Mathematical justification of the obstacle problem for a piezoelectric shallow shell

Abstract

In this paper we properly justify the modeling of a thin piezoelectric shallow shell in unilateral contact with a rigid plane. Starting from the three-dimensional nonlinear Signorini problem, we establish the convergence of the displacement field and of the electric potential as the thickness of the shell goes to zero. More precisely we obtain that the transverse mechanical displacement field coupled with the in-plane components solve an obstacle problem described new piezoelectric characteristics. We also investigate the very popular case of cubic crystals and show that, for two-dimensional shallow shells, the coupling piezoelectric effect disappears.

Keywords

Signorini problem obstacle problem asymptotic analysis shallow shell

1. Introduction

The objective of this work is two-fold. On the one hand we consider the so-called Signorini problem of an elastic body in contact with a rigid support, which is also called the unilateral contact problem. On the other hand we focus our attention on a body made of a piezoelectric material. One of the major interests of this modeling is to keep the full piezoelectricity tensor, namely, we do not assume any elastic isotropy (as in [20,21]).

The modeling of unilateral contact problems of elastic bodies was established by Signorini in 1933. Its mathematical analysis was due to Fichera [9] by means of an equivalent minimizing problem. Some existence results were established by Duvaut and Lions [8]. Later, Paumier gave, by an asymptotic approach, the model of an elastic Kirchhoff–Love plate with unilateral contact [18]. Léger and Miara generalized Paumier’s work to the elastic shallow shell and obtained the limit model written in terms of a variational inequality in the framework of Cartesian coordinates [13] and curvilinear coordinates [14], respectively.

Although the piezoelectricity has been predicted by Coulomb, and discovered by Becquerel in 1819, the piezoelectric effect was only experimented by Jacques and Pierre Curie in 1880. For further information about it’s discovery, see [1]. The piezoelectricity is an electromechanical interaction: piezoelectric materials are dielectrics on which the application of an electric field generates deformation, and which produce polarization under the effect of deformation. These phenomena are expressed by constitutive equations linking the stress tensor to the electric field on the one hand, and the polarization vector or the electric displacement vector to the strain tensor on the other hand (which will be expressed clearly in Section 2). The piezoelectric effect (coupling of elastic and electric effects) can be found in “natural” such as crystals, ceramics, or biological materials such as bone, DNA and various proteins. The piezoelectricity can be found in many applications such as production and detection of sound, generation of high voltages, electronic frequency generation, microbalances and ultrafine focusing of optical assemblies etc. It is also the basis of a number of scientific instrumental techniques with atomic resolution and the scanning probe microscopies such as STM, AFM, MTA, SNOM, etc., and everyday’s uses such as acting as the ignition source for cigarette lighters and push-start propane barbecues.

Bilateral models for plates and shells have been extensively studied by formal asymptotic methods or by variational analysis (see [2,4] and the references there in). As far as piezoelectricity is concerned we refer to the first justification of the two-dimensional (bilateral) model of piezoelectric plate by Rahmoune [19]. Miara and Suarez [17] investigated the two-dimensional model of dynamic thermopiezoelectric plates. For numerical aspects of unilateral contact for piezoelectric body we refer to the first formulation given by Maceri and Bisegna [16].

In this paper, we consider the obstacle problem for piezoelectric shallow shell. By means of asymptotic analysis, we will show that the solution to three dimensional Signorini problem of piezoelectric shallow shell converges to a two-dimensional obstacle problem of piezoelectric shallow shell as the thickness of the shell tends to zero, and the obstacle condition affect only the vertical limit displacement field. We also investigate the very popular case of cubic crystals (for example PZT) and show that, for two-dimensional shallow shells, the coupling piezoelectric effect disappears. In this paper, we generalize the work of G. Yan and B. Miara [22] to piezoelectric shallow shell in Cartesian coordinates. The corresponding consideration in curvilinear coordinates will be given in a forthcoming paper [23].

The plan of our work is as follows: In Section 2, we introduce piezoelectric constitutive equation and equilibrium equation. In Section 3, we study a Signorini problem arising in the case of three-dimensional shallow shells. In Section 4, by using appropriate scalings, we give the new scaled variational inequality problem. In Section 5, we prove the convergence of solutions when the thickness of the shallow shells tends to zero and establish the limit problem of a piezoelectric shallow shell in unilateral contact. In Section 6 we give the formulation of the problem in the physical domain, we also consider more general boundary conditions and investigate the interesting case of cubic crystals.

2. Piezoelectricity in a three dimensional body

In this paper, 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 inner product between two vectors a and b. Moreover, for simplicity c denotes different positive constants, $| \cdot |_{0, Ω} ‖ \cdot ‖_{1, Ω}$ denote the classical norms in $L^{2} (Ω)$ and $H^{1} (Ω)$ , respectively, for both scalar-valued and vector-valued functions.

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 $(C, d, P)$ can be represented by the following symmetric matrix (see [1,7,11] and [10]) $\begin{matrix} (\begin{matrix} C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} & P_{11} & P_{21} & P_{31} \\ C_{21} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} & P_{12} & P_{22} & P_{32} \\ C_{31} & C_{32} & C_{33} & C_{34} & C_{35} & C_{36} & P_{13} & P_{23} & P_{33} \\ C_{41} & C_{42} & C_{43} & C_{44} & C_{45} & C_{46} & P_{14} & P_{24} & P_{34} \\ C_{51} & C_{52} & C_{53} & C_{54} & C_{55} & C_{56} & P_{15} & P_{25} & P_{35} \\ C_{61} & C_{62} & C_{63} & C_{64} & C_{65} & C_{66} & P_{16} & P_{26} & P_{36} \\ P_{11} & P_{12} & P_{13} & P_{14} & P_{15} & P_{16} & d_{11} & d_{12} & d_{13} \\ P_{21} & P_{22} & P_{23} & P_{24} & P_{25} & P_{26} & d_{21} & d_{22} & d_{23} \\ P_{31} & P_{32} & P_{33} & P_{34} & P_{35} & P_{36} & d_{31} & d_{32} & d_{33} \end{matrix}), \end{matrix}$ where we use 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, $C_{1111}$ is denoted by $C_{11}$ , $C_{1131}$ by $C_{15} \dots$ , $P_{123}$ by $P_{14}$ , $P_{233}$ by $P_{23} \dots$ .

There exists a positive constant c such that, for every second order $3 \times 3$ symmetric tensor $M = (M_{i j})$ and every 3-dimensional vector $ϱ = (ϱ_{k})$ , we have $\begin{matrix} (2.1) & \{\begin{matrix} C_{i j k l} = C_{i j l k} = C_{k l i j}, & C_{i j k l} M_{k l} M_{i j} ⩾ c \sum_{i j = 1}^{3} M_{i j}^{2}, \\ d_{k l} = d_{l k}, & d_{k l} ϱ_{k} ϱ_{l} ⩾ c \sum_{k = 1}^{3} ϱ_{k}^{2}, \\ P_{m k l} = P_{m l k} . \end{matrix} \end{matrix}$ In the linear case, for static piezoelectric materials, the second order symmetric stress tensor $σ = (σ_{k l})$ and the displacement vector $D = (D_{k})$ , the second order symmetric strain tensor $e = (e_{k l})$ and the electric vector field $E = (E_{k})$ are described through the following relations: $\begin{matrix} (2.2) & \{\begin{matrix} σ_{i j} = C_{i j k l} e_{k l} - P_{k i j} E_{k}, \\ D_{k} = P_{k i j} e_{i j} + d_{k l} E_{l} . \end{matrix} \end{matrix}$ Let us recall that the dielectric tensor d is related to the electric susceptibility tensor $d^{'} = (d_{k}^{'})$ and the electric permitivity of the free space $d_{0}$ by $d = d_{0} I_{identity} + d^{'}$ . The polarization vector $X = (X_{k})$ is given by component-wise relation $X_{k} = P_{k i j} e_{i j} + d_{k l}^{'} E_{l}$ , so that the electric displacement D is related to the polarization vector $X = (X_{k})$ 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 $R^{3}$ . 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 following 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 $\begin{matrix} e (u) = \frac{1}{2} (\nabla u + {(\nabla u)}^{T}), E = - \nabla φ . \end{matrix}$

2.3. Boundary conditions

The equilibrium equations have to be supplemented by boundary conditions posed on $\partial Ω$ . Let $\partial Ω = Γ_{0} \cup \hat{Γ}$ be a partition of the boundary with area $Γ_{0} > 0$ and $ν$ the unit outward normal vector to $\partial Ω$ . Assume that on $Γ_{0}$ the body is clamped and is subjected to null electric potential, while, 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 are $\begin{matrix} (2.4) & \{\begin{matrix} u = 0 & on Γ_{0}, \\ φ = 0 & on Γ_{0}, \\ σ (u, φ) ν = g & on \hat{Γ}, \\ D (u, φ) \cdot ν = ϑ & on \hat{Γ} . \end{matrix} \end{matrix}$ The contact condition will be introduced in Section 3.2.

The existence and uniqueness of the solution to problem (2.3)–(2.4) will be adhered later in Section 4. More general boundary conditions are considered in Section 6.

3. Shallow shell

In order to establish properly the bi-dimensional model of a thin shallow shell in contact with a rigid plane, we recall that a general framework to carry out the construction of a shell model involves a three-dimensional Euclidian space $R^{3}$ , a two-dimensional vector space, identified with $R^{2}$ and containing a domain ω. We assume the boundary $γ_{0} = \partial ω$ is Lipschitz-continuous, and ω is locally on one side of $γ_{0}$ . For each $ε > 0$ , we define the sets $\begin{matrix} Ω^{ε} = ω \times (- ε, ε), Γ_{+}^{ε} = ω \times {ε}, Γ_{-}^{ε} = ω \times {- ε}, Γ_{0}^{ε} = γ_{0} \times [- ε, ε] . \end{matrix}$

Any given point in ${\bar{Ω}}^{ε}$ will be denoted by $x^{ε} = (x_{1}, x_{2}, x_{3}^{ε})$ , where $(x_{1}, x_{2}) \in ω$ and $x_{3}^{ε} \in (- ε, ε)$ . We let $\partial_{α} = \partial_{α}^{ε} : = \frac{\partial}{\partial x_{α}}$ , $\partial_{3} = \partial_{3}^{ε} : = \frac{\partial}{\partial x_{3}^{ε}}$ .

Assume that, for each $ε > 0$ , a function $θ^{ε} \in C^{3} (\bar{ω})$ is given. We define the surface $\begin{matrix} \hat{ω} = {(x_{1}, x_{2}, θ^{ε} (x_{1}, x_{2})) | (x_{1}, x_{2}) \in ω} \end{matrix}$ and introduce ${\hat{a}}^{ε}$ as the unit outward normal vector on $\hat{ω}$ . $\begin{matrix} (3.1) & {\hat{a}}^{ε} : = \frac{1}{\sqrt{a^{ε}}} (- \partial_{1} θ^{ε}, - \partial_{2} θ^{ε}, 1) \end{matrix}$ with $a^{ε} : = | \partial_{1} θ^{ε} |^{2} + | \partial_{2} θ^{ε} |^{2} + 1$ .

Introducing an application $Θ^{ε}$ which maps the cylinder $Ω^{ε}$ onto the shell ${\hat{Ω}}^{ε}$ , we shall write ${\hat{Ω}}^{ε} = Θ^{ε} (Ω^{ε})$ . $\begin{matrix} Θ^{ε} : = (x_{1}, x_{2}, θ^{ε} (x_{1}, x_{2})) + x_{3}^{ε} {\hat{a}}^{ε} for all x^{ε} \in {\bar{Ω}}^{ε} . \end{matrix}$ The boundary of ${\hat{Ω}}^{ε}$ is $\partial {\hat{Ω}}^{ε} = {\hat{Γ}}_{+}^{ε} \cup {\hat{Γ}}_{-}^{ε} \cup {\hat{Γ}}_{0}^{ε}$ , where ${\hat{Γ}}_{+}^{ε} = Θ^{ε} (Γ_{+}^{ε})$ , ${\hat{Γ}}_{0}^{ε} = Θ^{ε} (Γ_{0}^{ε})$ , ${\hat{Γ}}_{-}^{ε} = Θ^{ε} (Γ_{-}^{ε})$ , in particular $\begin{matrix} {\hat{Γ}}_{-}^{ε} = (x_{1}, x_{2}, θ^{ε} (x_{1}, x_{2})) - ε {\hat{a}}^{ε} (x_{1}, x_{2}), \forall (x_{1}, x_{2}) \in ω . \end{matrix}$ Of course, these definitions make sense only if $Θ^{ε}$ is globally injective, but it is precisely known that $Θ^{ε}$ is a $C^{1}$ -diffeomorphism and then $det {▽^{ε} Θ^{ε} (x^{ε})} > 0, \forall x^{ε} \in {\bar{Ω}}^{ε}$ , if the immersion is itself globally injective on ω and ε is small enough (a proof based on inverse function theorem is for instance given in [3]) provided that ε is small enough.

3.1. Three dimensional problem

We consider a family of shallow shells with reference configuration ${\hat{Ω}}^{ε}$ made of piezoelectric material with three characteristic tensors $C, P$ and d independent of ε. Let ${\hat{σ}}^{ε}, {\hat{e}}^{ε}, {\hat{D}}^{ε}, {\hat{E}}^{ε}$ be the stress tensor, the strain tensor, the electric displacement field and the electric vector field respectively and $({\hat{u}}^{ε}, {\hat{φ}}^{ε})$ be the mechanical displacement field and the electric potential.

The constitutive equations are given through the relations of the stress tensor ${\hat{σ}}^{ε} = ({\hat{σ}}_{i j}^{ε})$ and the electric displacement vector ${\hat{D}}^{ε}$ on the one hand, and of the linear stress tensor and the electric field on the other hand. The linear strain tensor ${\hat{e}}^{ε} ({\hat{u}}^{ε})$ is defined by $\begin{matrix} {\hat{e}}^{ε} ({\hat{u}}^{ε}) = \frac{1}{2} ({\hat{\nabla}}^{ε} {\hat{u}}^{ε} + {({\hat{\nabla}}^{ε} {\hat{u}}^{ε})}^{T}), \end{matrix}$ whose components read as $\begin{matrix} {\hat{e}}_{i j}^{ε} ({\hat{u}}^{ε}) = \frac{1}{2} ({\hat{\partial}}_{i}^{ε} {\hat{u}}_{j}^{ε} + {\hat{\partial}}_{j}^{ε} {\hat{u}}_{i}^{ε}), \end{matrix}$ and the electric field vector ${\hat{E}}^{ε} : Ω^{ε} \to R^{3}$ is given by $\begin{matrix} {\hat{E}}^{ε} = - {\hat{\nabla}}^{ε} {\hat{φ}}^{ε} . \end{matrix}$ Then the constitutive equations are $\begin{matrix} (3.2) & \{\begin{matrix} {\hat{σ}}^{ε} ({\hat{u}}^{ε}, φ^{ε}) = C {\hat{e}}^{ε} ({\hat{u}}^{ε}) + P {\hat{\nabla}}^{ε} {\hat{φ}}^{ε}, & in {\hat{Ω}}^{ε}, \\ {\hat{D}}^{ε} ({\hat{u}}^{ε}, {\hat{φ}}^{ε}) = P {\hat{e}}^{ε} ({\hat{u}}^{ε}) - d {\hat{\nabla}}^{ε} {\hat{φ}}^{ε}, & in {\hat{Ω}}^{ε}, \end{matrix} \end{matrix}$ which components read as $\begin{matrix} (3.3) & \{\begin{matrix} {\hat{σ}}_{i j}^{ε} ({\hat{u}}^{ε}, {\hat{φ}}^{ε}) = C_{i j k l} {\hat{e}}_{k l}^{ε} ({\hat{u}}^{ε}) + P_{k i j} {\hat{\partial}}_{k}^{ε} {\hat{φ}}^{ε}, & in {\hat{Ω}}^{ε}, \\ {\hat{D}}_{k}^{ε} ({\hat{u}}^{ε}, {\hat{φ}}^{ε}) = - d_{k j} {\hat{\partial}}_{j}^{ε} {\hat{φ}}^{ε} + P_{k i j} {\hat{e}}_{i j}^{ε} ({\hat{u}}^{ε}), & in {\hat{Ω}}^{ε} . \end{matrix} \end{matrix}$ For a shallow shell subjected to applied body forces with density ${\hat{f}}^{ε}$ and to electric charge with density ${\hat{ρ_{e}}}^{ε}$ , the equilibrium problem posed in ${\hat{Ω}}^{ε}$ reads $\begin{matrix} (3.4) & \{\begin{matrix} div {\hat{σ}}^{ε} = - {\hat{f}}^{ε}, \\ div {\hat{D}}^{ε} = {\hat{ρ}}_{e}^{ε} . \end{matrix} \end{matrix}$

We will consider the following cases:

the body is clamped on the whole lateral surface ${\hat{Γ}}_{0}^{ε}$ , this condition can be relaxed (see Section 6),

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

the body is in mechanical contact with the lower face ${\hat{Γ}}_{-}^{ε}$ .

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

We focus now on the unilateral contact with an horizontal plane $z = - ε$ . Let ${\hat{x}}_{-}^{ε}$ is a point of the part ${\hat{Γ}}_{-}^{ε}$ of the boundary. We write the Cartesian components of a vector $O {\hat{x}}_{-}^{ε}$ as follows: $\begin{matrix} (3.5) & O {\hat{x}}_{-}^{ε} = (\begin{matrix} x_{1}^{ε} + ε \frac{\partial_{1} θ^{ε}}{\sqrt{a^{ε}}} \\ x_{2}^{ε} + ε \frac{\partial_{2} θ^{ε}}{\sqrt{a^{ε}}} \\ θ^{ε} - ε \frac{1}{\sqrt{a^{ε}}} \end{matrix}) = (\begin{matrix} x_{1} + ε \frac{\partial_{1} θ^{ε}}{\sqrt{a^{ε}}} \\ x_{2} + ε \frac{\partial_{2} θ^{ε}}{\sqrt{a^{ε}}} \\ θ^{ε} - ε \frac{1}{\sqrt{a^{ε}}} \end{matrix}) . \end{matrix}$ Defining the following maps: $\begin{matrix} {\hat{θ}}^{ε} = θ^{ε} \circ {(Θ^{ε})}^{- 1}, {\hat{a}}^{ε} = a^{ε} \circ {(Θ^{ε})}^{- 1}, \end{matrix}$ then the unilateral contact conditions mean that the displacement on ${\hat{Γ}}_{-}^{ε}$ must satisfy a nonpenetrability condition $(O {\hat{x}}_{-}^{ε} + {\hat{u}}^{ε} ({\hat{x}}_{-}^{ε})) \cdot e_{3} ⩾ 0$ , which means that $\begin{matrix} (3.6) & \forall {\hat{x}}_{-}^{ε} \in {\hat{Γ}}_{-}^{ε}, ({\hat{θ}}^{ε} - \frac{ε}{\sqrt{{\hat{a}}^{ε}}}) ({\hat{x}}_{-}^{ε}) + {\hat{u}}_{3}^{ε} ({\hat{x}}_{-}^{ε}) ⩾ - ε, \end{matrix}$ here $\begin{matrix} e_{3} = (\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}) . \end{matrix}$ The so-called Signorini conditions are classically obtained by adding the following constraints to the nonpenetrability condition (3.6):

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

Either a point on the boundary ${\hat{Γ}}_{-}^{ε}$ is not in contact with the obstacle, so that the nonpenetrability inequality (3.6) is strictly satisfied, and the reaction of the obstacle at this point therefore vanish, or the point on ${\hat{Γ}}_{-}^{ε}$ is in contact with the obstacle, so that condition (3.6) is an equality, and the reaction of the obstacle can therefore differ from zero.

These constraints read as $\begin{array}{l} {\hat{σ}}_{3}^{ε} ≜ - ({\hat{σ}}^{ε} ({\hat{x}}_{-}^{ε}) {\hat{a}}^{ε}) \cdot e_{3} ⩾ 0 \forall {\hat{x}}^{ε} \in {\hat{Γ}}_{-}^{ε}, \\ {\hat{σ}}_{3}^{ε} ({\hat{θ}}^{ε} - \frac{ε}{\sqrt{{\hat{a}}^{ε}}} + {\hat{u}}_{3}^{ε} + ε) = 0 \forall {\hat{x}}^{ε} \in {\hat{Γ}}_{-}^{ε}, \end{array}$ so that the contact condition is $\begin{matrix} (3.7) & {\hat{θ}}^{ε} - \frac{ε}{\sqrt{{\hat{a}}^{ε}}} + {\hat{u}}_{3}^{ε} + ε ⩾ 0, {\hat{σ}}_{3}^{ε} ⩾ 0, {\hat{σ}}_{3}^{ε} ({\hat{θ}}^{ε} - \frac{ε}{\sqrt{{\hat{a}}^{ε}}} + {\hat{u}}_{3}^{ε} + ε) = 0 on {\hat{Γ}}_{-}^{ε} . \end{matrix}$ We gather all the conditions and get $\begin{matrix} (3.8) & \{\begin{matrix} {\hat{u}}^{ε} = 0 & on {\hat{Γ}}_{0}^{ε}, \\ {\hat{φ}}^{ε} = 0 & on {\hat{Γ}}_{0}^{ε}, \\ {\hat{σ}}^{ε} ({\hat{u}}^{ε}, {\hat{φ}}^{ε}) {\hat{a}}^{ε} = {\hat{g}}^{ε} & on {\hat{Γ}}_{+}^{ε}, \\ {\hat{D}}^{ε} ({\hat{u}}^{ε}, {\hat{φ}}^{ε}) \cdot {\hat{a}}^{ε} = {\hat{ϑ}}^{ε} & on {\hat{Γ}}_{+}^{ε}, \\ {\hat{D}}^{ε} \cdot {\hat{a}}^{ε} = 0 & on {\hat{Γ}}_{-}^{ε}, \\ {\hat{θ}}^{ε} - \frac{ε}{\sqrt{{\hat{a}}^{ε}}} + {\hat{u}}_{3}^{ε} + ε ⩾ 0, {\hat{σ}}_{3}^{ε} ⩾ 0, {\hat{σ}}_{3}^{ε} ({\hat{θ}}^{ε} - \frac{ε}{\sqrt{{\hat{a}}^{ε}}} + {\hat{u}}_{3}^{ε} + ε) = 0 & on {\hat{Γ}}_{-}^{ε} . \end{matrix} \end{matrix}$

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 but restrict the lateral surface to ${\hat{Γ}}_{0}^{ε} = {\hat{γ}}_{0} \times [- ε, ε]$ with ${\hat{γ}}_{0} = Θ (γ_{0})$ . In Section 6 we will use the “new definition of upper face” that includes the remaining part of the lateral surface ${\hat{Γ}}_{+}^{ε} \cup (\hat{γ} ∖ {\hat{γ}}_{0}) \times [- ε, ε]$ .

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

The natural functional framework for the equilibrium problem (3.4) and (3.8) is the convex set ${\hat{K}}^{ε} ({\hat{Ω}}^{ε})$ defined as $\begin{matrix} (3.9) & {\hat{K}}^{ε} ({\hat{Ω}}^{ε}) = {{\hat{v}}^{ε} \in H^{1} ({\hat{Ω}}^{ε}), {\hat{v}}^{ε} = 0 on {\hat{Γ}}_{0}^{ε}, {\hat{θ}}^{ε} - \frac{ε}{\sqrt{{\hat{a}}^{ε}}} + {\hat{v}}_{3}^{ε} + ε ⩾ 0 on {\hat{Γ}}_{-}^{ε}} \end{matrix}$ and the set $\begin{matrix} (3.10) & {\hat{Q}}^{ε} ({\hat{Ω}}^{ε}) = {{\hat{ψ}}^{ε} \in H^{1} ({\hat{Ω}}^{ε}), {\hat{ψ}}^{ε} = 0 on {\hat{Γ}}_{0}^{ε}} . \end{matrix}$ Then the weak solution $({\hat{u}}^{ε}, {\hat{φ}}^{ε})$ to problem (3.4) and (3.8), satisfies the following variational inequality: $\begin{matrix} (3.11) & \{\begin{matrix} Find ({\hat{u}}^{ε}, {\hat{φ}}^{ε}) \in {\hat{K}}^{ε} ({\hat{Ω}}^{ε}) \times {\hat{Q}}^{ε} ({\hat{Ω}}^{ε}) such that \\ \int_{{\hat{Ω}}^{ε}} C_{i j k l} {\hat{e}}_{k l}^{ε} ({\hat{u}}^{ε}) {\hat{e}}_{i j}^{ε} ({\hat{v}}^{ε} - {\hat{u}}^{ε}) d {\hat{x}}^{ε} - \int_{{\hat{Ω}}^{ε}} d_{i j} {\hat{\partial}}_{j}^{ε} {\hat{φ}}^{ε} {\hat{\partial}}_{i}^{ε} ({\hat{ψ}}^{ε} - {\hat{φ}}^{ε}) d {\hat{x}}^{ε} \\ + \int_{{\hat{Ω}}^{ε}} P_{k i j} {\hat{\partial}}_{k}^{ε} {\hat{φ}}^{ε} {\hat{e}}_{i j}^{ε} ({\hat{v}}^{ε} - {\hat{u}}^{ε}) d {\hat{x}}^{ε} + \int_{{\hat{Ω}}^{ε}} P_{k i j} {\hat{\partial}}_{k}^{ε} ({\hat{ψ}}^{ε} - {\hat{φ}}^{ε}) {\hat{e}}_{i j}^{ε} ({\hat{u}}^{ε}) d {\hat{x}}^{ε} \\ ⩾ \int_{{\hat{Ω}}^{ε}} {\hat{f}}^{ε} \cdot ({\hat{v}}^{ε} - {\hat{u}}^{ε}) d {\hat{x}}^{ε} + \int_{{\hat{Γ}}_{+}^{ε}} {\hat{g}}^{ε} \cdot ({\hat{v}}^{ε} - {\hat{u}}^{ε}) d {\hat{a}}^{ε} - \int_{{\hat{Ω}}^{ε}} {\hat{ρ}}_{e}^{ε} ({\hat{ψ}}^{ε} - {\hat{φ}}^{ε}) d {\hat{x}}^{ε} \\ + \int_{{\hat{Γ}}_{+}^{ε}} {\hat{ϑ}}^{ε} ({\hat{ψ}}^{ε} - {\hat{φ}}^{ε}) d {\hat{a}}^{ε} \\ \forall ({\hat{v}}^{ε}, {\hat{ψ}}^{ε}) \in {\hat{K}}^{ε} ({\hat{Ω}}^{ε}) \times {\hat{Q}}^{ε} ({\hat{Ω}}^{ε}) . \end{matrix} \end{matrix}$ Based on classical arguments [8] and [15], problem (3.4) and (3.8) and problem (3.11) are equivalent. Moreover, (3.11) has a unique solution for any fixed ε.

4. Scaling and equilibrium in the fixed domain Ω

4.1. The problem in $Ω^{ε}$

Using the one to one map ${(Θ^{ε})}^{- 1}$ , the reference configuration of the shell ${\hat{Ω}}^{ε}$ is mapped onto the cylindrical domain $Ω^{ε}$ . Because of the same diffeomorphism ${(Θ^{ε})}^{- 1}$ , the cone ${\hat{K}}^{ε} ({\hat{Ω}}^{ε})$ is mapped onto the cylinder domain $K^{ε} (Ω^{ε})$ defined as follows: $\begin{matrix} (4.1) & K^{ε} (Ω^{ε}) = {v^{ε} \in H^{1} (Ω^{ε}), v^{ε} = 0 on Γ_{0}^{ε}, v_{3}^{ε} ⩾ - θ^{ε} + \frac{ε}{\sqrt{a^{ε}}} - ε on Γ_{-}^{ε}}, \end{matrix}$ and the set ${\hat{Q}}^{ε} ({\hat{Ω}}^{ε})$ is mapped onto the set $\begin{matrix} (4.2) & Q^{ε} (Ω^{ε}) = {ψ^{ε} \in H^{1} (Ω^{ε}), ψ^{ε} = 0 on Γ_{0}^{ε}} . \end{matrix}$

In order to rewrite problem (3.11) over the domain $Ω^{ε}$ and, then, in a fixed domain Ω, we recall that the transformation of the volume element yields $\begin{matrix} d {\hat{x}}^{ε} = δ^{ε} d x^{ε}, where δ^{ε} = δ^{ε} (x^{ε}) = det \nabla^{ε} Θ^{ε} (x^{ε}), x^{ε} \in Ω^{ε} . \end{matrix}$ Similarly, a surface element of $Γ_{+}^{ε}$ is mapped onto a surface element of ${\hat{Γ}}_{+}^{ε}$ by $\begin{matrix} d {\hat{a}}^{ε} = δ^{ε} {b_{3 j}^{ε} b_{3 j}^{ε}}^{\frac{1}{2}} d a^{ε}, \end{matrix}$ here the matrix $b_{i j}^{ε} (x^{ε})$ is given by $\begin{matrix} b_{i j}^{ε} (x^{ε}) : = {({\nabla^{ε} Θ^{ε} (x^{ε})}^{- 1})}_{i j} . \end{matrix}$ Noting the chain rule ${\hat{\partial}}_{j}^{ε} {\hat{v}}_{i}^{ε} ({\hat{x}}^{ε}) = b_{k j}^{ε} (x^{ε}) \partial_{k}^{ε} v_{i}^{ε}$ , we obtain the expression for the linearized strain tensor $\begin{matrix} {\hat{e}}_{i j}^{ε} ({\hat{v}}^{ε}) = e_{i j}^{ε} (v^{ε}) ≜ \frac{1}{2} (b_{k i}^{ε} (x^{ε}) \partial_{k}^{ε} v_{j}^{ε} + b_{l j}^{ε} (x^{ε}) \partial_{l}^{ε} v_{i}^{ε}), \end{matrix}$ and ${\hat{\partial}}_{j}^{ε} {\hat{φ}}^{ε} ({\hat{x}}^{ε}) = b_{k j}^{ε} (x^{ε}) \partial_{k}^{ε} φ^{ε} ≜ τ_{j}^{ε} (φ^{ε})$ , which finally changes problem (3.11) into the following variational problem: $\begin{matrix} (4.3) & \{\begin{matrix} Find (u^{ε}, φ^{ε}) \in K^{ε} (Ω^{ε}) \times Q^{ε} (Ω^{ε}) such that \\ \int_{Ω^{ε}} C_{i j k l} e_{k l}^{ε} (u^{ε}) e_{i j}^{ε} (v^{ε} - u^{ε}) δ^{ε} d x^{ε} - \int_{Ω^{ε}} d_{i j} τ_{j}^{ε} (φ^{ε}) τ_{i}^{ε} (ψ^{ε} - φ^{ε}) δ^{ε} d x^{ε} \\ + \int_{Ω^{ε}} P_{k i j} τ_{k}^{ε} (φ^{ε}) e_{i j}^{ε} (v^{ε} - u^{ε}) δ^{ε} d x^{ε} + \int_{Ω^{ε}} P_{k i j} τ_{k}^{ε} (ψ^{ε} - φ^{ε}) e_{i j}^{ε} (u^{ε}) δ^{ε} d x^{ε} \\ ⩾ \int_{Ω^{ε}} (f^{ε} \cdot (v^{ε} - u^{ε}) - ρ_{e}^{ε} (ψ^{ε} - φ^{ε})) δ^{ε} d x^{ε} + \int_{Γ_{+}^{ε}} (g^{ε} \cdot (v^{ε} - u^{ε}) \\ + ϑ^{ε} (ψ^{ε} - φ^{ε})) δ^{ε} {b_{3 j}^{ε} b_{3 j}^{ε}}^{\frac{1}{2}} d a^{ε} \\ \forall (v^{ε}, ψ^{ε}) \in K^{ε} (Ω^{ε}) \times Q^{ε} (Ω^{ε}), \end{matrix} \end{matrix}$ here, the applied volume and surface forces $f^{ε}$ and $g^{ε}$ are defined by $f^{ε} = {\hat{f}}^{ε} \circ Θ^{ε}$ and $g^{ε} = {\hat{g}}^{ε} \circ Θ^{ε}$ , the applied volume and surface electric charge $ρ_{e}^{ε}$ and $ϑ^{ε}$ are defined by $ρ_{e}^{ε} = {\hat{ρ}}^{ε} \circ Θ^{ε}$ and $ϑ^{ε} = {\hat{ϑ}}^{ε} \circ Θ^{ε}$ , respectively.

4.2. Scaling

We now change the domain $Ω^{ε}$ with thickness $2 ε$ into a fixed domain Ω independent of ε via the simple geometrical transformation given by $\begin{matrix} (4.4) & \{\begin{matrix} Ω^{ε} ∋ x^{ε} = (x_{1}, x_{2}, x_{3}^{ε}) \to x = (x_{1}, x_{2}, x_{3}) \in Ω, \\ x_{3} = \frac{1}{ε} x_{3}^{ε} . \end{matrix} \end{matrix}$ The domain Ω is then a cylinder having cross-section ω and height 2, its boundary is $\partial Ω = Γ_{-} \cup Γ_{+} \cup Γ_{0}, Γ_{-} = ω \times {- 1}, Γ_{+} = ω \times {1}$ and $Γ_{0} = γ \times [- 1, 1]$ . We use the following scalings for the unknowns $(u^{ε}, φ^{ε})$ and test functions $(v^{ε}, ψ^{ε})$ defined as (see [17]): $\begin{matrix} (4.5) & \{\begin{matrix} u_{α}^{ε} = ε^{2} u_{α} (ε), & u_{3}^{ε} = ε u_{3} (ε), \\ v_{α}^{ε} = ε^{2} v_{α}, & v_{3}^{ε} = ε v_{3}, \\ φ^{ε} = ε^{2} φ (ε), & ψ^{ε} = ε^{2} ψ . \end{matrix} \end{matrix}$

In order to obtain a nontrivial limit problem by asymptotic analysis, it is essential to scale the data according to the scalings of the unknowns. 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 $\begin{matrix} (4.6) & \{\begin{matrix} f_{α}^{ε} = ε^{2} f_{α}, & f_{3}^{ε} = ε^{3} f_{3}, \\ g_{α}^{ε} = ε^{3} g_{α}, & g_{3}^{ε} = ε^{4} g_{3}, \\ ρ_{e}^{ε} = ε^{2} ρ_{e}, & ϑ = ε^{2} ϑ, \\ θ^{ε} (x_{1}, x_{2}) = ε θ (x_{1}, x_{2}) . \end{matrix} \end{matrix}$ The last assumption about $θ^{ε} (x_{1}, x_{2})$ comes from the exact mathematical definition of the shallowness in shell theory (see [6] where it is also shown that this assumption is the only one that gives a shell model in the limit).

Corresponding to the scaling procedure, we denote $e (ε) = (e_{i j} (ε))$ the scaled linearized strain tensor, the components of which are $\begin{matrix} (4.7) & \{\begin{matrix} e_{α β} (ε) (v) = ε^{2} {e_{α β}^{θ} (v) + ε^{2} e_{α β}^{♯} (ε, θ, v)}, \\ e_{α 3} (ε) (v) = ε {e_{α 3}^{θ} (v) + ε^{2} e_{α 3}^{♯} (ε, θ, v)}, \\ e_{3 α} (ε) (v) = ε {e_{3 α}^{θ} (v) + ε^{2} e_{3 α}^{♯} (ε, θ, v)}, \\ e_{33} (ε) (v) = e_{33}^{θ} (v) + ε^{2} (\partial_{α} θ \partial_{α} v_{3} + b_{33}^{♯} (ε, θ) \partial_{3} v_{3}) + ε^{4} e_{33}^{♭} (ε, θ, v), \end{matrix} \end{matrix}$ and $τ (ε) = (τ_{j} (ε))$ to denote the scaled vector fields: $\begin{matrix} (4.8) & \{\begin{matrix} τ_{α} (ε) (ψ (x)) = ε^{2} {τ_{α}^{θ} (ψ (x)) + ε^{2} τ_{α}^{♯} (ε, θ, ψ (x))}, \\ τ_{3} (ε) (ψ (x)) = ε {τ_{3}^{θ} (φ (x)) + ε^{2} τ_{3}^{♯} (ε, θ, ψ (x))}, \end{matrix} \end{matrix}$ where $\begin{matrix} (4.9) & \{\begin{matrix} τ_{α}^{θ} (ψ) = \partial_{α} ψ - \partial_{3} ψ \partial_{α} θ, \\ τ_{3}^{θ} (ψ) = \partial_{3} ψ, \end{matrix} \end{matrix}$ the explicit values of the terms $e^{θ}$ will be given later on, Following the procedure of [15], we get that the $L^{2}$ norms of $e^{♯}, τ^{♯}$ and the $L^{2}$ norm of $e_{33}^{♭}$ are bounded with respect to the $H^{1} (Ω)$ norms of v, ψ.

Moreover, we introduce the following notations: $\begin{matrix} (4.10) & \{\begin{matrix} b_{i j} (ε) (x) = b_{i j}^{ε} (x^{ε}), \\ δ (ε) (x) = δ^{ε} (x^{ε}) . \end{matrix} \end{matrix}$

4.3. The contact condition in the fixed domain $Ω = ω \times (- 1, 1)$

As we have seen in Section 3 the lower part $Γ_{-}^{ε}$ of the boundary is the image of ${\hat{Γ}}_{-}^{ε}$ by ${Θ^{ε}}^{- 1}$ . Noting (3.5), $Γ_{-}^{ε}$ is the set $\begin{matrix} Γ_{-}^{ε} = {(x_{1}, x_{2}, - ε), (x_{1}, x_{2}) \in ω} . \end{matrix}$ By the scaling process, the nonpenetrability condition satisfied now on $Γ_{-}$ is $\begin{matrix} (4.11) & v_{3} (x_{1}, x_{2}, - 1) ⩾ - θ (x_{1}, x_{2}) + \frac{1}{\sqrt{a^{ε}}} - 1, \forall (x_{1}, x_{2}) \in ω, \end{matrix}$ and the corresponding functional space is $\begin{matrix} K (ε) (Ω) = {v \in H^{1} (Ω), v = 0 on Γ_{0}, v_{3} (x_{1}, x_{2}, - 1) ⩾ - θ (x_{1}, x_{2}) + \frac{1}{\sqrt{a^{ε}}} - 1 on Γ_{-}} . \end{matrix}$ By the scaling process (4.4) and (4.5), $Q^{ε} (Ω^{ε})$ becomes $\begin{matrix} Q (Ω) = {ψ \in H^{1} (Ω), ψ = 0 on Γ_{0}} . \end{matrix}$ Replacing $(u^{ε}, φ^{ε})$ and $(v^{ε}, ψ^{ε})$ by their scalings $(u (ε), φ (ε))$ and $(v, ψ)$ , respectively, in (4.5), problem (4.3) becomes the following problem on the fixed domain Ω: $\begin{matrix} (4.12) & \{\begin{matrix} Find (u (ε), φ (ε)) \in K (ε) (Ω) \times Q (Ω) such that \\ ε \int_{Ω} C_{i j k l} e_{k l} (ε) (u (ε)) e_{i j} (ε) (v - u (ε)) δ (ε) d x \\ - ε \int_{Ω} d_{i j} τ_{j} (ε) (φ (ε)) τ_{i} (ε) (ψ - φ (ε)) δ (ε) d x \\ + ε \int_{Ω} P_{k i j} τ_{k} (ε) (φ (ε)) e_{i j} (ε) (v - u (ε)) δ (ε) d x \\ + ε \int_{Ω} P_{k i j} τ_{k} (ε) (ψ - φ (ε)) e_{i j} (ε) (u (ε)) δ (ε) d x \\ ⩾ ε \int_{Ω} f \cdot (v - u (ε)) δ (ε) d x + \int_{Γ_{+}} g \cdot (v - u (ε)) δ (ε) {b_{3 j} (ε) b_{3 j} (ε)}^{\frac{1}{2}} d a \\ - ε \int_{Ω} ρ_{e} (ψ - φ (ε)) δ (ε) d x + \int_{Γ_{+}} ϑ (ψ - φ (ε)) δ (ε) {b_{3 j} (ε) b_{3 j} (ε)}^{\frac{1}{2}} d a \\ \forall (v, ψ) \in K (ε) (Ω) \times Q (Ω), \end{matrix} \end{matrix}$ where $d a$ denotes the area element of the upper boundary $Γ_{+}$ , i.e., $d a^{ε} = d a$ .

Theorem 4.1.
For any fixed $ε > 0$ , problem ( 4.12 ) has a unique weak solution.
Proof.
The proof is a classical result for variational inequality [8,12]. □

5. Convergence

The aim of this section is to establish that when ε tends to zero, the sequence ${(u (ε), φ (ε))}$ 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 for shallow shells.

Lemma 5.1.
Let $θ \in C^{3} (\bar{Ω})$ be a given function, and let the functions $e_{i j}^{θ} (v)$ be defined by $\begin{matrix} \{\begin{matrix} e_{α β}^{θ} (v) = e_{α β} (v) - \frac{1}{2} (\partial_{β} θ \partial_{3} v_{α} + \partial_{α} θ \partial_{3} v_{β}), \\ e_{α 3}^{θ} (v) = e_{3 α}^{θ} (v) = e_{α 3} (v) - \frac{1}{2} \partial_{α} θ \partial_{3} v_{3}, \\ e_{33}^{θ} (v) = e_{33} (v) \end{matrix} \end{matrix}$ for any given $v \in H^{1} (Ω)$ . Then, for any given $ε > 0$ , the mapping $\begin{matrix} v \to {\sum_{i j} {| e_{i j}^{θ} (v) |}_{0, Ω}^{2}}^{\frac{1}{2}} \end{matrix}$ is a norm over the set $K (ε) (Ω)$ , which is equivalent to the norm induced by $‖ \cdot ‖_{1, Ω}$ .
Proof.
Similar in [5], we can get for any given $ε > 0$ , the norm ${\sum_{i j} | e_{i j}^{θ} (v) |_{0, Ω}^{2}}^{\frac{1}{2}}$ in the space ${v \in H^{1} (Ω), v = 0 on Γ_{0}}$ is equivalent to the norm induced by $‖ \cdot ‖_{1, Ω}$ . Then in the present case, the proof follows from the fact that the set $K (ε) (Ω)$ is a closed subset of the vector space ${v \in H^{1} (Ω), v = 0 on Γ_{0}}$ . □

5.1. Variational solutions

Theorem 5.1.
Assume that $f \in L^{2} (Ω), ρ_{e} \in L^{2} (Ω), g \in L^{2} (Γ_{+}), ϑ \in L^{2} (Γ_{+})$ and $θ \in C^{3} (\bar{Ω})$ . Then

(i) As ε tends to 0, the family ${(u (ε), φ (ε))}_{ε > 0}$ converges strongly to $(u, φ)$ in the set $K (Ω) \times Q (Ω)$ , where $\begin{matrix} \{\begin{matrix} K (Ω) = {v \in H^{1} (Ω), v = 0 on Γ_{0}, v_{3} (x_{1}, x_{2}, - 1) ⩾ - θ (x_{1}, x_{2}) on Γ_{-}}, \\ Q (Ω) = {ψ \in H^{1} (Ω), ψ = 0 on Γ_{0}} . \end{matrix} \end{matrix}$

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

(iii) The limit electric potential φ belongs to the bi-dimensional functional space $\begin{matrix} Q (ω) = {ψ \in H^{1} (Ω), ψ = 0 on γ_{0}} . \end{matrix}$

(iv) 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: $\begin{matrix} (5.1) & \{\begin{matrix} \frac{2}{3} \int_{ω} {\tilde{C}}_{α β σ τ} \partial_{σ τ} (ζ_{3}) \partial_{α β} (η_{3} - ζ_{3}) d ω + 2 \int_{ω} {\tilde{n}}_{α β}^{θ} (ζ, φ) \partial_{α} θ \partial_{β} (η_{3} - ζ_{3}) d ω \\ ⩾ \int_{ω} p_{3} (η_{3} - ζ_{3}) d ω - \int_{ω} s_{α} \partial_{α} (η_{3} - ζ_{3}) d ω \\ \forall η_{3} \in K_{3} (ω), \\ 2 \int_{ω} {\tilde{n}}_{α β}^{θ} (ζ, φ) e_{α β} (η_{H}) d ω + 2 \int_{ω} {\tilde{D}}_{γ}^{θ} (ζ, φ) \partial_{γ} ψ d ω \\ = \int_{ω} p_{H} \cdot η_{H} d ω - \int_{ω} (\tilde{ρ_{e}} - ϑ) ψ d ω \\ \forall (η_{H}, ψ) \in V_{H} (ω) \times Q (ω), \end{matrix} \end{matrix}$ where the new constitutive bi-dimensional law is given by $\begin{matrix} \{\begin{matrix} {\tilde{n}}_{α β}^{θ} (ζ, φ) = {\tilde{C}}_{α β σ τ} e_{σ τ}^{θ} (ζ) + {\tilde{P}}_{γ α β} \partial_{γ} φ, \\ {\tilde{D}}_{γ}^{θ} (ζ, φ) = {\tilde{P}}_{γ α β} e_{α β}^{θ} (ζ) - {\tilde{d}}_{γ β} \partial_{β} φ, \\ e_{α β}^{θ} (ζ) = \frac{1}{2} (\partial_{α} ζ_{β} + \partial_{β} ζ_{α}) + \frac{1}{2} (\partial_{α} θ \partial_{β} ζ_{3} + \partial_{β} θ \partial_{α} ζ_{3}), \end{matrix} \end{matrix}$ where the mechanical forces and electric charges are given by $\begin{array}{l} p_{i} = \int_{- 1}^{1} f_{i} d x_{3} + g_{i}, s_{α} = \int_{- 1}^{1} x_{3} f_{α} d x_{3} + g_{α}, {\tilde{ρ}}_{e} = \int_{- 1}^{1} ρ_{e} d x_{3}, \end{array}$ where the new bi-dimensional elasticity $\tilde{C}$ , piezoelectricity $\tilde{P}$ and dielectric $\tilde{d}$ are given by $\begin{matrix} (5.2) & \{\begin{matrix} {\tilde{C}}_{α β σ τ} = C_{α β σ τ}^{'} - \frac{1}{△} C_{α β k 3}^{'} △_{σ τ}^{k}, & C_{i j k l}^{'} = C_{i j k l} + \frac{P_{3 i j} P_{3 k l}}{d_{33}}, \\ {\tilde{P}}_{γ α β} = P_{γ α β}^{'} - \frac{1}{△} C_{α β k 3}^{'} △_{γ}^{k}, & P_{k i j}^{'} = P_{k i j} - \frac{P_{3 i j} d_{3 k}}{d_{33}}, \\ {\tilde{d}}_{γ β} = d_{γ β}^{'} + \frac{1}{△} P_{γ k 3}^{'} △_{β}^{k}, & d_{i j}^{'} = d_{i j} - \frac{d_{i 3} d_{3 j}}{d_{33}}, \end{matrix} \end{matrix}$ where $\begin{matrix} (5.3) & \{\begin{matrix} △ = ϵ_{p q r} C_{13 p 3}^{'} C_{23 q 3}^{'} C_{33 r 3}^{'}, \\ △_{γ}^{1} = ϵ_{p q r} P_{γ p 3}^{'} C_{23 q 3}^{'} C_{33 r 3}^{'}, & △_{γ}^{2} = ϵ_{p q r} C_{13 p 3}^{'} P_{γ q 3}^{'} C_{33 r 3}^{'}, \\ △_{γ}^{3} = ϵ_{p q r} C_{13 p 3}^{'} C_{23 q 3}^{'} P_{γ r 3}^{'}, \\ △_{ζ η}^{1} = ϵ_{p q r} C_{3 p ζ η}^{'} C_{23 q 3}^{'} C_{33 r 3}^{'}, & △_{ζ η}^{2} = ϵ_{p q r} C_{13 p 3}^{'} C_{3 q ζ η}^{'} C_{33 r 3}^{'}, \\ △_{ζ η}^{3} = ϵ_{p q r} C_{13 p 3}^{'} C_{23 q 3}^{'} C_{3 r ζ η}^{'} . \end{matrix} \end{matrix}$
Proof.
The proof is established through five steps. First, we introduce new scaled tensors $R^{θ} (ε)$ and $χ (ε)$ , and by means of 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. In the second step a new variational inequality problem is introduced from (4.12) thanks to the use of new scaled tensors $R^{θ} (ε)$ and $χ (ε)$ . The third step deals with some technical results about the components of these tensors. In the fourth step we show that the family ${(u (ε)), φ (ε)}$ converges strongly to $(u, φ)$ . The last step completes the proof by deducing the variational inequality problem with new piezoelectric characteristics.

∙ Step I. Introducing the following scaled symmetric tensor $R^{θ} (ε) = (R_{i j}^{θ} (ε)) \in L^{2} (Ω)$ and the scaled tensor $χ (ε) = (χ_{i} (ε)) \in L^{2} (Ω)$ : $\begin{matrix} \{\begin{matrix} R_{α β}^{θ} (ε) (v) = e_{α β}^{θ} (v), R_{α 3}^{θ} (ε) (v) = \frac{1}{ε} e_{α 3}^{θ} (v), R_{33}^{θ} (ε) (v) = \frac{1}{ε^{2}} (e_{33}^{θ} (v) + \partial_{α} θ \partial_{α} v_{3}), \\ χ_{α}^{θ} (ε) (ψ) = τ_{α}^{θ} (ψ), χ_{3}^{θ} (ε) (ψ) = \frac{1}{ε} τ_{3}^{θ} (ψ) = \frac{1}{ε} \partial_{3} ψ . \end{matrix} \end{matrix}$ We introduce $R^{θ} (ε) (u (ε))$ and $χ^{θ} (ε) (φ (ε))$ in the variational inequality (4.12) and get $\begin{matrix} (5.4) & \{\begin{matrix} \int_{Ω} C_{i j k l} R_{k l}^{θ} (ε) (u (ε)) R_{i j}^{θ} (ε) (v - u (ε)) d x - \int_{Ω} d_{i j} χ_{j}^{θ} (ε) (φ (ε)) χ_{i}^{θ} (ε) (ψ - φ (ε)) d x \\ + \int_{Ω} P_{k i j} χ_{k}^{θ} (ε) (φ (ε)) R_{i j}^{θ} (ε) (v - u (ε)) d x \\ + \int_{Ω} P_{k i j} χ_{k}^{θ} (ε) (ψ - φ (ε)) R_{i j}^{θ} (ε) (u (ε)) d x \\ + B^{♯} (ε, θ, u (ε), R^{θ} (ε), χ^{θ} (ε), v - u (ε), ψ - φ (ε)) \\ ⩾ \int_{Ω} f \cdot (v - u (ε)) d x + \int_{Γ_{+}} g \cdot (v - u (ε)) d a - \int_{Ω} ρ_{e} (ψ - φ (ε)) d x \\ + \int_{Γ_{+}} ϑ (ψ - φ (ε)) d a + ε^{2} L^{♯} (ε, θ, v - u (ε), ψ - φ (ε)) . \end{matrix} \end{matrix}$

Following the proof in [22], we get the uniform boundness of $| R^{θ} (ε) (u (ε)) |_{0, Ω}$ , $| χ^{θ} (ε) (φ (ε)) |_{0, Ω}$ and $‖ u (ε) ‖_{1, Ω}$ , $‖ φ (ε) ‖_{1, Ω}$ with respect to ε. There exist $u \in H^{1} (Ω), φ \in H^{1} (Ω)$ and $R^{θ} \in L^{2} (Ω), χ^{θ} \in L^{2} (Ω)$ such that when $ε \to 0$ , we have the following weak convergence: $\begin{matrix} \{\begin{matrix} u (ε) ⇀ u & in H^{1} (Ω), φ (ε) ⇀ φ in H^{1} (Ω), \\ R^{θ} (ε) (u (ε)) ⇀ R^{θ} (u) & in L^{2} (Ω), χ^{θ} (ε) (φ (ε)) ⇀ χ^{θ} (φ) in L^{2} (Ω) . \end{matrix} \end{matrix}$ Moreover, from the definition of $R^{θ} (ε)$ and $χ^{θ} (ε)$ , we have $\begin{matrix} {| e_{α 3}^{θ} (u (ε)) |}_{0, Ω} ⩽ c ε, {| e_{33}^{θ} (u (ε)) |}_{0, Ω} ⩽ c ε^{2}, {| \partial_{3} φ (ε) |}_{0, Ω} ⩽ c ε . \end{matrix}$ Hence $e_{i 3}^{θ} (u (ε)) ⇀ 0$ in $L^{2} (Ω)$ and $\partial_{3} φ (ε) ⇀ 0$ in $L^{2} (Ω)$ , then $\begin{matrix} {| e_{i 3}^{θ} (u) |}_{0, Ω} ⩽ \underset{ε \to 0}{lim inf} {| e_{i 3}^{θ} (u (ε)) |}_{0, Ω} = 0 and | \partial_{3} φ |_{0, Ω} ⩽ \underset{ε \to 0}{lim inf} {| \partial_{3} φ (ε) |}_{0, Ω} = 0, \end{matrix}$ following [2] (P233), we get $e_{i 3}^{θ} (u) = 0$ , i.e., there exist $ζ_{α} \in H^{1} (ω)$ and $ζ_{3} \in H^{2} (ω)$ such that $\begin{matrix} u_{α} = ζ_{α} - x_{3} \partial_{α} ζ_{3}, u_{3} = ζ_{3} and \partial_{3} φ = 0, so φ is independent of x_{3} . \end{matrix}$

The following lemma, the detailed proof of which can be found in [22], will be used several times in the next step. Lemma 5.2.
Let the operator $A (ε) : u (ε) ⟶ H^{1} (Ω)$ satisfy 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 $\begin{matrix} (5.5) & \int_{Ω} A (ε) (u (ε)) \cdot \partial_{3} (v - u (ε)) d x ⩾ ε \int_{Ω} f \cdot (v - u (ε)) d x \forall v \in K (Ω), \end{matrix}$ then $A (u) = 0$ .

∙ Step II. We first observe that for any given $v \in K (ε) (Ω)$ , we have $v_{3} (x_{1}, x_{2}, - 1) ⩾ - θ (x_{1}, x_{2}) + \frac{1}{\sqrt{a^{ε}}} - 1$ . Since ${lim}_{ε \to 0} a^{ε} = 1$ by the definition of $a^{ε}$ , we introduce the following subset of $K (ε) (Ω)$ defined by $\begin{matrix} K (Ω) = {v \in H^{1} (Ω), v = 0 on Γ_{0}, v_{3} (x_{1}, x_{2}, - 1) ⩾ - θ (x_{1}, x_{2}) on Γ_{-}}, \end{matrix}$ Substituting tensors $R^{θ} (ε)$ and $χ^{θ} (ε)$ into variational inequality problem (4.12), we get, $\begin{matrix} (5.6) & \{\begin{matrix} \int_{Ω} C_{α β σ τ} R_{σ τ}^{θ} (ε) (u (ε)) e_{α β}^{θ} (v - u (ε)) d x + 2 \int_{Ω} C_{α β σ 3} R_{σ 3}^{θ} (ε) (u (ε)) e_{α β}^{θ} (v - u (ε)) d x \\ + \int_{Ω} C_{α β 33} R_{33}^{θ} (ε) (u (ε)) e_{α β}^{θ} (v - u (ε)) d x \\ + \int_{Ω} P_{γ α β} χ_{γ}^{θ} (ε) (φ (ε)) e_{α β}^{θ} (v - u (ε)) d x + \int_{Ω} P_{3 α β} χ_{3}^{θ} (ε) (φ (ε)) e_{α β}^{θ} (v - u (ε)) d x \\ + \int_{Ω} C_{33 σ τ} R_{σ τ}^{θ} (ε) (u (ε)) \partial_{α} θ \partial_{α} (v_{3} - u_{3} (ε)) d x \\ + 2 \int_{Ω} C_{33 σ 3} R_{σ 3}^{θ} (ε) (u (ε)) \partial_{α} θ \partial_{α} (v_{3} - u_{3} (ε)) d x \\ + \int_{Ω} C_{3333} R_{33}^{θ} (ε) (u (ε)) \partial_{α} θ \partial_{α} (v_{3} - u_{3} (ε)) d x \\ + \int_{Ω} P_{γ 33} χ_{γ}^{θ} (ε) (φ (ε)) \partial_{α} θ \partial_{α} (v_{3} - u_{3} (ε)) d x + \int_{Ω} P_{333} χ_{3}^{θ} (ε) (φ (ε)) \partial_{α} θ \partial_{α} (v_{3} - u_{3} (ε)) d x \\ - \int_{Ω} d_{α β} χ_{β}^{θ} (ε) (φ (ε)) τ_{α}^{θ} (ψ - φ (ε)) d x - \int_{Ω} d_{α 3} χ_{3}^{θ} (ε) (φ (ε)) τ_{α}^{θ} (ψ - φ (ε)) d x \\ + \int_{Ω} P_{γ α β} τ_{γ}^{θ} (ψ - φ (ε)) R_{α β}^{θ} (ε) (u (ε)) d x \\ + 2 \int_{Ω} P_{γ α 3} τ_{γ}^{θ} (ψ - φ (ε)) R_{α 3}^{θ} (ε) (u (ε)) d x \\ + \int_{Ω} P_{γ 33} τ_{γ}^{θ} (ψ - φ (ε)) R_{33}^{θ} (ε) (u (ε)) d x + \frac{2}{ε} \int_{Ω} C_{α 3 σ τ} R_{σ τ}^{θ} (ε) (u (ε)) e_{α 3}^{θ} (v - u (ε)) d x \\ + \frac{4}{ε} \int_{Ω} C_{α 3 σ 3} R_{σ 3}^{θ} (ε) (u (ε)) e_{α 3}^{θ} (v - u (ε)) d x \\ + \frac{2}{ε} \int_{Ω} C_{α 333} R_{33}^{θ} (ε) (u (ε)) e_{α 3}^{θ} (v - u (ε)) d x + \frac{2}{ε} \int_{Ω} P_{γ α 3} χ_{γ}^{θ} (ε) (φ (ε)) e_{α 3}^{θ} (v - u (ε)) d x \\ + \frac{2}{ε} \int_{Ω} P_{3 α 3} χ_{3}^{θ} (ε) (φ (ε)) e_{α 3}^{θ} (v - u (ε)) d x - \frac{1}{ε} \int_{Ω} d_{3 α} χ_{α}^{θ} (ε) (φ (ε)) \partial_{3} (ψ - φ (ε)) d x \\ - \frac{1}{ε} \int_{Ω} d_{33} χ_{3}^{θ} (ε) (φ (ε)) \partial_{3} (ψ - φ (ε)) d x + \frac{1}{ε} \int_{Ω} P_{3 α β} \partial_{3} (ψ - φ (ε)) R_{α β}^{θ} (ε) (u (ε)) d x \\ + \frac{2}{ε} \int_{Ω} P_{3 α 3} \partial_{3} (ψ - φ (ε)) R_{α 3}^{θ} (ε) (u (ε)) d x + \frac{1}{ε} \int_{Ω} P_{333} \partial_{3} (ψ - φ (ε)) R_{33}^{θ} (ε) (u (ε)) d x \\ + \frac{1}{ε^{2}} \int_{Ω} C_{33 σ τ} R_{σ τ}^{θ} (ε) (u (ε)) \partial_{3} (v_{3} - u_{3} (ε)) d x \\ + \frac{2}{ε^{2}} \int_{Ω} C_{33 σ 3} R_{σ 3}^{θ} (ε) (u (ε) \partial_{3} (v_{3} - u_{3} (ε)) d x \\ + \frac{1}{ε^{2}} \int_{Ω} C_{3333} R_{33}^{θ} (ε) (u (ε)) \partial_{3} (v_{3} - u_{3} (ε)) d x \\ + \frac{1}{ε^{2}} \int_{Ω} P_{γ 33} χ_{γ}^{θ} (ε) (φ (ε)) \partial_{3} (v_{3} - u_{3} (ε)) d x + \frac{1}{ε^{2}} \int_{Ω} P_{333} χ_{3}^{θ} (ε) (φ (ε)) \partial_{3} (v_{3} - u_{3} (ε)) d x \\ + B^{♯} (ε, θ, u (ε), R^{θ} (ε), χ^{θ} (ε), v - u (ε), ψ - φ (ε)) \\ ⩾ \int_{Ω} (f \cdot (v - u (ε)) - ρ_{e} (ψ - φ (ε))) d x + \int_{Γ_{+}} (g \cdot (v - u (ε)) + ϑ (ψ - φ (ε)) d a \\ + ε^{2} L^{♯} (ε, θ, v - u (ε), ψ - φ (ε)) \forall (v, ψ) \in K (ε) (Ω) \times Q (Ω), \end{matrix} \end{matrix}$ where $B^{♯} (ε, θ, u (ε), R^{θ} (ε), χ^{θ} (ε), v, ψ)$ and $L^{♯} (ε, θ, v, ψ)$ stand for remainders which are given and estimated in Appendix. Remainders $B^{♯}$ is uniformly bounded, i.e., there exists a positive constant $c (θ)$ independent of ε such that for all $u \in K (Ω)$ , $v \in K (Ω)$ , $φ \in Q (Ω)$ , $ψ \in Q (Ω)$ , $R^{θ} \in L^{2} (Ω)$ , $χ^{θ} \in L^{2} (Ω)$ we have $\begin{array}{l} sup_{0 < ε ⩽ ε_{0}} B^{♯} (ε, θ, u (ε), R^{θ} (ε), χ^{θ} (ε), v, ψ) \\ ⩽ c (θ) ({| R^{θ} |}_{0, Ω} + | u |_{1, Ω} + {| χ^{θ} |}_{0, Ω} + | φ |_{1, Ω}) (| v |_{1, Ω} + | ψ |_{1, Ω}), \\ sup_{0 < ε ⩽ ε_{0}} L^{♯} (ε, θ, v, ψ) ⩽ c (| v |_{1, Ω} + | ψ |_{1, Ω}), \end{array}$

∙ Step III. Before we establish the strong convergence, we first compute $R_{α 3}^{θ} (u), R_{33}^{θ} (u)$ and $χ_{3}^{θ} (φ)$ .

Since $R_{α β}^{θ} (ε) (u (ε)) = e_{α β}^{θ} (u (ε))$ and $u (ε) ⇀ u$ in $H^{1} (Ω)$ , $φ (ε) ⇀ φ$ in $H^{1} (Ω)$ as $ε \to 0$ , we have $\begin{array}{l} R_{α β}^{θ} (ε) (u (ε)) ⇀ R_{α β}^{θ} (u) = e_{α β}^{θ} (u) in L^{2} (Ω), \\ χ_{α}^{θ} (ε) φ (ε) = \partial_{α}^{θ} φ (ε) ⇀ χ_{α}^{θ} (φ) = τ_{α}^{θ} (φ) in L^{2} (Ω) \end{array}$ as $ε \to 0$ .

Taking $v = u (ε)$ in (5.6) and multiplying it by ε, we get $\begin{matrix} \{\begin{matrix} \int_{Ω} (2 P_{3 α 3} R_{α 3}^{θ} (ε) (u (ε)) + P_{333} R_{33}^{θ} (ε) (u (ε)) - d_{3 α} χ_{α}^{θ} (ε) (φ (ε)) \\ - d_{33} χ_{3}^{θ} (ε) (φ (ε))) \partial_{3} (ψ - φ (ε)) d x + \int_{Ω} P_{3 α β} R_{α β}^{θ} (ε) (u (ε)) \partial_{3} (ψ - φ (ε)) d x \\ ⩾ ε \int_{Ω} d_{α β} χ_{β}^{θ} (ε) (φ (ε)) τ_{α}^{θ} (ψ - φ (ε)) d x + ε \int_{Ω} d_{α 3} χ_{3}^{θ} (ε) (φ (ε)) τ_{α}^{θ} (ψ - φ (ε)) d x \\ - ε \int_{Ω} P_{γ α β} τ_{γ}^{θ} (ψ - φ (ε)) R_{α β}^{θ} (ε) (u (ε)) d x - 2 ε \int_{Ω} P_{γ α 3} τ_{γ}^{θ} (ψ - φ (ε)) R_{α 3}^{θ} (ε) (u (ε)) d x \\ - ε \int_{Ω} P_{γ 33} τ_{γ}^{θ} (ψ - φ (ε)) R_{33}^{θ} (ε) (u (ε)) d x - ε \int_{Ω} ρ_{e} (ψ - φ (ε)) d x \\ + ε \int_{Γ_{+}} ϑ (ψ - φ (ε)) d a - ε B^{♯} (ε, θ, u (ε), R^{θ} (ε), χ^{θ} (ε), ψ - φ (ε)) \\ + ε^{3} L^{♯} (ε, θ, ψ - φ (ε)), \forall ψ \in Q (Ω), \end{matrix} \end{matrix}$ The weak convergences and Lemma 5.2 yield $\begin{matrix} (5.7) & 2 P_{3 α 3} R_{α 3}^{θ} (u) + P_{333} R_{33}^{θ} (u) - d_{33} χ_{3}^{θ} (φ) = d_{3 γ} τ_{γ}^{θ} (φ) - P_{3 α β} e_{α β}^{θ} (u) . \end{matrix}$

Taking $ψ = φ (ε) and v_{3} = u_{3} (ε)$ in (5.6) and multiplying it by ε, we get $\begin{matrix} \{\begin{matrix} 2 \int_{Ω} C_{α 3 σ τ} R_{σ τ}^{θ} (ε) (u (ε)) \partial_{3} (v_{α} - u_{α} (ε)) d x + 4 \int_{Ω} C_{α 3 σ 3} R_{σ 3}^{θ} (ε) (u (ε)) \partial_{3} (v_{α} - u_{α} (ε)) d x \\ + 2 \int_{Ω} C_{α 333} R_{33}^{θ} (ε) (u (ε)) \partial_{3} (v_{α} - u_{α} (ε)) d x + 2 \int_{Ω} P_{γ α 3} χ_{γ}^{θ} (ε) (φ (ε)) \partial_{3} (v_{α} - u_{α} (ε)) d x \\ + 2 \int_{Ω} P_{3 α 3} χ_{3}^{θ} (ε) (φ (ε)) \partial_{3} (v_{α} - u_{α} (ε)) d x \\ ⩾ - ε (\int_{Ω} C_{α β σ τ} R_{σ τ}^{θ} (ε) (u (ε)) e_{α β}^{θ} (v - u (ε)) d x + 2 \int_{Ω} C_{α β σ 3} R_{σ 3}^{θ} (ε) (u (ε)) e_{α β}^{θ} (v - u (ε)) d x \\ + \int_{Ω} C_{α β 33} R_{33}^{θ} (ε) (u (ε)) e_{α β}^{θ} (v - u (ε)) d x + \int_{Ω} P_{γ α β} χ_{γ}^{θ} (ε) (φ (ε)) e_{α β}^{θ} (v - u (ε)) d x \\ + \int_{Ω} P_{3 α β} χ_{3}^{θ} (ε) (φ (ε)) e_{α β}^{θ} (v - u (ε)) d x) + ε \int_{Ω} f_{α} (v_{α} - u_{α} (ε)) d x \\ + ε \int_{Γ_{+}} g_{α} (v_{α} - u_{α} (ε)) d a - ε B^{♯} (ε, θ, u (ε), R^{θ} (ε), χ^{θ} (ε), v_{α} - u_{α} (ε)) \\ + ε^{3} L^{♯} (ε, θ, v_{α} - u_{α} (ε)), \forall v_{α} \in H^{1} (Ω), v_{α} = 0 on Γ_{0}, \end{matrix} \end{matrix}$ then $\begin{array}{l} 2 C_{α 3 σ 3} R_{σ 3}^{θ} (u) + C_{α 333} R_{33}^{θ} (u) + P_{3 α 3} χ_{3}^{θ} (φ) = - P_{γ α 3} τ_{γ}^{θ} (φ) - C_{α 3 α β} e_{α β}^{θ} (u) . \end{array}$ Similarly taking $ψ = φ (ε)$ and $v_{α} = u_{α} (ε)$ in (5.6) and multiplying by $ε^{2}$ , we have $\begin{matrix} \{\begin{matrix} \int_{Ω} (C_{33 σ τ} R_{σ τ}^{θ} (ε) (u (ε)) + 2 C_{33 σ 3} R_{σ 3}^{θ} (ε) (u (ε)) + C_{3333} R_{33}^{θ} (ε) (u (ε))) \partial_{3} (v_{3} - u_{3} (ε)) d x \\ + \int_{Ω} (P_{γ 33} χ_{γ}^{θ} (ε) (φ (ε)) + P_{333} χ_{3}^{θ} (ε) (φ (ε))) \partial_{3} (v_{3} - u_{3} (ε)) d x \\ ⩾ - ε^{2} [\int_{Ω} (C_{33 σ τ} R_{σ τ}^{θ} (ε) (u (ε)) \partial_{α} θ + 2 C_{33 σ 3} R_{σ 3}^{θ} (ε) (u (ε)) \partial_{α} θ) \partial_{α} (v_{3} - u_{3} (ε))) d x \\ + \int_{Ω} (C_{3333} R_{33}^{θ} (ε) (u (ε)) \partial_{α} θ + P_{γ 33} χ_{γ}^{θ} (ε) (φ (ε)) \partial_{α} θ \\ + P_{333} χ_{3}^{θ} (ε) (φ (ε)) \partial_{α} θ) \partial_{α} (v_{3} - u_{3} (ε)) d x] \\ - ε \int_{Ω} C_{α 3 σ τ} R_{σ τ} (u (ε)) (ε) \partial_{α} (v_{3} - u_{3} (ε)) d x - 2 ε \int_{Ω} C_{α 3 σ 3} R_{σ 3} (ε) (u (ε)) \partial_{α} (v_{3} - u_{3} (ε)) d x \\ - ε \int_{Ω} C_{α 333} R_{33} (ε) (u (ε)) \partial_{α} (v_{3} - u_{3} (ε)) d x - ε \int_{Ω} P_{γ α 3} τ_{γ} (ε) (φ (ε)) \partial_{α} (v_{3} - u_{3} (ε)) d x \\ - ε \int_{Ω} P_{3 α 3} τ_{3} (ε) (φ (ε)) \partial_{α} (v_{3} - u_{3} (ε)) d x + ε^{2} \int_{Ω} f_{3} (v_{3} - u_{3} (ε)) d x \\ + ε^{2} \int_{Γ_{+}} g_{3} (v_{3} - u_{3} (ε)) d a, \\ - ε^{2} B^{♯} (ε, θ, u (ε), R^{θ} (ε), χ^{θ} (ε), v_{3} - u_{3} (ε)) + ε^{4} L^{♯} (ε, θ, v_{3} - u_{3} (ε)), \\ \forall v_{3} \in H^{1} (Ω), v_{3} = 0 on Γ_{0}, v_{3} ⩾ 0 on Γ_{-}, \end{matrix} \end{matrix}$ then $\begin{matrix} 2 C_{33 α 3} R_{α 3}^{θ} (u) + C_{3333} R_{33}^{θ} (u) + P_{333} χ_{3}^{θ} (φ) = - P_{γ 33} τ_{γ}^{θ} (φ) - C_{33 α β} e_{α β}^{θ} (u) . \end{matrix}$ Thus, $R_{i 3}^{θ} (u)$ satisfy the following linear system: $\begin{array}{l} (5.8) & 2 C_{i 3 α 3} R_{α 3}^{θ} (u) + C_{i 333} R_{33}^{θ} (u) + P_{3 i 3} χ_{3}^{θ} (φ) = - P_{γ i 3} τ_{γ}^{θ} (φ) - C_{i 3 α β} e_{α β}^{θ} (u) . \end{array}$ Getting $χ_{3}^{θ} (φ)$ from (5.7) and substituting it in (5.8), the unknowns $R_{i 3}^{θ} (u)$ solve the following linear system of three order: $\begin{array}{l} (5.9) & 2 C_{i 3 α 3}^{'} R_{α 3}^{θ} (u) + C_{i 333}^{'} R_{33}^{θ} (u) = - P_{γ i 3}^{'} τ_{γ}^{θ} (φ) - C_{i 3 α β}^{'} e_{α β}^{θ} (u), \end{array}$ where $C_{i j k l}^{'} = C_{i j k l} + \frac{P_{3 i j} P_{3 k l}}{d_{33}}$ and $P_{k i j}^{'} = P_{k i j} - \frac{P_{3 i j} d_{3 k}}{d_{33}}$ . Following [22], the linear system (5.9) admit a unique solution. So we get $\begin{matrix} (5.10) & \{\begin{matrix} R_{α 3}^{θ} (u) = - \frac{1}{2 △} (△_{γ}^{α} τ_{γ}^{θ} (φ) + △_{ζ η}^{α} e_{ζ η}^{θ} (u)), \\ R_{33}^{θ} (u) = - \frac{1}{△} (△_{γ}^{3} τ_{γ}^{θ} (φ) + △_{ζ η}^{3} e_{ζ η}^{θ} (u)), \\ χ_{3}^{θ} (φ) = - \frac{1}{d_{33}} [(\frac{1}{△} P_{3 k 3} △_{γ ρ}^{k} - P_{3 γ ρ}) e_{γ ρ}^{θ} (u) + (\frac{1}{△} P_{3 k 3} △_{γ}^{k} + d_{3 γ}) τ_{γ}^{θ} (φ)], \end{matrix} \end{matrix}$ where $\begin{matrix} \{\begin{matrix} △ = ϵ_{p q r} C_{13 p 3}^{'} C_{23 q 3}^{'} C_{33 r 3}^{'}, \\ △_{γ}^{1} = ϵ_{p q r} P_{γ p 3}^{'} C_{23 q 3}^{'} C_{33 r 3}^{'}, △_{γ}^{2} = ϵ_{p q r} C_{13 p 3}^{'} P_{γ q 3}^{'} C_{33 r 3}^{'}, △_{γ}^{3} = ϵ_{p q r} C_{13 p 3}^{'} C_{23 q 3}^{'} P_{γ r 3}^{'}, \\ △_{ζ η}^{1} = ϵ_{p q r} C_{3 p ζ η}^{'} C_{23 q 3}^{'} C_{33 r 3}^{'}, △_{ζ η}^{2} = ϵ_{p q r} C_{13 p 3}^{'} C_{3 q ζ η}^{'} C_{33 r 3}^{'}, △_{ζ η}^{3} = ϵ_{p q r} C_{13 p 3}^{'} C_{23 q 3}^{'} C_{3 r ζ η}^{'}, \end{matrix} \end{matrix}$ in which $\begin{matrix} ϵ_{i j k} = \{\begin{matrix} 1 & (i, j, k) is a even permutation of (1, 2, 3), \\ - 1 & (i, j, k) is an odd permutation of (1, 2, 3) . \end{matrix} \end{matrix}$

∙ Step IV. Let $\int_{Ω} C A : A d x$ denote $\int_{Ω} C_{i j k l} A_{k l} A_{i j} d x$ for any given second order symmetric tensor A and let $\int_{Ω} d v : v d x$ denote $\int_{Ω} d_{i j} v_{i} v_{j} d x$ for any given vector $v \in R^{3}$ .

We now show that the convergence is strong for the whole family ${u (ε), φ (ε)}_{ε}$ .

By (2.1) and $\int_{Ω} C A : A d x$ , we have $\begin{array}{rcl} c {| R^{θ} (ε) (u (ε)) - R^{θ} (u) |}_{0, Ω}^{2} & ⩽ & \int_{Ω} C (R^{θ} (ε) (u (ε)) - R^{θ} (u)) : (R^{θ} (ε) (u (ε)) - R^{θ} (u)) d x \\ ⩽ & \int_{Ω} C R^{θ} (u) : (R^{θ} (u) - 2 R^{θ} (ε) (u (ε))) d x \\ + \int_{Ω} C R^{θ} (ε) (u (ε)) : R^{θ} (ε) (u (ε)) d x . \end{array}$ Since $R^{θ} (ε) (u (ε)) ⇀ R^{θ} (u)$ in $L^{2} (Ω)$ as $ε \to 0$ , we have $\begin{array}{l} lim_{ε \to 0} c {| R^{θ} (ε) (u (ε)) - R^{θ} (u) |}_{0, Ω}^{2} \\ ⩽ - \int_{Ω} C R^{θ} (u) : R^{θ} (u) d x + lim_{ε \to 0} \int_{Ω} C R^{θ} (ε) (u (ε)) : R^{θ} (ε) (u (ε)) d x . \end{array}$ Taking $ψ = φ (ε)$ and $v = u$ in (5.6), by the boundness of $χ^{θ} (ε) (φ (ε))$ , as $ε \to 0$ we have $\begin{array}{l} \int_{Ω} C R^{θ} (u) : R^{θ} (u) d x - lim_{ε \to 0} \int_{Ω} C R^{θ} (ε) (u (ε)) : R^{θ} (ε) (u (ε)) d x \\ + ε B_{3}^{♯} (ε, θ, u (ε), R^{θ} (ε), χ^{θ} (ε), u - u (ε)) \\ ⩾ lim_{ε \to 0} \int_{Ω} f \cdot (u - u (ε)) d x + lim_{ε \to 0} \int_{Γ_{+}} g \cdot (u - u (ε)) d a \\ + lim_{ε \to 0} \int_{Ω} P_{k i j} χ_{k}^{θ} (ε) (φ (ε)) R_{i j}^{θ} (ε) (u - u (ε)) d x + ε^{2} L^{♯} (ε, θ, u - u (ε)) \\ ⩾ 0, \end{array}$ here $\begin{array}{rcl} B_{3}^{♯} (ε, θ, u (ε)), R^{θ} (ε), χ^{θ} (ε), v (ε), ψ (ε)) & = & B_{1}^{♯} (ε, θ, u (ε), R^{θ} (ε), χ^{θ} (ε), v (ε), ψ (ε)) \\ + ε B_{2}^{♯} (ε, θ, u (ε), R^{θ} (ε), χ^{θ} (ε), v (ε), ψ (ε)) . \end{array}$ Then $| R^{θ} (ε) (u (ε)) - R^{θ} (u) |_{0, Ω}^{2} ⩽ 0$ as $ε \to 0$ , namely, we get the strong convergence. By the definition of $R^{θ} (ε)$ , we have $\begin{array}{rcl} {| e^{θ} (u (ε)) - e^{θ} (u) |}_{0, Ω}^{2} & ⩽ & \sum_{α β} {| R_{α β}^{θ} (ε) (u (ε)) - R_{α β}^{θ} (u) |}_{0, Ω}^{2} + 2 ε^{2} \sum_{α} {| R_{α 3}^{θ} (ε) (u (ε)) - R_{α 3}^{θ} (u) |}_{0, Ω}^{2} \\ + ε^{4} {| R_{33}^{θ} (ε) (u (ε)) - R_{33}^{θ} (u) |}_{0, Ω}^{2} + ε^{4} {| \partial_{α} θ \partial_{α} u_{3} (ε) |}_{0, Ω}^{2}, \end{array}$ which implies that ${e^{θ} (u (ε))}_{ε}$ converges strongly to $e^{θ} (u)$ in $L^{2} (Ω)$ . Moreover, by Korn’s inequality, the sequence $u (ε)$ converges strongly to u in $H^{1} (Ω)$ .

Similarly, $\begin{array}{rcl} c {| χ^{θ} (ε) (φ (ε)) - χ^{θ} (φ) |}_{0, Ω}^{2} & ⩽ & \int_{Ω} d (χ^{θ} (ε) (φ (ε)) - χ^{θ} (φ)) : (χ^{θ} (ε) (φ (ε)) - χ^{θ} (φ)) d x \\ ⩽ & \int_{Ω} d χ^{θ} (φ) : (χ^{θ} (φ) - 2 χ^{θ} (ε) (φ (ε))) d x \\ + \int_{Ω} d χ^{θ} (ε) (φ (ε)) : χ^{θ} (ε) (φ (ε)) d x . \end{array}$ Since $χ^{θ} (ε) (φ (ε)) ⇀ χ^{θ} (φ)$ in $L^{2} (Ω)$ , as $ε \to 0$ , $\begin{array}{l} lim_{ε \to 0} c {| χ^{θ} (ε) (φ (ε)) - χ^{θ} (φ) |}_{0, Ω}^{2} \\ ⩽ - \int_{Ω} d χ^{θ} (φ) : χ^{θ} (φ) d x + lim_{ε \to 0} \int_{Ω} d χ^{θ} (ε) (φ (ε)) : χ^{θ} (ε) (φ (ε)) d x . \end{array}$ Taking $v = u (ε)$ in (5.6) yields $\begin{matrix} (5.11) & \{\begin{matrix} - \int_{Ω} d_{i j} χ_{j}^{θ} (ε) (φ (ε)) χ_{i}^{θ} (ε) (ψ - φ (ε)) d x + \int_{Ω} P_{k i j} χ_{k}^{θ} (ε) (ψ - φ (ε)) R_{i j}^{θ} (ε) (u (ε)) d x \\ + ε B_{3}^{♯} (ε, θ, u (ε), R^{θ} (ε), χ^{θ} (ε), ψ - φ (ε)) \\ ⩾ - \int_{Ω} ρ_{e} (ψ - φ (ε)) d x + \int_{Γ_{+}} ϑ (ψ - φ (ε)) d a + ε^{2} L^{♯} (ε, θ, ψ - φ (ε)), \\ \forall ψ \in Q (Ω) . \end{matrix} \end{matrix}$ Noting the fact that ψ belongs to a vector space, this inequality (5.11) can be rewritten as $\begin{matrix} \{\begin{matrix} - \int_{Ω} d_{i j} χ_{j}^{θ} (ε) (φ (ε)) χ_{i}^{θ} (ε) (φ (ε)) d x + \int_{Ω} P_{k i j} χ_{k}^{θ} (ε) (ψ - φ (ε)) R_{i j}^{θ} (ε) (u (ε)) d x \\ + ε B_{3}^{♯} (ε, θ, u (ε), R^{θ} (ε), χ^{θ} (ε), ψ - φ (ε)) \\ = - \int_{Ω} ρ_{e} φ (ε) d x + \int_{Γ_{+}} ϑ φ (ε) d a + ε^{2} L^{♯} (ε, θ, ψ - φ (ε)), \end{matrix} \end{matrix}$ then as $ε \to 0$ , we have $\begin{array}{l} - \int_{Ω} d χ^{θ} (φ) : χ^{θ} (φ) d x + lim_{ε \to 0} \int_{Ω} d χ^{θ} (ε) (φ (ε)) : χ^{θ} (ε) (φ (ε)) d x \\ = - \int_{Ω} d χ^{θ} (φ) : χ^{θ} (φ) d x + lim_{ε \to 0} \int_{Ω} ρ_{e} φ (ε) d x - lim_{ε \to 0} \int_{Γ_{+}} ϑ φ (ε) d x \\ + lim_{ε \to 0} \int_{Ω} P_{k i j} χ_{k}^{θ} (ε) (ψ - φ (ε)) R_{i j}^{θ} (ε) (u (ε)) d x \\ + lim_{ε \to 0} ε B_{3}^{♯} (ε, θ, u (ε), R^{θ} (ε), χ^{θ} (ε), ψ - φ (ε)) \\ - lim_{ε \to 0} ε^{2} L^{♯} (ε, θ, ψ - φ (ε)) = 0, \end{array}$ so $| χ^{θ} (ε) (φ (ε)) - χ^{θ} (φ) |_{0, Ω}^{2} ⩽ 0$ as $ε \to 0$ , namely, the sequence ${χ^{θ} (φ (ε))}_{ε}$ converges strongly to $χ^{θ} (φ)$ in $L^{2} (Ω)$ , then the sequence of ${φ (ε)}$ converges strongly to φ in $H^{1} (Ω)$ as $ε \to 0$ .

∙ Step V. Let us come back to variational inequality (5.6). In step I, we get $(u (ε), φ (ε))$ converges strongly towards $(u, φ)$ as $ε \to 0$ , the limit u a is Kirchhoff–Love displacement, φ is independent of $x_{3}$ and by (5.10), we can pass to the limit in (5.6) as $ε \to 0$ , which immediately gets $\begin{matrix} (5.12) & \{\begin{matrix} \int_{Ω} C_{α β σ τ} R_{σ τ}^{θ} (u) e_{α β}^{θ} (v - u) d x + 2 \int_{Ω} C_{α β σ 3} R_{σ 3}^{θ} (u) e_{α β}^{θ} (v - u) d x \\ + \int_{Ω} C_{α β 33} R_{33}^{θ} (u) e_{α β}^{θ} (v - u) d x \\ + \int_{Ω} P_{γ α β} χ_{γ}^{θ} (φ) e_{α β}^{θ} (v - u) d x + \int_{Ω} P_{3 α β} χ_{3}^{θ} (φ) e_{α β}^{θ} (v - u) d x \\ - \int_{Ω} d_{α β} χ_{β}^{θ} (φ) \partial_{α} (ψ - φ) d x - \int_{Ω} d_{α 3} χ_{3}^{θ} (φ) \partial_{α} (ψ - φ) d x \\ + \int_{Ω} P_{γ α β} \partial_{γ} (ψ - φ) R_{α β}^{θ} (u) d x + 2 \int_{Ω} P_{γ α 3} \partial_{γ} (ψ - φ) R_{α 3}^{θ} (u) d x \\ + \int_{Ω} P_{γ 33} \partial_{γ} (ψ - φ) R_{33}^{θ} (u) d x \\ ⩾ \int_{Ω} (f \cdot (v - u) - ρ_{e} (ψ - φ)) d x + \int_{Γ_{+}} (g \cdot (v - u) d a + ϑ (ψ - φ)) d a . \end{matrix} \end{matrix}$

Replacing the components $u_{i}$ and $v_{i}$ of the function $u and v \in K (Ω)$ by $u_{α} = ζ_{α} - x_{3} \partial_{α} ζ_{3}, u_{3} = ζ_{3}$ and $v_{α} = η_{α} - x_{3} \partial_{α} η_{3}, v_{3} = η_{3}$ , with $ζ_{H} = (ζ_{α}), η_{H} = (η_{α}) \in V_{H} (ω)$ and $ζ_{3}, η_{3} \in K_{3} (ω)$ , we obtain $\begin{matrix} e_{α β}^{θ} (u) = e_{α β}^{θ} (ζ_{H}) - x_{3} \partial_{α β} ζ_{3}, e_{α β}^{θ} (v) = e_{α β}^{θ} (η_{H}) - x_{3} \partial_{α β} η_{3}, \end{matrix}$ where $e_{α β}^{θ} (ζ_{H}) = \frac{1}{2} (\partial_{α} ζ_{β} + \partial_{β} ζ_{α}) + \frac{1}{2} (\partial_{α} θ \partial_{β} ζ_{3} + \partial_{β} θ \partial_{α} ζ_{3})$ . Then, following [22] (P299) and using (5.12), we get the desired variational problem. □
5.2. Strong formulation

Let us recall that $ν = (ν_{1}, ν_{2})$ is the unit outward normal vector along $\partial ω$ , and $ι = (- ν_{2}, ν_{1})$ denote one unit tangent vector along $\partial ω$ . From Theorem 5.1, we get the following strong formulations.

Theorem 5.2 (Flexural shell equations).

The limit two-dimensional vertical displacement field solves the Equilibrium equation of flexural shell problem $\begin{matrix} \frac{2}{3} \partial_{α β} {\tilde{σ}}_{α β} (ζ_{3}) - 2 \partial_{β} ({\tilde{n}}_{α β}^{θ} (ζ, φ) \partial_{α} θ) ⩾ p_{3} + \partial_{α} s_{α} in ω \end{matrix}$ with constitutive law $\begin{matrix} {\tilde{σ}}_{α β} (ζ_{3}) = {\tilde{C}}_{α β σ τ} \partial_{σ τ} ζ_{3} in ω, \end{matrix}$ and boundary conditions $\begin{matrix} \{\begin{matrix} \frac{2}{3} \partial_{α} {\tilde{σ}}_{α β} (ζ_{3}) ν_{β} + \frac{2}{3} \partial_{ι} ({\tilde{σ}}_{α β} ν_{α} ι_{β}) - 2 {\tilde{n}}_{α β}^{θ} (ζ, φ) \partial_{α} θ ν_{β} = s_{α} ν_{α} & on γ, \\ {\tilde{σ}}_{α β} (ζ_{3}) ν_{α} ν_{β} = 0 & on γ . \end{matrix} \end{matrix}$ The contact condition is now given in the interior of the domain ω as follows: $\begin{matrix} ζ_{3} ⩾ 0, {\tilde{σ}}_{α β} (ζ_{3}) ζ_{3} = 0, in ω . \end{matrix}$

Proof.
Follow [22], we get the desired results by using integration by parts. □
Theorem 5.3 (Membrane shell equations).

The two-dimensional limit $ζ_{H} = (ζ_{α})$ of the Kirchhoff–Love displacement on the middle surface, and the electric potential φ solve the following piezoelectric problem:

Equilibrium equation $\begin{matrix} (5.13) & \{\begin{matrix} - 2 \partial_{β} {\tilde{n}}_{α β}^{θ} (ζ, φ) = p_{α} & in ω, \\ 2 \partial_{γ} {\tilde{D}}_{γ}^{θ} (ζ, φ) = \tilde{ρ_{e}} - ϑ & in ω, \end{matrix} \end{matrix}$ with boundary conditions $\begin{matrix} (5.14) & \{\begin{matrix} {\tilde{n}}_{α β}^{θ} (ζ, φ) ν_{β} = 0 & on γ, \\ {\tilde{D}}_{γ}^{θ} (ζ, φ) ν_{γ} = 0 & on γ, \end{matrix} \end{matrix}$ where the reduced stress tensor $({\tilde{n}}_{α β}^{θ})$ and the reduced electric displacement vector $({\tilde{D}}_{γ}^{θ})$ are described by the coupled piezoelectric constitutive laws: $\begin{matrix} \{\begin{matrix} {\tilde{n}}_{α β}^{θ} (ζ, φ) = {\tilde{C}}_{α β σ τ} e_{σ τ}^{θ} (ζ) + {\tilde{P}}_{γ α β} \partial_{γ} φ, \\ {\tilde{D}}_{γ}^{θ} (ζ, φ) = {\tilde{P}}_{γ α β} e_{α β}^{θ} (ζ) - {\tilde{d}}_{γ β} \partial_{β} φ . \end{matrix} \end{matrix}$

Proof.
Follow [22], use integration by parts, we get the results. □

6. Some remarks

6.1. Return to the physical domain

Following Ciarlet [4], now we are going back to the physical domain. Let $\begin{matrix} \{\begin{matrix} ζ_{α}^{ε} = ε^{2} ζ_{α} & in \bar{ω}, \\ ζ_{3}^{ε} = ε ζ_{3} & in \bar{ω}, \\ φ^{ε} = ε^{2} φ & in \bar{ω}, \\ θ^{ε} = ε θ & in \bar{ω}, \end{matrix} \end{matrix}$ in which $(ζ^{ε}, φ^{ε})$ satisfies the following variational problem: $\begin{matrix} \{\begin{matrix} \frac{2}{3} \int_{ω} {\tilde{σ}}_{α β}^{ε} (ζ_{3}^{ε}) \partial_{α β} (η_{3} - ζ_{3}^{ε}) d ω + 2 \int_{ω} {\tilde{n}}_{α β}^{ε θ} (ζ^{ε}, φ^{ε}) \partial_{α} θ^{ε} \partial_{β} (η_{3} - ζ_{3}^{ε}) d ω \\ ⩾ \int_{ω} p_{3}^{ε} (η_{3} - ζ_{3}^{ε}) d ω - \int_{ω} s_{α}^{ε} \partial_{α} (η_{3} - ζ_{3}^{ε}) d ω \\ \forall η_{3} \in K_{3} (ω), \\ 2 \int_{ω} {\tilde{n}}_{α β}^{ε θ} (ζ^{ε}, φ^{ε}) e_{α β} (η_{H}) d ω + 2 \int_{ω} {\tilde{D}}_{γ}^{ε θ} (ζ^{ε}, φ^{ε}) \partial_{γ} ψ d ω \\ = \int_{ω} p_{H}^{ε} \cdot η_{H} d ω - \int_{ω} ({\tilde{ρ_{e}}}^{ε} - ϑ^{ε}) ψ d ω \\ \forall (η_{H}, ψ) \in V_{H} (ω) \times Q (ω), \end{matrix} \end{matrix}$ where $\begin{matrix} \{\begin{matrix} {\tilde{σ}}_{α β}^{ε} (ζ_{3}^{ε}) = ε^{3} {\tilde{C}}_{α β σ τ} \partial_{σ τ} ζ_{3}^{ε}, \\ {\tilde{n}}_{α β}^{ε θ} (ζ^{ε}, φ^{ε}) = ε {\tilde{n}}_{α β}^{θ} (ζ^{ε}, φ^{ε}), \\ {\tilde{D}}_{γ}^{ε θ} (ζ^{ε}, φ^{ε}) = ε {\tilde{D}}_{γ}^{θ} (ζ^{ε}, φ^{ε}), \\ e_{α β}^{ε θ} (ζ^{ε}) = \frac{1}{2} (\partial_{α} ζ_{β}^{ε} + \partial_{β} ζ_{α}^{ε}) + \frac{1}{2} (\partial_{α} θ^{ε} \partial_{β} ζ_{3}^{ε} + \partial_{β} θ^{ε} \partial_{α} ζ_{3}^{ε}), \end{matrix} \end{matrix}$ and the spaces $K_{3} (ω) V_{H} (ω)$ and $Q (ω)$ are defined as in Theorem 5.1.

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

∙ (Flexural shell equations) $\begin{matrix} \frac{2}{3} \partial_{α β} {\tilde{σ}}_{α β}^{ε} (ζ_{3}^{ε}) - 2 \partial_{β} ({\tilde{n}}_{α β}^{ε θ} (ζ^{ε}, φ^{ε}) \partial_{α} θ^{ε}) ⩾ p_{3}^{ε} + \partial_{α} s_{α}^{ε} in ω, \end{matrix}$ and boundary conditions $\begin{matrix} \{\begin{matrix} \frac{2}{3} \partial_{α} {\tilde{σ}}_{α β}^{ε} (ζ_{3}^{ε}) ν_{β} + \frac{2}{3} \partial_{ι} ({\tilde{σ}}_{α β}^{ε} ν_{α} ι_{β}) - 2 {\tilde{n}}_{α β}^{ε θ} (ζ^{ε}, φ^{ε}) \partial_{α} θ^{ε} ν_{β} = s_{α}^{ε} ν_{α} & on γ, \\ {\tilde{σ}}_{α β}^{ε} (ζ_{3}^{ε}) ν_{α} ν_{β} = 0 & on γ, \end{matrix} \end{matrix}$ with the constitutive law $\begin{matrix} {\tilde{σ}}_{α β}^{ε} (ζ_{3}^{ε}) = ε^{3} {\tilde{C}}_{α β σ τ} \partial_{σ τ} ζ_{3}^{ε} \end{matrix}$ and 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 shell equations) $\begin{matrix} \{\begin{matrix} - 2 \partial_{β} {\tilde{n}}_{α β}^{ε θ} (ζ^{ε}, φ^{ε}) = p_{α}^{ε} & in ω, \\ 2 \partial_{γ} {\tilde{D}}_{γ}^{ε θ} (ζ^{ε}, φ^{ε}) = {\tilde{ρ_{e}}}^{ε} - ϑ^{ε} & in ω, \end{matrix} \end{matrix}$ and boundary conditions $\begin{matrix} \{\begin{matrix} {\tilde{n}}_{α β}^{ε θ} (ζ^{ε}, φ^{ε}) ν_{β} = 0 & on γ, \\ {\tilde{D}}_{γ}^{ε θ} (ζ^{ε}, φ^{ε}) ν_{γ} = 0 & on γ, \end{matrix} \end{matrix}$ with $\begin{matrix} \{\begin{matrix} {\tilde{n}}_{α β}^{ε θ} ({\tilde{ζ}}^{ε}, φ^{ε}) = ε {\tilde{C}}_{α β σ τ} e_{σ τ}^{θ} (ζ^{ε}) + ε {\tilde{P}}_{γ α β} \partial_{γ} φ^{ε}, \\ {\tilde{D}}_{γ}^{ε θ} (ζ^{ε}, φ^{ε}) = ε {\tilde{P}}_{γ α β} e_{α β}^{θ} (ζ^{ε}) - ε {\tilde{d}}_{γ β} \partial_{β} φ^{ε} . \end{matrix} \end{matrix}$ In the previous formulations, $\begin{matrix} p_{i}^{ε} : = \int_{- ε}^{ε} f_{i}^{ε} d x_{3}^{ε} + g_{i}^{ε}, s_{α}^{ε} : = \int_{- ε}^{ε} x_{3}^{ε} f_{α}^{ε} d x_{3}^{ε} + g_{α}^{ε}, {\tilde{ρ_{e}}}^{ε} : = \int_{- ε}^{ε} ρ_{e}^{ε} d x_{3}^{ε} . \end{matrix}$ The descaled functions $u_{i}^{ε} (0)$ are then given by $\begin{array}{l} u_{α}^{ε} (0) = ζ_{α}^{ε} - x_{3}^{ε} \partial_{α} ζ_{3}^{ε}, u_{3}^{ε} (0) = ζ_{3}^{ε} \forall x^{ε} \in {\bar{Ω}}^{ε} . \end{array}$

6.2. Partial elastic clamping on the lateral surface

In this subsection we consider the following boundary conditions: $\begin{matrix} \{\begin{matrix} {\hat{u}}^{ε} = 0 & on {\hat{Γ}}_{0}^{ε}, \\ {\hat{φ}}^{ε} = 0 & on {\hat{Γ}}_{0}^{ε}, \\ {\hat{σ}}^{ε} ({\hat{u}}^{ε}, {\hat{φ}}^{ε}) {\hat{d}}^{ε} = {\hat{g}}^{ε} & on {\hat{Γ}}_{+}^{ε} \cup \hat{γ} \times [- ε, ε], \\ {\hat{D}}^{ε} ({\hat{u}}^{ε}, {\hat{φ}}^{ε}) \cdot {\hat{d}}^{ε} = {\hat{ϑ}}^{ε} & on {\hat{Γ}}_{+}^{ε} \cup \hat{γ} \times [- ε, ε], \\ {\hat{D}}^{ε} \cdot {\hat{d}}^{ε} = 0 & on {\hat{Γ}}_{-}^{ε}, \\ {\hat{θ}}^{ε} - \frac{ε}{\sqrt{{\hat{a}}^{ε}}} + {\hat{u}}_{3}^{ε} + ε ⩾ 0, {\hat{σ}}_{3}^{ε} ⩾ 0, {\hat{σ}}_{3}^{ε} ({\hat{θ}}^{ε} - \frac{ε}{\sqrt{{\hat{a}}^{ε}}} + {\hat{u}}_{3}^{ε} + ε) = 0 & on {\hat{Γ}}_{-}^{ε}, \end{matrix} \end{matrix}$ in which $γ_{0} \subset γ = \partial ω, \hat{γ} = γ - γ_{0}$ .

The limit solution $(ζ, φ) = (ζ_{H}, ζ_{3}, φ)$ solves the following coupled problem:

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

Remark 6.1.
Comparing with Theorem 5.1, the two dimensional problem (6.1) for partial elastic clamping on the lateral surface 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 $(C, d, P)$ is represented by a symmetric matrix of the following form (see [10]): $\begin{matrix} (\begin{matrix} C_{11} & C_{12} & C_{12} & 0 & 0 & 0 & 0 & 0 & 0 \\ C_{12} & C_{11} & C_{12} & 0 & 0 & 0 & 0 & 0 & 0 \\ C_{12} & C_{12} & C_{11} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 & P_{123} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{44} & 0 & 0 & P_{123} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{44} & 0 & 0 & P_{123} \\ 0 & 0 & 0 & P_{123} & 0 & 0 & d_{11} & 0 & 0 \\ 0 & 0 & 0 & 0 & P_{123} & 0 & 0 & d_{11} & 0 \\ 0 & 0 & 0 & 0 & 0 & P_{123} & 0 & 0 & d_{11} \end{matrix}), \end{matrix}$ i.e., $\begin{matrix} \{\begin{matrix} C_{1111} = C_{2222} = C_{3333} = C_{11}, C_{1212} = C_{2323} = C_{1313} = C_{44}, \\ C_{1122} = C_{1133} = C_{2233} = C_{12}, \\ P_{123} = P_{213} = P_{312} = P_{123}, \\ d_{11} = d_{22} = d_{33} = d_{11}, \\ all the other terms vanish. \end{matrix} \end{matrix}$ By (5.3) and (5.2), a direct computation gives that the disappearance of the coupling $\begin{matrix} \{\begin{matrix} △ = C_{11} C_{44}^{2}, \\ △_{1}^{1} = △_{2}^{2} = 0, △_{2}^{1} = △_{1}^{2} = C_{11} C_{44} P_{123}, △_{α}^{3} = 0, \\ △_{ζ η}^{1} △_{ζ η}^{2} = 0, △_{11}^{3} = △_{22}^{3} = C_{12} C_{44}^{2}, △_{ζ η}^{3} = 0 as ζ \neq η, \end{matrix} \end{matrix}$ and $\begin{matrix} \{\begin{matrix} C_{1212}^{'} = C_{1212} + \frac{P_{123}^{2}}{d_{11}}, C_{i j k l}^{'} = C_{i j k l} as i j k l \neq 1212, 2121, 1221, 2112, \\ P_{k i j}^{'} = 0, d_{i 3}^{'} = d_{3 i}^{'} = 0 . \end{matrix} \end{matrix}$ Then $(ζ, φ) = (ζ_{H}, ζ_{3}, φ)$ solves the following problem:

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

Remark 6.2.

Therefor for the cubic piezoelectric material, we find an interesting thing that, three dimensional Signorini problem of piezoelectric shallow shells converges to a two dimensional obstacle problem of shallow shell as the thickness of the piezoelectric shallow shell goes to zero, while in the two dimensional case, the piezoelectric terms disappear, we get three piezoelastic equations: flexural shell equation, membrane shell 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 second author wants to be grateful to Professor Li Tatsien and Professor Han Maoan for their kind support. The paper is supported by the Innovation Program of Shanghai Municipal Education Commission (11YZ80) and the program of Shanghai Normal University (SK201301).

The quantities $B^{♯}$ and $L^{♯}$ which appear in (5.6) correspond to the remainders resulting from the Taylor expansion of $\nabla^{ε} Θ^{ε}$ with respect to successive powers of ε. Noting $δ^{ε} = \nabla^{ε} Θ^{ε}$ and $b_{i j}^{ε} = {(\nabla^{ε} Θ^{ε})}_{i j}^{- 1}$ , we introduce the notations $δ^{♯}$ and $b_{33}^{♯}$ such that: $\begin{matrix} \{\begin{matrix} δ^{ε} (x^{ε}) = δ (ε) (x) = 1 + ε^{2} δ^{♯} (ε, θ), \\ b_{33}^{ε} (x^{ε}) = b_{33} (ε) (x) = 1 + ε^{2} b_{33}^{♯} (ε, θ) \end{matrix} \end{matrix}$ and there exist $ε_{0} > 0$ and a positive constant c independent of ε, such that: $\begin{matrix} sup_{0 ⩽ ε ⩽ ε_{0}} max_{x \in \bar{Ω}} (max_{i j} | b_{i j}^{♯} (ε, θ) (x) | + | δ^{♯} (ε, θ) (x) |) ⩽ c . \end{matrix}$ The functions indexed by ♯ depend on $(ε, θ)$ ; from now on, we will skip this dependence for the sake of simplification. Using the same kind of expansions, we introduce the notations $e_{α β}^{♯}$ , $e_{α 3}^{♯}$ , $e_{33}^{♯}$ , $e_{33}^{♭}$ , $τ_{α}^{♯}$ , and $τ_{3}^{♯}$ such that $\begin{matrix} \{\begin{matrix} e_{α β}^{ε} ({\hat{v}}^{ε}) ({\hat{x}}^{ε}) = ε^{2} e_{α β}^{θ} (v) (x) + ε^{4} e_{α β}^{♯} (v) (x), \\ e_{α 3}^{ε} ({\hat{v}}^{ε}) ({\hat{x}}^{ε}) = ε e_{α 3}^{θ} (v) (x) + ε^{3} e_{α 3}^{♯} (v) (x), \\ e_{33}^{ε} ({\hat{v}}^{ε}) ({\hat{x}}^{ε}) = e_{33}^{θ} (v) (x) + ε^{2} (\partial_{α} θ \partial_{α} v_{3} + b_{33}^{♯} (ε, θ) \partial_{3} v_{3}) (x) + ε^{4} e_{33}^{♭} (v) (x), \\ τ_{α}^{ε} (\hat{ψ^{ε}}) ({\hat{x}}^{ε}) = ε^{2} τ_{α}^{θ} (ψ) (x) + ε^{4} τ_{α}^{♯} (ψ) (x), \\ τ_{3}^{ε} (\hat{ψ^{ε}}) ({\hat{x}}^{ε}) = ε τ_{3}^{θ} (ψ) (x) + ε^{3} τ_{3}^{♯} (ψ) (x) . \end{matrix} \end{matrix}$ We have $\begin{array}{l} sup_{0 ⩽ ε ⩽ ε_{0}} max_{i j} {| e_{i j}^{♯} (v) (x) |}_{0, Ω} ⩽ c (θ) ‖ v ‖_{1, Ω}, \\ sup_{0 ⩽ ε ⩽ ε_{0}} {| e_{33}^{♭} (v) (x) |}_{0, Ω} ⩽ c (θ) ‖ v ‖_{1, Ω} \end{array}$ and $\begin{matrix} sup_{0 ⩽ ε ⩽ ε_{0}} max_{j} {| τ_{j}^{♯} (ψ) (x) |}_{0, Ω} ⩽ c (θ) ‖ ψ ‖_{1, Ω} . \end{matrix}$ We are now in a position to examine the quantity $B^{♯}$ . This quantity consists of three terms in a decomposition of the form $\begin{matrix} B^{♯} = B_{0}^{♯} + ε B_{1}^{♯} + ε^{2} B_{2}^{♯}, \end{matrix}$ where the complete expression of $B_{0}^{♯}$ reads $\begin{matrix} \{\begin{matrix} B_{0}^{♯} (ε, θ, u (ε), R^{θ} (ε) (u (ε)), χ^{θ} (φ (ε)), v, ψ) \\ = \int_{Ω} C_{α β 33} b_{33}^{♯} (ε, θ) \partial_{3} u_{3} (ε) e_{α β}^{θ} (v - u (ε)) d x + 4 \int_{Ω} C_{α 3 σ 3} e_{σ 3}^{♯} (u (ε)) e_{α 3}^{θ} (v - u (ε)) d x \\ + \int_{Ω} C_{33 σ τ} R_{σ τ}^{θ} (ε) (u (ε)) \partial_{3} (v_{3} - u_{3} (ε)) δ^{♯} d x \\ + \int_{Ω} C_{33 σ τ} R_{σ τ}^{θ} (ε) (u (ε)) b_{33}^{♯} (ε, θ) \partial_{3} (v_{3} - u_{3} (ε)) d x \\ + \int_{Ω} C_{33 σ τ} R_{σ τ}^{θ} (ε) (u (ε)) \partial_{α} θ \partial_{α} (v_{3} - u_{3} (ε)) d x + \int_{Ω} C_{33 σ τ} e_{σ τ}^{♯} (u (ε)) \partial_{3} (v_{3} - u_{3} (ε)) d x \\ + 2 \int_{Ω} C_{33 σ 3} R_{σ 3}^{θ} (ε) (u (ε) \partial_{3} (v_{3} - u_{3} (ε)) δ^{♯} d x \\ + 2 \int_{Ω} C_{33 σ 3} R_{σ 3}^{θ} (ε) (u (ε)) b_{33}^{♯} (ε, θ) \partial_{3} (v_{3} - u_{3} (ε)) δ^{♯} d x \\ + 2 \int_{Ω} C_{33 σ 3} R_{σ 3}^{θ} (ε) (u (ε)) \partial_{α} θ \partial_{α} (v_{3} - u_{3} (ε)) d x \\ + \int_{Ω} C_{3333} R_{33}^{θ} (ε) (u (ε)) b_{33}^{♯} (ε, θ) \partial_{3} (v_{3} - u_{3} (ε)) d x \\ + \int_{Ω} C_{3333} R_{33}^{θ} (ε) (u (ε)) \partial_{α} θ \partial_{α} (v_{3} - u_{3} (ε)) d x \\ + \int_{Ω} C_{3333} R_{33}^{θ} (ε) (u (ε)) (ε, θ) \partial_{3} (v_{3} - u_{3} (ε)) δ^{♯} d x \\ + \int_{Ω} C_{3333} b_{33}^{♯} (ε, θ) \partial_{3} u_{3} (ε) \partial_{3} (v_{3} - u_{3} (ε)) δ^{♯} d x \\ + \int_{Ω} C_{3333} b_{33}^{♯} (ε, θ) \partial_{3} u_{3} (ε) b_{33}^{♯} (ε, θ) (ε, θ) \partial_{3} (v_{3} - u_{3} (ε)) d x \\ + \int_{Ω} C_{3333} b_{33}^{♯} (ε, θ) \partial_{3} u_{3} (ε) \partial_{α} θ \partial_{α} (v_{3} - u_{3} (ε)) d x - \int_{Ω} d_{33} τ_{3}^{♯} (φ (ε)) \partial_{3} (ψ - φ (ε)) d x \\ + 2 \int_{Ω} P_{3 α 3} τ_{3}^{♯} (φ (ε)) e_{α 3}^{θ} (v - u (ε)) d x + \int_{Ω} P_{333} τ_{3}^{♯} (φ (ε) \partial_{3} (v_{3} - u_{3} (ε)) d x \\ + \int_{Ω} P_{γ 33} χ_{γ}^{θ} (ε) (φ (ε)) \partial_{3} (v_{3} - u_{3} (ε)) δ^{♯} d x \\ + \int_{Ω} P_{γ 33} χ_{γ}^{θ} (ε) (φ (ε)) b_{33}^{♯} (ε, θ) \partial_{3} (v_{3} - u_{3} (ε)) d x \\ + \int_{Ω} P_{γ 33} τ_{γ}^{♯} (φ (ε)) \partial_{α} θ \partial_{α} (v_{3} - u_{3} (ε)) d x + \int_{Ω} P_{γ 33} τ_{γ}^{♯} (φ (ε)) \partial_{3} (v_{3} - u_{3} (ε)) d x \\ + \int_{Ω} P_{333} (χ_{3}^{θ} (ε) (φ (ε)) \partial_{3} (v_{3} - u_{3} (ε)) δ^{♯} d x \\ + \int_{Ω} P_{333} (χ_{3}^{θ} (ε) (φ (ε)) b_{33}^{♯} (ε, θ) \partial_{3} (v_{3} - u_{3} (ε)) d x \\ + \int_{Ω} P_{333} (χ_{3}^{θ} (ε) (φ (ε)) \partial_{α} θ \partial_{α} (v_{3} - u_{3} (ε)) d x + 2 \int_{Ω} P_{3 α 3} \partial_{3} (ψ - φ (ε)) e_{α 3}^{♯} (u (ε))) d x . \end{matrix} \end{matrix}$ For any given $v \in K (ε) (Ω)$ and $ψ \in Q (Ω)$ , $R^{θ} (ε) (v), χ^{θ} (ε) (ψ) \in L^{2} (Ω)$ , we have $\begin{matrix} sup_{0 ⩽ ε ⩽ ε_{0}} {| B_{0}^{♯} (ε, θ, R^{θ} (ε) (v), χ^{θ} (ψ), v, ψ) |}_{0, Ω} ⩽ c ({| R^{θ} (ε) (v) |}_{0, Ω} + {| χ^{θ} (ε) (ψ) |}_{0, Ω}) (‖ v ‖_{1, Ω} + ‖ ψ ‖_{1, Ω}) \end{matrix}$ and similar result can be obtained for $B_{1}^{♯}$ and $B_{2}^{♯}$ , so that $\begin{array}{l} sup_{0 ⩽ ε ⩽ ε_{0}} {| B_{i}^{♯} (ε, θ, R^{θ} (ε) (v), χ^{θ} (ψ), v, ψ) |}_{0, Ω} \\ ⩽ c ({| R^{θ} (ε) (v) |}_{0, Ω} + {| χ^{θ} (ε) (ψ) |}_{0, Ω}) (‖ v ‖_{1, Ω} + ‖ ψ ‖_{1, Ω}), 0 ⩽ i ⩽ 2 . \end{array}$ We can also get $\begin{matrix} sup_{0 ⩽ ε ⩽ ε_{0}} {| L^{♯} (ε, θ, v, ψ) |}_{0, Ω} ⩽ c (‖ v ‖_{1, Ω} + ‖ ψ ‖_{1, Ω}) . \end{matrix}$

References

W.G.

Cady Piezoelectricity, An Introduction to the Theory and Applications of Electromechanical Phenomena in Crystals, McGraw-Hill, New York, 1946.

P.G.

Ciarlet, Mathematical Elasticity, Vol II, Theory of Plates, North-Holland, Amsterdam, 1997.

P.G.

Ciarlet, An introduction to differential geometry with applications to elasticity, J. Elast. 78–79 (2005), 1–215. doi:10.1007/s10659-005-4738-8.

P.G.

Ciarlet and

Destuynder, A justification of the two dimensional plate model, J. Mécanique 18 (1979), 315–344.

P.G.

Ciarlet and

Miara, Justification of the two-dimensional equations of a linearly elastic shallow shell, Comm. Pure Appl. Math. XLV (1992), 327–360. doi:10.1002/cpa.3160450305.

P.G.

Ciarlet and

Miara, Justification of the two-dimensional equations of a linearly elastic shallow shell, Comm. Pure Appl. Math. XLV (1992), 327–360. doi:10.1002/cpa.3160450305.

Dieulesaint, Ondes Élastiques dans les Solides, Masson, Paris, 1974.

Duvaut and

J.-L.

Lions, Les Inéquations en Mécanique et en Physique, Dunod, 1972.

Fichera, Problemi elastostatici con vincoli unilaterali: Il problema di Signorini con ambigue condizioni al contorno, Mem. Accad. Naz. Lincei Ser. VIII(7) (1964), 91–140.

10.

Fuxue and

Likun, Modern Piezoelectricity, Science Prese, 2001 (in Chinese).

11.

Ikeda, Fundamentals of Piezoelectricity, Oxford University Press, 1990.

12.

Kinderlehrer and

Stampacchia, An Introduction to Variational Inequalities and Their Applications, Society for Industrial and Applied Mathematics, 1987.

13.

Léger and

Miara, Mathematical justification of the obstacle problem in the case of a shallow shell, J. Elast. 90 (2008), 241–257. doi:10.1007/s10659-007-9141-1.

14.

Léger and

Miara, The obstacle problem for shallow shells: A curvilinear approach, Intl. J. of Numerical Analysis and Modeling, B 2 (2011), 1–26.

15.

J.L.

Lions, Quelques Méthodes de Résolution des Problêmes aux Limites Non Linéaires, Dunod–Gauthier-Villars, Paris, 1969.

16.

Maceri and

Bisegna, The unilateral frictionless contact of a piezoelectric body with a rigid support, Math. Comput. Modelling 28(4–8) (1998), 19–28. doi:10.1016/S0895-7177(98)00105-8.

17.

Miara and

J.S.

Suarez, Asymptotic pyroelectricity and pyroelasticity in thermopiezoelectric plates, Asymptot. Anal. 81 (2013), 211–250.

18.

J.C.

Paumier, Modélisation asymptotique d’un problème de plaque mince en contact unilatéral avec frottement contre un obstacle rigide, Prépublication L.M.C, 2002, http://www-lmc.imag.fr/~paumier/signoplaque.ps.

19.

Rahmoune, Plaques intelligentes piezoelectriques, modelisation et application au contrôle santé, Ph.D. Thesis, University of Paris VI, 1998.

20.

Sabu, Asymptotic analysis of piezoelectric shells with variable thickness, Asymptot. Anal. 54 (2007), 181–196.

21.

Sene, Modelling of piezoelectric static thin plates, Asymptot. Anal. 25 (2001), 1–20.

22.

Yan and

Miara, Mathematical jusitification of the obstacle problem in the case of piezoelectric plate, Asymptot. Anal. 96 (2016), 283–308. doi:10.3233/ASY-151339.

23.

Yan and

Miara, Mathmatical Jusitification of the Obstacle Problem for piezoelectric shallow, shell in curvilinear coordinate, manuscript.

Mathematical justification of the obstacle problem for a piezoelectric shallow shell

Abstract

Keywords

1. Introduction

2. Piezoelectricity in a three dimensional body

2.1. Constitutive equations of piezoelectricity

2.2. Equilibrium equations of piezoelectricity

2.3. Boundary conditions

3. Shallow shell

3.1. Three dimensional problem

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

3.3. The variational inequality in Ω ˆ ε

4. Scaling and equilibrium in the fixed domain Ω

4.1. The problem in Ω ε

4.2. Scaling

4.3. The contact condition in the fixed domain Ω = ω × ( − 1 , 1 )

Theorem 4.1. For any fixed ε > 0 , problem ( 4.12 ) has a unique weak solution. Proof. The proof is a classical result for variational inequality [8,12]. □ 5. Convergence

Theorem 5.2 (Flexural shell equations).

Proof. Follow [22], we get the desired results by using integration by parts. □ Theorem 5.3 (Membrane shell equations).

Proof. Follow [22], use integration by parts, we get the results. □ 6. Some remarks

6.1. Return to the physical domain

6.2. Partial elastic clamping on the lateral surface

Footnotes

Acknowledgements

References

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

4.1. The problem in $Ω^{ε}$

4.3. The contact condition in the fixed domain $Ω = ω \times (- 1, 1)$

Theorem 4.1.
For any fixed $ε > 0$ , problem ( 4.12 ) has a unique weak solution.
Proof.
The proof is a classical result for variational inequality [8,12]. □

5. Convergence

Proof.
Follow [22], we get the desired results by using integration by parts. □
Theorem 5.3 (Membrane shell equations).

Proof.
Follow [22], use integration by parts, we get the results. □

6. Some remarks