Abstract
We study the asymptotic behavior of solutions of the anisotropic heterogeneous linearized elasticity system in thin domain of which has a fixed cross-section in the plane with Tresca friction condition. The novelty here is that stress tensor has given by the most general form of Hooke’s law for anisotropic materials. We prove the convergence theorems for the transition 3D-2D when one dimension of the domain tends to zero. The necessary mathematical framework and (2D) equation model with a specific weak form of the Reynolds equation are determined. Finally, the properties of solution of the limit problem are given, in which it is confirmed that the limit problem is well defined.
1. Introduction
In this paper, we are interested of the asymptotic behavior of the linear elasticity system in a domain of 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 two-dimensional 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 [1–8] 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 [9–11] 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]): , where denotes the displacement vector, is the linearized strain tensor, and 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 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 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 . In section 4, we state the main results concerning the existence of a weak limit of , 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 is given.
2. Mathematical Formulation
Let be an open set in with Lipschitz boundary, and we consider a smooth function be a class such that , for all , where and are constants. We define the smooth bounded domain whose boundary has a flat part ,
We denote by is the upper boundary of the equation , and is the lateral boundary.
Let be a small parameter, and we define be the change of scale and the points of ,
We have which its boundary of and where is the upper surface defined by , and is the lateral boundary. The unit outward normal to is denoted by . It follows that there is correspondence between the functions and given by .
Let be the space of traces of functions on of functions from , and we use the vector function such that
We denote by the space of symmetric tensors on and the Euclidean norm on and . Here and below, the indices run between and , and the summation convention overrepeated indices is adopted.
The basic equations of frictionless contact problem for the anisotropic heterogeneous elastic body occupy the domain as follows:
The equations of equilibrium are as follows: where the vector represents the forces of density, is the displacement field, the elements denote the components of elasticity tensor , and is the rate of deformation operator,
On , the displacement is known:
On , we assume that the elastic body is held fixed:
On the surface , we assume that the contact is bilateral: and satisfies the Tresca boundary condition [7] with friction function ; where on . , , and are the tangential displacement, the tangential, and the normal stress tensor, respectively, with
Consider now the following closed convex subset of given by
Let us introduce the form and the functional defined by
In the study of the mechanical problem (3)–(10), we assume that all components belong to and satisfy the usual properties of symmetry and ellipticity [19], i.e., and there exists a constant such that
Remark 1. It follows from previous properties and by Korn’ s inequality (see [16], pp. 79), that the bilinear form is coercive and continuous, i.e., where and denoting a positive constant depends on , and .
Lemma 2. Assuming that and , the variational formulation of problem (3)–(10) is equivalent to
Find satisfying
for every small fixed.
Moreover, if the assumptions of (14) and (15) hold, then the variational inequality (18) has a unique solution .
Remark 3. A problem of the form (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.
3. Some Estimates in Fixed Domain
To be able to study the asymptotic behavior of the solutions of (18), we use the change of variable , to return to the fixed domain , and then we define the following functions in :
For the data , , and , we have the following relations:
(for ).
Let
is a Banach space for the following norm:
Everywhere in the sequel, the indexes and run from 1 to 2, and summation over repeated indices is implied. Follow the same steps as in , passing to the fixed domain , and using the symmetry of and , after multiplication by we have (18) that is equivalent to
Find such that where and is given by the relations
Lemma 4. Under the assumptions of Lemma 2, there exists a constant independent of , such that
Proof. Assume that is a solution of . As , then Using the Young’s inequality in (17) for , we find Also, by the Cauchy-Schwarz and Poincaré’s inequalities, we get then using Young’s inequality for to obtain Using (16), (29), and (31) in (27), we get Taking into account the function introduced in (5) and using [20] (lemma 2 pp.24), there exists a function such that Thus, choosing in , then multiplying product inequality by , and the fact that on , we obtain From , we can see the constant Korn contained in Remark 1 does not depend on and , for ; moreover, by changing the data of , remark that and are independent of . Therefore, passing to the fixed domain , we get with ☐
Lemma 5. Under the assumptions of Lemma 4, there exists such that
Proof. From (26), there exists a fixed constant such that Using Poincaré’s inequality in the domain we deduce that is bounded in . From the last two estimates, there exists and satisfies (37). From (26), we can extract a subsequence such that in ; on the other hand, from (37), we deduce (38). Also, (39)–(41) follow from (37) and (38).☐
4. Limit Problem and Main Result
At the limit , we give the satisfactory equations of and the properties of solution of the limit problem for the system (3)-(10).
Theorem 6. With the same assumptions as Lemma 5, satisfies where the symmetric matrix is given by Moreover, we have
Proof. As (23) can be written, Since the form is a symmetry and -elliptic, and the fact that is convex and lower semicontinuous, we deduce Using Lemma 5, we let tend to in (47), to obtain This completes the proof of (44) if we cross (49) in the matrix form We choose in the variational inequation (49) where (for ), and using Green’s formula, we find choosing and ; then, and , we get (59).☐
Theorem 7. Under the assumptions of Theorem 6 then, the solution of the limit problem (44)–(46) is unique in .
Proof. Suppose that there exists two solutions and of the variational inequality (44), and taking in (44), then in the inequality relating to By subtracting the two obtained inequalities, we have We must now check that is ellipticity. So, we return to the properties of mentioned in (14) and (15); in particular, we choose symmetric tensors that are given by (for ); otherwise, the rest of the components let it be whatever. Putting for we will get Consequently, and as , there exists a positive constant , and for all vectors in , we have So, is ellipticity. Thus, the relation (44) implies that Using Poincaré’s inequality, we obtain and the proof of uniqueness of is complete.☐
Theorem 8. Under the assumptions of Theorem 7, the traces with and defined by satisfy the following limit form of the Tresca boundary conditions: Moreover, if the coefficients for, depending only on the variable , we have the following weak form of the Reynolds equation: where denotes the inverse of and
Proof. We now choose in the variational inequality (49) where for and then using Green’s formula, we obtain
On the other hand, from (46), we have
By density theorems, we find (57). For (58), we use the analogue of [10].
To prove (59), we use those similar steps as in [2, 5, 9–11] by integrating (46) from to , and taking into account depending only on , we obtain
It follows from (51) that it is a invertible matrix , for almost every . Therefore,
By integrating between and , we obtain
As , we have
We integrate (67) from to , and we obtain
and by (68), we deduce that
such that is already defined in (61), and let us finally get the weak form (59) after multiplying (70) by and integrate it in .☐
5. Conclusions
We were able to find a framework to conclude that solving our original problem leads to solving a well-defined problem as in (44),(46) and (57)–(59) for the “small” parameter .
The key of the problem lies in the relation between the matrices and . Note that they have the same properties despite the difference in dimensions, therefore it played a key role in the transition from to .
Indeed, the special case where , are the Lamé coefficients (see [13] pp. 102-103) corresponds to the homogeneous and isotropic case of elastic materials, and has been studied in [2, 3, 5]. Thus also, the Stokes flow in [11] can be recovered when tends to 0.
Data Availability
No data were used.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to thank Professor A. Sili for suggesting the problem addressed in this work.