A Variational Approach for an Electro-Elastic Unilateral Contact Problem∗

We consider a mechanical model which describes the frictionless unilateral contact between an electro-elastic body and a rigid electrically non-conductive foundation. For this model, a mixed variational formulation is provided. Then, using elements of the saddle point theory and a fixed point technique, an abstract result is proved. Based on this result, the existence of a unique weak solution of the mechanical problem is established.


Introduction
In this paper we study a frictionless unilateral contact problem involving the piezoelectric effect. The piezoelectricity can be described as follows: when mechanical pressure is applied to a certain classes of crystalline materials (e.g ceramics BaT iO 3 , BiF eO 3 ), the crystalline structure produces a voltage proportional to the pressure. Conversely, when an electric field is applied, the structure changes his shape producing dimensional modifications in the material. Actually, there is a big interest into the study of piezoelectric materials, this type of materials being used in radioelectronics, electroacoustics and measuring equipments. In the same time, due to the fact that the parts of the equipments are in contact, the interest for the contact problems is increasing. The literature concerning this topic is very rich, see for example [10,11,14,15,16,22,23,24] for modelling in piezoelectricity and [5,6,13,17,20,25] for the modelling and the analysis of the contact processes. Some theoretical results for contact models taking into account the interaction between the electric and the mechanic fields have been obtained in [7,18,19,21].

A. Matei
In this paper we use a mixed variational formulation of the piezoelectric contact problem, which consists one of the traits of the novelty of the present paper. For details concerning the mathematical tools we refer to [1,2,3,4,6,12]. The main motivation of this approach is that in the last years this type of formulation in contact mechanics is preferred from numerical point of view, see for example [7,8,9]. The present paper follows [7] where a frictional bilateral contact problem for electro-elastic materials was treated.
The rest of this paper is organized as follows. In Section 2 we provide some notation and preliminaries. In Section 3 we describe the physical setting, formulate the mathematical problem and state the main result, Theorem 1. In Section 4 we provide an abstract result, Theorem 2, then we prove Theorem 1. The last section contains some conclusions and comments.

Notation and Preliminaries
Let us denote by S 3 the space of second order symmetric tensors on R 3 . Every element in R 3 or S 3 will be typeset in boldface and by " · " and | · | we denote the inner product and the Euclidean norm on R 3 and S 3 , respectively. Thus, Here and below, the indices i and j run between 1 and 3 and the summation convention over repeated indices is adopted.
Let Ω ⊂ R 3 be a bounded domain. We introduce the following functional spaces on Ω, Here and below the index that follows a comma indicates a partial derivative with respect to the corresponding component of the independent variable. The spaces H, H, H 1 and H 1 are real Hilbert spaces endowed with the inner products, The associated norms on the spaces H, H, H 1 and H 1 are denoted by · H , · H , · H1 and · H1 , respectively.
Let us assume that the boundary of Ω, denoted by Γ is Lipschitz continuous. We denote by n the unit outward normal vector on the boundary, defined a.e. In order to simplify the writing, everywhere below, for every field (scalar, vectorial or tensorial) we will use the same notation in order to indicate his Sobolev trace on Γ.
For a vectorial field v, we denote by v n and v τ the normal and the tangential components on the boundary, defined as follows: For a regular (say C 1 ) stress field σ, the application of its trace on the boundary to n is the Cauchy stress vector σn. Furthermore, we define the normal and tangential components of the Cauchy vector on the boundary by the formulas σ n = (σn) · n, σ τ = σn − σ n n and we note that the following identity takes place Finally, we recall the useful Green's formula For a proof of the formula (2.2) and for more details related to this section, we send the reader to [5].

The Model and the Statement of the Main Result
We consider an elasto-piezoelectric body that occupies the bounded domain Ω ⊂ R 3 , in contact with a rigid electrically non-conductive foundation. We assume that the boundary Γ is partitioned into three disjoint measurable parts Γ 1 , Γ 2 and Γ 3 , such that meas(Γ 1 ) > 0 and Γ 3 is a compact subset of Γ \Γ 1 . Let us denote by n 3 the restriction of n to Γ 3 . The body Ω is clamped on Γ 1 , body forces of density f 0 act on Ω and surface traction of density f 2 act on Γ 2 . Moreover, we assume that Γ 3 is the potential contact zone and we denote by g : Γ 3 → R the gap function. By gap in a given point of Γ 3 we understand the distance between the deformable body and the foundation measured along of the outward normal n. Let us consider a second partition of the boundary Γ in two disjoint measurable parts Γ a and Γ b such that meas(Γ a ) > 0 and On Γ a the electrical potential vanishes and on Γ b we assume electric charges of density q 2 . Since the foundation is electrically non-conductive, and assuming that the gap zone is also electrically non-conductive, q 2 must vanish on Γ 3 . By q 0 we will denote the density of the free electric charges on Ω.
We denote by u = (u i ) the displacement field, by σ = (σ ij ) the stress tensor, by ϕ the electric potential field and by D = (D i ) the electrical field. We start the modelling writing the universal equilibrium equations

A. Matei
In order to describe the behavior of the materials, we use the constitutive law is the piezoelectric tensor and β is the permitivity tensor. We use here E ⊤ to denote the transpose of the tensor E, and we notice that Such kind of electromechanic relations can be found in the literature, see, [24].
To complete the model, we prescribe the mechanical and the electrical boundary conditions. According to the physical setting we write To model the contact process, we use the Signorini condition with non-zero gap. In addition, we assume that the contact is frictionless. Consequently, we can express mathematically the frictionless contact condition as follows Knowing the displacement field u and the electric field ϕ we can compute the stress tensor σ and the electric displacement D using (3.3) and (3.4), respectively. Therefore, the displacement field u and the electric field ϕ are called the main unknowns.
To resume, we consider the following problem.

A. Matei
Since σ τ = 0 on Γ 3 , taking into account (2.1) we can write Let us consider a :Ṽ ×Ṽ → R the bilinear form, for allũ = (u, ϕ),ṽ = (v, ψ) ∈Ṽ . Moreover, using the Riesz representation theorem, we definef ∈Ṽ such that for allṽ = (v, ψ) ∈Ṽ , Consequently, using the Green formula (2.2), we deduce that In order to provide a mixed weak formulation, we define a dual Lagrange Using (3.19), keeping in mind that the Sobolev trace operator is linear and continuous and taking into account (3.14), we deduce that there exists In addition, using the properties of the Sobolev trace operator, it can be shown that there exists α > 0 such that Furthermore, we introduce a set as follows, We note that λ ∈ Λ and, using the assumption (3.13), we deduce that g ext n 3 ∈ V, gn 3 being the trace of g ext n 3 on Γ 3 . Taking into account the definition of λ, (3.18), the definition of b(·, ·), (3.19), and the definition of Λ, (3.22), we get: Thus, denoting byg ext := (g ext n 3 , 0 W ) ∈Ṽ we can write (3.23) Keeping in mind (3.17), (3.23) we obtain the following weak formulation of Problem 1.

Problem 2.
Findũ ∈Ṽ and λ ∈ Λ, such that The main result of this paper is the following.
The proof of this theorem will be presented in the next section; it follows by using an abstract result, Theorem 2.

An Abstract Result and Proof of Theorem 1
Let (X, (·, ·) X , · X ) and (Y, (·, ·) Y , · Y ) be two Hilbert spaces and let us consider two bilinear forms as follows: a(·, ·) : X × X → IR is non-symmetric and (a) there exists M a > 0 such that and b(·, ·) : X × Y → IR, for which (c) there exists M b > 0 such that

3)
A. Matei Let Λ ⊂ Y be a closed, convex set that contains 0 Y . We consider now the following problem: Problem 3. For given f, g ∈ X, find u ∈ X and λ ∈ Λ such that We emphasize that the bilinear form a(·, ·) is non-symmetric. Consequently, Problem 3 is not a saddle point problem. Moreover, we are interested here in the case g = 0 X . An analysis of the particular case g = 0 X can be found in [7].
The following result holds.
Theorem 2. Let f, g ∈ X and assume that (4.1)-(4.4) hold. Then, there exists a unique solution of Problem 3, (u, λ) ∈ X × Λ. Moreover, if (u 1 , λ 1 ) and (u 2 , λ 2 ) are two solutions of Problem 3, corresponding to the data functions f 1 , g 1 ∈ X and f 2 , g 2 ∈ X, then there exists Proof. Let a 0 (·, ·) and c(·, ·) be the symmetric and the antisymmetric part of a(·, ·), respectively, For a given r ∈ [0, 1], we introduce the following bilinear form We observe that for each r ∈ [0, 1], Furthermore, we note that for all u, v ∈ X, Let us consider the following auxiliary problem. For given f, g ∈ X, find u ∈ X and λ ∈ Λ, such that (4.10) The rest of the proof will be constructed in several steps.
Step 1. If r = 0, Problem (4.9)-(4.10) has a unique solution. Indeed, if r = 0, Problem (4.9)-(4.10) is equivalent to the saddle point problem: find u ∈ X and λ ∈ Λ such that where L : X × Λ → IR is the functional defined as follows: According to [4], the previous saddle point problem has at least one solution.

Conclusions and Comments
We provided a mixed variational formulation for a frictionless unilateral contact problem involving electro-elastic materials. The main advantage of this type of formulation consists in the fact that it offers the possibility to use modern numerical techniques in order to write efficient algorithms for the approximation of the weak solution. This approach allows to approximate simultaneously the displacement field, the electric potential and the normal stress field. A continuation of the study performed in this paper can be the writing of a discrete mortar formulation of Problem 1. Working on appropriate product spaces and following [9], mortar techniques with dual Lagrange multipliers can be applied in order to get an optimal a priori error estimate.