Frictional contact of an anisotropic piezoelectric plate

DOI:10.1051/cocv:2008022 www.esaim-cocv.org

FRICTIONAL CONTACT OF AN ANISOTROPIC PIEZOELECTRIC PLATE

Isabel N. Figueiredo

and Georg Stadler

Abstract. The purpose of this paper is to derive and study a new asymptotic model for the equi- librium state of a thin anisotropic piezoelectric plate in frictional contact with a rigid obstacle. In the asymptotic process, the thickness of the piezoelectric plate is driven to zero and the convergence of the unknowns is studied. This leads to two-dimensional Kirchhoff-Love plate equations, in which mechanical displacement and electric potential are partly decoupled. Based on this model numerical examples are presented that illustrate the mutual interaction between the mechanical displacement and the electric potential. We observe that, compared to purely elastic materials, piezoelectric bodies yield a significantly different contact behavior.

Mathematics Subject Classification.74K20, 78M35, 74M15, 74M10, 74F15 Received April 13, 2007. Revised November 7, 2007.

Published online March 6, 2008.

1. Introduction

The generation of electric charges in certain crystals when subjected to mechanical force was discovered in 1880 by Pierre and Jacques Curie and is nowadays known as piezoelectric effect (or direct piezoelectric effect).

The inverse phenomenon, that is, the generation of mechanical stress and strain in crystals when subjected to electric fields is called inverse piezoelectric effect and was predicted in 1881 by Lippmann (see [17]). Piezoelectric materials are solids exhibiting this kind of interaction between mechanical and electric properties. This provides them with sensor (direct effect) and actuator (inverse effect) capabilities making them extremely useful in a wide range of practical applications in aerospace, mechanical, electrical, civil and biomedical engineering (see [29]).

In many of these applications, additionally contact phenomena can occur or may be used on purpose,e.g., for measurement devices.

The aim of this paper is to derive, mathematically justify and numerically study a new bi-dimensional model for the equilibrium state of an anisotropic piezoelectric thin plate that is possibly in frictional contact with a rigid foundation. The derivation of the reduced (or lower-dimensional) model is done using an asymptotic procedure. It will turn out that the resulting equations are defined in the middle plane of the plate.

Keywords and phrases. Contact, friction, asymptotic analysis, anisotropic material, piezoelectricity, plate.

∗This work is partially supported by the project “Mathematical Analysis of Piezoelectric Problems” (POCTI/MAT/59502/2004, Funda¸c˜ao para a Ciˆencia e a Tecnologia, Portugal).

1 Centro de Matem´atica da Universidade de Coimbra (CMUC), Department of Mathematics, University of Coimbra, Apartado 3008, 3001-454 Coimbra, Portugal. isabelf@mat.uc.pt

2 Institute for Computational Engineering and Sciences (ICES), University of Texas at Austin, 1 University Station C0200, Austin, TX 78712, USA.georgst@ices.utexas.edu

Article published by EDP Sciences c EDP Sciences, SMAI 2008

Let us start with motivating our interest in this problem. It was observed that, if no contact and friction conditions have to be taken into account, for certain problems the mechanical and the electric parts of the equations decouple in an asymptotic process [8,11,26,28] (see [1,7,23,25,33] for related results where only partial or no decoupling occurs). In the presence of contact and friction it is not at all obvious if similar results hold true. We are also interested in numerically studying the behavior of piezoelectric materials, and by these means gain a better understanding for their properties and features.

Asymptotic methods have been widely used to deduce reduced models for plates, shells or rods. For the main ideas and bibliographic references see [3,5,6] for elastic plates, [4] for shells, and [32] for rods. For thin elastic plates, asymptotic analysis applies to the thickness variable and can be briefly summarized as follows: starting with the variational three-dimensional equations for a plate with thicknessh, these equations are scaled to a domain that is independent of h. Then, assuming appropriate scalings for the data and unknowns, one lets h→0 and studies the convergence of the unknowns as well as the properties of the limit variables. Rescaling to the original domain results in reduced model equations. For a general theory of asymptotic expansions for variational problems that depend on a small parameter we refer to [21].

In this paper we consider an anisotropic piezoelectric plate whose mechanical displacements are restricted due to possible contact with a rigid insulated foundation. The contact is unilateral (i.e., the contact region is not known in advance) and is modelled by the classical Signorini conditions. For the frictional behavior of the plate, the Tresca friction law is used. The variational formulation of this plate problem is a variational inequality of the second kind, see [9,18]. The unknowns are the mechanical displacement and the electric potential. The original, three-dimensional plate is subject to contact and friction on a part of its surface. While in the asymptotic procedure the system equations become two-dimensional, the contact and friction conditions remain similar to contact conditions occurring in three-dimensional elasticity.

Note that for modeling friction in physical applications, often the Coulomb rather than the Tresca friction law is used (see [9]). Since for the numerical realization of Coulomb friction usually a sequence of Tresca friction problems is used, Tresca friction is not only of theoretical but also practical relevance. In our numerical study, such a sequence of Tresca problems is used to solve a problem with Coulomb friction.

In the literature, several authors deal with asymptotic models for piezoelectric structures. We mention [26]

for piezoelectric plates including magnetic effects, [28] for piezoelectric thin plates with homogeneous isotropic elasticity coefficients, [8,11,12,23,25,33] for anisotropic piezoelectric plates and rods, [7] for geometrically non- linear thin piezoelectric shells, and [27] for the modelling of eigenvalue problems for thin piezoelectric shells.

However, these papers do not take into account the effects of possible contact or friction with a rigid founda- tion; nevertheless for elastic rods and shells, one finds asymptotic frictionless contact models in [32] and [20], respectively. On the other hand, there are papers dealing with contact and friction of piezoelectric materials that do not use an asymptotic procedure to reduce the model; we refer to [22], where two different variational formulations for the modelling of unilateral frictionless contact are established as well as [2] for primal and dual formulations of frictional contact problems. In [30,31], mathematical analysis of frictional contact problems with piezoelectric materials can be found; for error estimates and numerical simulations we refer to [15]. In all of the above references either none or only few numerical simulations can be found.

The main contributions of this paper are twofold: Firstly, the application of the asymptotic method to the variational inequality of the second kind that describes the anisotropic piezoelectric plate. Due to the presence of friction and contact conditions, the convergence proof in the asymptotic procedure is significantly more involved than in the unconstrained case (see [11,28]). Our second main contribution is the numerical study of the limit problem taking into account contact and friction. These conditions are similar to three-dimensional elasticity contact problems, where their numerical treatment is known to be a challenging task.

We finish this introduction with a sketch of the structure of this paper. In the next section, the three- dimensional plate model is described. In Section3, we apply the asymptotic analysis and prove strong conver- gence of the unknowns. Finally, in Section4we report on numerical tests for the asymptotic equations.

00000000 00000000 11111111 11111111

000000 000000 111111 111111

000000 000000 000000 000000 111111 111111 111111 111111

00000000000000 00000000000000 11111111111111 11111111111111

Γ_e x1

x3

x2

Γ₋ Γ₊

ω

Γ_D

−h h

Figure 1. Three-dimensional plate Ω with rectangular middle planeωand thickness 2h(note that Γ_s=

∂ω×(−h, h)

\Γ_e and Γ₁=

∂ω×(−h, h)

\Γ_D).

2. The 3D plate problem

Notations and geometry. Let ω ⊂R² be a bounded domain with Lipschitz continuous boundary ∂ω, γ₀, γ_e subsets of ∂ω with meas(γ₀)> 0. We denote γ₁ := ∂ω\γ₀, γ_s := ∂ω\γ_e. For 0 < h 1, we consider Ω =ω×(−h, h) a thin plate with middle planeω and thickness 2hand the boundary sets

Γ₊ = ω× {+h}, Γ₋ = ω× {−h}, Γ_± = Γ₊∪Γ₋, Γ_D = γ₀×(−h, h), Γ₁ = γ₁×(−h, h), Γ_N = Γ₁∪Γ₊,

Γ_s = γ_s×(−h, h), Γ_e = γ_e×(−h, h).

(2.1)

For a schematic visualization of Ω with the boundary sets (2.1) see Figure1.

We consider different disjoint partitions Γ_eN,Γ_eD of the boundary∂Ω. The splitting correspond to different electricboundaryconditions and are thus denoted by (ebci) fori= 1,2,3:

(ebc1) : Γ_eN = Γ_s and Γ_eD= Γ_±∪Γ_e, (ebc2) : Γ_eN = Γ_s∪Γ₊ and Γ_eD= Γ₋∪Γ_e, (ebc3) : Γ_eN = Γ_s∪Γ₋ and Γ_eD= Γ₊∪Γ_e.

(2.2)

Note that in all partitions, the set Γ_eD (where we will assume given Dirichlet data) contains Γ₋ or Γ₊. The case that neither on Γ₋nor on Γ₊ Dirichlet data for the electric potential are given requires a slightly different treatment than the one chosen in this paper, see Remark 3.4on page165.

Points of Ω are denoted by x = (x₁, x₂, x₃), where the first two components (x₁, x₂)∈ ω are independent ofhand x₃ ∈(−h, h). We denote byn= (n₁, n₂, n₃) the unit outward normal vector to∂Ω. Throughout the paper, the Latin indices i, j, k, l, . . .are taken from{1,2,3}, while the Greek indicesα, β, . . .from {1,2}. The summation convention with respect to repeated indices is employed, that is, a_ib_i = ₃

i=1a_ib_i. Moreover we denote bya·b=a_ib_i the inner product of the vectorsa= (a_i) andb= (b_i), byCe= (C_ijkle_kl) the contraction of a fourth order tensorC = (C_ijkl) with a second order tensore= (e_kl) and byCe:d=C_ijkle_kld_ij the inner product of the tensorsCeandd= (d_ij). Given a functionθ(x) defined in Ω we denote by∂_iθ= _∂x^∂θ

i its partial derivative with respect tox_i.

In the sequel, for an open subset Υ ⊂ Rⁿ, n = 2,3, we define D(Υ) to be the space of infinitely often differentiable functions with compact support on Υ. We denote byD(Υ) the dual space ofD(Υ), often called the space of distributions on Υ. Form= 1,2, the Sobolev spacesH^m(Υ) are defined by

H¹(Υ) =

v∈L²(Υ) : ∂_iv∈L²(Υ) fori= 1, . . . , n , H²(Υ) =

v∈L²(Υ) : ∂_iv, ∂_ijv∈L²(Υ) fori, j= 1, . . . , n ,

where L²(Υ) = {v : Υ → R,

Υ|v|²dΥ < ∞} and the partial derivatives are interpreted as distributional derivatives. Moreover, for v∈(H¹(Ω))³ we denote by v_n:=v·nandv_t:=v−v_nnthe normal and tangential components of v on the boundary of Ω, respectively. Similarly, for a second order symmetric tensor field τ = (τ_ij)∈(L²(Ω))⁹ we denote its normal and tangential components on the boundary of Ω asτ_n := (τ n)·n andτ_t:= (τ n)−τ_nn, respectively. Using the summation convention, this becomesτ_n:=τ_ijn_in_j andτ_t= (τ_ti) whereτ_ti:=τ_ijn_j−τ_nn_i. In addition, we denote by| · |the Euclidean norm in R³.

3D plate in frictional contact – classical formulation. We consider a piezoelectric anisotropic plate which in its reference configuration occupies the domain Ω. It is held fixed on Γ_D and submitted to a mechanical volume force of densityf in Ω and a mechanical surface traction of densityg on Γ_N. On its lower face Γ₋ it may be in frictional contact with the rigid foundation (which is assumed to be an insulator). We denote by s : Γ₋ −→ R⁺ the initial gap between the rigid foundation and the boundary Γ₋ measured in the direction of the outward unit normal vector n. To model the frictional contact we use the classical Signorini contact conditions and the Tresca friction law (see [9]).

We assume that the plate is subject to an electric volume charge of density r. Moreover, we suppose given an electric surface charge of density θ on Γ_eN and an electric potential equal to ϕ₀ applied to Γ_eD, where the pair Γ_eN,Γ_eD is defined as in one of the cases (ebci),i = 1,2,3 above. Note that the lower subscripts_eN and

eD in Γ_eN and Γ_eD refer to electric (e) Neumann (N) and Dirichlet (D) boundary conditions, respectively, while the lower subscripts_N and_Din Γ_N and Γ_D refer to mechanical (Neumann and Dirichlet) boundary conditions.

We now give the classical (i.e., strong) equations defining the mechanical and electric equilibrium state of the plate Ω. The equilibrium is described by the following five groups of equations and boundary conditions, whose unknowns are the mechanical displacement vectoru: Ω→R³ and the (scalar) electric potentialϕ: Ω→R.

Mechanical equilibrium equations and boundary conditions

⎡

⎢⎣

−divσ(u, ϕ) = f (i.e.,−∂_jσ_ij(u, ϕ) =f_i) in Ω, σ(u, ϕ)n = g (i.e., σ_ij(u, ϕ)n_j=g_i) on Γ_N,

u = 0 on Γ_D.

(2.3a)

Maxwell-Gauss equations and electric boundary conditions (ebci),i= 1,2,3

⎡

⎢⎢

⎣

divD(u, ϕ) = r (i.e., ∂_iD_i(u, ϕ) =r) in Ω, D(u, ϕ)n = θ (i.e., D_i(u, ϕ)n_i =θ) on Γ_eN,

ϕ = ϕ₀ on Γ_eD.

(2.3b)

Constitutive equations

σ_ij(u, ϕ) = C_ijkle_kl(u)−P_kijE_k(ϕ) in Ω,

D_k(u, ϕ) = P_kije_ij(u) +ε_klE_l(ϕ) in Ω. (2.3c)

Signorini’s contact conditions

u_n≤s, σ_n≤0, σ_n(u_n−s) = 0 on Γ₋. (2.3d)

Tresca’s law of friction

⎡

⎢⎣

|σt| ≤q, and

|σ_t|< q ⇒ u_t= 0,

|σt|=q ⇒ ∃c≥0 : u_t=−cσt

⎫⎪

⎬

⎪⎭ on Γ₋. (2.3e)

The mechanical equilibrium equations (2.3a) express the balance of mechanical loads and internal stresses.

The electric displacement field D is governed by the Maxwell-Gauss equations (2.3b), and the constitutive equations (2.3c) characterize piezoelectricity. They define the interaction between the stress tensorσ: Ω→R⁹, the electric displacement vectorD: Ω→R³, the linear strain tensore(u) and the electric field vectorE(ϕ), the latter two tensors given by

e(u) = ¹₂

∇u+ (∇u)

(i.e., e_ij(u) = ¹₂

∂_iu_j+∂_ju_i)) E(ϕ) = −∇ϕ (i.e., E_i(ϕ) =−∂_iϕ).

In (2.3c), C = (C_ijkl) is the elastic fourth order, P = (P_ijk) the piezoelectric third order and ε = (ε_ij) is the dielectric second order tensor field. The Signorini law (2.3d) describes the contact behavior of the plate with a rigid foundation. If the plate is not in contact with the rigid foundation (i.e., u_n < s), the normal stress vanishes, i.e., σ_n = 0. For u_n =s, that is, the plate is in contact with the obstacle, the normal stress component σ_n is nonpositive. These conditions are the complementarity conditions for contact. Finally, the conditions (2.3e) model the frictional behavior of the plate, whereq≥0 is a function representing the prescribed friction bound. Briefly, (2.3e) expresses the fact that on the contact boundary Γ₋ the Euclidean norm of the tangential stress component cannot exceed the given friction bound q, that slip occurs if this norm equals q, and stick if it is smaller than q. Moreover, slip can only occur in the negative direction of σ_t. Note that the regions where contact and slip or stick occur are not knowna priori. This makes contact problems with friction free boundary problems, which are theoretically and practically challenging.

We assume the following hypotheses on the data f ∈

L²(Ω)₃ , g∈

L²(Γ_N)₃

, r∈L²(Ω), θ∈L²(Γ_eN), ϕ₀∈H^1/2(Γ_eD), s∈H^1/2(Γ₋), q∈L²(Γ₋),

where θ = 0 on Γ₋ for (ebc3). Moreover, the tensor fields C = (C_ijkl), P = (P_ijk) and ε= (ε_ij) are defined on ¯ω×[−1,1] for x= (x₁, x₂,^x_h³). Defining them on the reference plate ¯ω×[−1,1] makes them independent ofhin the transformed variables also used in the next section. The tensorsC_ijkl,P_ijk, ε_ij are assumed to be sufficiently smooth functions that satisfy the following symmetriesC_ijkl =C_jikl=C_klij,P_ijk=P_ikj,ε_ij =ε_ji. Moreover,C andεare assumed to be coercive, that is there existc₁, c₂>0 such that

C_ijkl(x)M_klM_ij ≥c₁ 3 i,j=1

(M_ij)² and ε_ij(x)θ_iθ_j≥c₂ 3 i=1

θ²_i (2.4)

for every symmetric 3×3 real valued matrixM, every vectorθ∈R³ and everyx∈ω¯×[−1,1].

3D-plate in frictional contact – weak formulation. We now give the weak or variational formulation of (2.3a)–(2.3e). We define the space of admissible mechanical displacements

V :=

v∈

H¹(Ω)₃

: v_|ΓD = 0

that we endow with the norm vV =∇v_(L²_(Ω))⁹, which, due to the Poincar´e inequality is equivalent to the usualH¹-norm. Moreover, we introduce the convex cone

K:=

v∈V : v_n≤son Γ₋

, where v_n=−v₃, as well as the space of admissible electric potentials

Ψ :=

ψ∈H¹(Ω) : ψ_|ΓeD = 0 ,

in which we use the normψΨ=∇ψ_(L²_(Ω))³ (which is also equivalent to the usualH¹-norm).

Next, we briefly sketch how the variational formulation of (2.3a)–(2.3e) is obtained. Using the Green formula in (2.3a), we obtain for any v∈K

Ωσ_ije_ij(v−u) dx−

∂Ωσ_ijn_j(v_i−u_i) dΓ_N =

Ωf_i(v_i−u_i) dx. (2.5) Sincev=u= 0 on Γ_D,σ_ijn_j =g_i on Γ_N,∂Ω = Γ_D∪Γ_N ∪Γ₋ and due to

σ_ijn_j(v_i−u_i) = σ_t(v_t−u_t) +σ_n(v_n−u_n)

= σ_t(v_t−u_t) +σ_n(v_n−s)−σ_n(u_n−s), (2.5) becomes, using σ_n(v_n−s)≥0 andσ_n(u_n−s) = 0 on Γ₋,

Ωσ_ije_ij(v−u) dx−

Ωf_i(v_i−u_i) dx−

ΓN

g_i(v_i−u_i) dΓ_N ≥

Γ−

σ_t(v_t−u_t) dx. (2.6) Addingj(v)−j(u) to both sides of (2.6), where

j(v) :=

Γ−

q|v_t|dΓ₋, with v_t=v−v_nn= (v₁, v₂,0),

and using

Ω

σ_t(v_t−u_t) +q|v_t| −q|u_t|

dx≥0,

we obtain

Ωσ_ije_ij(v−u) dx+j(v)−j(u)−

Ωf_i(v_i−u_i) dx−

ΓN

g_i(v_i−u_i) dΓ_N ≥0. (2.7) Next, from (2.3b) we have for any ψ∈Ψ

−

Ω

D_i∂_iψdx+

ΓeN

θ ψdΓ_eN =

Ω

r ψdx, (2.8)

where D_in_i =θ on Γ_eN and ψ = 0 on Γ_eD have been used. Summing (2.7) and (2.8), using the constitutive equations (2.3c) and the transformationϕ= ¯ϕ+ϕ₀, we obtain as weak formulation of (2.3a)–(2.3e) the following elliptic variational inequality of the second kind [9,13,14,18]

⎧⎨

⎩

Find (u,ϕ)¯ ∈K×Ψ such that:

b

(u,ϕ),¯ (v−u, ψ)

+j(v)−j(u)≥l

(v−u, ψ)

∀(v, ψ)∈K×Ψ, (2.9) where

b

(u,ϕ),¯ (v, ψ)

:=

ΩCe(u) :e(v) dx+

Ωε_ij∂_iϕ ∂¯ _jψdx +

ΩP_ijk

∂_iϕe¯ _jk(v)−∂_iψe_jk(u) dx, and

l (v, ψ)

:=

Ωf·vdx+

ΓNg·vdΓ_N+

Ωr ψdx−

ΓeNθ ψdΓ_eN

−

Ωε_ij∂_iϕ₀∂_jψdx−

ΩP_ijk∂_iϕ₀e_jk(v) dx.

Remark 2.1. Note that b

(v, ψ),(v, ψ)

=

ΩCe(v) :e(v) dx+

Ωε_ij∂_iψ ∂_jψdx.

Thus, due to (2.4), the bilinear form b((·,·),(·,·)) is V ×Ψ- coercive. Since additionallyb((·,·),·,·)) and the linear form l((·,·)) are V ×Ψ-continuous, the set K is a nonempty, closed and convex subset of V and the functional j(·) is proper, convex and continuous on V, there exists a unique solution (u,ϕ) of (2.9) (see for¯ instance [9,13,14,18]).

3. Asymptotic analysis

In this section, we use an asymptotic method, which is mainly due to [3,5,6], to derive two-dimensional plate equations from the three-dimensional system of equations (2.3a)–(2.3e). The principal idea is letting the plate’s thicknesshtend to zero, after rescaling the 3D variational inequality (2.9) to a fixed reference domain that does not depend onh. We investigate the convergence of the unknowns ash→0 and analyze the resulting system of equations.

3.1. Scaling of the 3D-equations to a fixed domain

Here, we redefine the 3D variational problem (2.9) in the h-independent domain ˆΩ = ω×(−1,1). To each x = (x₁, x₂, x₃) ∈ Ω we associate the elementˆ x= (x₁, x₂, hx₃) ∈ Ω, through the isomorphism π(x) = (x₁, x₂, hx₃)∈Ω. We consider the subsets defined in (2.1) for the choiceh= 1, that is

Γˆ_± = ω× {±1}, Γˆ_D = γ₀×(−1,1), Γˆ₁ = γ₁×(−1,1), Γˆ_N = Γˆ₁∪Γˆ₊, ˆΓ_s = γ_s×(−1,1), Γˆ_e = γ_e×(−1,1),

and the disjoint partitions ˆΓ_eN,Γˆ_eD of∂Ω defined by consequently replacing Γ by ˆˆ Γ in (2.2).

We denote by n = (n₁, n₂) = (n_α) the unit outer normal vector along ∂ω, by t = (t₁, t₂) = (t_α), with t₁ =−n₂ and t₂ =n₁, the unit tangent vector along∂ω, by ^∂θ_∂ν =ν_α∂_αθ the outer normal derivative of the scalar functionθ along∂ω. For the asymptotic process we need the data to satisfy the following hypotheses

f_α◦π = h²fˆ_α, f₃◦π = h³fˆ₃ in Ω,ˆ g_α◦π = h²ˆg_α, g₃◦π = h³gˆ₃ in Γˆ₁, g_α◦π = h²ˆg_α, g₃◦π = h⁴gˆ₃ in Γˆ₊, ϕ₀◦π = h³ϕˆ₀, r◦π = hrˆ in Ω,ˆ

s◦π = hs,ˆ q◦π = h³qˆ in Γˆ₋, θ◦π = hθˆ in Γˆ_s∪Γˆ_e, θ◦π = h²θˆ in Γˆ_±.

(3.1)

Above, we assume that ˆf_α ∈ H¹(Ω), ˆf₃ ∈ L²(Ω), ˆg_α ∈ H¹(ˆΓ_N), ˆg₃ ∈ L²(ˆΓ_N), ˆr ∈ L²(Ω), ˆθ ∈ L²(ˆΓ_eN), ˆ

ϕ₀ ∈ H¹(Ω), ˆq ∈ L²(ˆΓ₋) and ˆs ∈ L²(ˆΓ₋) with ˆs ≥ 0. In addition, we denote by ˆg⁺₃ = ˆg_3|ˆ

Γ+, ˆϕ⁺₀ = ˆϕ_0|ˆ

Γ+, ˆ

ϕ⁻₀ = ˆϕ_0|ˆ

Γ− and we assume that ˆϕ⁺₀ −ϕˆ⁻₀ ∈H¹(ω). The unknowns are rescaled as follows

u_α◦π=h²u^h_α, u₃◦π=hu^h₃ and ϕ◦π=h³ϕ^hin ˆΩ, (3.2) and analogous scalings to (3.2) hold for the test functions vandψ in (2.9).

At this point, we briefly comment on the choice (3.1)–(3.2) for the scalings of data, unknowns and test functions. For the mechanical forces and displacements we assume the scalings also used in [3] for linearly elastic plates. In [24] it is shown that these are (up to a multiplicative power ofh) “the only possible scalings”

that lead to a linear Kirchhoff-Love theory in the asymptotic analysis. Regarding the scalings for the electric variablesϕ₀,r, θ,ϕandψ our scalings are chosen such that we are able to compute the limit ash→0 of the model (3.6) given below. We also refer to [11,25,28], where scalings as given by (3.1)–(3.2) are used as well.

However, different electric scalings have to be used if other electric boundary conditions are chosen, see the remark on page165.

The scaled spaces for the admissible mechanical and electric potential displacements are given by Vˆ :=

v∈

H¹( ˆΩ)₃ :v_|ˆΓ

D = 0 , Ψˆ :=

ψ∈H¹( ˆΩ) :ψ_|ˆΓeD = 0 .

As before, the spaces are endowed with the normsvVˆ =∇v_(L2(ˆΩ))⁹andψΨˆ =∇ψ_(L2(ˆΩ))³, respectively.

The scaled convex cone needed for the Signorini contact conditions is given by Kˆ :=

v∈Vˆ :v_n =−v₃≤son ˆΓ₋ .

For anyv∈Vˆ we define the second order symmetric tensor field κ^h(v) = (κ^h_ij(v)) by κ^h_αβ(v) :=e_αβ(v) = ¹₂(∂_βv_α+∂_αv_β),

κ^h_α3(v) :=_h¹e_α3(v) = _2h¹(∂₃v_α+∂_αv₃), κ^h₃₃(v) :=_h¹₂e₃₃(v) = _h¹₂∂₃v₃.

(3.3)

As a consequence of the scalings (3.2) we have for the strain tensorse(u),e(v) withv∈V e(u) =h²κ^h(u^h) and e(v) =h²κ^h(v◦π) with v◦π∈V .ˆ For the stress tensor and the electric displacement vector we obtain the scalings

σ_ij^h(u^h, ϕ^h) =h⁻²σ_ij(u, ϕ), D_i^h(u^h, ϕ^h) =h⁻²D_i^h(u, ϕ), (3.4) where

σ_ij^h(u^h, ϕ^h) =C_ijlmκ^h_lm(u^h) +h P_αij∂_αϕ^h+P_3ij∂₃ϕ^h,

D^h_i(u^h, ϕ^h) =P_ilmκ^h_lm(u^h)−h ε_iα∂_αϕ^h−ε_i3∂₃ϕ^h. (3.5) Weak formulation of the 3D scaled plate problem. Using the above scalings and assumptions (3.1) on the data, we obtain for eachh >0 a problem on the fixed domain ˆΩ that is equivalent to (2.9):

⎧⎨

⎩

Find (u^h,ϕ¯^h)∈Kˆ ×Ψˆ such that:

b^h

(κ^h(u^h),ϕ¯^h),(κ^h(v)−κ^h(u^h), ψ)

+j(v)−j(u^h)≥l^h

(v−u^h, ψ)

, ∀(v, ψ)∈Kˆ ×Ψ,ˆ (3.6)

where ¯ϕ^h=ϕ^h−ϕˆ₀, and forκ, ϑin (L²( ˆΩ))⁹ andϕ, ψ in Ψ,

b^h

(κ, ϕ),(ϑ, ψ)

:=

⎧⎪

⎪⎨

⎪⎪

⎩

ΩˆCκ:ϑdx+

Ωˆε₃₃∂₃ϕ ∂₃ψdx+

ΩˆP_3jk

∂₃ϕ ϑ_jk−∂₃ψ κ_jk dx +h

Ωˆε_3α

∂_αϕ ∂₃ψ+∂₃ϕ ∂_αψ

dx+h

ΩˆP_αjk

∂_αϕ ϑ_jk−∂_αψ κ_jk dx +h²

Ωˆε_αβ∂_αϕ ∂_βψdx, j(v) :=

ˆΓ−

ˆ

q|v_t|dˆΓ₋ with v_t:=v−v_nn= (v₁, v₂,0),

l^h (v, ψ)

:=

⎧⎪

⎪⎨

⎪⎪

⎩

Ωˆfˆ·vdx+

ˆΓNgˆ·vdˆΓ_N+

Ωˆr ψˆ dx−

ˆΓeNθ ψˆ dˆΓ_eN−

Ωˆε₃₃∂₃ϕˆ₀∂₃ψdx

−h

Ωˆε_α3

∂_αϕˆ₀∂₃ψ+∂₃ϕ₀∂_αψ

dx−h²

Ωˆε_αβ∂_αϕˆ₀∂_βψdx

−

ΩˆP_3ij∂₃ϕˆ₀κ^h_ij(v) dx−h

ΩˆP_αij∂_αϕˆ₀κ^h_ij(v) dx.

Using the relation (3.3) betweenκ^h(v) andv, in the sequel (mainly in the proof of Th.3.1), we abbreviate b^h

(κ^h(u^h), ϕ),(κ^h(v), ψ)

by b^h

(u^h, ϕ),(v, ψ)

.

In contrast to (2.9), where the dependence on the parameterhis implicit (by means of the domain Ω), prob- lem (3.6) now depends explicitly on h, but is defined on a domain independent ofh.

3.2. Convergence as h → 0

The aim of this section is to study the limit behavior of the sequences (u^h) and (ϕ^h) ash→0⁺. We are able to prove strong convergence of these sequences and give a limit problem that characterizes these limits. It will turn out that the limit displacement is an element of ˆV_KL, the Kirchhoff-Love mechanical displacement space defined by (see also [3])

Vˆ_KL :=

v= (v₁, v₂, v₃)∈

H¹( ˆΩ)₃ :v_|Γˆ

D = 0, e_i3(v) = 0

=

v= (v₁, v₂, v₃)∈

H¹( ˆΩ)₃

:∃η= (η₁, η₂, η₃)∈

H¹(ω)₂

×H²(ω), ∂_νη_3|γ0 = 0, η_1|γ0 =η_2|γ0 =η_3|γ0 = 0, v_α(x) =η_α(x₁, x₂)−x₃∂_αη₃(x₁, x₂), v₃(x) =η₃(x₁, x₂)

.

(3.7)

Moreover, ˆΨ_land ˆΨ_l0are the spaces for the admissible electric potentials defined by Ψˆ_l:=

ψ∈L²( ˆΩ) :∂₃ψ∈L²( ˆΩ)

and Ψˆ_l0:=

ψ∈L²( ˆΩ) :∂₃ψ∈L²( ˆΩ), ψ_|S = 0

, (3.8a)

where S=

⎧⎨

⎩

Γˆ_± for (ebc1), Γˆ₋ for (ebc2), Γˆ₊ for (ebc3).

(3.8b)

As usual, ˆV_KL is endowed with the normvVˆKL :=eαβ(v)_(L2(ˆΩ))⁴, which is equivalent to the

H¹( ˆΩ)₃ - norm for elements in ˆV_KL, see [3]. For the space ˆΨ_l we use the norm ψΨˆl :=

ψ²_L2(ˆΩ)+∂₃ψ²_L2(ˆΩ)1/2 . Finally, ˆΨ_l0 is endowed with the normψΨˆl0:=∂3ψ_L2(ˆΩ), which is equivalent to the norm defined in ˆΨ_lfor elements in ˆΨ_l0.

Theorem 3.1. As h → 0⁺, the sequence {(u^h, ϕ^h)}_h converges strongly to (u, ϕ) ∈Vˆ_KL×Ψˆ_l. This limit pair is characterized as the unique solution of the variational problem

⎧⎪

⎪⎨

⎪⎪

⎩

Find (u, ϕ)∈Vˆ_KL∩Kˆ ×Ψˆ_l such that:

a

(u, ϕ),(v−u, ψ)

+j(v)−j(u)≥l

(v−u, ψ)

∀(v, ψ)∈Vˆ_KL∩Kˆ ×Ψˆ_l0, ϕ= ˆϕ₀, on S,

(3.9)

whereS ⊂∂Ωis defined in (3.8b)anda

·,·

andl(·)are given by a

(u, ϕ),(v, ψ)

:=

ΩˆA_αβγρe_αβ(u)e_γρ(v) dx+

Ωˆp₃₃∂₃ϕ∂₃ψdx

−

Ωˆp_3αβ

e_αβ(u)∂₃ψ−e_αβ(v)∂₃ϕ dx, l

(v, ψ) :=

Ωˆ

fˆ·vdx+

ˆΓN

ˆ

g·vdˆΓ_N+

Ωˆr ψˆ dx−

ˆΓeN

θ ψˆ dˆΓ_N, (3.10) whereA_αβγρ, p_3αβ andp₃₃ are modified material parameters (see Part Aof the Appendix for details).

Proof. For convenience of the reader, we split the proof into five steps.

Step 1. Existence of weak limits u, κ andϕ of subsequences of (u^h), (κ^h(u^h))and (ϕ^h), respectively. We first choose (v, ϕ)∈Kˆ ×Ψ in (3.6) such thatˆ v_α=u^h_α,v₃= 0 andψ=−ϕ¯^h. This results in j(v)−j(u^h) = 0 and thus (3.6) becomes

b^h

(u^h,ϕ¯^h),((0,0, u^h₃),ϕ¯^h) ≤l^h

((0,0, u^h₃),ϕ¯^h)

. (3.11)

Next, choosing (v, ϕ)∈Kˆ ×Ψ asˆ v_α=−u^h_α,v₃=u^h₃ andψ=−2 ¯ϕ^h, againj(v)−j(u^h) = 0 and from (3.6) b^h

(u^h,ϕ¯^h),((−2u^h₁,−2u^h₂,0),−2 ¯ϕ^h) ≥l^h

((−2u^h₁,−2u^h₂,0),−2 ¯ϕ^h)

, which is equivalent to

b^h

(u^h,ϕ¯^h),((u^h₁, u^h₂,0),ϕ¯^h) ≤l^h

((u^h₁, u^h₂,0),ϕ¯^h)

. (3.12)

Adding (3.11) and (3.12) results in b^h

(u^h,ϕ¯^h),(u^h,ϕ¯^h) ≤l^h

(u^h,ϕ¯^h) , from which we can deduce

u^h2_Vˆ +

Ωˆ

κ^h(u^h) :κ^h(u^h)dx+h ∂₁ϕ¯^h2_L2(ˆΩ)+h ∂₂ϕ¯^h2_L2(ˆΩ)+∂₃ϕ¯^h2_L2(ˆΩ) < c, (3.13) where c > 0 is a constant independent ofh(see also [28]). Consequently, there are weakly convergent subse- quences of (u^h), (κ^h(u^h)) and (ϕ^h) with

u^h u in

H¹( ˆΩ)₃ ,

κ^h(u^h) κ in

L²( ˆΩ)₉ ,

¯

ϕ^h ϕ¯=ϕ−ϕˆ₀ in L²( ˆΩ), ϕ^h ϕ in L²( ˆΩ), (h∂₁ϕ^h, h∂₂ϕ^h, ∂₃ϕ^h) (0,0, ∂₃ϕ) in

L²( ˆΩ)₃ .

(3.14)

The first two weak convergences follow directly from (3.13). The existence of the third and fourth weak limit in (3.14) follows from (3.13) using the fact that

¯

ϕ^h(x₁, x₂, x₃) =

_x3

−1∂₃ϕ¯^h(x₁, x₂, y₃)dy₃ forS= ˆΓ_± or S= ˆΓ₋

−₊₁

x3 ∂₃ϕ¯^h(x₁, x₂, y₃)dy₃ forS= ˆΓ₊, (3.15) which results in

ϕ¯^h_L2(ˆΩ)≤√

2∂3ϕ¯^h(x₁, x₂, x₃)_L2(ˆΩ) ≤c (3.16) with c > 0 independent ofh. This implies the L²( ˆΩ)-boundedness of ¯ϕ^h and ϕ^h = ¯ϕ^h+ ˆϕ₀. In particular, ϕ = ˆϕ₀ onS because ¯ϕ^h = 0 on ˆΓ_eD ⊇S. Finally, the last convergence in (3.14) is a consequence of (3.13), which implies that

(h∂₁ϕ^h, h∂₂ϕ^h, ∂₃ϕ^h)(ϑ₁, ϑ₂, ϑ₃) in

L²( ˆΩ)₃ .

The weak convergence ofϕ^hto ϕ yields∂_iϕ^h ∂_iϕ inL²( ˆΩ), and thusϑ_α= 0 forα= 1,2, andϑ₃=∂₃ϕ. Moreover, from (3.14) we can also deduce thatu∈Vˆ_KL. In fact, from (3.13) we obtain boundedness of the sequenceκ^h_i3(u^h) inL²( ˆΩ). Consequently,e_α3(u^h) =hκ^h_α3(u^h) ande₃₃(u^h) =h²κ^h₃₃(u^h)→0 strongly inL²( ˆΩ).

Thus, e_i3(u) := ¹₂(∂_iu₃+∂₃u_i) = 0, which implies u ∈ Vˆ_KL. We also remark that, appropriate choice of subsequences guarantees that

κ_αβ=e_αβ(u) = 1

2(∂_αu_β+∂_βu_α) (3.17)

yielding that the weak limitκ depends explicitly onu.

Step 2. Auxiliary results. As a consequence of the weak convergences (3.14) we obtain for arbitrary (v, ψ)∈ Vˆ_KL×Ψ thatˆ

h→0lim⁺b^h

(u^h, ϕ^h),(v, ψ)

:= b

(u, ϕ),(v, ψ)

,

h→0lim⁺l^h (v, ψ

) := l (v, ψ

),

where for (v, ψ) in ˆV_KL×Ψˆ b

(u, ϕ),(v, ψ)

:=

ΩˆC_αβijκ_ije_αβ(v) dx+

Ωˆε₃₃∂₃ϕ∂₃ψdx +

ΩˆP_3αβ∂₃ϕe_αβ(v) dx−

ΩˆP_3lm∂₃ψ κ_lmdx,

(3.18)

and

l (v, ψ

) :=

Ωˆfˆ·vdx+

ˆΓNˆg·vdˆΓ_N +

Ωˆr ψˆ dx−

ˆΓeNθ ψˆ dˆΓ_N

−

Ωˆε₃₃∂₃ϕˆ₀∂₃ψdx−

ΩˆP_3αβ∂₃ϕˆ₀e_αβ(v) dx.

Moreover

h→0lim⁺

j(v)−j(u^h)

=j(v)−j(u).

Step 3. The weak limits u, κ and ϕ are also strong limits. Here, it suffices to prove that χ^h strongly converges toχ in (L²( ˆΩ))¹², with

χ^h = (κ^h(u^h), h∂₁ϕ^h, h∂₂ϕ^h, ∂₃ϕ^h)∈

L²( ˆΩ)₁₂ χ = (κ,0,0, ∂₃ϕ)∈

L²( ˆΩ)₁₂ .