A Study of the Anisotropic Static Elasticity System in Thin Domain

Department of Mathematics and Computer Sciences, University of El Oued, El Oued, PO Box 789 39000, Algeria Department of Mathematics, College of Sciences and Arts, ArRass, Qassim University, Saudi Arabia Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Oran, 31000 Oran, Algeria Applied Mathematics Laboratory, Department of Mathematics, Faculty of Sciences, University of Ferhat ABBAS-Sétif 1, 19000, Algeria


Introduction
In this paper, we are interested of the asymptotic behavior of the linear elasticity system in a domain of ℝ 3 with a Tresca friction condition where the boundary of this domain has a fixed cross-section in dimension 2 and a small thickness. One of the objectives of this study is to obtain twodimensional equation that allows a reasonable description of the phenomenon occurring in the three-dimensional domain by passing the limit to 0 on the small thickness of the domain (3D). Let us mention for example [1][2][3][4][5][6][7][8] in which the authors worked on the asymptotic behavior for the linearized elasticity system with different boundary conditions. Some problems of Newtonian or non-Newtonian fluids are considered in [9][10][11] where the authors proved a limit problem that gives a distribution of velocity and pressure through the weak form of the Reynolds equation. In [6,7], the authors demonstrate the transition 3D-1D in anisotropic heterogeneous linearized elasticity; so, we mention here that this phenomenon has been studied only about strong solutions, without friction law. Benseridi in [2] investigated the asymptotic analysis of a dynamical problem of linear elasticity with Tresca's friction. The static case with a nonlinear term for linear elastic materials has been considered in [3]. See another situation in [4] where the paper concerns asymptotic derivation of frictionless contact models for elastic rods on a foundation with normal compliance. Recently, the authors in [5,12] have proved the asymptotic behavior of a frictionless contact problem between two elastic bodies, when the vertical dimension of the two domain reaches zero. However, all these papers have been only restricted in a homogeneous and isotropic case of elastic materials.
The present work is a follow-up of [2,3,5] to study the heterogeneous and anisotropic situation with Tresca's friction. Here, the stress tensor with its components is given by the generalized Hooke's law (see [13]): σ ε = A ε eðu ε Þ, where u ε denotes the displacement vector, eðu ε Þ is the linearized strain tensor, and A ε is the fourth order tensor which describes the elastic properties of the material. Many materials that follow the linear elastic model, although they are well made, are not subject to the assumptions of isotropy, for example, wood, reinforced concrete, composite materials, and many biological materials, where the mechanical properties of these materials differ according to the directions of space; in that case, the elasticity operator depends on the location of the point (see [14,15]). Necas in [7] and Sofonea in [16] established the existence of a weak solution for the static frictional contact problem involving linearly elastic and viscoelastic materials, by using a results of convex optimization [17], and numerical approximation of this problem was studied in [18]. For the variational analysis of various contact problems, we mention excellent references in [14,15]. Mathematically, the asymptotic analysis is more difficult since in general, the limit problem involves an equation that takes into account the anisotropy of the medium, and it is thus important to identify the elastic components of A ε that appear in the (2D) equation model.
The paper is organized as follows; in section 2, the strong and weak formulation of the problem is given in terms of u ε and also the related existence and uniqueness of the weak solution. In section 3, we introduce a scaling, and we find some estimates on the displacement which are independent of the parameter ε. In section 4, we state the main results concerning the existence of a weak limit u * of u ε , the (2D) equation model with a specific weak form of the Reynolds equation is proved, the limit form of the Tresca boundary conditions is formulated, and finally, the uniqueness of u * is given.

Mathematical Formulation
Let ω be an open set in ℝ with Lipschitz boundary, and we consider a smooth function h : ω ⟶ ℝ be a class C 1 such that 0 < h min ≤ hðxÞ ≤ h max , for all x ∈ ω, where h min and h max are constants. We define the smooth bounded domain Ω whose boundary has a flat part ω, We denote by Γ 1 is the upper boundary of the equation z = hðxÞ, and Γ L is the lateral boundary.
Let ε > 0 be a small parameter, and we define Ω ε be the change of scale z = x 3 /ε and the points of Ω, We is the lateral boundary. The unit outward normal to Γ ε is denoted by ν. It follows that there is correspondence between the functions ϕ : Ω ε ⟶ ℝ n and b ϕ : Ω ⟶ ℝ n ðn = 1, 2, 3Þ given by b ϕðx, zÞ = ϕðx, x 3 Þ.
Let H 1/2 ðΓÞ 3 be the space of traces of functions on Γ of functions from H 1 ðΩÞ 3 , and we use the vector function g ∈ H 1/2 ðΓÞ 3 such that We denote by S n the space of symmetric tensors on ℝ n and j:j the Euclidean norm on ℝ n and S n . Here and below, the indices i, j, k, l run between 1 and 3, and the summation convention overrepeated indices is adopted.
The basic equations of frictionless contact problem for the anisotropic heterogeneous elastic body occupy the domain Ω ε as follows: The equations of equilibrium are as follows: where the vector ijkl denote the components of elasticity tensor A ε , and e ij ðu ε Þ is the rate of deformation operator, On Γ ε L , the displacement is known: On Γ ε 1 , we assume that the elastic body is held fixed: On the surface ω, we assume that the contact is bilateral: and satisfies the Tresca boundary condition [7] with friction function k ε ; where s = g ε on ω. u ε τ , σ ε τ , and σ ε ν are the tangential displacement, the tangential, and the normal stress tensor, respectively, with Consider now the following closed convex subset of H 1 ðΩ ε Þ 3 given by Let us introduce the form a : K ε × K ε ⟶ ℝ and the functional J ε : Journal of Function Spaces In the study of the mechanical problem (3)-(10), we assume that all components A ε ijkl belong to L ∞ ðΩ ε Þ and satisfy the usual properties of symmetry and ellipticity [19], i.e., and there exists a constant μ > 0 such that Remark 1. It follows from previous properties and by Korn' s inequality (see [16], pp. 79), that the bilinear form a is coercive and continuous, i.e., where M = max 1≤i,j,k,l≤3 kA ε ijkl k L ∞ ðΩ ε Þ and C K denoting a positive constant depends on Ω ε ,Γ ε 1 , and Γ ε L .
Moreover, if the assumptions of (14) and (15) hold, then the variational inequality (18) has a unique solution u ε ∈ K ε . (18) is called an elliptic variational inequality of the second kind ( [17]). The following theorem (see [19], Theorem 6) allows us to replace the variational inequality (18) by a minimization problem. Thus, we will not repeat the proof, but our goal is to study the asymptotic behavior.

Some Estimates in Fixed Domain
To be able to study the asymptotic behavior of the solutions of (18), we use the change of variable z = x 3 /ε, to return to the fixed domain Ω, and then we define the following functions in Ω:û For the dataÂ ijkl ,f i , andk, we have the following rela- (for 1 ≤ i, j, k, l ≤ 3). Let V z is a Banach space for the following norm: Everywhere in the sequel, the indexes α, β, γ and δ run from 1 to 2, and summation over repeated indices is implied. Follow the same steps as in ½6, 12, passing to the fixed domain Ω, and using the symmetry of σ ε ij and A ε ijkl , after multiplication by ε, we have (18) that is equivalent to Findû ε ∈ K, such that 3 Journal of Function Spaces andêðû ε Þ = ðê ij ðû ε ÞÞ ij is given by the relationŝ Lemma 4. Under the assumptions of Lemma 2, there exists a constant C > 0 independent of ε, such that Proof. Assume that u ε is a solution of ð2:12Þ. As J ε ðu ε Þ ≥ 0, then Using the Young's inequality in (17) for η = ffiffiffiffiffiffiffiffiffiffiffiffi ffi Also, by the Cauchy-Schwarz and Poincaré's inequalities, we get ð then using Young's inequality for η = ffiffiffiffiffiffiffiffiffiffiffiffi ffi Using (16), (29), and (31) in (27), we get Taking into account the g function introduced in (5) and using [20] (lemma 2 pp.24), there exists a functiong ∈ H 1 ðΩÞ 3 such that g = gonΓ L andg:ν = 0onω ∪ Γ 1 : Thus, choosing b ϕ =g in ð3:6Þ, then multiplying product inequality by ε, and the fact that g = s on ω, we obtain From ½6, we can see the constant Korn C K contained in Remark 1 does not depend on ε and ϕ, for ε ∈ 0, 1; moreover, by changing the data of A ε , remark that μ and M are independent of ε. Therefore, passing to the fixed domain Ω, we get with εê γ3û Proof. From (26), there exists a fixed constant C > 0 such that Journal of Function Spaces Using Poincaré's inequality in the domain Ω we deduce thatû ε is bounded in V z . From the last two estimates, there exists u * = ðu * 1 , u * 2 , u * 3 Þ ∈ V z and satisfies (37). From (26), we can extract a subsequence such that εð∂û ε i /∂ x α Þ ⇀ η in L 2 ðΩÞ; on the other hand, from (37), we deduce (38). Also, (39)-(41) follow from (37) and (38). ☐

Limit Problem and Main Result
At the limit ε = 0, we give the satisfactory equations of u * and the properties of solution of the limit problem for the system (3)-(10).

Theorem 6.
With the same assumptions as Lemma 5, u * satisfies where the symmetric matrix A * is given by Moreover, we have Proof. As (23) can be written, Since the formâð:, :Þ is a symmetry and K-elliptic, and the fact that b ϕ ⟶ Ð ωk j b ϕ − sjdx is convex and lower semicontinuous, we deduce Using Lemma 5, we let ε tend to 0 in (47), to obtain This completes the proof of (44) if we cross (49) in the matrix form A * : We choose in the variational inequation , and using Green's formula, we find choosing ψ 3 = 0 and ψ α ∈ H 1 0 ðΩÞ; then, ψ α = 0 and ψ 3 ∈ H 1 0 ðΩÞ, we get (59). ☐ Theorem 7. Under the assumptions of Theorem 6 then, the solution of the limit problem (44)-(46) is unique in V z .