Mathematical justification of the obstacle problem in the case of piezoelectric plate

Abstract

This paper aims at properly justifying the modeling of a thin piezoelectric plate in unilateral contact with a rigid plane. In order to do that we start from the three-dimensional non-linear Signorini problem which couples the elastic and the electric effects. By an asymptotic analysis we study the convergence of the displacement field and of the electric potential as the thickness of the plate goes to zero. We establish that, at the limit, the in-plane elastic components and the electric potential are coupled and solve a bilateral linear piezoelectric problem. However the transverse mechanical displacement field, is independent of the electric effect and solves a two-dimensional elastic obstacle problem. We also investigate the very popular case of cubic crystals and show that, for thin plates, the piezoelectric coupling effect disappears.

Keywords

Signorini problem obstacle problem asymptotic analysis piezoelectric plate

1. Introduction

This work is two-fold. From the one hand we consider the so-called Signorini problem of an elastic body in contact with a rigid support, also called unilateral contact problem. On the other hand we focus on a body made of a piezoelectric material.

The contact problem can be stated as the minimization of some energy functional under an inequality constraint. The first mathematical properties of the solution of such a problem can be found in Fichera [5] and Duvaut and Lions [4]. Later Paumier gave, by an asymptotic approach, the model of an elastic Kirchhoff–Love plate in unilateral contact [13]. Léger and Miara generalized Paumier’s work to elastic shallow shell. They obtained the limit model written in terms of a variational inequality in the framework of Cartesian [8] and curvilinear coordinates [9].

Piezoelectric materials are dielectrics in which the application of an electric field generates a mechanical deformation and, conversely, polarization is produced as the result of an elastic deformation. This electromechanical interaction appears in the constitutive equations which couple the mechanical stress tensor and the electric field on the one hand, and the polarization vector and the strain tensor on the other hand. As far as piezoelectricity is concerned we refer to the first justification of the two-dimensional (bilateral) model of piezoelectric plate provided by Rahmoune [14], Sene [15]. Miara and Suarez [12] investigated the two-dimensional model of dynamic thermopiezoelectric plates. An interesting feature of our modelling is to keep the full piezoelectricity tensor, namely we did not assume any elastic isotropy (as in [15]). For numerical aspects of unilateral contact for piezoelectric body we can refer to the first formulation given by Maceri and Bisegna [11].

The plan of our work is as follows: In Section 2 we introduce the equilibrium equations and the piezoelectric constitutive law in the framework of linearized elasticity for small deformations. In Section 3, we study the three-dimensional Signorini problem. In Section 4, in order to get a relevant limit model we follow Ciarlet and Destuynder [2] and assume appropriate forms of the data (applied forces and electric charge) and we scale the unknowns (elastic displacement and electric potential). This yields a new scaled variational inequality problem. In Section 5 we prove the convergence of the solution when the thickness of the plate tends to zero and establish the limit problem of a piezoelectric plate in unilateral contact. In Section 6 we give the formulation of the problem in the physical domain, we present more general boundary conditions and investigate the interesting case of cubic crystals.

2. Piezoelectricity in a three-dimensional body

Latin indices take their values in the set ${1, 2, 3}$ , Greek indices take their values in the set ${1, 2}$ ; and the Einstein summation convention is used. Bold letters are used for vectors or vector spaces. We denote by $a \cdot b$ the vector product between two vectors a and b. We note $| \cdot |_{0, Ω}$ , $‖ \cdot ‖_{1, Ω}$ be the classical norms in $L^{2} (Ω)$ , $H^{1} (Ω)$ , respectively, for both scalar-valued and vector-valued functions.

In this paper, for simplicity we let c denote different positive constants.

2.1. Constitutive equations of piezoelectricity

A piezoelectric material is described by three tensors: the fourth order symmetric positive definite stiffness tensor $C = (C_{i j k l})$ , the third order piezoelectric tensor $P = (P_{k i j})$ , and the second order symmetric positive definite dielectric tensor $d = (d_{i j})$ . The set of data $(C, P, d)$ can be represented by a symmetric matrix (see [6]) $\begin{matrix} (\begin{matrix} C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} & P_{11} & P_{21} & P_{31} \\ C_{21} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} & P_{12} & P_{22} & P_{32} \\ C_{31} & C_{32} & C_{33} & C_{34} & C_{35} & C_{36} & P_{13} & P_{23} & P_{33} \\ C_{41} & C_{42} & C_{43} & C_{44} & C_{45} & C_{46} & P_{14} & P_{24} & P_{34} \\ C_{51} & C_{52} & C_{53} & C_{54} & C_{55} & C_{56} & P_{15} & P_{25} & P_{35} \\ C_{61} & C_{62} & C_{63} & C_{64} & C_{65} & C_{66} & P_{16} & P_{26} & P_{36} \\ P_{11} & P_{12} & P_{13} & P_{14} & P_{15} & P_{16} & d_{11} & d_{12} & d_{13} \\ P_{21} & P_{22} & P_{23} & P_{24} & P_{25} & P_{26} & d_{21} & d_{22} & d_{23} \\ P_{31} & P_{32} & P_{33} & P_{34} & P_{35} & P_{36} & d_{31} & d_{32} & d_{33} \end{matrix}), \end{matrix}$ where we used the abbreviation $11 \to 1$ , $22 \to 2$ , $33 \to 3$ , $23 = 32 \to 4$ , $13 = 31 \to 5$ , $12 = 21 \to 6$ . For example, the component $C_{1111}$ is denoted by $C_{11}$ , $C_{1131}$ by $C_{15}$ , the component $P_{123}$ by $P_{14}$ , $P_{233}$ by $P_{23}$ , etc.

There exists a positive constant c such that the tensors C, P, d satisfy the following relations:

There exists positive c such that, for every second order $3 \times 3$ symmetric tensor $M = (M_{i j})$ and for every vector of dimension 3, $θ = (θ_{k})$ we have $\begin{matrix} (2.1) & \{\begin{matrix} C_{i j k l} = C_{i j l k} = C_{k l i j}, C_{i j k l} M_{k l} M_{i j} ⩾ c \sum_{i, j = 1}^{3} M_{i j}^{2}, \\ d_{k l} = d_{l k}, d_{k l} θ_{k} θ_{l} ⩾ c \sum_{k = 1}^{3} θ_{k}^{2}, \\ P_{m k l} = P_{m l k} . \end{matrix} \end{matrix}$ The constitutive laws that describe, in the linear case, the behavior of static piezoelectricity couple the second order symmetric stress tensor $σ = (σ_{k l})$ and the displacement vector $D = (D_{k})$ to the second order symmetric strain tensor $e = (e_{k l})$ and the electric vector field $E = (E_{k})$ through the relations (for a complete description we refer to [7] and [3]): $\begin{matrix} (2.2) & \{\begin{matrix} σ_{i j} = C_{i j k l} e_{k l} - P_{k i j} E_{k}, \\ D_{k} = P_{k i j} e_{i j} + d_{k l} E_{l} . \end{matrix} \end{matrix}$ Let us recall that the dielectric tensor d is related to the electric susceptibility tensor $d^{'} = (d_{k l}^{'})$ and to the electric permittivity of the free space $d_{0}$ by $d = d_{0} I_{identity} + d^{'}$ . The polarization vector $X = (X_{k})$ is given component-wise by $X_{k} = P_{k i j} e_{i j} + d_{k l}^{'} E_{l}$ , so that the electric displacement D is related to the polarization vector by $D = d_{0} E + X$ .

2.2. Equilibrium equations of piezoelectricity

Now let us write, in the linear case, the equilibrium equations of a piezoelectric body in the reference configuration $\overline{Ω}$ where Ω is a connected open subset of $R^{3}$ with smooth boundary. When subjected to the action of mechanical forces with density $f = (f_{k}) : \overline{Ω} \to R^{3}$ and electric charge of density $ρ_{e}$ , the body undergoes a mechanical displacement field $u = (u_{k}) : \overline{Ω} \to R^{3}$ and an electric potential $φ : \overline{Ω} \to R$ , which solve the system of partial differential equations $\begin{matrix} (2.3) & \{\begin{matrix} div σ (u, φ) = - f, \\ div D (u, φ) = ρ_{e}, \end{matrix} \end{matrix}$ here div stands for the divergence operator of a tensor or of a vector. In the static linearized case considered in this paper the strain tensor and the electric field depend upon the unknowns by the relations $\begin{matrix} e = \frac{1}{2} (\nabla u + {(\nabla u)}^{T}), E = - \nabla φ, \end{matrix}$ or component-wise by $e_{i j} = \frac{1}{2} (\partial_{i} u_{j} + \partial_{j} u_{i})$ , $E_{k} = - \partial_{k} φ$ .

2.3. Boundary conditions

The equilibrium equations have to be supplemented by conditions posed on the boundary $\partial Ω$ . Let $\partial Ω = Γ_{0} \cup \hat{Γ}$ be a partition of the boundary with area $Γ_{0} > 0$ . On $Γ_{0}$ the body is clamped and is subjected to null electric potential, and on $\hat{Γ}$ the body is subjected to applied surface force with density $g = (g_{k}) : \hat{Γ} \to R^{3}$ and is subjected to electric charge with density $ϑ : \hat{Γ} \to R^{3}$ . Then the boundary conditions read: $\begin{matrix} (2.4) & \{\begin{matrix} u = 0 & on Γ_{0}, \\ φ = 0 & on Γ_{0}, \\ σ (u, φ) ν = g & on \hat{Γ}, \\ D (u, φ) \cdot ν = ϑ & on \hat{Γ}, \end{matrix} \end{matrix}$ where $ν$ is the unit normal vector to $\partial Ω$ . The contact condition will be introduced later in Section 3.2. Another partition of the boundary with more general condition is considered in Section 6.

Let us recall that problem (2.3)–(2.4) has a unique weak solution in appropriate space [14].

3. Thin plate

In order to establish properly the bi-dimensional model of a thin plate in contact with a rigid plane we consider that the reference configuration is a cylinder with middle surface ω and thickness $2 ε$ , the case of a curved domain is considered in [16]. More precisely let $ε > 0$ be a small parameter and ω be an open bounded and connected subset of $R^{2}$ with Lipschitz-continuous boundary $\partial ω = γ$ . The reference configuration of the plate under study is denoted by ${\overline{Ω}}^{ε}$ where $Ω^{ε} = ω \times (- ε, ε)$ . We define a new partition of the boundary as $\partial Ω^{ε} = Γ_{+}^{ε} \cup Γ_{-}^{ε} \cup Γ_{0}^{ε}$ , with the upper and lower faces $Γ_{+}^{ε} = ω \times {ε}$ , $Γ_{-}^{ε} = ω \times {- ε}$ and the lateral boundary $Γ_{0}^{ε} = γ \times [- ε, ε]$ .

3.1. Three-dimensional problem

We consider a family of plates with reference configuration ${\overline{Ω}}^{ε}$ made of piezoelectric material where three characteristics tensors C, P and d are independent of ε. We denote by $σ^{ε}$ , $e^{ε}$ , $D^{ε}$ , $E^{ε}$ , respectively, the stress tensor, the strain tensor, the electric displacement field and the electric vector field and by $(u^{ε}, φ^{ε})$ the mechanical displacement field and the electric potential. The constitutive equations posed in $Ω^{ε}$ are: $\begin{matrix} (3.1) & \{\begin{matrix} σ^{ε} (u^{ε}, φ^{ε}) = C e^{ε} (u^{ε}) - P E^{ε} (φ^{ε}), \\ D^{ε} (u^{ε}, φ^{ε}) = P e^{ε} (u^{ε}) + d E^{ε} (φ^{ε}) . \end{matrix} \end{matrix}$ Let $x^{ε} = (x_{k}^{ε})$ be a generic point in ${\overline{Ω}}^{ε}$ , with $x_{α}^{ε} \in ω$ and $x_{3}^{ε} \in (- ε, ε)$ and let $\partial_{i}^{ε} = \frac{\partial}{\partial x_{i}^{ε}}$ . Then the linear strain tensor $e^{ε} (u^{ε})$ is defined by $e^{ε} (u^{ε}) = \frac{1}{2} (\nabla^{ε} u^{ε} + {(\nabla^{ε} u^{ε})}^{T})$ or component-wise by $\begin{matrix} e_{i j}^{ε} (u^{ε}) = \frac{1}{2} (\partial_{i}^{ε} u_{j}^{ε} + \partial_{j}^{ε} u_{i}^{ε}), \end{matrix}$ and the electric field vector $E^{ε} (φ^{ε})$ is given by $E_{k}^{ε} (φ^{ε}) = - \partial_{k}^{ε} φ^{ε}$ . For a plate subjected to applied body forces with density $f^{ε}$ and to electric charge with density $ρ_{e}^{ε}$ the equilibrium problem posed in $Ω^{ε}$ reads: $\begin{matrix} (3.2) & \{\begin{matrix} {div}^{ε} σ^{ε} (u^{ε}, φ^{ε}) = - f^{ε}, \\ {div}^{ε} D^{ε} (u^{ε}, φ^{ε}) = ρ_{e}^{ε} . \end{matrix} \end{matrix}$

We consider that the body

is fixed on the whole lateral surface $Γ_{0}^{ε}$ , this condition can be relaxed as indicated in Section 6,

is subjected to applied surface forces with density $g^{ε}$ , and to electric charge with density $ϑ^{ε}$ on the upper surface,

and is in mechanical contact with the lower face $Γ_{-}^{ε}$ .

3.2. The boundary conditions for a body in contact with a plane

We focus now on the unilateral contact with an horizontal plane set at level $- ε$ . For any point $x_{-}^{ε} = (x_{1}^{ε}, x_{2}^{ε}, - ε)$ on the lower face $Γ_{-}^{ε}$ the non-penetrability condition reads $(O x_{-}^{ε} + u^{ε} (x_{-}^{ε})) \cdot ν ⩾ 0$ , in other words $\begin{matrix} (3.3) & \forall x_{-}^{ε} \in Γ_{-}^{ε}, - ε + u_{3}^{ε} (x_{-}^{ε}) ⩾ - ε ⟹ u_{3}^{ε} (x_{-}^{ε}) ⩾ 0 . \end{matrix}$ The so-called Signorini conditions which give the full description of the unilaterality, are classically obtained by adding the following constraints to the non-penetrability condition:

No tensile forces but only compressive forces are exerted on the boundary by the obstacle;

All points in contact are on $Γ_{-}^{ε}$ , so that conditions (3.3) becomes an equality.

These constraints imply:

\begin{matrix} \{\begin{matrix} σ_{3}^{ε} (x_{-}^{ε}) = - (σ^{ε} (x_{-}^{ε}) ν) \cdot ν ⩾ 0 & \forall x^{ε} \in Γ_{-}^{ε}, \\ σ_{3}^{ε} (x_{-}^{ε}) u_{3}^{ε} (x_{-}^{ε}) = 0 & \forall x^{ε} \in Γ_{-}^{ε}, \end{matrix} \end{matrix}

so the contact condition reads:

\begin{matrix} (3.4) & u_{3}^{ε} (x_{-}^{ε}) ⩾ 0, σ_{3}^{ε} (x_{-}^{ε}) ⩾ 0, σ_{3}^{ε} (x_{-}^{ε}) u_{3}^{ε} (x_{-}^{ε}) = 0 on Γ_{-}^{ε}, \end{matrix}

and

σ_{3}^{ε} (x_{-})

is the Kuhn and Tucker multiplier associated to the contact condition. Eventually we gather all the conditions and get

\begin{matrix} (3.5) & \{\begin{matrix} u^{ε} = 0 & on Γ_{0}^{ε}, \\ φ^{ε} = 0 & on Γ_{0}^{ε}, \\ σ^{ε} (u^{ε}, φ^{ε}) ν = g^{ε} & on Γ_{+}^{ε}, \\ D^{ε} (u^{ε}, φ^{ε}) \cdot ν = ϑ^{ε} & on Γ_{+}^{ε}, \\ u_{3}^{ε} ⩾ 0, σ_{3}^{ε} ⩾ 0, σ_{3}^{ε} u_{3}^{ε} = 0, D^{ε} \cdot ν = 0 & on Γ_{-}^{ε} . \end{matrix} \end{matrix}

Let us note that in the case of a partial clamping on a part of the lateral surface, we can keep the same boundary conditions by restricting the lateral surface to $Γ_{0}^{ε} = γ_{0} \times [- ε, ε]$ with $γ_{0} = \partial ω$ . In Section 6 we use the “new definition of upper face” that includes the remaining part of the lateral surface $Γ_{+}^{ε} \cup (γ ∖ γ_{0}) \times [- ε, ε]$ .

3.3. The variational inequality in $Ω^{ε}$

The natural functional framework for problem (3.2), (3.5) is the product vector space $K^{ε} (Ω^{ε}) \times Q^{ε} (Ω^{ε})$ where $K^{ε} (Ω^{ε})$ is the convex set $\begin{matrix} (3.6) & K^{ε} (Ω^{ε}) = {v^{ε} \in H^{1} (Ω^{ε}), v^{ε} = 0 on Γ_{0}^{ε}, v_{3}^{ε} ⩾ 0 on Γ_{-}^{ε}} \end{matrix}$ and $Q^{ε} (Ω^{ε})$ is the vector space $\begin{matrix} (3.7) & Q^{ε} (Ω^{ε}) = {ψ^{ε} \in H^{1} (Ω^{ε}), ψ^{ε} = 0 on Γ_{0}^{ε}} . \end{matrix}$ Hence the weak solution $(u^{ε}, φ^{ε})$ to problem (3.2), (3.5), is given by the following variational inequality: $\begin{matrix} (3.8) & \{\begin{matrix} Find (u^{ε}, φ^{ε}) \in K^{ε} (Ω^{ε}) \times Q^{ε} (Ω^{ε}) such that: \\ \int_{Ω^{ε}} C_{i j k l} e_{k l}^{ε} (u^{ε}) e_{i j}^{ε} (v^{ε} - u^{ε}) d x^{ε} - \int_{Ω^{ε}} d_{i j} \partial_{j}^{ε} φ^{ε} \partial_{i}^{ε} (ψ^{ε} - φ^{ε}) d x^{ε} + \int_{Ω^{ε}} P_{k i j} \partial_{k}^{ε} φ^{ε} e_{i j}^{ε} (v^{ε} - u^{ε}) d x^{ε} \\ + \int_{Ω^{ε}} P_{k i j} \partial_{k}^{ε} (ψ^{ε} - φ^{ε}) e_{i j}^{ε} (u^{ε}) d x^{ε} \\ ⩾ \int_{Ω^{ε}} f^{ε} \cdot (v^{ε} - u^{ε}) d x^{ε} + \int_{Γ_{+}^{ε}} g^{ε} \cdot (v^{ε} - u^{ε}) d a^{ε} - \int_{Ω^{ε}} ρ_{e}^{ε} (ψ^{ε} - φ^{ε}) d x^{ε} \\ + \int_{Γ_{+}^{ε}} ϑ^{ε} (ψ^{ε} - φ^{ε}) d a^{ε} \\ \forall (v^{ε}, ψ^{ε}) \in K^{ε} (Ω^{ε}) \times Q^{ε} (Ω^{ε}), \end{matrix} \end{matrix}$ where $d a^{ε}$ is the area element of the boundary $\partial Ω^{ε}$ . Based on classical arguments ([10]).

For any fixed $ε > 0$ , the weak solution associated to problem (3.2), (3.5) is given by the unique solution (3.8) (for a proof see [10]).

Let us now introduce the scaling procedure in order to establish the convergence theorem.

4. Scaling and equilibrium equations in the fixed domain Ω

4.1. Scalings of the unknowns and test functions

We now change the domain $Ω^{ε}$ having middle surface ω and thickness $2 ε$ into a fixed domain Ω with the same middle surface and fixed thickness 2 via the simple geometrical transformation defined as follows: let $x^{ε} = (x_{k}^{ε})$ be a generic point in ${\overline{Ω}}^{ε}$ and the corresponding point $x = (x_{k})$ in $\overline{Ω}$ with $x_{α}^{ε} = x_{α} \in ω$ and $x_{3}^{ε} = ε x_{3} \in (- ε, ε)$ . This induces $\partial_{α}^{ε} = \frac{\partial}{\partial x_{α}^{ε}} = \frac{\partial}{\partial x_{α}}$ and $\partial_{3}^{ε} = \frac{\partial}{\partial x_{3}^{ε}} = \frac{1}{ε} \frac{\partial}{\partial x_{3}}$ . By analogy, the boundary of the domain Ω is divided into three parts: $\partial Ω = Γ_{-} \cup Γ_{+} \cup Γ_{0}$ , $Γ_{-} = ω \times {- 1}$ , $Γ_{+} = ω \times {1}$ , $Γ_{0} = γ \times [- 1, 1]$ .

We follow [12] and to any functions defined on ${\overline{Ω}}^{ε}$ we associate scaled functions defined on $\overline{Ω}$ as: $\begin{matrix} (4.1) & \{\begin{matrix} u_{α}^{ε} = ε^{2} u_{α} (ε), u_{3}^{ε} = ε u_{3} (ε), \\ v_{α}^{ε} = ε^{2} v_{α}, v_{3}^{ε} = ε v_{3}, \\ φ^{ε} = ε^{2} φ (ε), ψ^{ε} = ε^{2} ψ . \end{matrix} \end{matrix}$

Along with the scaling procedure, we let $e = (e_{i j})$ to denote the scaled linearized strain tensor, the components of which are $\begin{matrix} (4.2) & \{\begin{matrix} e_{α β}^{ε} (v^{ε}) = ε^{2} e_{α β} (v), \\ e_{α 3}^{ε} (v^{ε}) = ε e_{α 3} (v), \\ e_{33}^{ε} (v^{ε}) = e_{33} (v), \end{matrix} \end{matrix}$ where $e_{i j} (v) = \frac{1}{2} (\partial_{j} v_{i} + \partial_{i} v_{j})$ .

4.2. Assumptions on the data

In order to obtain a non-trivial limit problem by asymptotic analysis, it is essential, as it has been shown in [12] to scale the data in accordance with the scalings of the unknowns. More precisely we assume that there exist functions $f \in L^{2} (Ω)$ , $g \in L^{2} (Γ_{+})$ , $ρ_{e} \in L^{2} (Ω)$ , $ϑ \in L^{2} (Γ_{+})$ independent of ε such that $\begin{matrix} (4.3) & \{\begin{matrix} f_{α}^{ε} = ε^{2} f_{α}, f_{3}^{ε} = ε^{3} f_{3} & \forall x \in Ω, \\ g_{α}^{ε} = ε^{3} g_{α}, g_{3}^{ε} = ε^{4} g_{3} & \forall x \in Γ_{+}, \\ ρ_{e}^{ε} = ε^{2} ρ_{e} & \forall x \in Ω, \\ ϑ^{ε} = ε^{2} ϑ & \forall x \in Γ_{+} . \end{matrix} \end{matrix}$

4.3. The contact condition posed in the fixed domain

After the scaling process, the non-penetrability condition holds on $Γ_{-}$ and reads $\begin{matrix} (4.4) & v_{3} (x_{1}, x_{2}, - 1) ⩾ 0 \forall (x_{1}, x_{2}) \in ω, \end{matrix}$ and the corresponding functional space is $\begin{matrix} K (Ω) = {v \in H^{1} (Ω), v = 0 on Γ_{0}, v_{3} ⩾ 0 on Γ_{-}} . \end{matrix}$

4.4. The equilibrium problem in the fixed domain $Ω = ω \times (- 1, 1)$

Let us note $Q (Ω)$ the vector space associated to $Q^{ε} (Ω^{ε})$ : $\begin{matrix} Q (Ω) = {ψ \in H^{1} (Ω), ψ = 0 on Γ_{0}} . \end{matrix}$ Replacing $(u^{ε}, φ^{ε})$ by their scaled values $(u (ε), φ (ε))$ and $(v^{ε}, ψ^{ε})$ by $(v, ψ)$ given in (4.1) in the problem (3.8), yields the following problem posed over the fixed domain Ω: $\begin{matrix} (4.5) & \{\begin{matrix} Find (u (ε), φ (ε)) \in K (Ω) \times Q (Ω) such that: \\ ε^{5} \int_{Ω} C_{α β σ τ} e_{σ τ} (u (ε)) e_{α β} (v - u (ε)) d x - ε^{5} \int_{Ω} d_{α β} \partial_{β} φ (ε) \partial_{α} (ψ - φ (ε)) d x \\ + ε^{5} \int_{Ω} P_{γ α β} \partial_{γ} φ (ε) e_{α β} (v - u (ε)) d x + ε^{5} \int_{Ω} P_{γ α β} \partial_{γ} (ψ - φ (ε)) e_{α β} (u (ε)) d x \\ + 2 ε^{4} \int_{Ω} C_{α β σ 3} (e_{σ 3} (u (ε)) e_{α β} (v - u (ε)) + e_{α β} (u (ε)) e_{σ 3} (v - u (ε))) d x \\ - ε^{4} \int_{Ω} d_{α 3} (\partial_{3} φ (ε) \partial_{α} (ψ - φ (ε)) + \partial_{α} φ (ε) \partial_{3} (ψ - φ (ε))) d x \\ + 2 ε^{4} \int_{Ω} P_{γ α 3} \partial_{γ} φ (ε) e_{α 3} (v - u (ε)) d x \\ + ε^{4} \int_{Ω} P_{3 α β} \partial_{3} φ (ε) e_{α β} (v - u (ε)) d x + 2 ε^{4} \int_{Ω} P_{γ α 3} (\partial_{γ} (ψ - φ (ε)) e_{α 3} (u (ε)) \\ + \partial_{3} (ψ - φ (ε)) e_{α β} (u (ε))) d x \\ + 4 ε^{3} \int_{Ω} C_{α 3 σ 3} e_{σ 3} (u (ε)) e_{α 3} (v - u (ε)) d x \\ + ε^{3} \int_{Ω} C_{α β 33} (e_{33} (u (ε)) e_{α β} (v - u (ε)) + e_{α β} (u (ε)) e_{33} (v - u (ε))) d x \\ + 2 ε^{3} \int_{Ω} P_{3 α 3} (\partial_{3} φ (ε) e_{α 3} (v - u (ε)) + \partial_{3} (ψ - φ (ε)) e_{α 3} (u (ε))) d x \\ + ε^{3} \int_{Ω} P_{γ 33} (\partial_{γ} φ (ε) e_{33} (v - u (ε)) + \partial_{γ} (ψ - φ (ε)) e_{33} (u (ε))) d x \\ - ε^{3} \int_{Ω} d_{33} \partial_{3} φ (ε) \partial_{3} (ψ - φ (ε)) d x \\ + 2 ε^{2} \int_{Ω} C_{α 333} (e_{33} (u (ε)) e_{α 3} (v - u (ε)) + e_{α 3} (u (ε)) e_{33} (v - u (ε))) d x \\ + ε^{2} \int_{Ω} P_{333} (\partial_{3} φ (ε) e_{33} (v - u (ε)) + \partial_{3} (ψ - φ (ε)) e_{33} (u (ε))) d x \\ + ε \int_{Ω} C_{3333} e_{33} (u (ε)) e_{33} (v - u (ε)) d x \\ ⩾ ε^{5} \int_{Ω} (f \cdot (v - u (ε)) - ρ_{e} (ψ - φ (ε))) d x + ε^{5} \int_{Γ_{+}} (g \cdot (v - u (ε)) + ϑ (ψ - φ (ε))) d a \\ \forall (v, ψ) \in K (Ω) \times Q (Ω), \end{matrix} \end{matrix}$ where $d a$ is the area element of the boundary $\partial Ω$ .

Theorem 4.1.
For all fixed $ε > 0$ problem (4.5) has a unique weak solution.
Proof.
As pointed out before we use classical arguments ([10]) to prove the existence and uniqueness of the weak solution to the variational inequality problem (4.5). □

5. Two-dimensional limit scaled solution in the plate

The aim of this section is to establish that when ε tends to zero, the sequence ${(u (ε), φ (ε))}_{ε}$ converges to a limit $(u, φ)$ which solves a two-dimensional obstacle problem. An important preliminary point here is the following lemma, which is a new version of Korn’s inequality.

Lemma 5.1.
For all $v \in H^{1} (Ω)$ , the mapping $v \to {\sum_{i j} | e_{i j} (v) |_{0, Ω}^{2}}^{\frac{1}{2}}$ is a norm over the set $K (Ω)$ , which is equivalent to the norm $‖ \cdot ‖_{1, Ω}$ .
Proof.
The proof follows from the fact that the set $K (Ω)$ is a closed subset of the vector space $\begin{matrix} {v \in H^{1} (Ω), v = 0 on Γ_{0}} . □ \end{matrix}$

5.1. Variational solution

We recall the definition of the functional spaces $K (Ω)$ , $Q (Ω)$ $\begin{matrix} \{\begin{matrix} K (Ω) = {v \in H^{1} (Ω), v = 0 on Γ_{0}, v_{3} ⩾ 0 on Γ_{-}}, \\ Q (Ω) = {ψ \in H^{1} (Ω), ψ = 0 on Γ_{0}} . \end{matrix} \end{matrix}$ We are now in a position to prove the main result of the paper.

Theorem 5.1.
Assume $f \in L^{2} (Ω)$ , $ρ_{e} \in L^{2} (Ω)$ , $g \in L^{2} (Γ_{+})$ , $ϑ \in L^{2} (Γ_{+})$ . Then
As ε tends to 0, the family ${(u (ε), φ (ε))}_{ε > 0}$ converges strongly in the set $K (Ω) \times Q (Ω)$ to a limit ${(u, φ)}$ .

The limit u is a Kirchhoff–Love displacement field, namely there exists $ζ = (ζ_{H}, ζ_{3}) \in V_{H} (ω) \times K_{3} (ω)$ , such that $\begin{matrix} u_{α} = ζ_{α} - x_{3} \partial_{α} ζ_{3}, u_{3} = ζ_{3}, \end{matrix}$ with the definition of the bi-dimensional functional spaces $\begin{matrix} \{\begin{matrix} V_{H} (ω) = {η_{H} = (η_{α}) \in H^{1} (ω), η_{H} = 0 on γ}, \\ K_{3} (ω) = {η_{3} \in H^{2} (ω), η_{3} = \partial_{ν} η_{3} = 0 on γ, η_{3} ⩾ 0 in ω} . \end{matrix} \end{matrix}$

The limit electric potential φ belongs to the bi-dimensional functional space $\begin{matrix} Q (ω) = {ψ \in H^{1} (Ω), ψ = 0 on γ}, \end{matrix}$ and the limit solution $(ζ, φ) = (ζ_{H}, ζ_{3}, φ)$ solves the following coupled problem:

Find $(ζ_{H}, ζ_{3}, φ) \in V_{H} (ω) \times K_{3} (ω) \times Q (ω)$ such that: $\begin{matrix} (5.1) & \{\begin{matrix} \frac{2}{3} \int_{ω} {\tilde{C}}_{α β σ τ} \partial_{σ τ} ζ_{3} \partial_{α β} (η_{3} - ζ_{3}) d ω ⩾ \int_{ω} (p_{3} (η_{3} - ζ_{3}) - s_{α} \partial_{α} (η_{3} - ζ_{3})) d ω \\ \forall η_{3} \in K_{3} (ω), \\ 2 \int_{ω} {\tilde{n}}_{α β} (ζ_{H}, φ) e_{α β} (η_{H}) d ω + 2 \int_{ω} {\tilde{D}}_{γ} (ζ_{H}, φ) \partial_{γ} ψ d ω = \int_{ω} (p_{α} η_{α} - ({\tilde{ρ}}_{e} - ϑ ψ)) d ω \\ \forall (η_{H}, ψ) \in V_{H} (ω) \times Q (ω), \end{matrix} \end{matrix}$ where the new constitutive bi-dimensional law is given by $\begin{matrix} \{\begin{matrix} {\tilde{n}}_{α β} (ζ_{H}, φ) = {\tilde{C}}_{α β σ τ} e_{σ τ} (ζ_{H}) + {\tilde{P}}_{γ α β} \partial_{γ} φ, \\ {\tilde{D}}_{γ} (ζ_{H}, φ) = {\tilde{P}}_{γ α β} e_{α β} (ζ_{H}) - {\tilde{d}}_{γ β} \partial_{β} φ \end{matrix} \end{matrix}$ and the mechanical forces and electric charges are given by $\begin{matrix} p_{i} (x_{1}, x_{2}) : = \int_{- 1}^{1} f_{i} d x_{3} + g_{i}, s_{α} (x_{1}, x_{2}) : = \int_{- 1}^{1} x_{3} f_{α} d x_{3} + g_{α}, {\tilde{ρ}}_{e} (x_{1}, x_{2}) : = \int_{- 1}^{1} ρ_{e} d x_{3}, \end{matrix}$ where the new bi-dimensional elasticity tensor $\tilde{C}$ , piezoelectricity tensor $\tilde{P}$ and dielectric tensor $\tilde{d}$ are given in (5.13) .

Proof.
The proof is broken into five steps. In the first one, we introduce a new scaled strain tensor $R (ε)$ and with some boundness result we establish that the sequence ${(u (ε)), φ (ε)}$ converges weakly to a limit $(u, φ)$ , where u is a Kirchhoff–Love field. The second one deals with technical results about the components of this strain tensor. In the third step we show that the convergence of the family ${(u (ε)), φ (ε)}$ towards $(u, φ)$ is strong. The fourth step completes the proof by deducing the limit de-coupled transverse and membrane variational problems. Finally in the last step, we show that the membrane problem is of a piezoelectric form. Step I.
Let us introduce the following scaled symmetric tensor $R (ε) = (R_{i j} (ε)) \in L^{2} (Ω)$ and the scaled vector $τ (ε) = (τ_{i} (ε)) \in L^{2} (Ω)$ by $\begin{matrix} \{\begin{matrix} R_{α β} (ε) (v) = e_{α β} (v), R_{α 3} (ε) (v) = \frac{1}{ε} e_{α 3} (v), R_{33} (ε) (v) = \frac{1}{ε^{2}} e_{33} (v), \\ τ_{α} (ε) (ψ) = \partial_{α} ψ, τ_{3} (ε) (ψ) = \frac{1}{ε} \partial_{3} ψ . \end{matrix} \end{matrix}$ We introduce $R (ε) (u (ε))$ and $τ (ε) (φ (ε))$ in the variational inequality (4.5) and get $\begin{matrix} (5.2) & \{\begin{matrix} \int_{Ω} C_{α β σ τ} R_{σ τ} (ε) (u (ε)) e_{α β} (v - u (ε)) d x + 2 \int_{Ω} C_{α β σ 3} R_{σ 3} (ε) (u (ε)) e_{α β} (v - u (ε)) d x \\ + \int_{Ω} C_{α β 33} R_{33} (ε) (u (ε)) e_{α β} (v - u (ε)) d x \\ - \int_{Ω} d_{α β} τ_{β} (ε) (φ (ε)) \partial_{α} (ψ - φ (ε)) d x - \int_{Ω} d_{α 3} τ_{3} (ε) (φ (ε)) \partial_{α} (ψ - φ (ε)) d x \\ + \int_{Ω} P_{γ α β} τ_{γ} (ε) (φ (ε)) e_{α β} (v - u (ε)) d x + \int_{Ω} P_{γ α β} \partial_{γ} (ψ - φ (ε)) R_{α β} (ε) (u (ε)) d x \\ + \int_{Ω} P_{3 α β} τ_{3} (ε) (φ (ε)) e_{α β} (v - u (ε)) d x + 2 \int_{Ω} P_{γ α 3} \partial_{γ} (ψ - φ (ε)) R_{α 3} (ε) (u (ε)) d x \\ + \int_{Ω} P_{γ 33} \partial_{γ} (ψ - φ (ε)) R_{33} (ε) (u (ε)) d x \\ + \frac{2}{ε} \int_{Ω} C_{α 3 σ τ} R_{σ τ} (u (ε)) (ε) e_{α 3} (v - u (ε)) d x + \frac{4}{ε} \int_{Ω} C_{α 3 σ 3} R_{σ 3} (ε) (u (ε)) e_{α 3} (v - u (ε)) d x \\ + \frac{2}{ε} \int_{Ω} C_{α 333} R_{33} (ε) (u (ε)) e_{α 3} (v - u (ε)) d x \\ - \frac{1}{ε} \int_{Ω} d_{3 α} τ_{α} (ε) (φ (ε)) \partial_{3} (ψ - φ (ε)) d x - \frac{1}{ε} \int_{Ω} d_{33} τ_{3} (ε) (φ (ε)) \partial_{3} (ψ - φ (ε)) d x \\ + \frac{2}{ε} \int_{Ω} P_{γ α 3} τ_{γ} (ε) (φ (ε)) e_{α 3} (v - u (ε)) d x + \frac{2}{ε} \int_{Ω} P_{3 α 3} τ_{3} (ε) (φ (ε)) e_{α 3} (v - u (ε)) d x \\ + \frac{1}{ε} \int_{Ω} P_{3 α β} \partial_{3} (ψ - φ (ε)) R_{α β} (ε) (u (ε)) d x + \frac{2}{ε} \int_{Ω} P_{3 α 3} \partial_{3} (ψ - φ (ε)) R_{α 3} (ε) (u (ε)) d x \\ + \frac{1}{ε} \int_{Ω} P_{333} \partial_{3} (ψ - φ (ε)) R_{33} (ε) (u (ε)) d x \\ + \frac{1}{ε^{2}} \int_{Ω} C_{33 σ τ} R_{σ τ} (ε) (u (ε)) e_{33} (v - u (ε)) d x + \frac{2}{ε^{2}} \int_{Ω} C_{33 σ 3} R_{σ 3} (ε) (u (ε)) e_{33} (v - u (ε)) d x \\ + \frac{1}{ε^{2}} \int_{Ω} C_{3333} R_{33} (ε) (u (ε)) e_{33} (v - u (ε)) d x + \frac{1}{ε^{2}} \int_{Ω} P_{γ 33} τ_{γ} (ε) (φ (ε)) e_{33} (v - u (ε)) d x \\ + \frac{1}{ε^{2}} \int_{Ω} P_{333} τ_{3} (ε) (φ (ε)) e_{33} (v - u (ε)) d x \\ ⩾ \int_{Ω} f \cdot (v - u (ε)) d x + \int_{Γ_{+}} g \cdot (v - u (ε)) d a - \int_{Ω} ρ_{e} (ψ - φ (ε)) d x + \int_{Γ_{+}} ϑ (ψ - φ (ε)) d a, \end{matrix} \end{matrix}$ and next by introducing the new tensors $R (ε) (v - u (ε))$ and $τ (ε) (ψ - φ (ε))$ into inequality (5.2) we obtain, $\begin{matrix} (5.3) & \{\begin{matrix} \int_{Ω} C_{i j k l} R_{k l} (ε) (u (ε)) R_{i j} (ε) (v - u (ε)) d x - \int_{Ω} d_{i j} τ_{j} (ε) (φ (ε)) τ_{i} (ε) (ψ - φ (ε)) d x \\ + \int_{Ω} P_{k i j} τ_{k} (ε) (φ (ε)) R_{i j} (ε) (v - u (ε)) d x + \int_{Ω} P_{k i j} τ_{k} (ε) (ψ - φ (ε)) R_{i j} (ε) (u (ε)) d x \\ ⩾ \int_{Ω} f \cdot (v - u (ε)) d x + \int_{Γ_{+}} g \cdot (v - u (ε)) d a - \int_{Ω} ρ_{e} (ψ - φ (ε)) d x + \int_{Γ_{+}} ϑ (ψ - φ (ε)) d a . \end{matrix} \end{matrix}$ In inequality (5.3) we take $v = 0$ and $ψ = φ (ε)$ and get the inequality $\begin{matrix} \begin{matrix} - \int_{Ω} C_{i j k l} R_{k l} (ε) (u (ε)) R_{i j} (ε) (u (ε)) d x - \int_{Ω} P_{k i j} τ_{k} (ε) (φ (ε)) R_{i j} (ε) (u (ε)) d x \\ ⩾ - \int_{Ω} f \cdot u (ε) d x - \int_{Γ_{+}} g \cdot u (ε) d a . \end{matrix} \end{matrix}$ Next we let $v = u (ε)$ and $ψ = 2 φ (ε)$ in inequality (5.3), and we get: $\begin{matrix} \begin{matrix} - \int_{Ω} d_{i j} τ_{j} (ε) (φ (ε)) τ_{i} (ε) (φ (ε)) d x + \int_{Ω} P_{k i j} τ_{k} (ε) (φ (ε)) R_{i j} (ε) (u (ε)) d x \\ ⩾ - \int_{Ω} ρ_{e} φ (ε) d x + \int_{Γ_{+}} ϑ φ (ε) d a . \end{matrix} \end{matrix}$ Adding the previous two inequalities yields, $\begin{array}{l} \int_{Ω} C_{i j k l} R_{k l} (ε) (u (ε)) R_{i j} (ε) (u (ε)) d x + \int_{Ω} d_{i j} τ_{j} (ε) (φ (ε)) τ_{i} (ε) (φ (ε)) d x \\ ⩽ \int_{Ω} [f \cdot u (ε) + ρ_{e} φ (ε)] d x + \int_{Γ_{+}} [g \cdot u (ε) - ϑ φ (ε)] d a . \end{array}$ From this inequality, using the coerciveness properties of tensors C and d, we get $\begin{array}{l} {| R (ε) (u (ε)) |}_{0, Ω}^{2} + {| τ (ε) (φ (ε)) |}_{0, Ω}^{2} & ⩽ c ({‖ u (ε) ‖}_{0, Ω} + {‖ φ (ε) ‖}_{0, Ω}) \\ ⩽ c ({‖ u (ε) ‖}_{1, Ω} + {‖ φ (ε) ‖}_{1, Ω}) . \end{array}$ We recall Korn inequality and Poincare inequality: there exists $c > 0$ such that $\begin{matrix} {‖ u (ε) ‖}_{1, Ω}^{2} ⩽ c {| e (u (ε)) |}_{0, Ω}^{2}, {‖ φ (ε) ‖}_{1, Ω}^{2} ⩽ c {| τ (ε) (φ (ε)) |}_{0, Ω}^{2} . \end{matrix}$ Therefore for $ε ⩽ 1$ , there exist $c > 0$ : $\begin{matrix} \begin{matrix} {‖ u (ε) ‖}_{1, Ω}^{2} + {‖ φ (ε) ‖}_{1, Ω}^{2} & ⩽ c ({| e (u (ε)) |}_{0, Ω}^{2} + {‖ τ (ε) (φ (ε)) ‖}_{0, Ω}^{2}) \\ ⩽ c ({| R (ε) (u (ε)) |}_{0, Ω}^{2} + {| τ (ε) (φ (ε)) |}_{0, Ω}^{2}) \\ ⩽ c ({‖ u (ε) ‖}_{1, Ω} + {‖ φ (ε) ‖}_{1, Ω}), \end{matrix} \end{matrix}$ by ${(a + b)}^{2} ⩽ 2 (a^{2} + b^{2})$ , these inequalities imply that the norms $‖ u (ε) ‖_{1, Ω}$ , $‖ φ (ε) ‖_{1, Ω}$ , $| R (ε) (u (ε)) |_{0, Ω}$ , $| τ (ε) (φ (ε)) |_{0, Ω}$ are uniformly bounded. Therefore there exist $u \in H^{1} (Ω)$ , $φ \in H^{1} (Ω)$ and $R \in L^{2} (Ω)$ , $τ \in L^{2} (Ω)$ such that we have the weak convergences: $\begin{matrix} \{\begin{matrix} u (ε) ⇀ u in H^{1} (Ω), φ (ε) ⇀ φ in H^{1} (Ω), \\ R (ε) (u (ε)) ⇀ R (u) in L^{2} (Ω), τ (ε) (φ (ε)) ⇀ τ (φ) in L^{2} (Ω) . \end{matrix} \end{matrix}$ Moreover, from the definition of $R (ε)$ and $τ (ε)$ , we have the bounds: $\begin{matrix} {| e_{α 3} (u (ε)) |}_{0, Ω} ⩽ c ε, {| e_{33} (u (ε)) |}_{0, Ω} ⩽ c ε^{2}, {| \partial_{3} φ (ε) |}_{0, Ω} ⩽ c ε . \end{matrix}$ Hence $e_{i 3} (u (ε)) ⇀ 0$ in $L^{2} (Ω)$ , $\partial_{3} φ (ε) ⇀ 0$ in $L^{2} (Ω)$ , thus $\begin{matrix} {| e_{i 3} (u) |}_{0, Ω} ⩽ \underset{ε \to 0}{lim inf} {| e_{i 3} (u (ε)) |}_{0, Ω} = 0 \end{matrix}$ and $\begin{matrix} | \partial_{3} φ |_{0, Ω} ⩽ \underset{ε \to 0}{lim inf} {| \partial_{3} φ (ε) |}_{0, Ω} = 0 . \end{matrix}$ From the one hand, since $e_{i 3} (u) = 0$ , we deduce that there exist a bi-dimensional field $ζ = (ζ_{i})$ such that $ζ_{α} \in H^{1} (ω)$ and $ζ_{3} \in H^{2} (ω)$ and u is a Kirchhoff–Love displacement field $\begin{matrix} u_{α} = ζ_{α} - x_{3} \partial_{α} ζ_{3}, u_{3} = ζ_{3} . \end{matrix}$ From the other hand, since $\partial_{3} φ = 0$ , the limit φ is independent of $x_{3}$ .

The following lemma will be used several times in the next step. Lemma 5.2.
Let the operator $A (ε) : u (ε) \to H^{1} (Ω)$ which satisfies the weak convergence $\begin{matrix} A (ε) (u (ε)) ⇀ A (u) \in L^{2} (Ω) . \end{matrix}$ If $u (ε) \in K (Ω)$ solves the variational inequality $\begin{matrix} (5.4) & \int_{Ω} A (ε) (u (ε)) \cdot \partial_{3} (v - u (ε)) ⩾ ε \int_{Ω} f \cdot (v - u (ε)) \forall v \in K (Ω) \end{matrix}$ then $A (u) = 0$ .
Proof.
In (5.4) we first let $v = 0$ and then $v = 2 u (ε)$ . This yields $\begin{matrix} \int_{Ω} A (ε) (u (ε)) \cdot \partial_{3} u (ε) = ε \int_{Ω} f \cdot u (ε) . \end{matrix}$ And (5.4) now reads $\begin{matrix} \int_{Ω} A (ε) (u (ε)) \cdot \partial_{3} v ⩾ ε \int_{Ω} f \cdot v \forall v \in K (Ω) \end{matrix}$ which in turn implies $\begin{matrix} \int_{Ω} A (u) \cdot \partial_{3} v ⩾ 0 \forall v \in K (Ω) . \end{matrix}$ We consider the 2 cases of the in-plane components and the vertical component of the displacement fields.
Since $v_{α}$ belongs to a vector space, the previous inequality becomes an equality $\begin{matrix} \int_{Ω} A_{α} (u) \partial_{3} v_{α} = 0 \forall v_{α} \in H^{1} (Ω) \end{matrix}$ and following [8] we get $A_{α} (u) = 0$ .

For the transverse component we consider $z, t \in D (Ω)$ , $z > 0$ and select $v_{3} ⩾ 0$ under the form: $\begin{matrix} v_{3} (x) = z (x_{1}, x_{2}) + max_{- 1 < s < 1} | t (x_{1}, x_{2}, s) | + \int_{- 1}^{x_{3}} t (x_{1}, x_{2}, s) d s . \end{matrix}$ A direct computation $\int_{Ω} A_{3} (u) \partial_{3} v_{3} = \int_{Ω} A_{3} (u) t (x_{1}, x_{2}, x_{3}) = 0$ for all $t \in D (Ω)$ implies $A_{3} (u)$ . □

Step II.
Before we establish the strong convergence, let us compute $τ_{3} (φ)$ and $R_{i 3} (u)$ .

The weak convergences established in Step I imply: $\begin{matrix} \begin{matrix} R_{α β} (ε) (u (ε)) ⇀ R_{α β} (u) = e_{α β} (u) in L^{2} (Ω), \\ τ_{α} (ε) φ (ε) = \partial_{α} φ (ε) ⇀ τ_{α} (φ) = \partial_{α} φ in L^{2} (Ω) . \end{matrix} \end{matrix}$ We let $v = u (ε)$ in (5.2), and multiply the inequality by ε, we obtain: $\begin{matrix} \{\begin{matrix} \int_{Ω} (2 P_{3 α 3} R_{α 3} (ε) (u (ε)) + P_{333} R_{33} (ε) (u (ε)) - d_{33} τ_{3} (ε) (φ (ε)) - d_{3 γ} \partial_{γ} φ (ε) \\ + P_{3 α β} e_{α β} (ε) (u (ε))) \partial_{3} (ψ - φ (ε)) d x \\ ⩾ ε \int_{Ω} d_{α β} τ_{β} (ε) (φ (ε)) \partial_{α} (ψ - φ (ε)) d x + ε \int_{Ω} d_{α 3} τ_{3} (ε) (φ (ε)) \partial_{α} (ψ - φ (ε)) d x \\ - ε \int_{Ω} P_{γ α β} \partial_{γ} (ψ - φ (ε)) R_{α β} (ε) (u (ε)) d x - 2 ε \int_{Ω} P_{γ α 3} \partial_{γ} (ψ - φ (ε)) R_{α 3} (ε) (u (ε)) d x \\ - ε \int_{Ω} P_{γ 33} \partial_{γ} (ψ - φ (ε)) R_{33} (ε) (u (ε)) d x - ε \int_{Ω} ρ_{e} (ψ - φ (ε)) d x \\ + ε \int_{Γ_{+}} ϑ (ψ - φ (ε)) d a \\ \forall ψ \in Q (Ω) . \end{matrix} \end{matrix}$ The weak convergences and Lemma 5.2 yields $\begin{matrix} (5.5) & 2 P_{3 α 3} R_{α 3} (u) + P_{333} R_{33} (u) - d_{33} τ_{3} (φ) = d_{3 γ} \partial_{γ} φ - P_{3 α β} e_{α β} (u) . \end{matrix}$ Similarly in (5.2), we let $ψ = φ (ε)$ , $v_{3} = u_{3} (ε)$ , we multiply by ε and get: $\begin{matrix} \{\begin{matrix} \int_{Ω} (2 C_{α 3 σ τ} R_{σ τ} (u (ε)) (ε) + 4 C_{α 3 σ 3} R_{σ 3} (ε) (u (ε)) + 2 C_{α 333} R_{33} (ε) (u (ε))) \partial_{3} (v_{α} - u_{α} (ε)) d x \\ + 2 \int_{Ω} (P_{γ α 3} τ_{γ} (ε) (φ (ε)) + P_{3 α 3} τ_{3} (ε) (φ (ε))) \partial_{3} (v_{α} - u_{α} (ε)) d x \\ ⩾ - ε \int_{Ω} C_{α β σ τ} R_{σ τ} (ε) (u (ε)) e_{α β} (v - u (ε)) d x - 2 ε \int_{Ω} C_{α β σ 3} R_{σ 3} (ε) (u (ε)) e_{α β} (v - u (ε)) d x \\ - ε \int_{Ω} C_{α β 33} R_{33} (ε) (u (ε)) e_{α β} (v - u (ε)) d x - ε \int_{Ω} P_{γ α β} τ_{γ} (ε) (φ (ε)) e_{α β} (v - u (ε)) d x \\ - ε \int_{Ω} P_{3 α β} τ_{3} (ε) (φ (ε)) e_{α β} (v - u (ε)) d x + ε \int_{Ω} f_{α} (v_{α} - u_{α} (ε)) d x \\ + ε \int_{Γ_{+}} g_{α} (v_{α} - u_{α} (ε)) d a \\ \forall v_{α} \in H^{1} (Ω), v_{α} = 0 on Γ_{0}, \end{matrix} \end{matrix}$ from this, we obtain: $\begin{matrix} 2 C_{α 3 σ 3} R_{σ 3} (u) + C_{α 333} R_{33} (u) + P_{3 α 3} τ_{3} (φ) = - P_{γ α 3} \partial_{γ} φ - C_{α 3 α β} e_{α β} (u) . \end{matrix}$ Finally, in (5.2), we let $ψ = φ (ε)$ , $v_{α} = u_{α} (ε)$ , multiply by $ε^{2}$ , we have: $\begin{array}{l} \{\begin{matrix} \int_{Ω} (C_{33 σ τ} R_{σ τ} (ε) (u (ε)) + 2 C_{33 σ 3} R_{σ 3} (ε) (u (ε)) + C_{3333} R_{33} (ε) (u (ε))) \partial_{3} (v_{3} - u_{3} (ε)) d x \\ + \int_{Ω} (P_{γ 33} τ_{γ} (ε) (φ (ε)) + P_{333} τ_{3} (ε) (φ (ε))) \partial_{3} (v_{3} - u_{3} (ε)) d x \\ ⩾ - ε \int_{Ω} C_{α 3 σ τ} R_{σ τ} (u (ε)) (ε) \partial_{α} (v_{3} - u_{3} (ε)) d x - 2 ε \int_{Ω} C_{α 3 σ 3} R_{σ 3} (ε) (u (ε)) \partial_{α} (v_{3} - u_{3} (ε)) d x \\ - ε \int_{Ω} C_{α 333} R_{33} (ε) (u (ε)) \partial_{α} (v_{3} - u_{3} (ε)) d x - ε \int_{Ω} P_{γ α 3} τ_{γ} (ε) (φ (ε)) \partial_{α} (v_{3} - u_{3} (ε)) d x \\ - ε \int_{Ω} P_{3 α 3} τ_{3} (ε) (φ (ε)) \partial_{α} (v_{3} - u_{3} (ε)) d x + ε^{2} \int_{Ω} f_{3} (v_{3} - u_{3} (ε)) d x \\ + ε^{2} \int_{Γ_{+}} g_{3} (v_{3} - u_{3} (ε)) d a \\ \forall v_{3} \in H^{1} (Ω), v_{3} = 0 on Γ_{0}, v_{3} ⩾ 0 on Γ_{-}, \end{matrix} \\ 2 C_{33 α 3} R_{α 3} (u) + C_{3333} R_{33} (u) + P_{333} τ_{3} (φ) = - P_{γ 33} \partial_{γ} φ - C_{33 α β} e_{α β} (u) . \end{array}$ Thus $R_{i 3} (u)$ satisfy the following linear system $\begin{matrix} (5.6) & 2 C_{i 3 α 3} R_{α 3} (u) + C_{i 333} R_{33} (u) + P_{3 i 3} τ_{3} (φ) = - P_{γ i 3} \partial_{γ} φ - C_{i 3 α β} e_{α β} (u) . \end{matrix}$ We get $τ_{3} (φ)$ from (5.5) and substitute it in (5.6), so the three unknowns $R_{i 3} (u)$ solve the linear system of three equations: $\begin{matrix} (5.7) & 2 C_{i 3 α 3}^{'} R_{α 3} (u) + C_{i 333}^{'} R_{33} (u) = - P_{γ i 3}^{'} \partial_{γ} φ - C_{i 3 α β}^{'} e_{α β} (u), \end{matrix}$ where $C_{i j k l}^{'} = C_{i j k l} + \frac{P_{3 i j} P_{3 k l}}{d_{33}}$ and $P_{k i j}^{'} = P_{k i j} - \frac{P_{3 i j} d_{3 k}}{d_{33}}$ . To show that this system has a unique solution we first note that since $d_{33} > 0$ , the symmetry and positivity of tensor C yield the symmetry and positivity of tensor $C^{'}$ , more precisely: $\begin{matrix} C_{i j k l}^{'} A_{k l} A_{i j} = C_{i j k l} A_{k l} A_{i j} + \frac{P_{3 i j} P_{3 k l}}{d_{33}} A_{k l} A_{i j} ⩾ c A_{k l} A_{i j} \forall A_{i j} \in R, A_{i j} = A_{j i} . \end{matrix}$ Next we note that system (5.7) can be written as $2 C_{i 3 α 3}^{'} x_{α} + C_{i 333}^{'} x_{3} = f_{i}$ , the determinant of this linear system is $\begin{matrix} |\begin{matrix} 2 C_{1313}^{'} & 2 C_{1323}^{'} & C_{1333}^{'} \\ 2 C_{2313}^{'} & 2 C_{2323}^{'} & C_{2333}^{'} \\ 2 C_{3313}^{'} & 2 C_{3323}^{'} & C_{3333}^{'} \end{matrix}| = 4 |\begin{matrix} C_{1313}^{'} & C_{1323}^{'} & C_{1333}^{'} \\ C_{2313}^{'} & C_{2323}^{'} & C_{2333}^{'} \\ C_{3313}^{'} & C_{3323}^{'} & C_{3333}^{'} \end{matrix}| . \end{matrix}$ With $A_{k α} = 0$ we get $C_{i j k l}^{'} A_{k l} A_{i j} = C_{i 3 j 3}^{'} A_{i 3} A_{j 3} > 0$ , therefore system (5.7) has a unique solution, which reads: $\begin{matrix} (5.8) & \{\begin{matrix} R_{α 3} (u) = - \frac{1}{2 △} (△_{γ}^{α} \partial_{γ} φ + △_{ζ η}^{α} e_{ζ η} (u)), \\ R_{33} (u) = - \frac{1}{△} (△_{γ}^{3} \partial_{γ} φ + △_{ζ η}^{3} e_{ζ η} (u)), \\ τ_{3} (φ) = - \frac{1}{d_{33}} [(\frac{1}{△} P_{3 k 3} △_{γ ρ}^{k} - P_{3 γ ρ}) e_{γ ρ} (u) + (\frac{1}{△} P_{3 k 3} △_{γ}^{k} + d_{3 γ}) \partial_{γ} φ], \end{matrix} \end{matrix}$ where $\begin{matrix} \{\begin{matrix} △ = ϵ_{p q r} C_{13 p 3}^{'} C_{23 q 3}^{'} C_{33 r 3}^{'}, \\ △_{γ}^{1} = ϵ_{p q r} P_{γ p 3}^{'} C_{23 q 3}^{'} C_{33 r 3}^{'}, △_{γ}^{2} = ϵ_{p q r} C_{13 p 3}^{'} P_{γ q 3}^{'} C_{33 r 3}^{'}, △_{γ}^{3} = ϵ_{p q r} C_{13 p 3}^{'} C_{23 q 3}^{'} P_{γ r 3}^{'}, \\ △_{ζ η}^{1} = ϵ_{p q r} C_{3 p ζ η}^{'} C_{23 q 3}^{'} C_{33 r 3}^{'}, △_{ζ η}^{2} = ϵ_{p q r} C_{13 p 3}^{'} C_{3 q ζ η}^{'} C_{33 r 3}^{'}, △_{ζ η}^{3} = ϵ_{p q r} C_{13 p 3}^{'} C_{23 q 3}^{'} C_{3 r ζ η}^{'}, \end{matrix} \end{matrix}$ and ϵ is the Levi-Civitta symbol $\begin{matrix} ϵ_{i j k} = \{\begin{matrix} 1, & (i, j, k) is even permutation of (1, 2, 3), \\ - 1, & (i, j, k) is odd permutation of (1, 2, 3) . \end{matrix} \end{matrix}$
Step III.
We now show that the whole family ${u (ε), φ (ε)}_{ε}$ converges strongly.

Let us introduce the notation $\int_{Ω} C A : A d x = \int_{Ω} C_{i j k l} A_{k l} A_{i j} d x$ for all second order symmetric tensor A and let $\int_{Ω} d v : v d x = \int_{Ω} d_{i j} v_{i} v_{j} d x$ for all vector $v \in R^{3}$ , $\begin{array}{l} c {| R (ε) (u (ε)) - R (u) |}_{0, Ω}^{2} \\ ⩽ \int_{Ω} C (R (ε) (u (ε)) - R (u)) : (R (ε) (u (ε)) - R (u)) d x \\ ⩽ \int_{Ω} C R (u) : (R (u) - 2 R (ε) (u (ε))) d x + \int_{Ω} C R (ε) (u (ε)) : R (ε) (u (ε)) d x . \end{array}$ Since we have already established the weak convergences $R (ε) ⇀ R$ in $L^{2} (Ω)$ as $ε \to 0$ , then $\begin{matrix} lim_{ε \to 0} c {| R (ε) (u (ε)) - R (u) |}_{0, Ω}^{2} ⩽ - \int_{Ω} C R (u) : R (u) d x + lim_{ε \to 0} \int_{Ω} C R (ε) (u (ε)) : R (ε) (u (ε)) d x . \end{matrix}$ In (5.3), let $ψ = φ (ε)$ , $v = u$ , and pass to the limit, we get: $\begin{matrix} \int_{Ω} C R (u) : R (u) d x - lim_{ε \to 0} \int_{Ω} C R (ε) (u (ε)) : R (ε) (u (ε)) d x ⩾ 0 . \end{matrix}$ Then we have ${lim}_{ε \to 0} | R (ε) (u (ε)) - R (u) |_{0, Ω}^{2} ⩽ 0$ , which combined with the previous inequality as $ε \to 0$ , so we get the strong convergence.

By the definition of $R (ε)$ , we get: $\begin{array}{l} {| e (u (ε)) - e (u) |}_{0, Ω}^{2} ⩽ & \sum_{α, β} {| R_{α β} (ε) (u (ε)) - R_{α β} (u) |}_{0, Ω}^{2} + 2 ε^{2} \sum_{α} {| R_{α 3} (ε) (u (ε)) - R_{α 3} (u) |}_{0, Ω}^{2} \\ + ε^{4} {| R_{33} (ε) (u (ε)) - R_{33} (u) |}_{0, Ω}^{2}, \end{array}$ which implies that the sequence ${e (u (ε))}_{ε}$ converges strongly in $L^{2} (Ω)$ to $e (u)$ . Therefore by Korn’s inequality the sequence $u (ε)$ converges strongly in $H^{1} (Ω)$ to u.

Similarly, $\begin{array}{l} c {| τ (ε) (φ (ε)) - τ (φ) |}_{0, Ω}^{2} \\ ⩽ \int_{Ω} d (τ (ε) (φ (ε)) - τ (φ)) : (τ (ε) (φ (ε)) - τ (φ)) d x \\ ⩽ \int_{Ω} d τ (φ) : (τ (φ) - 2 τ (ε) (φ (ε))) d x + \int_{Ω} d τ (ε) (φ (ε)) : τ (ε) (φ (ε)) d x . \end{array}$ Since we have already established the weak convergence $τ (ε) (φ (ε)) ⇀ τ (φ)$ in $L^{2} (Ω)$ as $ε \to 0$ , we get $\begin{matrix} lim_{ε \to 0} c {| τ (ε) (φ (ε)) - τ (φ) |}_{0, Ω}^{2} ⩽ - \int_{Ω} d τ (φ) : τ (φ) d x + lim_{ε \to 0} \int_{Ω} d τ (ε) (φ (ε)) : τ (ε) (φ (ε)) d x . \end{matrix}$ We let $v = u (ε)$ in (5.3), which yields: $\begin{matrix} (5.9) & \{\begin{matrix} - \int_{Ω} d_{i j} τ_{j} (ε) (φ (ε)) τ_{i} (ε) (ψ - φ (ε)) d x + \int_{Ω} P_{k i j} τ_{k} (ε) (ψ - φ (ε)) R_{i j} (ε) (u (ε)) d x \\ ⩾ - \int_{Ω} ρ_{e} (ψ - φ (ε)) d x + \int_{Γ_{+}} ϑ (ψ - φ (ε)) d a \\ \forall ψ \in Q (Ω) . \end{matrix} \end{matrix}$ Using the fact that ψ belongs to a vector space this inequality can be rewritten, tacking $ψ = φ (ε)$ , as the equality: $\begin{matrix} \begin{matrix} \int_{Ω} d_{i j} τ_{j} (ε) (φ (ε)) τ_{i} (ε) (φ (ε)) d x \\ = \int_{Ω} P_{k i j} τ_{k} (ε) (φ (ε)) R_{i j} (ε) (u (ε)) d x - \int_{Ω} ρ_{e} φ (ε) d x + \int_{Γ_{+}} ϑ φ (ε) d a, \end{matrix} \end{matrix}$ then as $ε \to 0$ , $\begin{array}{l} - \int_{Ω} d τ (φ) : τ (φ) d x + lim_{ε \to 0} \int_{Ω} d τ (ε) (φ (ε)) : τ (ε) (φ (ε)) d x \\ = - \int_{Ω} d τ (φ) : τ (φ) d x \\ + lim_{ε \to 0} (\int_{Ω} P_{k i j} τ_{k} (ε) (φ (ε)) R_{i j} (ε) (u (ε)) d x + \int_{Ω} ρ_{e} φ (ε) d x - \int_{Γ_{+}} ϑ φ (ε) d x) \\ = 0, \end{array}$ so $| τ (ε) (φ (ε)) - τ (φ) |_{0, Ω}^{2} ⩽ 0$ , we get the strong convergence, the sequence ${τ (φ (ε))}_{ε}$ converges strongly in $L^{2} (Ω)$ to $τ (φ)$ , then the sequence of $φ (ε)$ converges strongly to φ in $H^{1} (Ω)$ .
Step IV.
For any Kirchhoff–Love vector field $v = (ζ_{H}, ζ_{3}) \in V_{H} (ω) \times K_{3} (ω)$ we have $\begin{matrix} e_{α β} (v) = e_{α β} (η_{H}) - x_{3} \partial_{α β} η_{3}, e_{i 3} (v) = 0 . \end{matrix}$ We pass to the limit in (5.3) and get for all vector field $v = (ζ_{H}, ζ_{3}) \in V_{H} (ω) \times K_{3} (ω)$ the variational inequality: $\begin{matrix} \{\begin{matrix} \int_{Ω} C_{α β σ τ} (e_{σ τ} (ζ_{H}) - x_{3} \partial_{σ τ} ζ_{3}) (e_{α β} (η_{H} - ζ_{H}) - x_{3} \partial_{α β} (η_{3} - ζ_{3})) d x \\ + 2 \int_{Ω} C_{α β σ 3} [- \frac{1}{2 △} (△_{γ}^{σ} \partial_{γ} φ + △_{σ τ}^{σ} (e_{σ τ} (ζ_{H}) - x_{3} \partial_{σ τ} ζ_{3}))] \\ \times (e_{α β} (η_{H} - ζ_{H}) - x_{3} \partial_{α β} (η_{3} - ζ_{3})) d x \\ + \int_{Ω} C_{α β 33} [- \frac{1}{△} (△_{γ}^{3} \partial_{γ} φ + △_{σ τ}^{3} (e_{σ τ} (ζ_{H}) - x_{3} \partial_{σ τ} ζ_{3}))] \\ \times (e_{α β} (η_{H} - ζ_{H}) - x_{3} \partial_{α β} (η_{3} - ζ_{3})) d x \\ - \int_{Ω} d_{α β} \partial_{β} φ \partial_{α} (ψ - φ) d x \\ - \int_{Ω} d_{α 3} [- \frac{1}{d_{33}} [(\frac{1}{△} P_{3 k 3} △_{σ τ}^{k} - P_{3 σ τ}) (e_{σ τ} (ζ_{H}) - x_{3} \partial_{σ τ} ζ_{3}) \\ + (\frac{1}{△} P_{3 k 3} △_{γ}^{k} + d_{3 γ}) \partial_{γ} φ]] \partial_{α} (ψ - φ) d x \\ + \int_{Ω} P_{γ α β} \partial_{γ} φ (e_{α β} (η_{H} - ζ_{H}) - x_{3} \partial_{α β} (η_{3} - ζ_{3})) d x \\ + \int_{Ω} P_{γ α β} \partial_{γ} (ψ - φ) (e_{α β} (ζ_{H}) - x_{3} \partial_{α β} ζ_{3}) d x \\ + \int_{Ω} P_{3 α β} [- \frac{1}{d_{33}} (\frac{1}{△} P_{3 k 3} △_{σ τ}^{k} - P_{3 σ τ}) (e_{σ τ} (ζ_{H}) - x_{3} \partial_{σ τ} ζ_{3}) \\ \times (e_{α β} (η_{H} - ζ_{H}) - x_{3} \partial_{α β} (η_{3} - ζ_{3}))] d x \\ + 2 \int_{Ω} P_{γ σ 3} \partial_{γ} (ψ - φ) [- \frac{1}{2 △} (△_{γ}^{σ} \partial_{γ} φ + △_{σ τ}^{σ} (e_{σ τ} (ζ_{H}) - x_{3} \partial_{σ τ} ζ_{3}))] d x \\ + \int_{Ω} P_{γ 33} \partial_{γ} (ψ - φ) [- \frac{1}{△} (△_{γ}^{3} \partial_{γ} φ + △_{σ τ}^{3} (e_{σ τ} (ζ_{H}) - x_{3} \partial_{σ τ} ζ_{3}))] d x \\ ⩾ \int_{Ω} [f_{α} (η_{α} - ζ_{α} - x_{3} \partial_{α} (η_{3} - ζ_{3})) + f_{3} (η_{3} - ζ_{3})] d x \\ + \int_{Γ_{+}} [g_{α} (η_{α} - ζ_{α} - x_{3} \partial_{α} (η_{3} - ζ_{3})) + g_{3} (η_{3} - ζ_{3})] d a \\ - \int_{Ω} ρ_{e} (ψ - φ) d x + \int_{Γ_{+}} ϑ (ψ - φ) d a . \end{matrix} \end{matrix}$ Then we get $\begin{matrix} \{\begin{matrix} 2 \int_{ω} {\tilde{C}}_{α β σ τ} e_{σ τ} (ζ_{H}) e_{α β} (η_{H} - ζ_{H}) d ω + \frac{2}{3} \int_{ω} {\tilde{C}}_{α β σ τ} \partial_{σ τ} ζ_{3} \partial_{α β} (η_{3} - ζ_{3}) d ω \\ + 2 \int_{ω} {\tilde{P}}_{γ α β} \partial_{γ} φ e_{α β} (η_{H} - ζ_{H}) d ω + 2 \int_{ω} {\tilde{P}}_{γ α β}^{♠} \partial_{γ} (ψ - φ) e_{α β} (ζ_{H}) d ω \\ - 2 \int_{ω} {\tilde{d}}_{γ β} \partial_{β} φ \partial_{γ} (ψ - φ) d ω \\ ⩾ \int_{ω} p_{H} \cdot (η_{H} - ζ_{H}) d ω - \int_{ω} ({\tilde{ρ}}_{e} - ϑ) (ψ - φ) d ω \\ + \int_{ω} p_{3} (η_{3} - ζ_{3}) d ω - \int_{ω} s_{α} \partial_{α} (η_{3} - ζ_{3}) d ω . \end{matrix} \end{matrix}$ This inequality can be decoupled as: $\begin{matrix} (5.10) & \{\begin{matrix} \frac{2}{3} \int_{ω} {\tilde{C}}_{α β σ τ} \partial_{σ τ} ζ_{3} \partial_{α β} (η_{3} - ζ_{3}) d ω ⩾ \int_{ω} p_{3} (η_{3} - ζ_{3}) d ω - \int_{ω} s_{α} \partial_{α} (η_{3} - ζ_{3}) d ω \\ \forall η_{3} \in K_{3} (ω), \\ 2 \int_{ω} {\tilde{n}}_{α β} (ζ_{H}, φ) e_{α β} (η_{H}) d ω + 2 \int_{ω} {\tilde{D}}_{γ} (ζ_{H}, φ) \partial_{γ} ψ d ω = \int_{ω} p_{H} \cdot η_{H} d ω - \int_{ω} ({\tilde{ρ}}_{e} - ϑ) ψ d ω \\ \forall (η_{H}, ψ) \in V_{H} (ω) \times Q (ω), \end{matrix} \end{matrix}$ where the constitutive law is $\begin{matrix} \{\begin{matrix} {\tilde{n}}_{α β} (ζ_{H}, φ) = {\tilde{C}}_{α β σ τ} e_{σ τ} (ζ_{H}) + {\tilde{P}}_{γ α β} \partial_{γ} φ, \\ {\tilde{D}}_{γ} (ζ_{H}, φ) = {\tilde{P}}_{γ α β}^{♠} e_{α β} (ζ_{H}) - {\tilde{d}}_{γ β} \partial_{β} φ \end{matrix} \end{matrix}$ and the mechanical forces and electric charges are given by $\begin{matrix} p_{i} : = \int_{- 1}^{1} f_{i} d x_{3} + g_{i}, s_{α} : = \int_{- 1}^{1} x_{3} f_{α} d x_{3} + g_{α}, {\tilde{ρ}}_{e} : = \int_{- 1}^{1} ρ_{e} d x_{3}, \end{matrix}$ where the new characteristics of the piezoelectric plate are $\begin{matrix} (5.11) & \{\begin{matrix} {\tilde{C}}_{α β σ τ} = C_{α β σ τ}^{'} - \frac{1}{△} C_{α β k 3}^{'} △_{σ τ}^{k}, C_{i j k l}^{'} = C_{i j k l} + \frac{P_{3 i j} P_{3 k l}}{d_{33}}, \\ {\tilde{P}}_{γ α β} = P_{γ α β}^{'} - \frac{1}{△} C_{α β k 3}^{'} △_{γ}^{k}, {\tilde{P}}_{γ α β}^{♠} = P_{γ α β}^{'} - \frac{1}{△} P_{γ k 3}^{'} △_{α β}^{k}, P_{k i j}^{'} = P_{k i j} - \frac{P_{3 i j} d_{3 k}}{d_{33}}, \\ {\tilde{d}}_{γ β} = d_{γ β}^{'} + \frac{1}{△} P_{γ k 3}^{'} △_{β}^{k}, d_{i j}^{'} = d_{i j} - \frac{d_{i 3} d_{3 j}}{d_{33}}, \end{matrix} \end{matrix}$ and $\begin{matrix} (5.12) & \{\begin{matrix} △ = ϵ_{p q r} C_{13 p 3}^{'} C_{23 q 3}^{'} C_{33 r 3}^{'}, \\ △_{γ}^{1} = ϵ_{p q r} P_{γ p 3}^{'} C_{23 q 3}^{'} C_{33 r 3}^{'}, △_{γ}^{2} = ϵ_{p q r} C_{13 p 3}^{'} P_{γ q 3}^{'} C_{33 r 3}^{'}, △_{γ}^{3} = ϵ_{p q r} C_{13 p 3}^{'} C_{23 q 3}^{'} P_{γ r 3}^{'}, \\ △_{ζ η}^{1} = ϵ_{p q r} C_{3 p ζ η}^{'} C_{23 q 3}^{'} C_{33 r 3}^{'}, △_{ζ η}^{2} = ϵ_{p q r} C_{13 p 3}^{'} C_{3 q ζ η}^{'} C_{33 r 3}^{'}, △_{ζ η}^{3} = ϵ_{p q r} C_{13 p 3}^{'} C_{23 q 3}^{'} C_{3 r ζ η}^{'}, \end{matrix} \end{matrix}$ with the Levi-Civitta symbol $\begin{matrix} ϵ_{i j k} = \{\begin{matrix} + 1, & (i, j, k) is an even permutation of (1, 2, 3), \\ - 1, & (i, j, k) is an odd permutation of (1, 2, 3) . \end{matrix} \end{matrix}$ We note that the constitutive law which relates ${\tilde{D}}_{γ}$ and ${\tilde{n}}_{α β}$ to $(ζ_{H}, φ)$ has not the symmetric property of piezoelectricity. The next step will tackle this incoherence.
Step V.
According to the definition given in (5.12), we have $\begin{array}{l} {\tilde{P}}_{γ α β} = P_{γ α β}^{'} - \frac{1}{△} C_{α β k 3}^{'} △_{γ}^{k}, \\ C_{α β k 3}^{'} △_{γ}^{k} = C_{α β 13}^{'} △_{γ}^{1} + C_{α β 23}^{'} △_{γ}^{2} + C_{α β 33}^{'} △_{γ}^{3} \\ = C_{α β 13}^{'} P_{γ p 3}^{'} C_{23 q 3}^{'} C_{33 r 3}^{'} + C_{α β 23}^{'} C_{13 p 3}^{'} P_{γ q 3}^{'} C_{33 r 3}^{'} + C_{α β 33}^{'} C_{13 p 3}^{'} C_{23 q 3}^{'} P_{γ r 3}^{'} \\ = P_{γ 13}^{'} (C_{α β 13}^{'} C_{2323}^{'} C_{3333}^{'} - C_{α β 13}^{'} C_{2333}^{'} C_{3323}^{'} + C_{α β 23}^{'} C_{1333}^{'} C_{3323}^{'} - C_{α β 23}^{'} C_{1323}^{'} C_{3333}^{'} \\ + C_{α β 33}^{'} C_{1323}^{'} C_{2333}^{'} - C_{α β 33}^{'} C_{1333}^{'} C_{2323}^{'}) \\ + P_{γ 23}^{'} (C_{α β 13}^{'} C_{2333}^{'} C_{3313}^{'} - C_{α β 13}^{'} C_{2313}^{'} C_{3333}^{'} + C_{α β 23}^{'} C_{1313}^{'} C_{3333}^{'} - C_{α β 23}^{'} C_{1333}^{'} C_{3313}^{'} \\ + C_{α β 33}^{'} C_{1333}^{'} C_{2313}^{'} - C_{α β 33}^{'} C_{1313}^{'} C_{2333}^{'}) \\ + P_{γ 33}^{'} (C_{α β 13}^{'} C_{2313}^{'} C_{3323}^{'} - C_{α β 13}^{'} C_{2323}^{'} C_{3313}^{'} + C_{α β 23}^{'} C_{1313}^{'} C_{3333}^{'} - C_{α β 23}^{'} C_{1333}^{'} C_{3313}^{'} \\ + C_{α β 33}^{'} C_{1313}^{'} C_{2323}^{'} - C_{α β 33}^{'} C_{1323}^{'} C_{2313}^{'}) \\ = ϵ_{p q r} (P_{γ 13}^{'} C_{α β p 3}^{'} C_{23 q 3}^{'} C_{33 r 3}^{'} + P_{γ 23}^{'} C_{13 p 3}^{'} C_{α β q 3}^{'} C_{33 r 3}^{'} + P_{γ 33}^{'} C_{13 p 3}^{'} C_{23 q 3}^{'} C_{α β r 3}^{'}), \end{array}$ and similarly $\begin{array}{l} {\tilde{P}}_{γ α β}^{♠} & = P_{γ α β}^{'} - \frac{1}{△} P_{γ k 3}^{'} △_{α β}^{k} = P_{γ α β}^{'} - \frac{1}{△} P_{γ 13}^{'} △_{α β}^{1} - \frac{1}{△} P_{γ 23}^{'} △_{α β}^{2} - \frac{1}{△} P_{γ 33}^{'} △_{α β}^{3} \\ = P_{γ α β}^{'} - \frac{1}{△} ϵ_{p q r} (P_{γ 13}^{'} C_{3 p α β}^{'} C_{23 q 3}^{'} C_{33 r 3}^{'} + P_{γ 23}^{'} C_{13 p 3}^{'} C_{3 q α β}^{'} C_{33 r 3}^{'} + P_{γ 33}^{'} C_{13 p 3}^{'} C_{23 q 3}^{'} C_{3 r α β}^{'}) . \end{array}$ By the properties of the determinant and symmetry (2.1), we get ${\tilde{P}}_{γ α β} = {\tilde{P}}_{γ α β}^{♠}$ . So finally the new bi-dimensional piezoelectric tensors are given by $\begin{matrix} (5.13) & \{\begin{matrix} {\tilde{C}}_{α β σ τ} = C_{α β σ τ}^{'} - \frac{1}{△} C_{α β k 3}^{'} △_{σ τ}^{k}, C_{i j k l}^{'} = C_{i j k l} + \frac{P_{3 i j} P_{3 k l}}{d_{33}}, \\ {\tilde{P}}_{γ α β} = P_{γ α β}^{'} - \frac{1}{△} C_{α β k 3}^{'} △_{γ}^{k}, P_{k i j}^{'} = P_{k i j} - \frac{P_{3 i j} d_{3 k}}{d_{33}}, \\ {\tilde{d}}_{γ β} = d_{γ β}^{'} + \frac{1}{△} P_{γ k 3}^{'} △_{β}^{k}, d_{i j}^{'} = d_{i j} - \frac{d_{i 3} d_{3 j}}{d_{33}}, \end{matrix} \end{matrix}$ and △, $△_{γ}^{k}$ , $△_{ζ η}^{k}$ are given in (5.12).

Hence we get the desired variational inequalities. □

5.2. Strong formulation

Let us recall that $ν = (ν_{1}, ν_{2})$ is the unit outer normal vector along $\partial ω$ , and we denote $ι = (- ν_{2}, ν_{1})$ one unit tangent vector along $\partial ω$ . From Theorem 5.1, we get the following strong formulations as two decoupled problems, the flexural one with a contact condition and without electric part and the membrane one which contains the piezoelectric effect.

Theorem 5.2 (Flexural equations).

The limit two-dimensional vertical displacement field solves the flexural problem.

Equilibrium equation $\begin{matrix} \frac{2}{3} \partial_{α β} {\tilde{σ}}_{α β} (ζ_{3}) ⩾ (p_{3} + \partial_{α} s_{α}) in ω . \end{matrix}$ Constitutive law $\begin{matrix} {\tilde{σ}}_{α β} (ζ_{3}) = {\tilde{C}}_{α β σ τ} \partial_{σ τ} ζ_{3} in ω . \end{matrix}$ Boundary conditions $\begin{matrix} \{\begin{matrix} \frac{2}{3} \partial_{α} {\tilde{σ}}_{α β} (ζ_{3}) ν_{β} + \frac{2}{3} \partial_{ι} ({\tilde{σ}}_{α β} ν_{β} ι_{α}) = s_{α} ν_{α} & on γ, \\ {\tilde{σ}}_{α β} (ζ_{3}) ν_{α} ν_{β} = 0 & on γ . \end{matrix} \end{matrix}$ The contact condition is now in the interior of the domain ω $\begin{matrix} ζ_{3} ⩾ 0, {\tilde{σ}}_{α β} (ζ_{3}) ζ_{3} = 0 in ω . \end{matrix}$

Proof.
Let $\hat{σ} : = \tilde{σ} (ζ_{3})$ . An integration by parts in (5.1) yields $\begin{matrix} \{\begin{matrix} \int_{ω} {\tilde{C}}_{α β σ τ} \partial_{σ τ} ζ_{3} \partial_{α β} (η_{3} - ζ_{3}) d ω = \int_{ω} {\hat{σ}}_{α β} \partial_{α β} (η_{3} - ζ_{3}) \\ = - \int_{ω} \partial_{β} {\hat{σ}}_{α β} \partial_{α} (η_{3} - ζ_{3}) d ω + \int_{γ} {\hat{σ}}_{α β} \partial_{α} (η_{3} - ζ_{3}) ν_{β} d γ \\ = \int_{ω} \partial_{α β} {\hat{σ}}_{α β} (η_{3} - ζ_{3}) d ω - \int_{γ} \partial_{β} {\hat{σ}}_{α β} (η_{3} - ζ_{3}) ν_{α} d γ \\ + \int_{γ} {\hat{σ}}_{α β} \partial_{α} (η_{3} - ζ_{3}) ν_{β} d γ, \\ noting that \int_{γ} {\hat{σ}}_{α β} \partial_{α} (η_{3} - ζ_{3}) ν_{β} d γ = \int_{γ} {\hat{σ}}_{α β} ν_{β} ν_{α} \partial_{ν} (η_{3} - ζ_{3}) d γ + \int_{γ} {\hat{σ}}_{α β} ν_{β} ι_{α} \partial_{ι} (η_{3} - ζ_{3}) d γ \\ and also we use that \int_{γ} \partial_{ι} ({\hat{σ}}_{α β} ν_{β} ι_{α} (η_{3} - ζ_{3})) d γ = 0 \\ then \\ \int_{ω} {\tilde{C}}_{α β σ τ} \partial_{σ τ} ζ_{3} \partial_{α β} (η_{3} - ζ_{3}) d ω = \int_{ω} \partial_{α β} {\hat{σ}}_{α β} (η_{3} - ζ_{3}) d ω - \int_{γ} \partial_{β} {\hat{σ}}_{α β} (η_{3} - ζ_{3}) ν_{α} d γ \\ + \int_{γ} {\hat{σ}}_{α β} ν_{β} ν_{α} \partial_{ν} (η_{3} - ζ_{3}) d γ - \int_{γ} \partial_{ι} ({\hat{σ}}_{α β} ν_{β} ι_{α}) (η_{3} - ζ_{3}) d γ, \\ \int_{ω} s_{α} \partial_{α} (η_{3} - ζ_{3}) d ω = \int_{γ} s_{α} ν_{α} (η_{3} - ζ_{3}) d γ - \int_{ω} \partial_{α} s_{α} (η_{3} - ζ_{3}) d ω . \end{matrix} \end{matrix}$ So we get: $\begin{matrix} \{\begin{matrix} \frac{2}{3} \int_{ω} {\tilde{C}}_{α β σ τ} \partial_{σ τ} ζ_{3} \partial_{α β} (η_{3} - ζ_{3}) d ω ⩾ \int_{ω} p_{3} (η_{3} - ζ_{3}) d ω - \int_{ω} s_{α} \partial_{α} (η_{3} - ζ_{3}) d ω, \\ ⟹ \frac{2}{3} \int_{ω} \partial_{α β} {\hat{σ}}_{α β} (η_{3} - ζ_{3}) d ω - \frac{2}{3} \int_{γ} \partial_{β} {\hat{σ}}_{α β} (η_{3} - ζ_{3}) ν_{α} d γ \\ + \frac{2}{3} \int_{γ} {\hat{σ}}_{α β} ν_{β} ν_{α} \partial_{ν} (η_{3} - ζ_{3}) d γ - \frac{2}{3} \int_{γ} \partial_{ι} ({\hat{σ}}_{α β} ν_{β} ι_{α}) (η_{3} - ζ_{3}) d γ \\ ⩾ \int_{ω} p_{3} (η_{3} - ζ_{3}) d ω - \int_{γ} s_{α} ν_{α} (η_{3} - ζ_{3}) d γ + \int_{ω} \partial_{α} s_{α} (η_{3} - ζ_{3}) d ω \end{matrix} \end{matrix}$ $\forall η_{3} \in K_{3} (ω)$ , the last inequality holds, from the Lemma 5.2 we get the desired flexural equations. □
Theorem 5.3 (Membrane equations).

The limit two-dimensional components of $ζ_{H} = (ζ_{α})$ of the Kirchhoff–Love displacement on the middle surface and the electric potential φ solve the following system of partial differential equations, which is a piezoelectric problem without contact:

Equilibrium equation $\begin{matrix} \{\begin{matrix} - 2 \partial_{β} {\tilde{n}}_{α β} (ζ_{H}, φ) = p_{α} & in ω, \\ 2 \partial_{γ} {\tilde{D}}_{γ} (ζ_{H}, φ) = {\tilde{ρ}}_{e} - ϑ & in ω . \end{matrix} \end{matrix}$ Boundary conditions $\begin{matrix} \{\begin{matrix} {\tilde{n}}_{α β} (ζ_{H}, φ) ν_{β} = 0 & on γ, \\ {\tilde{D}}_{γ} (ζ_{H}, φ) ν_{γ} = 0 & on γ . \end{matrix} \end{matrix}$ The reduced stress tensor ${\tilde{n}}_{α β}$ , the reduced electric displacement vector ${\tilde{D}}_{γ}$ are described by the coupled piezoelectric constitutive laws $\begin{matrix} \{\begin{matrix} {\tilde{n}}_{α β} (ζ_{H}, φ) = {\tilde{C}}_{α β σ τ} e_{σ τ} (ζ_{H}) + {\tilde{P}}_{γ α β} \partial_{γ} φ, \\ {\tilde{D}}_{γ} (ζ_{H}, φ) = {\tilde{P}}_{γ α β} e_{α β} (ζ_{H}) - {\tilde{d}}_{γ β} \partial_{β} φ . \end{matrix} \end{matrix}$

Proof.
Similar to Theorem 5.2, by integration by parts, we get $\begin{matrix} \{\begin{matrix} \int_{ω} {\tilde{n}}_{α β} (ζ_{H}, φ) \partial_{β} η_{α} d ω = - \int_{ω} \partial_{β} {\tilde{n}}_{α β} (ζ_{H}, φ) η_{α} d ω + \int_{γ} {\tilde{n}}_{α β} (ζ_{H}, φ) η_{α} ν_{β} d γ \\ \int_{ω} {\tilde{D}}_{γ} (ζ_{H}, φ) \partial_{γ} ψ d ω = - \int_{ω} \partial_{γ} {\tilde{D}}_{γ} (ζ_{H}, φ) ψ d ω + \int_{γ} {\tilde{D}}_{γ} (ζ_{H}, φ) ψ ν_{γ} d γ, \end{matrix} \end{matrix}$ then $\begin{matrix} \{\begin{matrix} 2 \int_{ω} {\tilde{n}}_{α β} (ζ_{H}, φ) \partial_{β} η_{α} d ω + 2 \int_{ω} {\tilde{D}}_{γ} (ζ_{H}, φ) \partial_{γ} ψ d ω = \int_{ω} p_{H} \cdot η_{H} d ω - \int_{ω} ({\tilde{ρ}}_{e} - ϑ) ψ d ω \\ ⟹ 2 \int_{γ} {\tilde{n}}_{α β} (ζ_{H}, φ) ν_{β} η_{α} d ω - 2 \int_{ω} \partial_{β} {\tilde{n}}_{α β} (ζ_{H}, φ) η_{α} d ω \\ + 2 \int_{γ} {\tilde{D}}_{γ} (ζ_{H}, φ) ν_{γ} ψ d ω - 2 \int_{ω} \partial_{γ} {\tilde{D}}_{γ} (ζ_{H}, φ) ψ d ω \\ = \int_{ω} p_{H} \cdot η_{H} d ω - \int_{ω} ({\tilde{ρ}}_{e} - ϑ) ψ d ω \end{matrix} \end{matrix}$ $\forall (η_{H}, ψ) \in V_{H} (ω) \times Q (ω)$ , the last equality holds, and we get the desired result. □

6. Some complements

6.1. Return to the physical domain

Following Ciarlet [1,2], now we are going back to the physical domain, let $\begin{matrix} \{\begin{matrix} ζ_{α}^{ε} = ε^{2} ζ_{α} & in \bar{ω}, \\ ζ_{3}^{ε} = ε ζ_{3} & in \bar{ω}, \\ φ^{ε} = ε^{2} φ & in \bar{ω} \end{matrix} \end{matrix}$ and $(ζ^{ε}, φ^{ε}) = (ζ_{H}^{ε}, ζ_{3}^{ε}, φ^{ε})$ satisfy the variational problem $\begin{matrix} \{\begin{matrix} \frac{2}{3} \int_{ω} {\tilde{σ}}_{α β}^{ε} (ζ_{3}^{ε}) \partial_{α β} (η_{3} - ζ_{3}^{ε}) d ω \\ ⩾ \int_{ω} p_{3}^{ε} (η_{3} - ζ_{3}^{ε}) d ω - \int_{ω} s_{α}^{ε} \partial_{α} (η_{3} - ζ_{3}^{ε}) d ω \forall η_{3} \in K_{3} (ω), \\ 2 \int_{ω} {\tilde{n}}_{α β}^{ε} (ζ_{H}^{ε}, φ^{ε}) e_{α β} (η_{H}) d ω + \int_{ω} {\tilde{D}}_{γ}^{ε} (ζ_{H}^{ε}, φ^{ε}) \partial_{γ} ψ d ω \\ = \int_{ω} p_{H}^{ε} \cdot η_{H} d ω - \int_{ω} ({\tilde{ρ}}_{e}^{ε} - ϑ^{ε}) ψ d ω \forall (η_{H}, ψ) \in V_{H} (ω) \times Q (ω), \end{matrix} \end{matrix}$ and $\begin{matrix} \{\begin{matrix} {\tilde{σ}}_{α β}^{ε} (ζ_{3}^{ε}) = ε^{3} {\tilde{C}}_{α β σ τ} \partial_{σ τ} ζ_{3}^{ε}, \\ {\tilde{n}}_{α β}^{ε} (ζ_{H}^{ε}, φ^{ε}) = ε {\tilde{C}}_{α β σ τ} e_{σ τ} (ζ_{H}^{ε}) + ε {\tilde{P}}_{γ α β} \partial_{γ} φ^{ε}, \\ {\tilde{D}}_{γ}^{ε} (ζ_{H}^{ε}, φ^{ε}) = ε {\tilde{P}}_{γ α β} e_{α β} (ζ_{H}^{ε}) - ε {\tilde{d}}_{γ β} \partial_{β} φ^{ε} . \end{matrix} \end{matrix}$

$(ζ^{ε}, φ^{ε})$ satisfies the following strong formulations.

Flexural equations $\begin{matrix} \frac{2}{3} \partial_{α β} {\tilde{σ}}_{α β}^{ε} (ζ_{3}^{ε}) ⩾ (p_{3}^{ε} + \partial_{α} s_{α}^{ε}) in ω . \end{matrix}$ Constitutive law $\begin{matrix} {\tilde{σ}}_{α β}^{ε} (ζ_{3}^{ε}) = ε^{3} {\tilde{C}}_{α β σ τ} \partial_{σ τ} ζ_{3}^{ε} in ω . \end{matrix}$ Boundary conditions $\begin{matrix} \{\begin{matrix} \frac{2}{3} \partial_{α} {\tilde{σ}}_{α β}^{ε} (ζ_{3}^{ε}) ν_{β} + \frac{2}{3} \partial_{ι} ({\tilde{σ}}_{α β}^{ε} ν_{β} ι_{α}) = s_{α}^{ε} ν_{α} & on γ, \\ {\tilde{σ}}_{α β}^{ε} (ζ_{3}^{ε}) ν_{α} ν_{β} = 0 & on γ . \end{matrix} \end{matrix}$ The contact condition $\begin{matrix} ζ_{3}^{ε} ⩾ 0, {\tilde{σ}}_{α β}^{ε} (ζ_{3}^{ε}) ζ_{3}^{ε} = 0 in ω . \end{matrix}$

(Membrane equations) $\begin{matrix} \{\begin{matrix} - 2 \partial_{β} {\tilde{n}}_{α β}^{ε} (ζ_{H}^{ε}, φ^{ε}) = p_{α}^{ε} & in ω, \\ 2 \partial_{γ} {\tilde{D}}_{γ}^{ε} (ζ_{H}^{ε}, φ^{ε}) = {\tilde{ρ}}_{e}^{ε} - ϑ^{ε} & in ω . \end{matrix} \end{matrix}$ Constitutive law $\begin{matrix} \{\begin{matrix} {\tilde{n}}_{α β} (ζ_{H}^{ε}, φ^{ε}) = ε {\tilde{C}}_{α β σ τ} e_{σ τ} (ζ_{H}^{ε}) + ε {\tilde{P}}_{γ α β} \partial_{γ} φ^{ε}, \\ {\tilde{D}}_{γ} (ζ_{H}^{ε}, φ^{ε}) = ε {\tilde{P}}_{γ α β}^{ε} e_{α β} (ζ_{H}^{ε}) - ε {\tilde{d}}_{γ β}^{ε} \partial_{β} φ^{ε} . \end{matrix} \end{matrix}$ Boundary conditions $\begin{matrix} \{\begin{matrix} {\tilde{n}}_{α β}^{ε} (ζ_{H}^{ε}, φ^{ε}) ν_{β} = 0 & on γ, \\ {\tilde{D}}_{γ}^{ε} (ζ_{H}^{ε}, φ^{ε}) ν_{γ} = 0 & on γ, \end{matrix} \end{matrix}$ where $\begin{matrix} p_{i}^{ε} : = \int_{- ε}^{ε} f_{i}^{ε} d x_{3}^{ε} + g_{i}^{ε}, s_{α}^{ε} : = \int_{- ε}^{ε} x_{3}^{ε} f_{α}^{ε} d x_{3}^{ε} + g_{α}^{ε}, {\tilde{ρ}}_{e}^{ε} : = \int_{- ε}^{ε} ρ_{e}^{ε} d x_{3}^{ε} . \end{matrix}$ The de-scaled solution $u_{i}^{ε} (0)$ is then given by $\begin{matrix} u_{α}^{ε} (0) = ζ_{α}^{ε} - x_{3}^{ε} \partial_{α} ζ_{3}^{ε}, u_{3}^{ε} (0) = ζ_{3}^{ε} \forall x^{ε} \in {\bar{Ω}}^{ε} . \end{matrix}$

6.2. Partial elastic clamping on the lateral surface

In this part we consider the following boundary conditions: $\begin{matrix} \{\begin{matrix} u^{ε} = 0 & on Γ_{0}^{ε}, \\ φ^{ε} = 0 & on Γ_{0}^{ε}, \\ σ^{ε} (u^{ε}, φ^{ε}) ν = g^{ε} & on Γ_{+}^{ε} \cup \hat{γ} \times [- ε, ε], \\ D^{ε} (u^{ε}, φ^{ε}) \cdot ν = ϑ^{ε} & on Γ_{+}^{ε} \cup \hat{γ} \times [- ε, ε], \\ u_{3}^{ε} ⩾ 0, σ_{3}^{ε} ⩾ 0, σ_{3}^{ε} u_{3}^{ε} = 0, D^{ε} \cdot ν = 0 & on Γ_{-}^{ε}, \end{matrix} \end{matrix}$ here $γ_{0} \subset γ$ , $\hat{γ} = γ - γ_{0}$ . The limit solution $(ζ, φ) = (ζ_{H}, ζ_{3}, φ)$ solves the following coupled problem:

Find $(ζ_{H}, ζ_{3}, φ) \in V_{H} (ω) \times K_{3} (ω) \times Q (ω)$ , with the bi-dimensional functional spaces $\begin{matrix} \{\begin{matrix} V_{H} (ω) = {η_{H} = (η_{α}) \in H^{1} (ω), η_{H} = 0 on γ_{0}}, \\ K_{3} (ω) = {η_{3} \in H^{2} (ω), η_{3} = \partial_{ν} η_{3} = 0 on γ_{0}, η_{3} ⩾ 0 in ω}, \\ Q (ω) = {ψ \in H^{1} (Ω), ψ = 0 on γ_{0}} \end{matrix} \end{matrix}$ such that $\begin{matrix} (6.1) & \{\begin{matrix} \frac{2}{3} \int_{ω} {\tilde{C}}_{α β σ τ} \partial_{σ τ} ζ_{3} \partial_{α β} (η_{3} - ζ_{3}) d ω \\ ⩾ \int_{ω} p_{3} (η_{3} - ζ_{3}) d ω - \int_{ω} s_{α} \partial_{α} (η_{3} - ζ_{3}) d ω + \int_{\hat{γ}} [{\tilde{g}}_{3} (η_{3} - ζ_{3}) - {\tilde{g}}_{α}^{'} \partial_{α} (η_{3} - ζ_{3})] d γ \\ \forall η_{3} \in K_{3} (ω), \\ 2 \int_{ω} {\tilde{n}}_{α β} (ζ_{H}, φ) e_{α β} (η_{H}) d ω + 2 \int_{ω} {\tilde{D}}_{γ} (ζ_{H}, φ) \partial_{γ} ψ d ω \\ = \int_{ω} p_{H} \cdot η_{H} d ω - \int_{ω} ({\tilde{ρ}}_{e} - ϑ^{+}) ψ d ω + \int_{\hat{γ}} \tilde{ϑ} ψ d γ \\ \forall (η_{H}, ψ) \in V_{H} (ω) \times Q (ω), \end{matrix} \end{matrix}$ where the constitutive bi-dimensional law is given by $\begin{matrix} \{\begin{matrix} {\tilde{n}}_{α β} (ζ_{H}, φ) = {\tilde{C}}_{α β σ τ} e_{σ τ} (ζ_{H}) + {\tilde{P}}_{γ α β} \partial_{γ} φ, \\ {\tilde{D}}_{γ} (ζ_{H}, φ) = {\tilde{P}}_{γ α β} e_{α β} (ζ_{H}) - {\tilde{d}}_{γ β} \partial_{β} φ \end{matrix} \end{matrix}$ and the mechanical forces and electric charges are given by $\begin{array}{l} p_{i} : = \int_{- 1}^{1} f_{i} d x_{3} + g_{i}^{+}, s_{α} : = \int_{- 1}^{1} x_{3} f_{α} d x_{3} + g_{α}^{+}, {\tilde{ρ}}_{e} : = \int_{- 1}^{1} ρ_{e} d x_{3}, \\ {\tilde{g}}_{i} : = \int_{- 1}^{1} g_{i} d x_{3}, {\tilde{g}}_{i}^{'} : = \int_{- 1}^{1} x_{3} g_{i} d x_{3}, \tilde{ϑ} : = \int_{- 1}^{1} ϑ d x_{3}, \end{array}$ where the bi-dimensional elasticity $\tilde{C}$ , piezoelectricity $\tilde{P}$ , dielectric $\tilde{d}$ are the same as in Theorem 5.1 $g_{i}^{+} = g (x_{1}, x_{2}, 1)$ , $ϑ^{+} = ϑ (x_{1}, x_{2}, 1)$ .

Remark 6.1.
Comparing with Theorem 5.1, the two-dimensional problem of partial elastic clamping on the lateral surface case in (6.1) has the term $\int_{\hat{γ}} [{\tilde{g}}_{3} (η_{3} - ζ_{3}) - {\tilde{g}}_{α}^{'} \partial_{α} (η_{3} - ζ_{3})] d γ$ in the transverse variational inequality and the term $\int_{\hat{γ}} \tilde{ϑ} ψ d γ$ in the in-plane variational equality.

6.3. Two-dimensional cubic piezoelectricity

For cubic system which is the simplest piezoelectric case, the set of data $(C, d, P)$ is represented by a symmetric matrix of the following form (see [6]): $\begin{matrix} (\begin{matrix} C_{11} & C_{12} & C_{12} & 0 & 0 & 0 & 0 & 0 & 0 \\ C_{12} & C_{11} & C_{12} & 0 & 0 & 0 & 0 & 0 & 0 \\ C_{12} & C_{12} & C_{11} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 & P_{123} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{44} & 0 & 0 & P_{123} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{44} & 0 & 0 & P_{123} \\ 0 & 0 & 0 & P_{123} & 0 & 0 & d_{11} & 0 & 0 \\ 0 & 0 & 0 & 0 & P_{123} & 0 & 0 & d_{11} & 0 \\ 0 & 0 & 0 & 0 & 0 & P_{123} & 0 & 0 & d_{11} \end{matrix}), \end{matrix}$ i.e. $\begin{matrix} \{\begin{matrix} C_{1111} = C_{2222} = C_{3333} = C_{11}, C_{1212} = C_{2323} = C_{1313} = C_{44}, \\ C_{1122} = C_{1133 =} C_{2233} = C_{12}, \\ P_{123} = P_{213} = P_{312} = P_{123}, \\ d_{11} = d_{22} = d_{33} = d_{11}, \\ all the other terms vanish. \end{matrix} \end{matrix}$ After a direct computation using (5.12) and (5.13) we obtain: $\begin{matrix} \{\begin{matrix} △ = C_{11} C_{44}^{2}, \\ △_{1}^{1} = △_{2}^{2} = 0, △_{2}^{1} = △_{1}^{2} = C_{11} C_{44} P_{123}, △_{α}^{3} = 0, \\ △_{ζ η}^{1} = △_{ζ η}^{2} = 0, △_{11}^{3} = △_{22}^{3} = C_{12} C_{44}^{2}, △_{ζ η}^{3} = 0 as ζ \neq η, \end{matrix} \end{matrix}$ and $\begin{matrix} \{\begin{matrix} C_{1212}^{'} = C_{1212} + \frac{P_{123}^{2}}{d_{11}}, C_{i j k l}^{'} = C_{i j k l} as i j k l \neq 1212, 2121, 1221, 2112, \\ P_{k i j}^{'} = 0, d_{i 3}^{'} = d_{3 i}^{'} = 0 . \end{matrix} \end{matrix}$ So the solution $(ζ, φ)$ to the cubic piezoelastic system solves the following three de-coupled problems:

Find $(ζ_{H}, ζ_{3}, φ) \in V_{H} (ω) \times K_{3} (ω) \times Q (ω)$ $\begin{matrix} \{\begin{matrix} \frac{2}{3} \int_{ω} {\tilde{C}}_{α β σ τ} \partial_{σ τ} ζ_{3} \partial_{α β} (η_{3} - ζ_{3}) d ω ⩾ \int_{ω} p_{3} (η_{3} - ζ_{3}) d ω - \int_{ω} s_{α} \partial_{α} (η_{3} - ζ_{3}) d ω \\ \forall η_{3} \in K_{3} (ω), \\ 2 \int_{ω} {\tilde{n}}_{α β} (ζ_{H}) e_{α β} (η_{H}) d ω = \int_{ω} p_{H} \cdot η_{H} d ω \\ \forall η_{H} \in V_{H} (ω), \\ 2 \int_{ω} {\tilde{D}}_{γ} (φ) \partial_{γ} ψ d ω = \int_{ω} ({\tilde{ρ}}_{e} - ϑ) ψ d ω \\ \forall ψ \in Q (ω), \end{matrix} \end{matrix}$ where the new set of data $(\tilde{C}, \tilde{d})$ is represented by the following symmetric matrix $\begin{matrix} (\begin{matrix} {\tilde{C}}_{11} & {\tilde{C}}_{12} & 0 & 0 & 0 \\ {\tilde{C}}_{21} & {\tilde{C}}_{22} & 0 & 0 & 0 \\ 0 & 0 & {\tilde{C}}_{33} & 0 & 0 \\ 0 & 0 & 0 & {\tilde{d}}_{11} & 0 \\ 0 & 0 & 0 & 0 & {\tilde{d}}_{11} \end{matrix}) . \end{matrix}$ We use the abbreviation $11 \to 1$ , $22 \to 2$ , $12 \to 3$ , and the constitutive bi-dimensional law is $\begin{matrix} {\tilde{n}}_{α β} (ζ_{H}) = {\tilde{C}}_{α β σ τ} e_{σ τ} (ζ_{H}), {\tilde{D}}_{γ} (φ) = (d_{11} + \frac{P_{123}^{2}}{C_{44}}) \partial_{γ} φ, \end{matrix}$ with the new tensors $\begin{matrix} \{\begin{matrix} {\tilde{C}}_{11} = {\tilde{C}}_{22} = C_{11} - \frac{C_{12}^{2}}{C_{11}}, {\tilde{C}}_{12} = {\tilde{C}}_{21} = C_{12} - \frac{C_{12}^{2}}{C_{11}}, {\tilde{C}}_{33} = C_{44} + \frac{P_{123}^{2}}{d_{11}}, \\ {\tilde{d}}_{11} = (d_{11} + \frac{P_{123}^{2}}{C_{44}}), \end{matrix} \end{matrix}$ with $p_{i}$ , $s_{α}$ , ${\tilde{ρ}}_{e}$ , ϑ, $V_{H} (ω)$ , $K_{3} (ω)$ , $Q (ω)$ as Theorem 5.1.

Remark 6.2.

In the cubic piezoelectric material, we find an interesting thing that, three-dimensional Signorini problem of piezoelectric plate converges to a two-dimensional obstacle problem of plate as the thickness of the piezoelectric plate goes to zero, but in the two-dimensional case, the piezoelectric terms disappear. Here is three decoupled piezoelastic equation flexural inequality, membrane equation and dielectric equation.

Footnotes

Acknowledgements

The first author is greatly indebted to Professor Li Tatsien and Professor Zhou Yi for their guidance, encouragement and help. The paper is supported by the Innovation Program of Shanghai Municipal Education Commission (11YZ80) and the program of Shanghai Normal University (SK201301). The second author acknowledge Professor Han’s hospitality at Shanghai Normal University.

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