGA/DSP/ STTC/MA for the partial financial support of this work.

APPENDIX Modified Constants for the Insulated Piezoelectric Plate

Modified material constant used in Equations ( 58) and ( 59) are defined by, where