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Thibaut Weller, Christian Licht

To cite this version:

Thibaut Weller, Christian Licht. Asymptotic modeling of thin piezoelectric plates. Annals of Solid

and Structural Mechanics, Springer Berlin Heidelberg, 2010, 2 (2-4), pp.87-98. �10.1007/s12356-010-

0013-1�. �hal-00742903�

Thibaut Weller , Christian Licht

Laboratoire de Mécanique et Génie Civil, UMR-CNRS 5508, c.c. 048, Université Montpellier II, Place Eugène Bataillon 34095 Montpellier Cedex 5, France

{weller, licht}@lmgc.univ-montp2.fr

Abstract. We study the asymptotic behavior of a three dimensional flat, heterogeneous and anisotropic piezoelectric body when its thickness - seen as a parameter - goes to zero. Depending on the type of the electrical loading two models are obtained which are related to the plate used as a sensor or as an actuator. These models are explicitly derived in any piezoelectric crystal symmetry class. For some of them, astriking structural switch-off of the piezoelectric effect occurs. The static case is solved through a unifying approach using techniques of singular perturbation while the transient situation is formulated in terms of evolution equations in Hilbert spaces of possible states with finite electromechanical energy, so that the study of these transient problems areeasily deduced from the static casethrough the Trotter theory of convergence of semi-groups of operators acting on variable spaces.

1 Introduction

The interest of an efficient modeling of the dynamic response of piezoelectric plates lies in the fact that a major technological application of piezoelectric effect is the control of vibrations of structures through very thin patches. But, if many studies have been devoted to their static behavior (see for instance the introduction in [2]), much less attention have been paid to the dynamic response. We are only aware of the substantial but rather complex modeling of [11]. It seems to us that this complexity stems from both the taking into account of the magnetic effects and its mathematical treatment. The latter involves a variational evolution equation in terms of auxiliary variables like the electrical and the magnetic vector potentials and not in terms of the root variables, i.e. the electric and magnetic fields. Moreover, because of the large discrepancy between the celerities of the mechanical and electromechanical waves, magnetic effects can indeed be ignored.

That is why we propose a modeling in the appropriate framework of the quasi-electrostatic approximation through the theory of semi-groups of linear operators acting on variable spaces. Since the Trotter result of convergence of semi-groups claims that the study of the convergence of the transient problems reduces to the static case, we have choosed to first revisit the static modeling of thin piezoelectric plates in an unified way. To this aim, we extend the mathematical derivation of the asymptotic behavior of a linearly elastic plate exposed in [4] to the linearly piezoelectric case. The keypoint is to consider that the thickness of the flat piezoelectric body is aparameter. We study the behavior of the solution of the physical problem when this parameter goes to zero. Our modeling is derived from the limit behavior, so that the thinner the plate the sharper the modeling. It is also efficient from the computational point of view because it involves two-dimensional problems.

Indeed, depending on the boundary conditions, two models are obtained which correspond to the cases when the plate acts as a sensor or as an actuator. Because the piezoelectric effect is significantly increased in some composite materials, our motivation is to treat realistic situations (general heterogeneous and anisotropic piezoelectric materials) in order that our mathematical analysis be applicable, even if some of its aspects have already been studied in some papers for unrealistic cases (see Section 3.1).

2 The static case

As usual we make no difference between the physical space andR³and, for allξ= (ξ1, ξ2, ξ3)inR³, we define

ξb:= (ξ1, ξ2). (1)

In all this paper, greek indices for coordinates take their values in1,2whereas latin indices run from1to3. Let

H:=S³×R³, (2)

whereS³ denotes the set of all3×3real and symmetric matrices. For the sake of simplicity, the classical symbol^′ ·^′ will stand for the scalar product inH, S³ andR³. The set of all linear mappings from a space V into a spaceW is denoted

L(V, W). IfV =W, we simply writeL(V). In the sequel, for every domainGinR^N, the subset of the Sobolev spaceH¹(G) whose elements vanish onΓ, included in the boundary∂GofG, will be denoted by:

HΓ¹(G) :={v∈H¹(G) :v_|Γ = 0, Γ ⊂∂G}. (3)

2.1 Setting the problem

The reference configuration of a linearly piezoelectric thin plate is the closure in R³ of the set Ω^ε :=ω×(−ε, ε), where εis a small positive number and ωa bounded domain ofR² with a Lipschitz boundary∂ω. The lateral part of the plate

∂ω×(−ε, ε)is denoted Γlat^ε , while the set constituted by its lower and upper faces isΓ_±^ε :=ω× {±ε}. Let (ΓmD^ε , ΓmN^ε ), (ΓeD^ε , ΓeN^ε )two suitable partitions of ∂Ω^ε with bothΓmD^ε andΓeD^ε of strictly positive Lebesgue measure. The plate is, on one hand, clamped alongΓ_mD^ε and at an electric potential ϕ^ε₀ onΓ_eD^ε and, on the other hand, subjected to body forces f^ε and electrical loadingsF^ε inΩ^ε. ActuallyF^ε vanishes, the material being an insulator, anyway the following analysis stands withF^εdifferent from0. Moreover, the plate is subjected to surface forcesg^ε and electrical loadingsd^εonΓ_mN^ε and ΓeN^ε respectively. We noten^ε the outward unit normal to∂Ω^ε and assume thatΓmD^ε =γ0×(−ε, ε), withγ0⊂∂ω. Then the equations determining the electromechanical states^ε:= (u^ε, ϕ^ε)at equilibrium are:







divσ^ε+f^ε= 0inΩ^ε, σ^εn^ε=g^ε onΓmN^ε , u^ε= 0onΓmD^ε , divD^ε+F^ε= 0inΩ^ε, D^ε·n^ε=d^εonΓeN^ε , ϕ^ε=ϕ^ε0 onΓeD^ε , (σ^ε, D^ε) =M^ε(x)(e(u),∇ϕ)inΩ^ε,

(4)

where u^ε, ϕ^ε, σ^ε, e(u^ε)andD^ε respectively denote the displacement and electrical potential fields, the stress tensor, the linearized strain tensor and the electrical displacement. In the linearly piezoelectric framework which is studied here, we recall that the operatorM^ε is an element ofL(H)such that:

(σ^ε=Mmm^ε e(u^ε)−Mme^ε ∇ϕ^ε,

D^ε=Mme^εTe(u^ε) +Mee^ε∇ϕ^ε, (5)

where(Mmm^ε , Mme^ε , Mee^ε)∈ L(S³)× L(R³, S³)× L(R³)are respectively the elastic, piezoelectric and dielectric tensors while Mme^εT =:Mem^ε denotes the transpose ofMme^ε . We recall thatMmm^ε andMee^ε are symmetric and positive.

To give a variational formulation of (4), we first make the following regularity hypothesis on the exterior loading:

(H1) :

((f^ε, g^ε, F^ε, d^ε)∈L²(Ω^ε)³×L²(Γ_mN^ε )³×L²(Ω^ε)×L²(Γ_eN^ε ), ϕ^ε₀has anH¹(Ω^ε)³ extension intoΩ^ε still denoted byϕ^ε₀, and on the space of electromechanical states

V^ε:={r= (v, ψ)∈HΓ¹_mD^ε (Ω^ε)³×HΓ¹_eD^ε (Ω^ε)} (6)

we define a bilinear formm^ε: m^ε(r, q) =m^ε((v, ψ),(w, φ)) :=

Z

Ω^ε

M^ε(e(v),∇ψ))·(e(w),∇φ)dx^ε, (7)

and a linear formL^ε:

L^ε(r) =L^ε((v, ψ)) :=

Z

Ω^ε

(f^ε·v+F^εψ)dx^ε+ Z

Γ_mN^ε

g^ε·v ds^ε+ Z

Γ_eN^ε

d^εψ ds^ε. (8)

Thephysical problem, set on thereal plate, then takes the form

P(Ω^ε) : Finds^ε= (u^ε, ϕ^ε)∈(0, ϕ^ε₀) +V^εsuch thatm^ε(s^ε, r) = L^ε(r), ∀r∈V^ε.

Thus, with the additional and realistic assumptions of boundedness ofMmm^ε ,Mme^ε ,Mee^ε and of uniform ellipticity ofMmm^ε

andM_ee^ε:

(H2) : M^ε∈L^∞(Ω^ε,L(H)),∃κ^ε>0 : M^ε(x^ε)h·h≥κ^ε|h|²H,∀h∈ H, a.e. x^ε∈Ω^ε, the theorem of Stampacchia (cf.[3]) implies the

Theorem 1. Under assumptions(H₁)-(H₂), the problemP(Ω^ε) has a unique solution.

To derive a simplified and accurate model, the very question is to study the behavior ofs^ε when ε,considered as a parameter, tends to zero. We will show that, depending on the type of boundary conditions, two limit behaviors, indexed bypwith value 1 or 2, can be obtained (see [17]).

2.2 The scaling operation

Classically (see [4]), we come down to a fixed open setΩ:=ω×(−1,1)through the mappingπ^ε:

x= (bx, x3)∈Ω 7→ π^εx= (bx, εx3)∈Ω^ε. (9)

Also, we drop the indexεfor the image by(π^ε)⁻¹ of the previous geometric sets. To get physically meaningful results, we have to make various kinds of assumptions. They deal with

1. the electromechanical coefficients:

(H3) : M^ε(π^εx) =M(x)withM ∈L^∞(Ω,L(H)),∃κ >0 : M(x)h·h≥κ|h|²H,∀h∈ H, a.e. x∈Ω, 2. the electromechanical loading:

(H4) :





















there exists(f, F, g, d)∈L²(Ω)³×L²(Ω)×L²(ΓmN)³×L²(ΓeN);

fα^ε(π^εx) =ε fα(x), f3^ε(π^εx) =ε²f3(x), F^ε(π^εx) =ε^2−pF(x),∀x∈Ω, gα^ε(π^εx) =ε²gα(x), g^ε₃(π^εx) =ε³g3(x),∀x∈ΓmN∩Γ_±,

gα^ε(π^εx) =ε gα(x), g₃^ε(π^εx) =ε²g3(x),∀x∈ΓmN ∩Γlat

d^ε(π^εx) =ε^3−pd(x),∀x∈ΓeN∩Γ_±, d^ε(π^εx) =ε^2−pd(x),∀x∈ΓeN∩Γlat, ϕ^ε₀(π^εx) =ε^pϕ0(x),∀x∈ΓeD.

3. the boundedness of the "work of the exterior loading":

(H5) :







p= 1: the extension ofϕ0 intoΩdoes not depend onx3.

p= 2: the closureδof the projection ofΓeD onωcoincides withω, and eitherd= 0onΓeN∩ΓlatorΓeN∩Γlat=∅.

Also, we associate a scaled electromechanical state sp(ε) := (up(ε), ϕp(ε)) =: Πp^εs^ε defined on Ω with the true physical electromechanical states^ε= (u^ε, ϕ^ε)defined onΩ^ε:

c

u^ε(x^ε) =ε(\up(ε))(x), u^ε₃(x^ε) = (up(ε))3(x), ϕ^ε(x^ε) =ε^pϕp(ε)(x),∀x^ε=π^εx∈Ω^ε. (10) Assumptions(H3),(H4)together with the scaling operation (10) are classical. Actually, they are justified by the con- vergence results they lead to. If we just consider the displacement, these hypotheses are the ones made in [4] and supply a mathematical justification of the Kirchhoff-Love theory of thin linearly elastic plates.

Remark 1. In the following, ifEstands for any function space defined onΩ, the same function space - but defined onΩ^ε - will be denoted byE^εand vice-versa.

2.3 Variational formulation of the scaled problem LetVbe the space of scaled electromechanical states:

V:={r= (v, ψ)∈HΓ¹_mD(Ω)³×HΓ¹_eD(Ω)}. (11)

Of course,r ∈V^ε ⇐⇒Πp^εr ∈ V. Now, for all r = (v, ψ)∈ V, we define the scaled strain tensor e(ε, v)and the scaled electrical potential gradient∇(p)(ε, ψ)by:

eαβ(ε, v) :=eαβ(v), eα3(ε, v) :=ε⁻¹eα3(v), e33(ε, v) :=ε⁻²e33(v),∇^(p)(ε, ψ)α:=ε^p−1∂αψ,∇^(p)(ε, ψ)3:=ε^p−2∂3ψ. (12)

To simplify the notations, we set

k(r) := (e(v),∇ψ), kp(ε, r) := (e(ε, v),∇(p)(ε, ψ)), (13)

and, as in (7) and (8) we introduce a bilinear formmp(ε)and a linear formLonV:

mp(ε)(r, q) :=

Z

Ω

M(x)kp(ε, r)·kp(ε, q)dx, L(r) :=

Z

Ω

(f·v+F ψ)dx+ Z

Γ_mN

g·v ds+ Z

Γ_eN

d ψ ds, (14)

so that under asumptions(H1)−(H4), the scaled electromechanical state sp(ε) = (up(ε), ϕp(ε))is the unique solution of the mathematical problem:

P(ε, Ω)p : Findsp(ε)∈(0, ϕ0) +Vsuch thatmp(ε)(sp(ε), r) =L(r),∀r∈V.

2.4 Asymptotic behavior of the scaled electromechanical state The process

To generalize the method described in [4], we will show that some components of kp(ε, sp(ε))and ofM kp(ε, sp(ε))have vanishing limits when ε goes to 0. In the purely mechanical case, we recall that the classical result is that the limit displacementv is of Kirchhoff-Love type, i.e. such that ei3(v) = 0. In fact it is possible to generalize this idea to multi- physical couplings by observing that a fundamental role is played by thealgebraicstructure of the spaceHdefined in (2):

with the scaling operation, various powers of ε appear in kp(ε, .) (see (13)); their signs suggest a suitable orthogonal decomposition ofHin the following subspaces:

H⁻1 :={h= (e, g) : eαβ = 0, gα= 0},H⁻2 :={h= (e, g) : eαβ= 0, gi= 0}, H⁰1 :={h= (e, g) : ei3= 0, g3= 0} ,H⁰2 :={h= (e, g) : ei3= 0, gα= 0}, H⁺1 :={h= (e, g) : eij= 0, gi= 0} ,H⁺2 :={h= (e, g) : eij= 0, g3= 0} .

(15) This process is similar to the one of [14]. For a givenp∈ {1,2},Mcan then be decomposed in nine elementsMp^⋆⋄∈ L(H^⋄p,H^⋆p) with⋆,⋄ ∈ {−,0,+}. Hypothesis(H3)on the electromechanical coefficients implies thatMp⁰⁰etMp⁻⁻are positive operators onH^0pandH^−p. Therefore, the Schur complement

Mfp:=Mp⁰⁰−Mp⁰⁻(Mp⁻⁻)⁻¹Mp⁻⁰ (16)

is an element ofL(H⁰p). It is important to note that neitherMp⁰⁰norMfpare necessarily symmetric, but nevertheless κ|h⁰p|²H6Mfp(x)h⁰p·h⁰p, ∀h⁰p∈ H⁰p, a.e. x∈Ω. (17) This is implied by the coercivity ofM (see(H3)) and by thefundamental relation:

(M h)⁻p =h⁺p = 0⇒

(Mfph⁰p= (M h)⁰p,

Mfph⁰p·h⁰p=M h·h. (18)

The key point of the asymptotic study is to show that if kp is the limit (in a suitable topology) of kp(ε, sp(ε)), then (M kp)⁻p = (kp)⁺p = 0! This will enable us to exhibitMfp as the operator governing the limit constitutive equations.

Functional framework

We will show that the limit displacements live in the spaceVKL of Kirchhoff-Love displacements

VKL:={v∈HΓ¹_mD(Ω)³:ei3(v) = 0}, (19)

while the limit electrical potential belongs to

Φ1:={ψ∈HΓ¹_eD(Ω) :∂3ψ= 0}orΦ2:={ψ∈H∂¹₃(Ω) :ψ_|ΓeD∩Γ±≡0}. (20)

We recall that for allv∈VKL, there exists a unique couple(v^M, v^F)∈Hγ¹₀(ω)²×Hγ²₀(ω)such that:

b

v(x) =v^M(x)b −x3∇bv^F(x),b v3(x) =v^F(x).b (21)

and we introduce

V^MKL:={v∈VKL:v^F = 0}, V^FKL:={v∈VKL:v^M = 0}. (22) The space

H∂¹3(Ω) :={ψ∈L²(Ω) :∂3ψ∈L²(Ω)} (23)

equipped with the scalar product:

(ϕ, ψ)7→

Z

Ω

ϕ ψ dx+ Z

Ω

∂3ϕ ∂3ψ dx. (24)

is an Hilbert space. The trace mapping being linear and continuous fromH_∂¹₃(Ω) to L²(Γ_±), the definition (20) of Φ2 is meaningful. Thus, with the assumption(H5),Φ2 can be equipped with the scalar product:

(ϕ, ψ)7→

Z

Ω

∂3ϕ ∂3ψ dx (25)

equivalent to the one defined by (24). Finally, let

Sp:=VKL×Φp, Xp:=

(H_Γ¹_mD(Ω)³×H¹(Ω), ifp= 1,

HΓ¹_mD(Ω)³×H_∂3¹ (Ω), ifp= 2. (26)

The Korn and Poincaré inequalities allow us to define onSpandXpthe hilbertian norms

|(v, ψ)|^2S1 :=|e(v)|²L²(Ω)+|∇ψ|²L²(Ω) ,|(v, ψ)|^2S2 :=|e(v)|²L²(Ω)+|∂3ψ|²L²(Ω),

|(v, ψ)|^2X1 :=|e(v)|²L²(Ω)+|ψ|²L²(Ω)+|∇ψ|²L²(Ω),|(v, ψ)|^2X2 :=|e(v)|²L²(Ω)+|ψ|²L²(Ω)+|∂3ψ|²L²(Ω). (27) The set(0, ϕ0) +Spwill appear to be the limit set of electromechanical states.

Remark 2. Becausek(r)is not rigorously defined whenr= (v, ψ)belongs toX2 (see definitions (13), (15), (23) and (26)), we are led to slighty abuse of the notations by letting

k(r)⁰₂= ((eαβ(v),0),(0, ∂3ψ)). (28)

The two limit scaled problems We have the following convergence result:

Proposition 1. Under assumptions(H₃)−(H₅), and whenε→0, the family(sp(ε))ε>0 of the unique solutions ofP(ε, Ω)p

strongly converges inX_pto the unique solutionspof:

P(Ω)p:Finds∈(0, ϕ0) +S_psuch thatmep(s, r) :=

Z

Ω

Mfpk(s)⁰p·k(r)⁰pdx=L(r),∀r∈S_p. Furthermore,limε→0mp(ε)(sp(ε), sp(ε)) =mep(sp, sp).

Proof. It is divided in five steps.

First step : the family(sp(ε))ε is bounded inXp.

We may assumeε≤1. For allr∈V,esp(ε) :=sp(ε)−(0, ϕ0)satisfies:

mp(ε)(sep(ε), r) =L(r) :=e Z

Ω

(f·v+F ψ)dx+ Z

Γ_mN

g·v ds+ Z

Γ_eN

d ψ ds−mp(ε)((0, ϕ0), r). (29) Thus, assumption(H3)on the electromechanical coefficients implies:

κ|esp(ε))|^2Xp≤κ|k(ε,esp(ε))|²L²(Ω,H)≤mp(ε)(esp(ε),esp(ε)) =L(esep(ε)). (30) To establish the boundedness ofesp(ε) inXp, it suffices to show that there exists a constantcwhich does not depend onε such that

L(e esp(ε))≤c|k(ε,esp(ε))|L²(Ω,H). (31)

Bounding the four terms ofL(e esp(ε))- linked to the work of the exterior loading - is obvious except the third in the casep= 2 and the fourth ifp= 1. Indeed,(H5)implies ¹_ε∂3ϕ0 = 0, therefore |k1(ε,(0, ϕ0))|^L2(Ω) is uniformly bounded with respect toε; assumption(H5)deletes the problem of boundingR

Γ_eNd(ϕ(ε)−ϕ0)ds. Hence,ϕ0being fixed, the family(sp(ε))_0<ε≤1 is bounded in the Hilbert spaceXpand so there exists a subsequence, not relabelled, such that:

(sp(ε), k(ε, sp(ε))⇀(sp, kp)inXp×L²(Ω,H), (32)

k(sp(ε))⁻p →0inL²(Ω,H), (33)

k(sp)⁰p = (kp)⁰p, (34)

where the⇀and→symbols respectively denote weak and strong convergences.

Second step : to identify the operator providing the limit constitutive law, we establish that(M kp)⁻p = (kp)⁺p = 0.

To get(M kp)⁻p = 0, we generalize the method introduced in [4] in the case of linearly isotropic elastic plates. In the equation associated withP(ε, Ω)p, we choose:

i) rsuch thatv3=ψ= 0and multiply byε, ii) rsuch thatvα= 0andψ= 0and multiply byε², iii) rsuchv= 0and multiply byεifp= 1only,

and, by going to the limit, we conclude with the lemma of [4]:

Lemma 1. Ifw∈L²(Ω)satisfiesR

Ωw ∂3v dx= 0, for allvinC^∞(Ω)such thatv= 0onγ×[−1,1], thenw= 0.

From its very definition(k1)⁺₁ vanishes, while(k2)⁺₂ = 0stems from a classical argument (see [15] for the details) in going to the limit in the identity

Z

Ω

ε∂αϕ(ε)g dx=−ε Z

Ω

ϕ(ε)∂αg dx,∀g∈C₀^∞(Ω). (35)

Therefore, using (18), we have

M kp·kp=M(kf p)⁰p·(kp)⁰p, (M kp)⁰p=Mfp(kp)⁰p. (36) Third step : the limit problem.

For allr∈Sp∩C^∞(Ω,R⁴),kp(ε, r)strongly converges tok(r)⁰pinL²(Ω,H). Therefore, by passing to the limit inP(ε, Ω)p

with (32), (36) and (34), we get

L(r) = Z

Ω

M kp·k(r)⁰pdx= Z

Ω

(M kp)⁰p·k(r)⁰pdx= Z

Ω

Mfp(kp)⁰p·k(r)⁰pdx= Z

Ω

Mfpk(sp)⁰p·k(r)⁰pdx. (37)

The density of Sp∩C^∞(Ω,R⁴) inSp implies that sp solvesP(0, Ω)p. The bilinear form mep beingSp-elliptic (see (17)), the problem P(0, Ω)p has aunique solution. Thus, by a classical compacity argument, the whole sequencesp(ε) weakly converges tosp.

Fourth step : strong convergence.

By (32)-(34), the strong convergence ofsp(ε)tospinXpis equivalent to the one ofk(sp(ε))⁰ptok(sp)⁰pinL²(Ω,H). But we have

κ|k(sp(ε))⁰p−k(sp)⁰p|²L²(Ω,H) ≤ κ|k(ε, sp(ε))−kp|²L²(Ω,H)≤ Z

Ω

M(k(ε, sp(ε))−kp)·(k(ε, sp(ε))−kp)dx

= Z

Ω

M(k(ε, sp(ε)))·(k(ε, sp(ε)))dx− Z

Ω

M kp·(k(ε, sp(ε))dx− Z

Ω

M(k(ε, sp(ε))−kp)·kpdx

=L(esp(ε)) + Z

Ω

M(k(ε, sp(ε)))·(k(ε,(0, ϕ0)))dx− Z

Ω

M kp·(k(ε, sp(ε))dx− Z

Ω

M(k(ε, sp(ε))−kp)·kpdx, (38) and from (32) and (36), we deduce that, asε→0, the right member converges to

l=L(sp−(0, ϕ0))+

Z

Ω

M kp·k(0, ϕ0)dx− Z

Ω

M kp·kpdx=L(sp−(0, ϕ0))+

Z

Ω

Mfpk⁰p·k(0, ϕ0)⁰pdx− Z

Ω

Mfpk⁰p·k⁰pdx= 0. (39) Fifth step :limε→0mp(ε)(sp(ε), sp(ε)) =mep(sp, sp).

It suffices to observe that (32),(H5)and (36) imply

εlim→0mp(ε)(sp(ε),(0, ϕ0)) = Z

Ω

M kp·k(0, ϕ0)⁰pdx=mep(sp,(0, ϕ0)). (40)

2.5 Variants to Proposition 1

It is interesting to consider the cases when the elementsf, F, g, dandϕ0are depending onεor when the linear formLdefined in (14) is a more abstract mathematical object than the electromechanical loading. In this direction, a careful examination of the preceding proof leads to the

Corollary 1. Let(Λε)ε>0 a family of continuous linear forms onX_p which converges weakly toΛand(ϕ0ε)ε>0 ⊂H¹(Ω) such that

(H₅)^′:

(∃ϕ0∈H¹(Ω)if p= 1, H_∂3¹ (Ω)if p= 2,∃k0 ∈L²(Ω,H);

(ϕ0ε, k(ε, ϕ0ε))⇀(ϕ0, k0) inL²(Ω)×L²(Ω,H) with(k0)⁰p= (k(0, ϕ0))⁰p, then, whenε→0, the family(s^′p(ε))ε>0 of the unique solutions of

Finds∈(0, ϕ0ε) +Vsuch thatmp(ε)(s, r) =Λε(r),∀r∈V. weakly converges inX_pto the unique solutions^′pof

Finds∈(0, ϕ0) +S_p such thatmep(s, r) =Λ(r),∀r∈S_p.

In addition, if the previous convergences of the data are strong then the familly(s^′p(ε))ε>0 converges strongly inX_p and

ε→0limmp(ε)(s^′p(ε), s^′p(ε)) =mep(s^′p, s^′p). (41)

For instance, whenΛε is the scaled work:

Λε(r) =Lε(r) = Z

Ω

(fε·v+Fεψ)dx+ Z

Γ_mN

gε·v ds+ Z

Γ_eN

dεψ ds, (42)

withdε= 0onΓeN∩Γlat orΓeN∩Γlat=∅ifp= 2, the weak convergence ofΛεis implied by

(fε, Fε, gε, dε)⇀(f, F, g, d)inL²(Ω)³×L²(Ω)×L²(ΓmN)³×L²(ΓeN). (43) Actually, whenp= 1, this assumption yields the strong convergence ofΛε which is obtained ifp= 2with the additional assumption of strong convergence of(Fε, dε)inL²(Ω)×L²(ΓeN). It is worthwile to note that(H^′5)implies the weak (resp.

strong) convergence ofϕ0ε inH¹(Ω)with∂3ϕ0= 0ifp= 1 or inH_∂3¹ (Ω)with(k0)⁺p = 0ifp= 2.

It is also interesting to consider a perturbation of the bilinear formmp(ε). In fact, the following corollary will play a crucial role in the study of the dynamic case. Let us introduce the assumption

(H^decouplp ) : Z +1

−1

x3Mf1dx3= 0, Mf2(x, xb 3) =Mf2(bx)withΓ^±⊂ΓeD, and the bilinear forms

(v, w)∈L²(Ω)² 7→kl(ε)(v, w) :=ε^2(l−1)R

Ω(bv·wb+ε⁻²v3w3)dx, l= 1,2, (v, w)∈V^M_KL² 7→ ek1(v, w) :=R

Ωvb·w dx,b (v, w)∈V^F_KL² 7→ ek2(v, w) :=R

Ωv3w3dx,

(44)

and let

V1:=V^MKL, W1:=V^FKL, V2:=V^FKL, W2:=V^MKL. (45) We have the

Corollary 2. Letλ∈R⁺ andΛa continuous linear form onX_psuch thatΛ(W_l) = 0. Then, under assumptions(H₃)and (H^decoupl_p ), the familly(s^′′p,l(ε))ε>0 of the unique solutions of

Finds= (u, ϕ)∈Vsuch thatkl(ε)(u, w) +λmp(ε)(s, r) =Λ(r),∀r= (w, ψ)∈V, converges strongly inX_pto the unique solutions^′′p,lof

Finds= (u, φ)∈V_l×Φp such thatekl(u, w) +λmep(s, r) =Λ(r),∀r= (w, ψ)∈V_l, and, of course,

εlim→0kl(ε)(u^′′p,l(ε), u^′′p,l(ε)) +λmp(ε)(s^′′p,l(ε), s^′′p,l(ε)) =ekl(ε)(u^′′p,l, u^′′p,l) +λmep(s^′′p,l, s^′′p,l). (46)

Actually, the additional propertys^′′p,l∈Vlis deduced from assumption(H^decouplp )which permits (see section 2.7) to exploit Λ(Wl) = 0.

2.6 Back to the problem P(Ω^ε): a proposal of simplified and acurate modeling

We now come back to the reference configurationΩ^ε of the real plate of thickness2εthrough the operatorsπ^εand(Πp^ε)⁻¹ (see (9) and (10)). With the solutionspofP(Ω)pis associated aphysical electromechanical states^εp defined onΩ^εby:

s^εp(π^εx) := (Πp^ε)⁻¹sp(x),∀x∈Ω. (47)

This electromechanical state is the solution of a problem posed over Ω^ε which is the transportation byπ^ε of the (limit scaled) problemP(Ω)p. This transported problem, set onΩ^ε, is our proposal to model thin linearly piezoelectric plates of thickness2ε. The functions^εp represents an approximation of the electromechanical states^ε inside the plate. It remains to show that this approximation is acurate. In this direction, we let

Mfp^ε(π^εx) :=Mfp(x),∀x∈Ω, (s, r)∈V^ε2 7→me^εp(s, r) :=

Z

Ω^ε

Mfp^ε(x)k(s)⁰p·k(r)⁰pdx. (48) Proposition 1 implies the

Theorem 2. Under assumptions(H₃)−(H₅), the couples^εp=: (u^εp, ϕ^ε_p)constituted by the limit ("descaled") displacement and electrical potential defined over the physical plateΩ^ε is the unique solution of the problem:

P(Ω^ε)p : Finds∈(0, ϕ^ε0) +S^ε_p such thatme^εp(s, r) =L(r),∀r∈S^ε_p.

Furthermore, the electromechanical state s^εp is asymptotically equivalent to the unique solution s^ε of the genuine physical problemP(Ω^ε)in the sense that:

εlim→0ε⁻¹ Z

Ω^ε

ε⁻²|(u^εp)α−u^εα|²+|(u^εp)3−u^ε₃|²dx^ε= 0, lim

ε→0ε⁻³ Z

Ω^ε|e^εαβ((u^εp)α)−e^εαβ(u^ε)|²dx^ε= 0,

ε→0limε⁻³ Z

Ω^ε|ϕ^ε₁−ϕ^ε|²+|∂α^εϕ^ε₁−∂^εαϕ^ε|²dx^ε= 0, lim

ε→0ε⁻⁵ Z

Ω^ε|ϕ^ε₂−ϕ^ε|²+ε²|∂₃^εϕ^ε₂−∂^ε₃ϕ^ε|²dx^ε= 0, ε⁻³

Z

Ω^ε

|ei3(u^ε)|²dx^ε, ε⁻³ Z

Ω^ε

|∂α^εϕ^ε|²dx^ε and ε⁻³ Z

Ω^ε

|∂3^εϕ^ε|²dx^ε are bounded.

We emphasize on the following points (see [17]):

1. the first model, withϕ^ε₀= 0, deals with the physical situation when the plate is used as asensor, 2. the second model corresponds to anactuator.

2.7 Bidimensional limit equations. Decoupling

To simplify the notations (and to be realistic), we make the additional assumption

(H6) : F^ε= 0and there existsγeN⊂∂ωsuch thatΓeN =γeN×(−1,+1).

LetγmN =γ\γ0, we consider the functionsg_i^ε±,d^ε±,p^εi,q^εα,ri^ε,s^εα,p^′ε,r^′ε∈L²(ω)defined by

g^ε±_i (x) :=b

(gi^ε(x,b±ε)ifx∈Γ_mN^ε ∩Γ_±^ε

0in the other cases , d^ε±(bx) :=

(d^ε(bx,±ε)ifx∈Γ_eN^ε ∩Γ_±^ε

0in the other cases , (49)

and

p^εi :=R+ε

−εfi^εdx+g_i^ε++g^ε−_i , q^εα :=R+ε

−ε xfα^εdx+g^ε+α −g^ε−α , r^εi :=R+ε

−ε g^εidx, s^εα:=R+ε

−εxg^εαdx , p^′ε:=d^ε++d^ε− , r^′ε:=R+ε

−ε d^εdx. (50)

Case p= 1 We define

L^MKL^ε(v^M) :=R

ωp^εαv^Mα dbx+R

γ_mN[r^εαvα^M−s^εαv^Mα ]dbx, L^FKL^ε(v^F) :=R

ωp^ε₃v^Fdxb−R

ωqα^ε∂αv^Fdxb+R

γ_mNr₃^εv^Fdbx, L^εe(ψ) :=R

ωp^′εψ dxb−R

γ_eNr^′εψ dbx, (51)

for allv= (v^M, v^F)∈V^ε_KL and allψ∈Φ^ε₁. The limit spaceS^ε₁ is the direct sum of the two subspaces

S^M₁ ^ε :=V^MKL^ε×Φ^ε₁, S^F₁^ε:=V^FKL^ε × {0}, (52) and for allr= (v, ψ)∈S^ε₁, we have

k(r)⁰₁(x) =









e11(v^M)−x3∂₁₁² v^F e12(v^M)−x3∂²₁₂v^F 0 e12(v^M)−x3∂₁₂² v^F e22(v^M)−x3∂²₂₂v^F 0

0 0 0



,





∂1ψ

∂2ψ 0









=









e11(v^M)e12(v^M) 0 e12(v^M)e22(v^M) 0

0 0 0



,





∂1ψ

∂2ψ 0







−x3









∂₁₁² v^F ∂²₁₂v^F 0

∂₁₂² v^F ∂²₂₂v^F 0

0 0 0



,



 0 0 0







=:k(v^M, ψ)⁰₁(x)b −x3D²(v^F)⁰₁(x).b (53)

Letting

Mf⁰1=Mf⁰1(x) :=b Z +ε

−ε

Mf₁^ε(x^ε)dx^ε₃, Mf¹1=Mf¹1(bx) :=

Z +ε

−ε

x^ε₃Mf₁^ε(x^ε)dx^ε₃, Mf²1 =Mf²1(x) :=b Z +ε

−ε

x^ε₃²Mf₁^εdx^ε₃, (54) the problemP(Ω^ε)1 takes the following form

P(Ω^ε)1







Find(u, ϕ)∈(0, ϕ0) +S^ε₁ such that R

ω[Mf⁰1k(u^M, ϕ)⁰₁·k(v^M, ψ)⁰₁−Mf¹1k(u^M, ϕ)⁰₁·D²(v^F)⁰₁−Mf¹1D²(u^F)⁰₁·k(v^M, ψ)⁰₁+Mf²1D²(u^F)⁰₁·D²(v^F)⁰₁]dbx

=L^M_KL^ε(v^M) +L^F_KL^ε(v^F) +L^εe(ψ),∀(v, ψ)∈S^ε₁.

It is important to note thatP(Ω^ε)1is abidimensionalproblem in the sense that it is posed overωand thatS^ε₁only involves functions defined onω. We then remark that hypothesis(H^decoupl_p ) (which is true if the electromechanical coefficients are even functions ofx3) implies a decoupling between membrane displacements and flexural displacements in the sense that they solve two independent variational equations:

P(Ω^ε)1

(Find(u^M, ϕ)∈(0, ϕ0) +S^M₁ ^ε such that R

ωMf⁰1k(u^M, ϕ)⁰₁·k(v^M, ψ)⁰₁dxb=L^MKL^ε(v^M) +L^εe(ψ),∀(v^M, ψ)∈S^M₁ ^ε, Findu^F ∈V^F_KL^ε such that R

ωMf²1D²(u^F)⁰1·D²(v^F)⁰1dbx=L^FKL^ε(v^F),∀v^F ∈S^F₁^ε,

where the second problem does not involve the electrical potential. The decisive aspect of assumption(H^decouplp ) is that it implies theme^ε₁-polarity ofS^M₁ ^ε andS^F₁^ε,i.e.:

e

m^ε₁(s^M, r^F) =me1(r^F, s^M) = 0, ∀(s^M, r^F)∈S^M₁ ^ε×S^F₁^ε. (55) Case p= 2

From its very definition,ϕ^ε₂ satisfies

∂3(Mf₂^ε_ee∂3ϕ^ε₂) =−∂3(Mf₂^ε_mebe(u^εM₂ )(x)b −x3D²(u^εF₂ )(bx)), (56) so thatϕ^ε₂(x,b·)is a second order polynomial as soon asMf₂^εdoes not depend onx3. WhenMf₂^εdepends onx3 butΓ_±^ε ⊂Γ_eD^ε , ϕ^ε₂ is alocal function ofbe(u^εM₂ )andD²(u^εF₂ ):

ϕ^ε₂=A^ε(ϕ^ε₀⁺(x)b −ϕ^ε₀⁻(bx)) +B^εe(ub ^εM₂ ) +C^εD²(u^εF₂ ), (57) where

A^ε:=a^ε−1(bx, ε)∂3a^ε, B^ε:=∂3b^ε−a^ε−1b^ε(bx, ε)∂3a^ε, C^ε:=∂3c^ε−a^ε−1c^ε(x, ε)∂b 3a^ε, a^ε(bx, x3) :=Rx3

−ε(Mf₂^ε_ee)⁻¹(bx, z)dz, b^ε(bx, x3) :=Rx3

−ε(Mf₂^ε_ee)⁻¹Mf₂^ε_me^T (bx, z)dz, ϕ^ε₀^±(bx) :=ϕ^ε₀(bx,±ε), c^ε(bx, x3) :=Rx3

−εz(Mf₂^ε_ee)⁻¹Mf₂^ε_me^T (bx, z)dz.

(58)

Hence,u^ε2satisfies Z

Ω^ε

[Mf₂^ε_mm(e(ub ^εM)−x3D²(u^εF)) +Mf₂^ε_me(B^εe(ub ^εM) +C^εD²(u^εF)]·[e(vb ^M)−x3D²v^F]dx

=− Z

Ω^ε

(Mf₂^ε_meA^ε(ϕ^ε+−ϕ^ε−))dx+L^MKL^ε(v^M) +L^FKL^ε(v^F),∀v∈V^εKL. (59) Actually, it is abidimensionnal non symmetric variational problem set on ω^ε. Moreover, with the additional hypothesis (H^decoupl₂ ), there is a decoupling between the membrane and the flexural parts of the displacement. If we let

Φ^o₂^ε:={x3-odd parts ofΦ^ε₂ elements}, Φ^e₂^ε:={x3-even parts ofΦ^ε₂ elements}, (60)

the two subspaces

S^M₂ ^ε :=V^MKL^ε×Φ^o₂^ε, S^F₂^ε:=V^FKL^ε ×Φ^e₂^ε, (61) areme2-polar in the sense of (55) where the index1is replaced by2.

These facts, presented in [17] and [19], already noted under stronger symmetry hypothesis in [10], [15] and [9], have also been observed latter in [5].

2.8 Some properties of the limit constitutive laws

It is interesting to give some properties of the operatorsMfp^ε which supply the constitutive equations of the piezoelectric plate. For a detailed discussion on this point, see [19]. Similarly to (5), we associate withMf_p^ε the sub-operatorsMf_p^ε_mm, Mfp^εem,Mfp^εme, Mfp^εee. Due to the fact that the projections fromH to H^p0 commutes with the involution idS³−id_R³, the fundamental coupling property ofM^ε remains true for the Schur complementMfp^ε:

Mfp^ε_em =−(Mfp^ε_me)^T. (62)

A handmade proof of this nice property can be found in [19]. Considering the influence of crystalline symmetries (see for example [13]), we deduce that in the case of a polarization normal to the plate:

- Mf₂^ε_mm involves mechanical terms only,

- Mf₁^ε_mm =Mf₂^ε_mm for the crystalline classesm,32,422,6,622and6m2, - Mf₁^ε_mm involves electrical terms except for these previous classes,

- whenp= 1, there is an electromechanical decoupling (Mf1^εme = 0) for the classes2,222,2mm,4,4,422,4mm,42m,6, 622,6mm and23, while whenp= 2, this decoupling occurs with the classesm,32,422,6,622and6m2, nevertheless the operators Mf_p^ε_mm and Mf_p^ε_ee involve a mixture of elastic, piezoelectric and dielectric coefficients. In these cases, Mfp^ε is symmetric which involves a quadratic convex energy.For plates made of these piezoelectric monocrystals, the piezoelectric effect disappears at the structural level!

3 Application and example: 222 crystalline class

Let’s consider a thin piezoelectric plate constituted by a material whose crystalline symmetry class is222. Then (5) takes the form:















 σ11

σ22

σ33

√2σ23

√2σ31

√2σ12

D1

D2

D3

















=

















a11a12a13 0 0 0 0 0 0 a12a22a23 0 0 0 0 0 0 a13a23a33 0 0 0 0 0 0 0 0 0 a44 0 0 −b41 0 0 0 0 0 0 a55 0 0 −b52 0

0 0 0 0 0 a66 0 0 −b63

0 0 0 b41 0 0 c11 0 0

0 0 0 0 b52 0 0 c22 0

0 0 0 0 0 b63 0 0 c33

















·















 e11(u) e22(u) e33(u)

√2e23(u)

√2e31(u)

√2e12(u) ϕ,1

ϕ,2

ϕ,3

















. (63)

Therefore, (16) leads to







 σ11

σ22

√2σ12

D1

D2









=











a11−^aa²¹³33 a12−^a13a33^a23 0 0 0 a12−^a13a33^a23 a22−^aa²²³33 0 0 0

0 0 a66+^b_c²⁶³₃₃ 0 0

0 0 0 c11+^b_a²⁴¹

44 0

0 0 0 0 c22+_a^b252

55











·







 e11(u) e22(u)

√2e12(u) ϕ,1

ϕ,2









(64)

in the sensor case (p= 1) and to