Mohamed Boussaa
email: mohamed.boussaa@inria.fr

Olivier Barais
email: olivier.barais@inria.fr

Gerson Sunyé
email: gerson.sunye@inria.fr

Benoît Baudry
email: benoit.baudry@inria.fr

A Novelty Search Approach for Automatic Test Data Generation

In search-based structural testing, metaheuristic search techniques have been frequently used to automate the test data generation. In Genetic Algorithms (GAs) for example, test data are rewarded on the basis of an objective function that represents generally the number of statements or branches covered. However, owing to the wide diversity of possible test data values, it is hard to find the set of test data that can satisfy a specific coverage criterion. In this paper, we introduce the use of Novelty Search (NS) algorithm to the test data generation problem based on statement-covered criteria. We believe that such approach to test data generation is attractive because it allows the exploration of the huge space of test data within the input domain. In this approach, we seek to explore the search space without regard to any objectives. In fact, instead of having a fitness-based selection, we select test cases based on a novelty score showing how different they are compared to all other solutions evaluated so far.

I. INTRODUCTION

In general, manually creating test cases for testing software systems is time consuming and error-prone, making necessary the automation of this process. In fact, metaheuristic search techniques such as Genetic Algorithms (GAs) are frequently used in order to automate the test data generation process and gather relevant test cases through the wide search space [START_REF] Roper | Computer aided software testing using genetic algorithms[END_REF], [START_REF] Watkins | The automatic generation of test data using genetic algorithms[END_REF]. These techniques are especially applied for structural white-box testing. For coverage-oriented approaches, applying Evolutionary Algorithms (EAs) to test data generation has been focused on finding input data for a specific path of the program in accordance with a coverage criterion (e.g., longest path executed). The problem with coverage-oriented approaches is that search-based techniques cannot exploit the huge space of possible test data. In fact, some structures of the system cannot be reached since they are executed only by a small portion of the input domain. Applying GAs for test data generation consists in searching for relevant test data according to an objective function that tries for example to maximize the number of statements or branches covered. The use of a fitness function as a coverage criterion to guide the search to detect relevant test data usually create many local optima to which search may converge. Thus, if the relevant test data, that could coverage the longest path of the program, lie far from the search space defined by the gradient of the fitness function, then some promising search areas may not be reached. The issue of premature convergence to local optima has been a common problem in GAs. Many methods are proposed to avoid this problem [START_REF] Banzhaf | The effect of extensive use of the mutation operator on generalization in genetic programming using sparse data sets[END_REF], [START_REF] Gathercole | An adverse interaction between crossover and restricted tree depth in genetic programming[END_REF]. However, all these alternatives use a fitness-based selection to guide the search.

In this paper, we introduce the use of Novelty Search (NS) algorithm to the test data generation problem. In this approach, we seek to explore the search space of possible test input values without regarding to any objectives (there is no fitness function). In fact, instead of having a fitness-based selection, we rather select test cases based on a novelty score showing how different they are compared to all other test data evaluated so far. So during the evolutionary process, we use to select test data that remain in sparse regions of the search space in order to guide the search through novelty. We choose the statement coverage metric as a coverage criterion to our NS-based test data generation.

The primary contribution of this paper can be summarized as follows: the paper introduces a novel formulation of the test data generation problem using NS and, to the best of our knowledge, this is the first paper in the literature to use NS algorithm to generate test data.

The paper is organized as follows: section II describes the related work. The approach overview and the NS adaptation are presented in Section III. Finally, concluding remarks and future work are provided in Section IV.

II. RELATED WORK A. Generating Test Data

Most forms of automatic test data generation have been focused on finding specific input values that meet a specific testing criteria. In search-based test data generation, Harman and McMinn [START_REF] Harman | A theoretical and empirical study of searchbased testing: Local, global, and hybrid search[END_REF] conducted a large empirical study that compares the behavior of both global and local search-based optimization on real-world problems. The results show that the use of EAs is suitable in many cases. However, it can be outperformed by simpler search techniques. They presented as well, a Memetic Algorithm (Hybrid Algorithm) that combines the global and local search.

Furthermore, a lot of research work have been conducted in the field of structural code coverage. In the work of Roper for example [START_REF] Roper | Computer aided software testing using genetic algorithms[END_REF], GAs are used to generate test data based on the number of structures executed in accordance with a coverage criterion. Whereas, Watkins [START_REF] Watkins | The automatic generation of test data using genetic algorithms[END_REF] attempts to obtain full path coverage for programs.

For test case selection strategies, Chen et al. [START_REF] Chen | Adaptive random testing: The art of test case diversity[END_REF] presented a synthesis of the most important results in Adaptive Random Testing (ART) (an extension of Random Testing). They outlined the importance of diversity in test case selection. In fact, they argue that diversity among test cases should be rewarded because failing test cases tend to be clustered in contiguous regions of the input domain. The success of ART motivated us to introduce a new technique that maximizes input diversity.

B. Novelty Search

In the literature, EAs are often applied to the test data generation problem [START_REF] Harman | Improving evolutionary testing by flag removal[END_REF]. These techniques use basically a fitness function to guide the search e.g., gather fittest solutions over generations. These techniques are good since they reward individuals with high score but they do not favor diversity and the search may converge to many local optima [START_REF] Banzhaf | The effect of extensive use of the mutation operator on generalization in genetic programming using sparse data sets[END_REF], [START_REF] Gathercole | An adverse interaction between crossover and restricted tree depth in genetic programming[END_REF]. The idea of NS, introduced by Lehman and Stanley in 2008 [START_REF] Lehman | Exploiting open-endedness to solve problems through the search for novelty[END_REF], represents an alternative solution for this issue. In fact, individuals in an evolving population are selected based on how different they are compared to other solutions evaluated so far. They also argue that objective fitness functions can be deceptive, leading the evolutionary search to local maxima rather than towards the goal. In contrast, NS ignores the objective and simply searches for novel behavior, and therefore cannot be deceived. So mainly, NS acts like GAs. However, NS needs extra changes. First, a measure of individuals behavior. This depends on the context of the search and the way we represent individuals. Then, a new novelty metric, which rewards individuals with different behavior from pastdiscovered solutions. Finally, an archive must be added to the algorithm which is a kind of a data base that remembers individuals that were highly novel when they were discovered in past generations. NS has been often evaluated in deceptive tasks and applied to evolutionary robotics (in the context of neuroevolution) [START_REF] Risi | Evolving plastic neural networks with novelty search[END_REF], [START_REF] Krčah | Solving deceptive tasks in robot body-brain co-evolution by searching for behavioral novelty[END_REF].

III. NOVELTY SEARCH FOR TEST DATA GENERATION

A. Approach Overview

Our test data generation framework aims to fully automate the test data generation process and avoid any tester's implication. We aim to use a new optimization evolutionary technique, namely Novelty Search, to the test data generation problem.

To automate the test data generation process, we have considered that our System Under Test (SUT) is like a gray/semi-transparent box in where the internal structure is partially known. Thus, we can design test cases through the exposed interfaces and conduct a code coverage analysis from the general structure of our target SUT. For example, within apache.commons.math library, Methods Under Test (MUTs) are accessed through some specific Java interfaces. Each interface belongs to a sub-package of the whole library and exhibits a set of methods. Our test data generation framework will rely on this concept to generate automatically test cases. In fact, as shown in Figure 1, starting from an input interface and a source code package, the testing framework is able to: (1) automatically generate sequences of method invocation, (2) generate relative test data, (3) execute test cases on target Classes Under Test (CUTs), and then (4) analyze the code coverage. The process is iterated until a termination criterion is met (e.g., number of iterations). The main task of our proposal is to use an alternative approach for test data generation based on the NS algorithm. To do so, we adapt the general idea of NS, presented earlier in the related work, to the test data generation problem. In fact, instead of using a fitness function to evaluate generated test cases, we define a new measure of novelty to maximize. We replace, as well, the fitness-based selection (for fittest test cases) with a novelty-based one. This may favor the diversity of generated test data. In addition, although we are exploring the search space without regarding to any objective, we keep a measure of statements coverage for each set of test cases so that, by the end of the evolutionary process, we can gather only test cases with high coverage value. Finally, we add an archive that acts as a memory of previously generated test cases. This archive is used to calculate the novelty metric. Otherwise, we keep the same logic as GAs, namely the crossover and mutations operators. In the following, we detail the main tasks of our automatic test data generation framework:

1) Method Sequence Generator: The method sequence generator is used to automatically generate the test scenario. It takes as an input a Java interface and a source code package. The generated sequence represents the methods names and their parameters types as declared in the Java interface. Indeed, we keep the same number and order of methods signatures as defined in the interface. In this way, we test each method of the target classes in isolation without considering dependencies among methods.

2) Novelty-based Test Data Generation: To generate test cases, we produce input data that will be passed as arguments to the MUTs. Depending on parameters types, we generate randomly input values for Java primitive types and strings. In fact, we did not define any constraint on test data values and we have considered all possible input values for each data type. This process applies the NS approach to generate test data.

3) Test Case Executor: Initially, the test case executor explores the source code package and catches all classes (CUTs) implementing the input interface. Once this is done, we execute automatically our set of test cases on top of these classes except for abstract classes.

4) Code Coverage Analyzer: Our testing framework defines an analyzer able to instrument and analyze all Java classes within a source code package. Thereby, after the execution of a set of test cases, instrumented code is analyzed and a coverage value is assigned to each set of test cases. For instance, we check the number of executed statements. We use Jacoco 1 as a tool to instrument classes and compute the coverage value.

B. Novelty Search Adaptation

In this section, the main contribution is presented, namely, a method for generating automatically test data using NS. In order to ease the understanding of this new approach, we first describe the pseudo code of our adaptation. Then, we present the solution structure and the novelty function used to evaluate solutions. generation ← generation + 1 21: until generation = N 22: return bestSolutions Algorithm 1 describes the overall idea of our NS adaptation for test data generation: The algorithm takes as input a Java interface and a source code package. We initiate the number of iterations N, the population size and the minimum coverage 1 http://www.eclemma.org/jacoco/index.html value. This latter defines the threshold of covered statements that should be reached. Test cases that exceed this threshold are automatically added to the set of relevant test cases. Since we want to compare our NS approach to the fitness-based and random approaches, we are going to set the minimum coverage value based on the maximum coverage value reached by these two approaches. So that, we can record only test cases that may overcome those generated by fitness-based and random approaches. Same thing for the novelty threshold T that defines the threshold for how novel a test suite has to be before it is added to the archive. In addition, we define a maximum size limit for the archive L and a k number that will be used in calculating the distance from k-nearest neighbors. k is a fixed parameter that is determined experimentally.

First, we load the set of classes (CUTs) that implement the given interface I (Line 1). Before starting the test data generation process, we define our set of test cases that will be used to generate test data (Line 2). Line 3-21 encode the main NS loop, which searches for the best set of test cases. During each iteration, we generate a new population that stands for a set of test suites. Then, we generate random test data (Line 4) and we execute our set of test cases on target classes (Line 7-9). The coverage value is computed and assigned to each solution (the executed test cases). This value corresponds to the total number of executed statements by the total number of statement being in the source package.

In the same way, for each set of test cases in the population, we calculate the average distance from its k-nearest neighbors (Line 10). If the novelty is sufficiently high (higher than the given threshold T), then the set of test cases is selected (Line 13) and entered into the permanent archive (Line 12). Genetic operators (mutation and crossover) are applied later to these selected test cases in order to produce offspring individuals and fulfill the next population (Line 19). Finally, we save relevant test cases that reach a coverage value higher than the minimum coverage threshold defined initially (Line 15-17).

1) Solution Representation: For our case study, a candidate solution represents the set of methods declared in the input interface (see Figure 2). Thereby, we represent this solution as a vector where each dimension has a method name, a list of parameters types and a list of test data. The test scenario remains the same for all individuals. In fact, we always run the same number of test cases with the same order as defined in the Java interface. This is important since the order in which test cases are executed may affect the rate of code coverage [START_REF] Srivastava | Test case prioritization[END_REF]. Therefore, we may focus only on the test data generation problem instead of focusing on the test case prioritization issue. Additional information should be added to each candidate solution such as the CUTs where the test cases should be executed and also required constructors needed to create objects from the CUT. In fact, we save for each set of test cases the list of CUTs where the test cases have to be executed and the list of constructors required for each CUT. The constructor representation is the same as test cases. The test case executor presented earlier in the sub-section A.3 chooses arbitrary a constructor from the list of constructors 2) Novelty Metric: The Novelty metric measures the distance of a test suite (solution) to all other test suites in the current population and to those that were discovered in the past. The distance between two test suites is computed as a Manhattan distance between the input parameter values of all methods tested in the test suite. Formally, we define this distance as follows:

distance(t1, t2) = m i=1 p j=1 |t1(M i , P j ) -t1(M i , P j )| (1)
where t1 et t2 are two selected test suites (solutions), m is the number of methods composing the test suite, p is the number of parameters composing a method. The couple (M i , P j ) returns the j th parameter value of the i th method M relative to a test suite (t1 or t2). Since we are using Java primitive types for test data such as floats, integers, doubles, etc, it is easy to calculate this distance for numerical parameters values. However, for string data types we use the Levenshtein Algorithm 2 to measure the strings distance. Furthermore, we have normalized this distance in the range [0-100].

To measure the sparseness of a test suite, we will use the previously defined distance to compute the average distance of a test suite to its k-nearest neighbors. In this context, we define the novelty metric of a particular solution as follows:

N M (S) = 1 k k i=1 distance(S, µ i ) (2)
where µ i is the i th nearest neighbor of the solution S within the population and the archive of novel individuals. In order to test the applicability of our approach for test data generation, we conduct initial implementation and experimentation of our NS Algorithm based on Java Libraries. Thus, we have opted to generate test cases based on apache.commons.math interfaces. For instance, we choose the 'UnivariateFunction' interface from the sub-package apache.commons.math.analysis to generate test data. This interface is implemented by 42 classes within apache.commons.math. We are going to target only classes within the sub-package apache.commons.math.analysis.function. To compute the novelty, we used k=20 (PopSize/5) and T = 50 (max(NM)/2). The 2 http://www.levenshtein.net/ population size is fixed to 100 and the number of generations to 1000. In this way, the algorithm performs 100000 evaluations. Since we haven't yet the minimum coverage value (the max fitness value from GAs), we are trying with this initial implementation, to record the maximum coverage value that may be reached by applying the NS. Initial project and results can be found at the project website3 .

IV. CONCLUSION AND FUTURE WORK

In this paper, we proposed a new search-based approach for test data generation. In our NS adaptation, we try to exploit the large search space of input values and catch relevant test cases that may cover as much as possible the executed statements. However, the limitation of this approach lies in the use of continuous data to compute the novelty score. If we assume that the input values may have only certain values or categorical data, the distance score will not be straightforward and we may use other similarity measures for categorical data.

As future work, we aim to conduct an empirical evaluation of our NS approach by comparing it to fitness-based and random approaches. In addition, we can optimize our approach by adding the diversity as an addition goal to a traditional objective driven approach to form a multi-objective optimization problem. Finally, since we are testing gray-box systems, we can also apply this approach for black-box testing. In this case, we will be able to measure some non-functional properties such as memory usage and CPU consumption.

Fig. 1 .

 1 Fig. 1. APPROACH OVERVIEW

Fig. 2 .

 2 Fig. 2. SOLUTION REPRESENTATION relative to the target CUT in order to instantiate objects and run test cases.2) Novelty Metric: The Novelty metric measures the distance of a test suite (solution) to all other test suites in the current population and to those that were discovered in the past. The distance between two test suites is computed as a Manhattan distance between the input parameter values of all methods tested in the test suite. Formally, we define this distance as follows:

http://goo.gl/T3U0v2


Michel Bellieud
email: michel.bellieud@univ-montp2.fr

Françoise Giuseppe Geymonat
email: giuseppe.geymonat@lms.polytechnique.fr

Françoise Krasucki
email: krasucki@math.univ-montp2.fr

Asymptotic analysis of a linear isotropic elastic composite reinforced by a thin layer of periodically distributed isotropic parallel stiff fibres

Keywords: AMS subject classifications: 35B27, 35B40, 49J45, 74A40, 74Q05 linear elasticity, fibres reinforced structures, two-scale convergence, homogenization

published or not. The documents may come

Asymptotic analysis of a linear isotropic elastic composite reinforced by a thin layer of periodically

distributed isotropic parallel stiff fibres.

Introduction

Due to its importance for many applied problems, the study of boundary value problems where the domain and/or the coefficients have a large number of heterogeneities has attracted the interest of many researchers of the mathematical, of the physical and of the engineering communities. Hence the corresponding literature is huge. When the heterogeneities (inclusions, holes, layers, fibres) are periodically distributed in all the volume (the ratio between a characteristic size of the period and a characteristic size of the whole structure being traditionally denoted ε) and (still with respect to the whole structure) the characteristic size of the heterogeneities is a power of ε, in order to obtain numerically efficient (i.e. precise and not too much computationally expensive) simplified models, it is now classical to use an homogenization method. An advantage of the homogenization methods is that they have been developed and fully mathematically justified for many different geometrical and/or mechanical situations (in particular when also the ratio between the mechanical characteristics of the heterogeneities and of the surrounding material can depend on ε), see, e.g., [START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF], [START_REF] Palencia | Non-homogeneous media and vibration theory[END_REF], [START_REF] Khruslov | Homogenized models of composite media[END_REF], [START_REF] Oleinik | Mathematical problems in elasticity and homogenization[END_REF], [START_REF] Pideri | A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium[END_REF], [START_REF] Bellieud | Bouchitté: Homogenization of a soft elastic material reinforced by fibres[END_REF], [START_REF] Cioranescu | An introduction to homogenization[END_REF], and also [START_REF] Cherednichenko | Two-scale Γ -convergence of integral functionals and its application to homogenisation of nonlinear high-contrast periodic composites[END_REF][START_REF] Neukamm | Rigorous derivation of a homogenized bending-torsion theory for inextensible rods from 3d elasticity[END_REF][START_REF] Stelzig | On problems in homogenization and two-scale convergence[END_REF] in the context of nonlinear elasticity.

Another situation of interest for the engineering community arises when there exists only one layer of heterogeneity located in the interior of the whole domain. This is for instance the situation when two bodies of different (or equal) nature are pasted together using an homogeneous thin layer "centred" around a surface Σ and whose transversal depth has a ratio to the whole structure generally denoted ε. In this case many numerically efficient simplified models have been constructed and mathematically fully justified for various mechanical and /or geometrical circumstances; see, e.g., [START_REF] Geymonat | Bonded joint with a soft thin adhesive[END_REF], [20], [START_REF] Bessoud | Multi-materials with strong interface: variational modelings[END_REF], [START_REF] Bessoud | Asymptotic analysis of shell-like inclusions with high rigidity[END_REF], and also [START_REF] Cherednichenko | Two-scale Γ -convergence of integral functionals and its application to homogenisation of nonlinear high-contrast periodic composites[END_REF][START_REF] Neukamm | Rigorous derivation of a homogenized bending-torsion theory for inextensible rods from 3d elasticity[END_REF][START_REF] Stelzig | On problems in homogenization and two-scale convergence[END_REF] in the context of nonlinear elasticity.

Still another situation arises when the heterogeneities are periodically distributed in a thin layer "centred" around a surface Σ . In some cases of inclusions one can use matched asymptotic expansion methods (see, e.g., [START_REF] Nguetseng | Stress concentration for defects distributed near a surface, Local Effects in the Analysis of Structures[END_REF], [START_REF] Abdelmoula | The effective behavior of a fiber bridged crack[END_REF], [START_REF] Marigo | The effective behavior of elastic bodies containing microcracks or microholes localized on a surface[END_REF], ...). The aim of this paper is to study the situation when the heterogeneities in the layer "centred" around a surface Σ are linearly elastic isotropic parallel fibres periodically distributed and whose rigidity is higher with respect of the rigidity of the surrounding linearly elastic material.

In Section 2 we describe the problem and we state the main results; in Section 3 we introduce the notion of two-scale convergence with respect to a family of measures m ε essentially connected with the total volume of the fibres. This is a variant of the two-scale convergence introduced by Nguetseng [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF] and developed by G. Allaire in [START_REF] Allaire | Homogenization and two-scale convergence[END_REF]. In Section 4 we prove the a priori estimates and we identify the limits obtained with the two-scale method; these results are fundamental for the proof of the weak convergence (Section 5.1) and the strong convergence (Section 5.2).

Notations. In the sequel, {e e e 1 , e e e 2 , e e e 3 } stands for the canonical basis of R 3 . Points in R 3 or in Z 3 and real-valued functions are represented by symbols beginning by a light-face minuscule (example x, i, det A A A...), vectors and vector-valued functions by symbols beginning by a boldface minuscule (examples: x x x, x x x S , i i i, u u u, f f f , g g g, divΨ Ψ Ψ ,...). Matrices and matrix-valued functions are represented by symbols beginning by a capital boldface with the following exceptions: ∇ ∇ ∇u u u (displacement gradient), e e e(u u u) (linearised strain tensor) . We denote by u i or (u u u) i the components of a vector u u u and by A ij or (A A A) ij those of a matrix A A A (that is u u u = 3 i=1 u i e e e i = 3 i=1 (u u u) i e e e i ; A A A = 3 i,j=1 A ij e e e i ⊗e e e j = 3 i,j=1 (A A A) ij e e e i ⊗e e e j ). t A A A denotes the transposed matrix of A A A. We do not employ the usual repeated index convention for summation. We denote by A A A :B B B = 3 i,j=1 A ij B ij the inner product of two matrices, by S N the set of all real symmetric matrices of order N , and by I I I N the N × N identity matrix. The symbols L N and H k represent, respectively, the Lebesgue outer measure on R N and the k-dimensional Hausdorff outer measure on R N . For any x = (x 1 , x 2 , x 3 ) ∈ R 3 , we set x = (x 1 , x 2 ). Given an open subset A of R 2 we note -A f f f dy the mean value of f on A . Given an open subset Ω of R 3 and a 2-dimensional Lipschitz sub-manifold Σ of Ω, the trace on Σ of an element ϕ ϕ ϕ of H 1 (Ω; R k ) is denoted by γ γ γ Σ (ϕ ϕ ϕ) (resp. γ Σ (ϕ) if k = 1), or occasionally by the same symbol ϕ ϕ ϕ when ambiguity is not possible. If A is a subset of Ω, the symbol 1 A represents the characteristic function of A. The letter C denotes different constants whose precise values may vary.

Description of the problem and statement of the result

Let Ω := (-L, L) 3 , let S be a bounded Lipschitz domain of R 2 satisfying

D ⊂ S ⊂ S ⊂ Y, Y := Å - 1 2 , 1 2
ã 2 , - S y y ydy = 0, (1)
for some open disk D of R 2 centred at the origin and let T := S × (-L, L). For ε > 0 we set (see fig. 1):

Y i ε := ε((0, i) + Y ), I ε := {i ∈ Z, Y i ε × (-L, L) ⊂ Ω} S i ε := ε((0, i) + S), T i ε = S i ε × (-L, L), T ε = i∈Iε T i ε , (2)
i min := min I ε , i max := max I ε , Σ := Ω ∩ {0} × R 2 , Σ ε := Σ ∩ T ε = x ∈ Σ, εi min - ε 2 < x 2 < εi max + ε 2 , Ω -:= Ω ∩ ((-∞, 0) × R 2 ), Ω + := Ω ∩ ((0, +∞) × R 2 ). (3)
Let us remark that

L 3 (T ε ) ε|S||Σ| as ε → 0, (4)
where, for simplicity, |S| (resp. |Σ|) denotes the area, that is the two-dimensional Hausdorff measure of S (resp. Σ). Hence from a geometrical point of view there are two natural length scales: the first is a global one (i.e. the 3D-diameter of Ω ) the other one is a local one connected with the heterogeneities (the diameter of the cross section of every fiber). The ratio between these two scales will be denoted by ε. More precisely, the parameter ε is a non-dimensional parameter characterizing the geometrical distribution of the heterogeneities in the structure since, at the same time, it characterizes the ratio between the diameter of the cross section of the fibres and the diameter of Ω and the ratio between the diameter of the cross section of the fibres and the length in the transversal direction e e e 2 of the planar set Σ supporting the heterogeneities.

Fig. 1 The layer of fibres

We consider the case of a linear isotropic elastic structure occupying the set Ω reinforced by the linear isotropic elastic parallel fibres T i ε of very high stiffness surrounded by a matrix of constant stiffness. More precisely we assume that the Young's modulus E ε depends on ε as follows :

   E ε (x) = 1 ε m E T if x ∈ T ε , E ε (x) = E if x ∈ Ω \ T ε , (5)
with m > 0. Instead the Poisson's coefficient denoted by ν ε is independent from ε :

® ν ε (x) = ν 1 if x ∈ T ε , ν ε (x) = ν if x ∈ Ω \ T ε . (6)
We also assume that they satisfy the usual conditions E, E T > 0 and 0 < ν, ν 1 < 1 2 . Let us stress that, with this choice, the parameter ε characterizes at the same time the geometrical distribution of the heterogeneities in the structure and the ratio between the rigidity of the heterogeneities and the rigidity of the structure. We are concerned with the behaviour for ε -→ 0 of the following elasticity problem where we have taken for simplicity homogeneous Dirichlet boundary data:

         -divσ σ σ ε = f f f in Ω, u u u ε ∈ H 1 0 (Ω, R 3 ), f f f ∈ L 2 (Ω, R 3 ), σ σ σ ε = E ε 1 + ν ε Å ν ε 1 -2ν ε
tr(e e e(u u u ε ))I I I + e e e(u u u ε )

ã , e e e(u u u

ε ) = 1 2 (∇ ∇ ∇u u u ε + t ∇ ∇ ∇u u u ε ). (7)
The limit problem we derive in Theorem 1 depends on the value of m. Case 0 < m < 1. The limit problem reads:

         -divσ σ σ 0 = f f f in Ω, σ σ σ 0 = E 1 + ν Å ν 1 -2ν
tr(e e e(u u u))I I I + e e e(u u u)

ã , u u u ∈ D m , (8)
where

D m = H 1 0 (Ω; R 3 ). ( 9
)
Case m = 1. The limit problem reads:

                   -divσ σ σ 0 = f f f in Ω + ∪ Ω -, σ σ σ 0 = E 1 + ν Å ν 1 -2ν
tr(e e e(u u u))I I I + e e e(u u u) ã ,

(σ σ σ + 0 -σ σ σ - 0 )e e e 1 + |S|E T ∂ 2 γ Σ (u 3 ) ∂x 2 3
e e e 3 = 0 on Σ,

u u u ∈ D 1 , (10)
where σ σ σ - 0 (resp. σ σ σ + 0 ) denotes the restriction of σ σ σ 0 to Ω -(resp. Ω + ) and

D 1 = ß u u u ∈ H 1 0 (Ω; R 3 ); ∂γ Σ (u 3 ) ∂x 3 ∈ L 2 (Σ), γ Σ (u 3 ) = 0 on ∂Σ ∩ {x ∈ R 3 , x 3 ∈ {-L; L}} ™ . (11)
Note that the effective superficial density of forces exerted by the fibres on the matrix along Σ, given by

(σ σ σ + 0 -σ σ σ - 0 )e e e 1 = -|S|E T ∂ 2 γ Σ (u 3 ) ∂x 2 3 e e e 3 ,
is parallel to the fibres.

Case 1 < m < 3. The limit problem reads:

         -divσ σ σ 0 = f f f in Ω + ∪ Ω -, σ σ σ 0 = E 1 + ν Å ν 1 -2ν
tr(e e e(u u u))I I I + e e e(u u u)

ã , u u u ∈ D m , (12)
where

D m = u u u ∈ H 1 0 (Ω; R 3 ); γ Σ (u 3 ) = 0 . ( 13
)
Case m = 3. The limit problem reads:

                     -divσ σ σ 0 = f f f in Ω + ∪ Ω -, σ σ σ 0 = E 1 + ν Å ν 1 -2ν
tr(e e e(u u u))I I I + e e e(u u u) ã ,

(σ σ σ - 0 -σ σ σ + 0 )e e e 1 + |S|E T 2 α,β=1 J αβ ∂ 4 γ Σ (u β ) ∂x 4
3 e e e α = 0 on Σ,

u u u ∈ D 3 , (14)
where

J αβ := - S y α y β dy, (15)
and

D 3 =        u u u ∈ H 1 0 (Ω; R 3 ); γ Σ (u 3 ) = 0, ∂ 2 γ Σ (u 1 ) ∂x 2 3 , ∂ 2 γ Σ (u 2 ) ∂x 2 3 ∈ L 2 (Σ), γ Σ (u α ) = ∂γ Σ (u α ) ∂x 3 = 0 on Σ ∩ {x ∈ R 3 , x 3 ∈ {-L; L}} ∀α ∈ {1, 2}        . ( 16
)
Unlike the case m = 1, the effective superficial density of forces exerted by the fibres on the matrix along Σ, given by

h h h = (σ σ σ + 0 -σ σ σ - 0 )e e e 1 = |S|E 1 2 α,β=1 J αβ ∂ 4 γ Σ (u β ) ∂x 4
3 e e e α , is orthogonal to the direction of the fibres.

Case m > 3. The limit problem reads:

         -divσ σ σ 0 = f f f in Ω + ∪ Ω -, σ σ σ 0 = E 1 + ν Å ν 1 -2ν
tr(e e e(u u u))I I I + e e e(u u u)

ã , u u u ∈ D m , (17)
where

D m = u u u ∈ H 1 0 (Ω; R 3 ); γ γ γ Σ (u u u) = 0 . ( 18
)
Theorem 1 Assume ( 5) and ( 6), then for ε → 0 the solution u u u ε of (7) strongly converges in H 1 0 (Ω; R 3 )) to the unique solution u u u of (8

) if 0 < m < 1, of (10) if m = 1, of (12) if 1 < m < 3, of (14) if m = 3, and of (17) if m > 3.
Remark 1 For every m > 0 the limit problem does not depend on the value ν 1 of the Poisson's coefficient of the material of the fibres.

The problem [START_REF] Bellieud | Bouchitté: Homogenization of a soft elastic material reinforced by fibres[END_REF] has a unique solution u u u ε that realizes the minimum in H 1 0 (Ω, R 3 ) of the functional

F ε,m (v v v) : = 1 2 a ε,m (v v v, v v v) - Ω f f f .v v vdx, with 1 2 a ε,m (v v v, v v v) := 1 2 a ε (v v v, v v v) + 1 2ε m a ε,T (v v v, v v v), (19)
where a ε (., .) and a ε,T (., .) are the symetric bilinear forms on H 1 0 (Ω, R 3 ) × H 1 0 (Ω, R 3 ) given by :

a ε (v v v, φ φ φ) = Ω\Tε { E 1 + ν Å ν 1 -2ν
tr(e e e(v v v))I I I + e e e(v v v)

ã :e e e(φ φ φ)}dx,

a ε,T (v v v, φ φ φ) = Tε { E T 1 + ν 1 Å ν 1 1 -2ν 1
tr(e e e(v v v))I I I + e e e(v v v)

ã :e e e(φ φ φ)}dx.

(20)

Let us point out that the limit problems have a unique solution u u u ∈ D m that realizes the minimum in D m of the functional F m defined on D m by

F m (v v v) := 1 2 a m (v v v, v v v) - Ω f f f .v v v, (21)
where a m (., .) is the bilinear form on D m × D m given by:

a 0 (v v v, φ φ φ) := Ω E 1 + ν Å ν 1 -2ν
tr(e e e(v v v))I I I + e e e(v v v)

ã : e e e(φ φ φ)dx,

a 1 (v v v, φ φ φ) := a 0 (v v v, φ φ φ) + |S|E T Σ ∂γ Σ (v 3 ) ∂x 3 ∂γ Σ (φ 3 ) ∂x 3 dH 2 , a 3 (v v v, φ φ φ) := a 0 (v v v, φ φ φ) + |S|E T 2 α,β=1 Σ J αβ ∂ 2 γ Σ (v α ) ∂x 2 3 ∂ 2 γ Σ (φ β ) ∂x 2 3 dH 2 (x), a m (v v v, φ φ φ) := a 0 (v v v, φ φ φ) if m ∈ (0, +∞) \ {1; 3}. ( 22
)
The proof of Theorem 1 is structured as follows:

i) In Sect.3, we introduce a two-scale convergence method adapted to the particular geometry of our problem (layer of fibres) and we prove some useful properties of this method. ii) In Sect.4, we study the behaviour when ε → 0 of the solution (u u u ε ) of [START_REF] Bellieud | Bouchitté: Homogenization of a soft elastic material reinforced by fibres[END_REF]. The most delicate task resides in the study of the behaviour of (u u u ε ) in the fibres and here we use the properties of the two-scale method introduced in Sect.3 iii) In Sect.5.1, we multiply [START_REF] Bellieud | Bouchitté: Homogenization of a soft elastic material reinforced by fibres[END_REF] by an appropriate sequence of oscillating test fields (φ φ φ ε ) and, by passing to the limit as ε → 0 in accordance with the convergences established in ii), we obtain a variational formulation of the limit problem satisfied by u u u. iv) We prove in Sect.5.2 the strong convergence using the linearity of the problem.

3 Two-scale convergence with respect to (m ε )

As a mean to particularize the oscillatory behaviour of the displacement in the fibres, we consider a variant of the two-scale convergence introduced by G. Nguetseng in [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF]. This seminal idea has been further developed by G. Allaire in [START_REF] Allaire | Homogenization and two-scale convergence[END_REF] (see also a more general presentation in [START_REF] Lukkassen | Wall: Two-scale convergence[END_REF]) and extended in various ways (e.g. for the two-scale convergence with respect to a sequence of measures see [START_REF] Bouchitté | Fragalà: Homogenization of thin structures by two-scale method with respect to measures[END_REF], [START_REF] Lukkassen | Wall: Two-scale convergence with respect to measures and homogenization of monotone operators[END_REF]), [START_REF] Zhikov | On an extension and an application of the two-scale convergence method[END_REF],...). Here the two-scale convergence is adapted to the geometry of the problem i.e. that the heterogeneities are a layer of fibres.

Let us at first introduce the following shortened notation for R-valued functions of L 1 (Ω) (the analogous notation will also be used for R k -valued functions):

f (x)dm ε := 1 ε|S| Tε f (x 1 , x 2 , x 3 )dx 1 dx 2 dx 3 = 1 ε|S| i∈Iε L -L dx 3 S i ε f (x 1 , x 2 , x 3 )dx 1 dx 2 , ( 23
)
where |S| denotes the two-dimensional Lebesgue measure of S. Notice that by (4) there holds

dm ε |Σ|. (24)
In the study of the behaviour when ε → 0 of the solution (u u u ε ) of ( 7) a delicate task resides in the study of the behaviour of the fibres. For this we use the operator

v ε : L 2 (Ω) → L 2 (Ω) defined by v ε (ϕ)(x) := i∈Iε Ç - S i ε ϕ(s, x 3 )dH 2 (s) å 1 Y i ε (x 1 , x 2 ) (25)
(in the case of R k -valued functions the analogous definition will be used). We also define the R 2 -valued function (see [START_REF] Allaire | Homogenization and two-scale convergence[END_REF])

y ε (x ) = y ε (x 1 , x 2 ) := i∈Iε (x 1 e e e 1 + (x 2 -εi)e e e 2 )1 Y i ε (x 1 , x 2 ). (26)
A sequence (f f f ε ) in L 2 (Ω; R k ) will be said to two-scale converge with respect to (m ε ) to some

f f f 0 ∈ L 2 (Σ × S; R k ) (notation: f f f ε mε f f f 0 ) if for all ψ ψ ψ ∈ C(Ω × S; R k ) lim ε→0 f f f ε (x).ψ ψ ψ Å x, y ε (x ) ε ã dm ε (x) = 1 |S| Σ×S f f f 0 (x 2 , x 3 ; y 1 , y 2 ).ψ ψ ψ(0, x 2 , x 3 ; y 1 , y 2 )dH 2 (x)dy, (27)
where y ε (x ) is given by [START_REF] Marigo | The effective behavior of elastic bodies containing microcracks or microholes localized on a surface[END_REF] and |S| denotes the two-dimensional Lebesgue measure of S. Similar notions have been considered in [START_REF] Bouchitté | Fragalà: Homogenization of thin structures by two-scale method with respect to measures[END_REF], [START_REF] Lukkassen | Wall: Two-scale convergence with respect to measures and homogenization of monotone operators[END_REF], [START_REF] Zhikov | On an extension and an application of the two-scale convergence method[END_REF]. As shown in the next proposition, the two-scale convergence with respect to (m ε ) enjoys a compactness property for sequences satisfying an uniform bound of the type

sup ε>0 |f f f ε | 2 dm ε ≤ C.
Proposition 1 (i) There holds

lim ε→0 ψ ψ ψ Å x, y ε (x ) ε ã dm ε (x) = 1 |S| Σ×S ψ ψ ψ(x, y)dH 2 (x)dy ∀ψ ψ ψ ∈ C(Ω × S; R k ). ( 28
) (ii) Let (f f f ε ) be a sequence in L 2 (Ω; R k ) such that sup ε>0 |f f f ε | 2 dm ε < +∞. Then, the sequence (f f f ε ) two-scale converges with respect to (m ε ), up to a subsequence, to some f f f 0 ∈ L 2 (Σ × S; R k ). (iii) Let (f f f ε ) be a sequence in L 2 (Ω; R k ) such that sup ε>0 |f f f ε | 2 dm ε < +∞ and that (f f f ε ) two-scale converges with respect to (m ε ) to f f f 0 ∈ L 2 (Σ × S; R k ). Then: a) the sequence (v v v ε (f f f ε )) two-scale converges with respect to (m ε ) to 1 |S| S f f f 0 (x, y)dy; b) the sequence of traces (γ γ γ Σ (v v v ε (f f f ε ))) is well defined and weakly converges in L 2 (Σ; R k ) to 1 |S| S f f f 0 (x, y)dy; c) if f f f ε takes constant values for a.e. x 3 ∈ (-L, L) on each set Y i ε × {x 3 }
, and vanishes elsewhere, then the sequence of traces

(γ γ γ Σ (f f f ε )) is well defined, f f f ε = v v v ε (f f f ε ), f f f 0 (x, y) = -S f f f 0 (x, y )dy a.e. on Σ × S, and (γ γ γ Σ (f f f ε )) weakly converges in L 2 (Σ; R k ), up to a subsequence, to 1 |S| S f f f 0 (x, y)dy.
Proof. 1 In analogy to [START_REF] Arbogast | Derivation of the double porosity model of single phase flow via homogenization theory[END_REF][START_REF] Neukamm | Rigorous derivation of a homogenized bending-torsion theory for inextensible rods from 3d elasticity[END_REF][START_REF] Stelzig | On problems in homogenization and two-scale convergence[END_REF][START_REF] Visintin | Toward a two-scale calculus[END_REF] (and in the spirit of the periodic unfolding method [START_REF] Cioranescu | The periodic unfolding method in homogenization[END_REF]), denoting by [s] the integer part of a real s, we define the transformation S ε : Σ ε × S → T ε as follows

S ε (x, y) := (εy 1 , ε[x 2 /ε + 1/2] + εy 2 , x 3 ). ( 29
)
For

arbitrary f ∈ L 1 (T ε ) the function (f • S ε )(x, y) ∈ L 1 (Σ ε × S). Its extension by zero to Σ × S will be indicated f ε • S ε 1 Σ ε ×S or simply (f • S ε
) when there are no ambiguities. By the following isometry property

f dm ε := 1 ε|S| Tε f dx = 1 |S| Σ ε ×S (f • S ε )(x, y)dH 2 (x)dy , (30)
any sequence

f ε ∈ L 2 (T ε ) with sup ε>0 |f ε | 2 dm ε < +∞ yields a subsequence with f ε • S ε 1 Σ ε ×S f 0 weakly in L 2 (Σ × S). ( 31
) If ψ ∈ C(Ω × S) and ψ ε (x) := ψ Ä x, yε(x ) ε ä then ψ ε • S ε 1 Σ ε ×S (x, y) = ψ (εy 1 , ε[x 2 /ε + 1/2] + εy 2 , x 3 , y) 1 Σ ε ×S (x, y) (32)
is uniformly bounded in Σ × S and converges uniformly to ψ on Σ ε0 × S for each fixed ε 0 > 0, therefore, since

H 2 (Σ \ Σ ε ) → 0, ψ ε • S ε 1 Σ ε ×S → ψ strongly in L p (Σ × S), ∀p ∈ [1, +∞[. ( 33
) Assertion (i) (resp. (ii)) follows from ( 30) and (33) (resp. ( 30), ( 31) and ( 33)). Moreover, the combination of ( 30), ( 31) and [START_REF] Stelzig | On problems in homogenization and two-scale convergence[END_REF] shows that

î f f f ε mε f f f 0 ó ⇔ f ε • S ε 1 Σ ε ×S f 0 weakly in L 2 (Σ × S) . (34)
In order to prove the assertions (iii) we note at first that

v v v ε (f f f ε )(x)1 Tε (x) = Å 1 |S| S f f f ε (εy 1 , ε[x 2 /ε + 1/2] + εy 2 , x 3 )dy ã 1 Tε (x),
and we deduce from ( 30) and ( 32) that

v v v ε (f f f ε )(x).ψ ψ ψ Å x, y ε (x ) ε ã dm ε = 1 |S| Σ ε ×S f f f ε (εy 1 , [x 2 /ε + 1/2] + εy 2 , x 3 ) ψ ψ ψ ε (x)dH 2 (x)dy, (35)
where

ψ ψ ψ ε (x) := 1 |S| S ψ ψ ψ (εy 1 , ε[x 2 /ε + 1/2] + εy 2 , x 3 , y) dy.
By the uniform continuity of ψ ψ ψ, we have

ψ ψ ψ ε (x) -ψ ψ ψ(εy 1 , [x 2 /ε + 1/2] + εy 2 , x 3 ) 1 Σ ε ×S ≤ Cε, ψ ψ ψ(x) := 1 |S| S ψ ψ ψ(x, y)dH 2 (y). (36)
Testing the two-scale convergence with respect to (m ε ) of (f

f f ε ) to f f f 0 with the test function ψ ψ ψ(x), taking (34)
into account, we infer from ( 35) and ( 36) that

lim ε→0 v v v ε (f f f ε )(x).ψ ψ ψ Å x, y ε (x ) ε ã dm ε = lim ε→0 1 |S| Σ ε ×S f f f ε (εy 1 , [x 2 /ε + 1/2] + εy 2 , x 3 ).ψ ψ ψ(εy 1 , [x 2 /ε + 1/2] + εy 2 , x 3 )dydH 2 (x) = 1 |S| Σ×S f f f 0 (x, y).ψ ψ ψ(x)dH 2 (x)dy = 1 |S| Σ×S Å 1 |S| Σ×S f f f 0 (x, y)dH 2 (y) ã .ψ ψ ψ(x, y)dH 2 (x)dy. ( 37
) Assertion (iii) a) is proved. For each (x 2 , x 3 ) ∈ Σ the mapping x 1 → v v v ε (f f f ε )(x 1 , x 2 , x 3 ) is constant on the set (-ε, ε) × (x 2 , x 3 ), and so in particular equal to v v v ε (f f f ε )(0, x 2 , x 3 ). Hence the field v v v ε (f f f ε )(0, x 2 , x 3 )
is a well defined element of L 2 (Σ; R k ) which will also be denoted by γ γ γ Σ (v v v ε (f f f ε )), using for simplicity the same notation as for the trace operator from

H 1 (Ω; R k ) to L 2 (Σ; R k ). One can easily check that Σ |γ γ γ Σ (v v v ε (f f f ε ))| 2 dH 2 = |v v v ε (f f f ε )| 2 dm ε ≤ |f f f ε | 2 dm ε , hence γ γ γ Σ (v v v ε (f f f ε )) is bounded in L 2 (Σ; R k ). Given ϕ ϕ ϕ ∈ C(Ω; R k ), a straightforward computation yields Σ γ γ γ Σ (v v v ε (f f f ε )).ϕ ϕ ϕ(0, x 2 , x 3 )dH 2 = f f f ε .Û ϕ ϕ ϕ ε dm ε , Û ϕ ϕ ϕ ε (x) := i∈Iε Ç - εi+ ε 2 εi-ε 2 ϕ ϕ ϕ(0, t, x 3 )dt å 1 S i ε (x ). ( 38
)
Noticing that, by the uniform continuity of ϕ ϕ ϕ on Ω, there holds sup

x∈Tε |ϕ ϕ ϕ(x) -Û ϕ ϕ ϕ ε (x)| ≤ Cε, we deduce from the two-scale convergence with respect to (m ε ) of (f f f ε ) to f f f 0 that lim ε→0 Σ γ γ γ Σ (v v v ε (f f f ε ))ϕ ϕ ϕ(x)dH 2 = lim ε→0 f f f ε .ϕ ϕ ϕdm ε = 1 |S| Σ×S f f f 0 (x, y)ϕ ϕ ϕ(x)dH 2 (x)dy = Σ Å - S f f f 0 (x, y)dy ã .ϕ ϕ ϕdH 2 . Assertion (iii) b) is proved. The proof of Assertion (iii) c) is straightforward.
The next proposition states a lower semicontinuity property related to the two-scale convergence with respect to (m ε ) when dealing with convex integrands.

Proposition 2 Let (f f f ε ) be a sequence in L 2 (Ω; R k ) such that (f f f ε ) two-scale converges to f f f 0 with respect to (m ε )
, and let j : R k → R be a convex function. Then

lim inf ε→0 j(f f f ε )dm ε ≥ 1 |S| Σ×S j(f f f 0 )dH 2 ⊗ dL 2 . ( 39
)
Proof. Let us fix ψ ψ ψ ∈ C(Ω × S; R k ). Denoting by j the Fenchel transform of j, by applying Fenchel inequality and then [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF] to the function j

(ψ ψ ψ) ∈ C(Ω × S), we get lim inf ε→0 j(f f f ε )dm ε (x) ≥ lim ε→0 f f f ε ψ ψ ψ Å x, y ε (x ) ε ã -j Å ψ ψ ψ Å x, y ε (x ) ε ãã dm ε (x) = 1 |S| Σ×S f f f 0 ψ ψ ψ -j (ψ ψ ψ)dH 2 (x)dy. ( 40
)
Taking the supremum of the right-hand side of (40) when ψ ψ ψ varies in C(Ω × S; R k ), using a classical localization argument and the convexity assumption on j, we get

lim inf ε→0 j(f f f ε )dm ε (x) ≥ sup ψ ψ ψ∈C(Ω×S;R k ) 1 |S| Σ×S f f f 0 ψ ψ ψ -j (ψ ψ ψ)dH 2 (x)dy = 1 |S| Σ×S sup ψ ψ ψ∈R k {f f f 0 ψ ψ ψ -j (ψ ψ ψ)}dH 2 (x)dy = 1 |S| Σ×S j (f f f 0 )dH 2 (x)dy = 1 |S| Σ×S j(f f f 0 )dH 2 (x)dy.
The estimates established in the next lemma will be employed to study the link between the two-scale limit with respect to (m ε ) of a bounded sequence in H 1 (Ω; R k ) and the strong limit of its traces in L 2 (Σ; R k ). Also, they will take a crucial part in the proof of the apriori estimates established in Proposition 4. In what follows, for each ϕ ϕ ϕ ∈ H 1 (Ω; R 3 ), we denote by Û ϕ ϕ ϕ and ϕ ϕ ϕ ε the elements of H 1 (Ω; R 3 ) defined by

ϕ ϕ ϕ ε (x) := - (-ε 2 , ε 2 ) ϕ ϕ ϕ(s 1 , x 2 , x 3 )ds 1 , Û ϕ ϕ ϕ(x) := ϕ ϕ ϕ(0, x 2 , x 3 ). ( 41
)
Lemma 1 There exists a constant C such that, for all ϕ ϕ ϕ ∈ H 1 (Ω; R 3 ), see (3),

Σε |γ γ γ Σ (v v v ε (ϕ ϕ ϕ)) -γ γ γ Σ (ϕ ϕ ϕ)| 2 dH 2 ≤ Cε ε/2 -ε/2 Σ |∇ ∇ ∇ϕ ϕ ϕ| 2 dx, |ϕ ϕ ϕ -v v v ε (ϕ ϕ ϕ)| 2 dm ε ≤ Cε 2 |∇ ∇ ∇ϕ ϕ ϕ| 2 dm ε , |ϕ ϕ ϕ -Û ϕ ϕ ϕ| 2 dm ε ≤ Cε Σε |∇ ∇ ∇ϕ ϕ ϕ| 2 dx, (42)
and

|Û ϕ ϕ ϕ| 2 dm ε ≤ C Σ |γ γ γ Σ (ϕ ϕ ϕ)| 2 dH 2 , Σ ∂γ Σ (v ε3 (ϕ ϕ ϕ)) ∂x 3 2 dH 2 ≤ C |e e e(ϕ ϕ ϕ)| 2 dm ε . (43)
Moreover for all ϕ ϕ ϕ ∈ L 2 (Ω; R 3 ) and k ∈ {1, 2, 3}

Σ |γ Σ (v εk (ϕ ϕ ϕ))| 2 dH 2 = |v εk (ϕ ϕ ϕ)| 2 dm ε . (44)
Proof. We use the following estimate, whose proof is postponed to the end:

1 2 -1 2 - S ψ ψ ψds -ψ ψ ψ(0, y 2 ) 2 dy 2 ≤ C Y |∇ ∇ ∇ψ ψ ψ| 2 dy 1 dy 2 ∀ψ ψ ψ ∈ H 1 (Y ; R 3 ). ( 45
)
By making suitable changes of variables in (45), we infer that for all i ∈ I ε , there holds

ε(i+1/2) ε(i-1/2) - S i ε ψ ψ ψds -ψ ψ ψ(0, x 2 ) 2 dx 2 ≤ Cε Y i ε |∇ ∇ ∇ψ ψ ψ| 2 dx 1 dx 2 ∀ψ ψ ψ ∈ H 1 (Y i ε ; R 3 ). ( 46
)
Fixing ϕ ϕ ϕ ∈ H 1 (Ω; R 3 ) and applying (46) for a.e. x 3 ∈ (-L, L) to ψ ψ ψ(x 1 , x 2 ) := ϕ ϕ ϕ(x 1 , x 2 , x 3 ), then integrating with respect to x 3 over (-L, L) and summing over i ∈ I ε , we infer

Σε |γ γ γ Σ (v v v ε (ϕ ϕ ϕ)) -γ γ γ Σ (ϕ ϕ ϕ)| 2 dH 2 ≤ Cε ε/2 -ε/2 Σ |∇ ∇ ∇ϕ ϕ ϕ| 2 dx.
The proof of the first line of (42) is achieved.

Let us prove the second line of (42). By making suitable changes of variables in the classical Poincaré-Wirtinger inequality

S ψ ψ ψ -- S ψ ψ ψ 2 dy ≤ C S |∇ ∇ ∇ψ ψ ψ| 2 dy ∀ψ ψ ψ ∈ H 1 (S; R 2 ),
we infer

S i ε ψ ψ ψ -- S i ε ψ ψ ψ 2 dx ≤ Cε 2 S i ε |∇ ∇ ∇ψ ψ ψ| 2 dx ∀i ∈ I ε , ∀ψ ψ ψ ∈ H 1 (S i ε ; R 2 ). ( 47
)
By applying (47) for a.e. x 3 ∈ (-L, L) to ψ ψ ψ(x 1 , x 2 ) = ϕ ϕ ϕ(x 1 , x 2 , x 3 ), integrating with respect to x 3 over (-L, L) and summing with respect to i over I ε , we deduce the second line of (42).

By Jensen's inequality we have

|ϕ ϕ ϕ -Û ϕ ϕ ϕ| 2 dm ε ≤ C ε Σε |ϕ ϕ ϕ -Û ϕ ϕ ϕ| 2 dx ≤ C ε (-ε 2 , ε 2 ) dx 1 Σ |ϕ ϕ ϕ(x 1 , x 2 , x 3 ) -ϕ ϕ ϕ(0, x 2 , x 3 )| 2 dx 2 dx 3 ≤ Σ (-ε 2 , ε 2 ) ∂ϕ ϕ ϕ ∂x 1 (t 1 , x 2 , x 3 ) dt 1 2 dx 2 dx 3 ≤ Cε Σε |∇ ∇ ∇ϕ ϕ ϕ| 2 dx,
which proves the third line of (42).

By [START_REF] Murat | Effets non locaux dans le passage 3d-1d en élasticité linéarisée anisotrope hétérogène[END_REF] and Jensen's inequality, there holds

Σ ∂γ Σ (v ε3 (ϕ ϕ ϕ)) ∂x 3 2 dH 2 = i∈Iε L -L dx 3 ε(i+ 1 2 ) ε(i-1 2 ) dx 2 - S i ε ∂ϕ 3 ∂x 3 (s 1 , s 2 , x 3 )ds 1 ds 2 2 ≤ i∈Iε L -L dx 3 ε(i+ 1 2 ) ε(i-1 2 ) dx 2 - S i ε ∂ϕ 3 ∂x 3 2 (s 1 , s 2 , x 3 )ds 1 ds 2 = ε i∈Iε L -L dx 3 - S i ε ∂ϕ 3 ∂x 3 2 (s 1 , s 2 , x 3 )ds 1 ds 2 = 1 ε|S| Tε ∂ϕ 3 ∂x 3 2 dx ≤ C |e e e(ϕ ϕ ϕ)| 2 dm ε ,
yielding the second line of (43). Moreover, we have

|Û ϕ ϕ ϕ| 2 dm ε = 1 ε|S| Tε |ϕ ϕ ϕ(0, x 2 , x 3 )| 2 dx ≤ 1 ε|S| (-ε 2 , ε 2 )×Σ |ϕ ϕ ϕ(0, x 2 , x 3 )| 2 dx = 1 |S| Σ |γ γ γ Σ (ϕ ϕ ϕ)| 2 dH 2 .
The estimates (43) are proved. The estimate (44) results from the next computation, holding for k ∈ {1, 2, 3}:

Σ |γ Σ (v εk (ϕ ϕ ϕ))| 2 dH 2 = i∈Iε L -L dx 3 ε(i+ 1 2 ) ε(i-1 2 ) dx 2 - S i ε ϕ k (s, x 3 )dH 2 (s) 2 = i∈Iε L -L dx 3 ε - S i ε ϕ k (s, x 3 )dH 2 (s) 2 = i∈Iε L -L dx 3 ε 1 |S i ε | S i ε - S i ε ϕ k (s, x 3 )dH 2 (s) 2 dx 1 dx 2 = 1 ε|S| Tε |v εk (ϕ ϕ ϕ)| 2 dx = |v εk (ϕ ϕ ϕ)| 2 dm ε . (48)
Proof of (45). If the first line of (45) is not satisfied, there exists (ψ

ψ ψ n ) ⊂ H 1 (Y ; R 3 ) such that ( -1 2 , 1 2 ) - S ψ ψ ψ n ds -ψ ψ ψ n (0, y 2 ) 2 dy 2 = 1, Y |∇ ∇ ∇ψ ψ ψ n | 2 dy 1 dy 2 ≤ 1 n . (49)
After possibly substracting a constant vector to ψ ψ ψ n , we can assume that

- S ψ ψ ψ n dy = 0. ( 50
)
From the Poincaré-Wirtinger inequality

Y ψ ψ ψ -- S ψ ψ ψds 2 dy ≤ C Y |∇ ∇ ∇ψ ψ ψ| 2 dy ∀ψ ψ ψ ∈ H 1 (Y ; R 3 ),
we deduce that (ψ ψ ψ n ) is bounded in H 1 (Y ; R 3 ) and weakly converges, thanks to (50) and up to a subsequence (still denoted by ψ n ), to 0. By the continuity of the trace application from H 1 (Y ; R 3 ) weak to L 2 {0} × -1 2 , 1 2 ; R 3 strong, the trace ψ ψ ψ n (0, y 2 ) converges strongly to 0 in L 2 {0} × -1 2 , 1 2 ; R 3 . This is in contradiction with the fact that by (49) and (50), the norm in L 2 {0} × -1 2 , 1 2 ; R 3 of ψ ψ ψ n (0, y 2 ) is equal to 1.

We are now in a position to investigate the link between the two-scale limit with respect to (m ε ) of a bounded sequence in H 1 (Ω; R k ) and the strong limit of its traces in L 2 (Σ; R k ).

Proposition 3 Assume that (f f f ε ) weakly converges in H 1 (Ω; R k ) to some f f f ∈ H 1 (Ω; R k ). Then, (f f f ε ) two-scale converges with respect to (m ε ) to the field f f f 0 ∈ L 2 (Σ × S; R k ) defined by f f f 0 (x, y) := γ γ γ Σ (f f f )(x) for H 2 ⊗ L 2 -a.e. (x, y) ∈ Σ × S.
(51) In addition, the sequence (v v v ε (f f f ε )) defined by [START_REF] Murat | Effets non locaux dans le passage 3d-1d en élasticité linéarisée anisotrope hétérogène[END_REF] two-scale converges with respect to (m ε ) to f f f 0 . Furthermore, the sequence

(γ γ γ Σ (v v v ε (f f f ε ))) strongly converges in L 2 (Σ; R k ) to γ γ γ Σ (f f f ).
Proof. By the first line of (42), we have

Σ |γ γ γ Σ (v v v ε (f f f ε )) -γ γ γ Σ (f f f )| 2 dH 2 ≤ C Σε |γ γ γ Σ (v v v ε (f f f ε )) -γ γ γ Σ (f f f ε )| 2 dH 2 + C Σε |γ γ γ Σ (f f f ε ) -γ γ γ Σ (f f f )| 2 dH 2 + C Σ\Σε |γ γ γ Σ (f f f )| 2 dH 2 ≤ Cε ε/2 -ε/2 Σ |∇ ∇ ∇f f f ε | 2 dx + C Σ |γ γ γ Σ (f f f ε ) -γ γ γ Σ (f f f )| 2 dH 2 + C Σ\Σε |γ γ γ Σ (f f f )| 2 dH 2 . ( 52
) Since (f f f ε ) weakly converges to f f f in H 1 (Ω; R k ), the sequence (γ γ γ Σ (f f f ε )) strongly converges to γ γ γ Σ (f f f ) in L 2 (Σ; R k ). Since the sequence Ä |γ γ γ Σ (f f f )| 2 1 Σ\Σ ε ä is dominated by |γ γ γ Σ (f f f )| 2 ∈ L 1 (Σ) and H 2 -a.e. converges
to 0 on Σ, by the Dominated Convergence Theorem, there holds:

lim ε→0 Σ\Σ ε |γ γ γ Σ (f f f )| 2 dH 2 = 0. It then follows from (52) that the sequence (γ γ γ Σ (v v v ε (f f f ε ))) strongly converges in L 2 (Σ; R k ) to γ γ γ Σ (f f f ). Given ψ ψ ψ ∈ C(Ω × S; R k ), we consider the fields ψ ψ ψ Σ ε ∈ L ∞ (Σ; R k ) and ψ ψ ψ Σ ∈ C(Σ; R k ) defined by ψ ψ ψ Σ ε (x 2 , x 3 ) := i∈Iε Ç - S i ε ψ ψ ψ Å s 1 , s 2 , x 3 , y ε (s) ε ã ds 1 ds 2 å 1 (εi-ε 2 ,εi+ ε 2 ) (x 2 ), ψ ψ ψ Σ (x 2 , x 3 ) := - S ψ ψ ψ(0, x 2 , x 3 , y)dy. (53)
By the change of variables formula, there holds (see [START_REF] Marigo | The effective behavior of elastic bodies containing microcracks or microholes localized on a surface[END_REF])

ψ ψ ψ Σ (x 2 , x 3 ) = - S i ε ψ ψ ψ Å 0, x 2 , x 3 , y ε (s) ε ã ds ∀i ∈ I ε .
Therefore, for all (x 2 , x 3 ) ∈ εi -ε 2 , εi + ε 2 × (-L, L) we have

ψ ψ ψ Σ (x 2 , x 3 ) -ψ ψ ψ Σ ε (x 2 , x 3 ) ≤ sup (s,y)∈S i ε ×S
|ψ ψ ψ(0, x 2 , x 3 , y) -ψ ψ ψ(s, x 3 , y)|.

By (53) we have

ψ ψ ψ Σ ε L ∞ (Σ;R k ) ≤ ||ψ ψ ψ|| L ∞ (Ω×S;R k ) , hence the sequence (ψ ψ ψ Σ ε ) is bounded in L ∞ (Σ; R k ).
On the other hand, one can check by using the uniform continuity of ψ ψ ψ on Ω × S, that the sequence (ψ ψ ψ Σ ε ) uniformly converges to ψ ψ ψ Σ on each compact subset of Σ. We deduce that

ψ ψ ψ Σ ε → ψ ψ ψ Σ strongly in L q (Σ; R k ) ∀q ∈ [1, +∞). ( 54
)
By ( 25) and (53) there holds

v v v ε (f f f ε ).ψ ψ ψ Å x, y ε (x ) ε ã dm ε = 1 ε|S| i∈Iε L -L dx 3 S i ε Ç - S i ε f f f ε (s 1 , s 2 , x 3 )ds å .ψ ψ ψ Å x, y ε (x ) ε ã dx 1 dx 2 = 1 ε|S| i∈Iε L -L dx 3 Ç - S i ε f f f ε (s 1 , s 2 , x 3 )ds å . S i ε ψ ψ ψ Å s 1 , s 2 , x 3 , y ε (s) ε ã ds = i∈Iε (εi-ε 2 ,εi+ ε 2 )×(-L,L) γ γ γ Σ (v v v ε (f f f ε )).ψ ψ ψ Σ ε dH 2 = Σ γ γ γ Σ (v v v ε (f f f ε )).ψ ψ ψ Σ ε dH 2 .
(55)

By passing to the limit as ε → 0, thanks to the strong convergences of (γ

γ γ Σ (v v v ε (f f f ε ))) to γ Σ (f f f ) in L 2 (Σ; R k ) and of (ψ ψ ψ Σ ε ) to ψ ψ ψ Σ in L 2 (Σ; R k ) (see (54)), we get lim ε→0 v v v ε (f f f ε ).ψ ψ ψ Å x, y ε (x ) ε ã dm ε = Σ γ γ γ Σ (f f f ).ψ ψ ψ Σ dH 2 . ( 56
)
On the other hand, by (53) there holds

Σ γ γ γ Σ (f f f ).ψ ψ ψ Σ dH 2 = 1 |S| Σ×S γ γ γ Σ (f f f )(x).ψ ψ ψ(x, y)dH 2 (x)dy. ( 57
)
By the arbitrary choice of ψ ψ ψ, we infer from ( 56) and (57) that the sequence (v v v ε (f f f ε )) two-scale converges with respect to (m ε ) to the field f f f 0 ∈ L 2 (Σ; R k ) defined by (51).

The proof of the first statement is achieved provided we show that (f f f ε ) two-scale converges to f f f 0 with respect to (m ε ). To that aim, let us fix ψ ψ ψ ∈ C(Ω × S; R k ). By ( 25), Hölder's inequality and Jensen's inequality, we have

(f f f ε -v v v ε (f f f ε )).ψ ψ ψ Å x, y ε (x ) ε ã dm ε 2 ≤ C |f f f ε -v v v ε (f f f ε )| 2 dm ε ≤ C 1 ε|S| i∈Iε L -L dx 3 S i ε f f f ε -- S i ε f f f ε (s 1 , s 2 , x 3 )ds 2 dx 1 dx 2 .
(58) By making suitable changes of variables in the following Poincaré-Wirtinger inequality

S ϕ ϕ ϕ -- S ϕ ϕ ϕds 2 dx ≤ C S |∇ ∇ ∇ϕ ϕ ϕ| 2 dx ∀ϕ ϕ ϕ ∈ H 1 (S; R k ),
we deduce that for a. e. x 3 ∈ (-L, L), there holds

S i ε f f f ε -- S i ε f f f ε (s 1 , s 2 , x 3 )ds 2 dx 1 dx 2 ≤ Cε 2 S i ε |∇ ∇ ∇f f f ε (x 1 , x 2 , x 3 )| 2 dx 1 dx 2 . (59)
Joining ( 58) and (59), we infer

(f f f ε -v v v ε (f f f ε )).ψ ψ ψ Å x, y ε (x ) ε ã dm ε 2 ≤ Cε Tε |∇ ∇ ∇f f f ε | 2 dx. (60)
We deduce from (56), ( 57) and (60) that

lim ε→0 f f f ε .ψ ψ ψ Å x, y ε (x ) ε ã dm ε = 1 |S| Σ×S γ γ γ Σ (f f f )(x).ψ ψ ψ(x, y)dH 2 (x)dy.
Therefore, by the arbitrary choice of ψ ψ ψ, the sequence (f f f ε ) two-scale converges with respect to (m ε ) to f f f 0 .

A priori estimates, identification relations

The following section is mainly devoted to the study of the asymptotic behaviour of the solution (u u u ε ) of ( 7) and of the sequences (v v v ε (u u u ε )m ε ) and (u u u ε m ε ). The main results of this section are stated in Proposition 4. Their proofs rely on some basic inequalities established in the previous section in Lemma 1 and also on those established in the next lemma.

Lemma 2

The following estimates hold for all i ∈ I ε :

T i ε |εϕ 1 | 2 + |εϕ 2 | 2 + |ϕ 3 | 2 dx ≤ C T i ε |e e e(ϕ ϕ ϕ)| 2 dx ∀ ϕ ϕ ϕ ∈ H 1 0 (Ω; R 3 ), T i ε |εv ε1 (ϕ ϕ ϕ)| 2 + |εv ε2 (ϕ ϕ ϕ)| 2 + |v ε3 (ϕ ϕ ϕ)| 2 dx ≤ C T i ε |e e e(ϕ ϕ ϕ)| 2 dx ∀ ϕ ϕ ϕ ∈ H 1 0 (Ω; R 3 ). (61)
Proof. We consider the set

V := {ψ ψ ψ ∈ H 1 (T, R 3 ), ψ ψ ψ = 0 on S × {-L}}.
The space V is a closed linear subset of H 1 (T, R 3 ) satisfying V ∩ R = {0}, where R denotes the space of rigid displacements, therefore by Korn inequality there holds

T |ψ ψ ψ| 2 dz ≤ C T |e e e(ψ ψ ψ)| 2 dz ∀ ψ ψ ψ ∈ V. (62)
Let us fix ϕ ϕ ϕ ∈ H 1 0 (Ω; R 3 ). By applying (62) to the field ψ ψ ψ defined by

ψ α (z 1 , z 2 , z 3 ) := ϕ α (ε(z 1 - (y y y i ε (z)) 1 ), ε(z 2 -(y y y i ε (z)) 2 ), z 3 ) for α ∈ {1, 2} and 3 (z) := 1 ε ϕ 3 (ε(z 1 -(y y y i ε (z)) 1 ), ε(z 2 -(y y y i ε (z))
2 ), z 3 ), and then by making a suitable change of variable, we get

T i ε |εϕ 1 | 2 + |εϕ 2 | 2 + |ϕ 3 | 2 dx ≤ C T i ε |e e e(ϕ ϕ ϕ)| 2 dx. (63)
On the other hand, it easily follows from ( 25), Fubini Theorem and Jensen's inequality that

T i ε |εv ε1 (ϕ ϕ ϕ)| 2 + |εv ε2 (ϕ ϕ ϕ)| 2 + |v ε3 (ϕ ϕ ϕ)| 2 dx ≤ C T i ε |εϕ 1 | 2 + |εϕ 2 | 2 + |ϕ 3 | 2 dx. (64)
The estimates (61) result from ( 63) and (64).

The next proposition specifies the asymptotic behavior of the solution u u u ε of (7) and of some associated auxiliary sequences. The strong relative compactness of (u u u ε ) in H 1 0 (Ω; R 3 ) is established at the end of the paper in Section 5.2.

Proposition 4 Let u u u ε be the solution of [START_REF] Bellieud | Bouchitté: Homogenization of a soft elastic material reinforced by fibres[END_REF] and let v v v ε (u u u ε ) be defined by [START_REF] Murat | Effets non locaux dans le passage 3d-1d en élasticité linéarisée anisotrope hétérogène[END_REF]. Then: (i) the next estimates hold true

|u u u ε | 2 H 1 0 (Ω;R 3 ) ≤ C, |e e e(u u u ε )| 2 dm ε ≤ Cε m-1 , {|u u u ε | 2 + |v v v ε (u u u ε )| 2 }dm ε ≤ C, {|u ε1 | 2 + |u ε2 | 2 + u ε3 ε 2 + |v ε1 (u u u ε )| 2 + |v ε2 (u u u ε )| 2 + v ε3 (u u u ε ) ε 2 }dm ε ≤ Cε m-3 , |u u u ε -v v v ε (u u u ε )| 2 dm ε ≤ Cε, Σ |γ γ γ Σ (v v v ε (u u u ε ))| 2 dH 2 ≤ C. (65)
(ii) If m ≥ 1, the next convergences take place, up to a subsequence, for some

u u u ∈ H 1 0 (Ω; R 3 ), u u u 0 , v v v 0 ∈ L 2 (Σ × S; R 3 ), Ξ Ξ Ξ 0 ∈ L 2 (Σ × S; S 3 ): u u u ε u u u weakly in H 1 0 (Ω; R 3 ), γ γ γ Σ (u u u ε ) → γ γ γ Σ (u u u), γ γ γ Σ (v v v ε (u u u ε )) → γ γ γ Σ (u u u) strongly in L 2 (Σ; R 3 ), u u u ε mε u u u 0 , v v v ε (u u u ε ) mε v v v 0 , e e e(u u u ε ) mε Ξ Ξ Ξ 0 two-scale with respect to (m ε ), (66)
where the symbol mε is defined by [START_REF] Neukamm | Rigorous derivation of a homogenized bending-torsion theory for inextensible rods from 3d elasticity[END_REF] and the following identification relations hold true:

v v v 0 (x, y) = u u u 0 (x, y) = γ γ γ Σ (u u u)(x) for H 2 ⊗ L 2 -a.e. (x, y) ∈ Σ × S, (67)
∂γ Σ (u 3 ) ∂x 3 ∈ L 2 (Σ), γ Σ (u 3 ) = 0 on {x ∈ Σ, x 3 ∈ {-L, L}}, ∂γ Σ (u 3 ) ∂x 3 (x) = (Ξ Ξ Ξ 0 ) 33 (x, y) for H 2 ⊗ L 2 -a.e. (x, y) ∈ Σ × S, (68)
γ Σ (u 3 ) = 0 if m > 1. ( 69
)
(iii) If m ≥ 3, then besides (66), the next convergences take place up to a subsequence, for some

w 0 ∈ L 2 (Σ × S), Υ Υ Υ 0 ∈ L 2 (Σ × S; S 3 ), w ∈ L 2 (Σ): u ε3 ε mε w 0 , 1 ε
e e e(u u u ε ) mε Υ Υ Υ 0 two-scale with respect to (m ε ),

γ Σ (v ε3 (u u u ε )) ε w weakly in L 2 (Σ). ( 70
)
Moreover there holds

γ Σ (u 3 ) = 0, ∂ 2 γ γ γ Σ (u u u) ∂x 2 3 ∈ L 2 (Σ; R 3 ), γ γ γ Σ (u u u) = ∂γ γ γ Σ (u u u) ∂x 3 = 0 on ∂Σ ∩ {x ∈ R 3 , x 3 ∈ {-L, L}}, w 0 (x, y) = w(x) - 2 α=1 ∂γ Σ (u α ) ∂x 3 (x)y α , in Σ × S, ∂w ∂x 3 ∈ L 2 (Σ), Υ 033 = ∂w ∂x 3 - ∂ 2 γ Σ (u 1 ) ∂x 2 3 y 1 - ∂ 2 γ Σ (u 2 ) ∂x 2 3 y 2 .
(71)

Furthermore,

γ γ γ Σ (u u u) = 0 if m > 3. ( 72
)
(iv) We have u u u ∈ D m , where D m is given by ( 9), ( 11), [START_REF] Cioranescu | An introduction to homogenization[END_REF], or [START_REF] Bessoud | Asymptotic analysis of shell-like inclusions with high rigidity[END_REF], in accordance with the choice of m.

Proof. (i) We multiply [START_REF] Bellieud | Bouchitté: Homogenization of a soft elastic material reinforced by fibres[END_REF] by u u u ε , integrate by parts over Ω, and use Cauchy Schwarz inequality to find

Ω σ σ σ ε : e e e(u u u ε )dx = Ω f f f .u u u ε ≤ Å Ω |f f f | 2 dx ã 1 2 Å Ω |u u u ε | 2 dx ã 1 2 . ( 73
)
By applying Poincaré and Korn inequalities in H 1 0 (Ω; R 3 ), we get

Ω |u u u ε | 2 dx ≤ C Ω |e e e(u u u ε )| 2 dx ≤ C Ω\Tε |e e e(u u u ε )| 2 dx + C ε m Tε |e e e(u u u ε )| 2 dx ≤ C Ω σ σ σ ε : e e e(u u u ε )dx. ( 74
)
We infer from ( 73) and (74) that

|u u u ε | 2 H 1 0 (Ω;R 3 ) ≤ C, |e e e(u u u ε )| 2 dm ε ≤ Cε m-1 . ( 75
)
By the third line of (42), the first line of (43), (74), and the continuity of the trace application from H 1 (Ω; R 3 ) to L 2 (Ω; R 3 ), there holds

|u u u ε | 2 dm ε ≤ C |u u u ε -Û u u u ε | 2 dm ε + C |Û u u u ε | 2 dm ε ≤ Cε Σε |∇ ∇ ∇u u u ε | 2 dx + C Σ |γ γ γ Σ (u u u ε )| 2 dH 2 ≤ C|u u u ε | 2 H 1 (Ω;R 3 ) ≤ C. ( 76
)
We deduce from ( 61) and (75) that

{|u ε1 | 2 + |u ε2 | 2 + u ε3 ε 2 }dm ε ≤ C ε 2 |e e e(u u u ε )| 2 dm ε = Cε m-3 (77)
and that

{|v ε1 (u u u ε )| 2 + |v ε2 (u u u ε )| 2 + v ε3 (u u u ε ) ε 2 }dm ε ≤ C ε 2 |e e e(u u u ε )| 2 dm ε = Cε m-3 . ( 78
)
By ( 42) and (75), we have

|u u u ε -v v v ε (u u u ε )| 2 dm ε ≤ Cε ε/2 -ε/2 Σ |∇ ∇ ∇u u u ε | 2 dx ≤ Cε|u u u ε | 2 H 1 0 (Ω;R 3 ) ≤ Cε. ( 79
)
By the first line of ( 42), ( 74), ( 75), (79), and by the continuity of the trace application from H 1 (Ω; R 3 ) to L 2 (Ω; R 3 ), there holds

Σ |γ γ γ Σ (v v v ε (u u u ε ))| 2 dH 2 = Σ ε |γ γ γ Σ (v v v ε (u u u ε ))| 2 dH 2 ≤ C Σ ε |γ γ γ Σ (u u u ε ) -γ γ γ Σ (v v v ε (u u u ε ))| 2 dH 2 + C Σ |γ γ γ Σ (u u u ε )| 2 dH 2 ≤ Cε + C|u u u ε | 2 H 1 (Ω;R 3 ) ≤ C. (80)
Collecting ( 75)-( 80), the estimates (65) are proved.

(ii) By ( 65), the sequence (u u u ε ) is bounded in H 1 0 (Ω; R 3 ) hence weakly converges, up to a subsequence, to some u u u ∈ H 1 0 (Ω; R 3 ). From the compactness of the trace operator from H 1 0 (Ω; R 3 ) to L 2 (Σ; R 3 ), we deduce that the sequence (γ γ γ Σ (u u u ε )) strongly converges to γ γ γ Σ (u u u) in L 2 (Σ; R 3 ), and from Proposition 3 that the sequence (γ γ γ Σ (v v v ε (u u u ε ))) strongly converges to γ γ γ Σ (u u u) in L 2 (Σ; R 3 ). The convergences stated up to a subsequence in the last line of (66) result from Proposition 1 (ii) and from the fact that by (65), the estimate sup

ε>0 |f f f ε | 2 dm ε < +∞ takes place for f f f ε ∈ {u u u ε , v v v ε (u u u ε )
, e e e(u u u ε )}. Assertion (67) is a consequence of (66) and Proposition 3. Let us fix ψ ∈ C ∞ (Ω; D(S)). Since u ε ∈ H 1 0 (Ω; R 3 ), we have

∂u ε3 ∂x 3 ψ Å x, y ε (x ) ε ã dm ε = -u ε3 ∂ψ ∂x 3 Å x, y ε (x ) ε ã dm ε .
By passing to the limit as ε → 0, thanks to the two-scale convergences with respect to (m ε ) of (e e e(u u u ε )) to Ξ Ξ Ξ 0 and of (u u u ε ) to γ γ γ Σ (u u u) (see (66), ( 67)), we infer

1 |S| Σ×S (Ξ Ξ Ξ 0 ) 33 (x, y)ψ(x, y)dH 2 (x)dy = - 1 |S| Σ×S γ Σ (u 3 )(x) ∂ψ ∂x 3 (x, y)dH 2 (x)dy. ( 81
)
Since this equality is satisfied in particular for all ψ ∈ D(Ω × S), we deduce

∂γ Σ (u 3 ) ∂x 3 (x) = (Ξ Ξ Ξ 0 ) 33 (x, y) for H 2 ⊗ L 2 -a.e. (x, y) ∈ Σ × S. ( 82
) As Ξ Ξ Ξ 0 ∈ L 2 (Σ × S; S 3 ), we infer that ∂γ Σ (u3) ∂x3 ∈ L 2 (Σ). Choosing then ψ ∈ C ∞ (Ω; D(S)
) and integrating (81) by parts with respect to x 3 , taking (82) into account, we obtain {x∈Σ, x3∈{-L,L}}×S γ Σ (u 3 )(x)ψ(x, y)dH 1 (x)dy = 0, yielding, by the arbitrary choice of ψ,

γ Σ (u 3 ) = 0 on {x ∈ Σ, x 3 ∈ {-L, L}}.
Assertion (68) is proved. Assertion (69) follows from the second line of (65) and the second line of (66).

(iii) If m ≥ 3, then by (65) we have

sup ε>0 v ε3 (u u u ε ) ε 2 dm ε < +∞, sup ε>0 u ε3 ε 2 dm ε < +∞, sup ε>0 1 ε e e e(u u u ε ) 2 dm ε < +∞.
The convergences stated, up to a subsequence, in the first line of (70), then follow from Proposition 1 (ii). As regards those stated in the second line of (70), they result from the fact that by (44) and by the second line of (65), there holds

Σ γ Σ (v ε3 (u u u ε )) ε 2 dH 2 ≤ C Ç v ε3 (u u u ε ) ε 2 dm ε å ≤ C.
We deduce from the two-scale convergence of uε3 ε to w 0 with respect to (m ε ) that (u ε3 ) two-scale converges to 0 with respect to (m ε ). It follows then from Proposition 3 that

γ Σ (u 3 ) = 0. ( 83
)
To prove (71), we fix a test field Ψ Ψ Ψ ∈ C ∞ (Ω, D(S; S 3 )) such that Ψ αβ = 0 for all (α, β) ∈ {1, 2} 2 . By integration by parts, we have:

ε Tε 1 ε e e e(u u u ε )(x) : Ψ Ψ Ψ Å x, y ε (x ) ε ã dm ε = 3 i=1 -ε Tε u ε3 (x) ε ∂Ψ 3i ∂x i Å x, y ε (x ) ε ã dm ε - 2 α=1 Tε u εα (x) ∂Ψ α3 ∂x 3 Å x, y ε (x ) ε ã dm ε - 2 α=1 Tε u ε3 (x) ε ∂Ψ 3α ∂y α Å x, y ε (x ) ε ã dm ε . (84)
By passing to the limit as ε → 0, thanks to the two-scale convergences stated in (70), we obtain

2 α=1 - Σ×S γ γ γ Σ (u α )(x) ∂Ψ α3 ∂x 3 (x, y)dH 2 (x)dy - Σ×S w 0 (x, y) ∂Ψ 3α ∂y α (x, y)dH 2 (x)dy = 0. ( 85
)
Fixing α ∈ {1, 2} and choosing Ψ Ψ Ψ such that Ψ ij = 0 if {i, j} = {α, 3}, we get

- Σ×S γ γ γ Σ (u α )(x) ∂Ψ α3 ∂x 3 (x, y)dH 2 (x)dy - Σ×S w 0 (x, y) ∂Ψ 3α ∂y α (x, y)dH 2 (x)dy = 0. ( 86
)
Choosing at first arbitrary fields Ψ α3 in D(Ω; D(S)), we deduce that

∂w 0 ∂y α (x, y) = - ∂γ γ γ Σ (u α ) ∂x 3 (x) in D (Σ × S), (87)
hence w 0 , as a distribution on Σ × S, can be written for a suitable c ∈ D (Σ) under the following form:

w 0 (x, y) = -y 1 ∂γ Σ (u 1 ) ∂x 3 (x) -y 2 ∂γ Σ (u 2 ) ∂x 3 (x) + c(x). (88)
As w 0 belongs to L 2 (Σ × S), we infer

c ∈ L 2 (Σ), ∂γ Σ (u α ) ∂x 3 ∈ L 2 (Σ) (α ∈ {1, 2}),
and (87) holds a.e. on Σ × S. We then deduce from Proposition 1 (iii) b), ( 70) and (88) that

w(x) = - S w 0 (x, y)dy = c(x). (89)
Next, we multiply (84) by 1 ε and choose a field Ψ Ψ Ψ such that Ψ ij = 0 if (i, j) = (3, 3). Passing to the limit as ε → 0, taking into account the two-scale convergences with respect to (m ε ) of 1 ε e e e(u u u ε ) to Υ Υ Υ 0 and of uε3 ε to w 0 given by ( 88) and (89), we obtain

Σ×S Υ 033 (x, y)Ψ 33 (x, y)dH 2 (x)dy = - Σ×S Å -y 1 ∂γ Σ (u 1 ) ∂x 3 (x) -y 2 ∂γ Σ (u 2 ) ∂x 3 (x) + w(x) ã ∂Ψ 33 ∂x 3 (x, y)dH 2 (x)dy. (90)
Choosing at first an arbitrary Ψ 33 ∈ D(Ω; D(S)), we conclude that

∂w ∂x 3 ∈ L 2 (Σ), ∂ 2 γ γ γ Σ (u u u) ∂x 2 3 ∈ L 2 (Σ; R 3 ), (Υ Υ Υ 0 ) 33 = ∂w ∂x 3 (x) - 2 α=1 ∂ 2 γ Σ (u α ) ∂x 2 3 (x)y α , in Σ × S, (91)
then, choosing an arbitrary Ψ 33 ∈ C ∞ (Ω; D(S)) in (90) and integrating by parts, we obtain

0 = ∂Σ∩{x∈R 3 , x3∈{-L,L}}×S Å -y 1 ∂γ Σ (u 1 ) ∂x 3 (x) -y 2 ∂γ Σ (u 2 ) ∂x 3 (x) + w(x) ã Ψ 33 (x, y)dH 1 (x)dy, (92)
yielding

w = 0 on ∂Σ ∩ {x ∈ R 3 , x 3 ∈ {-L, L}}, ∂γ γ γ Σ (u u u) ∂x 3 = 0 on ∂Σ ∩ {x ∈ R 3 , x 3 ∈ {-L, L}}. (93)
Assertion ( 71) is proved. Finally, if m > 3, then by the second line of (65) the sequence |u u u ε | 2 dm ε converges to 0. We deduce from Proposition 2 (applied to j(.) := |.| 2 ) that u u u 0 = 0. The assertion (72) then results from (67). (iv) Assertion (iv) follows from (66), (71), and (72).

5 Proof of Theorem 1

Weak convergence

Let us briefly outline the proof of Theorem 1. In the spirit of Tartar's method [START_REF] Tartar | Cours Peccot[END_REF], we will multiply (7) by an appropriate sequence of oscillating test fields (φ φ φ ε ) and, by passing to the limit as ε → 0 in accordance with the convergences established in proposition 4, obtain a variational formulation of the limit problem satisfied by u u u. The test field φ φ φ ε is constructed in terms of some fixed φ φ φ ∈ D(Ω; R 3 ) such that φ φ φ ∈ D m (see [START_REF] Bellieud | Homogenization of an elastic material reinforced by very stiff or heavy fibers. Non local effects. Memory effects[END_REF], ( 11), ( 16), ( 18)). We fix a smooth domain S of R 2 such that

S ⊂ S ⊂ S ⊂ Y, (94)
and

a function ∈ D(Y ) such that = 0 in Y \ S , = 1 in S, 0 ≤ ≤ 1.
The function ε defined on Ω by (see [START_REF] Marigo | The effective behavior of elastic bodies containing microcracks or microholes localized on a surface[END_REF])

ε (x) := i∈Iε Å y ε (x ) ε ã 1 Y i ε (x ), satisfies ε ∈ C ∞ (Ω), ε = 0 in Ω \ T ε , ε = 1 in T ε , |∇ ε | ≤ C ε , (95)
where T ε is defined by substituting S for S in (2). The test field will then be defined by

φ φ φ ε := (1 -ε )φ φ φ + ε χ χ χ ε , (96)
in terms of χ χ χ ε given by ( 102), (110) (assuming φ 3 = 0 if 1 < m < 3), or (120), depending on the choice of m.

We multiply equation ( 7) by φ φ φ ε and integrate it by parts over Ω. We get (see [START_REF] Grisvard | Singularities in Boundary Values Problems[END_REF]):

a ε,m (u u u ε , φ φ φ ε ) = Ω f f f .φ φ φ ε dx. (97)
It is easy to check that lim

ε→0 Ω f f f .φ φ φ ε dx = Ω f f f .φ φ φdx. (98)
In order to compute the limit of the left hand side of (97), we split it into a sum of three terms:

a ε,m (u u u ε , φ φ φ ε ) = I 1ε + I 2ε + I 3ε ; I 1ε = Ω\T ε E 1 + ν Å ν 1 -2ν
tr(e e e(u u u ε ))I I I + e e e(u u u ε )

ã : e e e(φ φ φ)dx,

I 2ε = T ε \Tε E 1 + ν Å ν 1 -2ν
tr(e e e(u u u ε ))I I I + e e e(u u u ε )

ã : e e e(φ φ φ ε )dx,

I 3ε = |S| ε m-1 E T 1 + ν Å ν 1 -2ν
tr(e e e(u u u ε ))I I I + e e e(u u u ε )

ã : e e e(χ χ χ ε )dm ε .

(99)

After possibly extracting a subsequence, we can assume that the convergences stated in Proposition 4 take place. In particular, the sequence E

The χ χ χ ε will be chosen such that

|φ φ φ -χ χ χ ε | L ∞ (T ε ) ≤ Cε,
≤ lim sup ε→0 C|T ε \ T ε | 1 2 Ç T ε \Tε | E 1 + ν Å ν 1 -2ν
tr(e e e(u u u ε ))I I I + e e e(u u u ε )

ã | 2 dx å 1 2 ≤ C lim sup ε→0 C|T ε \ T ε | 1 2 = 0. ( 101
)
We distinguish then different cases.

Case m = 1. We set (see [START_REF] Murat | Effets non locaux dans le passage 3d-1d en élasticité linéarisée anisotrope hétérogène[END_REF] and Remark 2 below)

χ χ χ ε := v v v ε (φ φ φ) -ε Ñ 0 0 ∂vε1(φ φ φ) ∂x3 yε1(x ) ε + ∂vε2(φ φ φ) ∂x3 yε2(x ) ε é + εq q q Å x, y ε (x ) ε ã , (102)
where y y y ε (x) is given by ( 26) and q q q(x, y) :=

Ö -ν 1 ∂vε3(φ φ φ) ∂x3 y 1 -ν 1 ∂vε3(φ φ φ) ∂x3 y 2 0 è . (103)
By ( 25) and (103) there holds in T ε :

e e e(χ χ χ

ε ) = ∂v ε3 (φ φ φ) ∂x 3 Ñ -ν 1 0 0 0 -ν 1 0 0 0 1 é + εe e e x (q q q) Å x, y ε (x ) ε ã , (104)
hence, by ( 7) and ( 99)

I 3ε = |S|E T ∂u ε3 ∂x 3 ∂v ε3 (φ φ φ) ∂x 3 dm ε +|S|ε E T 1 + ν 1 Å ν 1 1 -2ν 1
tr(e e e(u u u ε ))I I I + e e e(u u u ε )

ã :e e e x (q q q) Å x, y ε (x ) ε ã dm ε .

(105) By Proposition 4, we have

ε E T 1 + ν 1 Å ν 1 1 -2ν 1
tr(e e e(u u u ε ))I I I + e e e(u u u ε )

ã :e e e x (q q q) Å x,

y ε (x ) ε ã dm ε ≤ Cε | E T 1 + ν 1 Å ν 1
where a 1 (., .) is the bilinear form on D 1 × D 1 defined by [START_REF] Lukkassen | Wall: Two-scale convergence[END_REF]. Joining (98) and (108), we obtain

a 1 (u u u, φ φ φ) = Ω f f f .φ φ φdx ∀φ φ φ ∈ D(Ω; R 3 ). ( 109
)
This variational formulation is equivalent to [START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF].

Case m = 3. We choose φ 3 = 0 and set (see Remark 2)

χ χ χ ε (x) = Ñ v ε1 (φ φ φ)(x) v ε2 (φ φ φ)(x) 0 é + ε Ñ 0 0 -∂vε1(φ φ φ) ∂x3 yε(x )1 ε -∂vε2(φ φ φ) ∂x3 yε(x )2 ε é + ε 2 q q q Å x, y ε (x ) ε ã , (110)
where

q q q (x, y) = ν 1 à ∂ 2 v ε1 (φ φ φ) ∂x 2 3 y 2 1 -y 2 2 2 + ∂ 2 v ε2 (φ φ φ) ∂x 2 3 y 1 y 2 ∂ 2 v ε2 (φ φ φ) ∂x 2 3 y 2 2 -y 2 1 2 + ∂ 2 v ε1 (φ φ φ) ∂x 2 3 y 1 y 2 0 í .
We have in T ε e e e(χ χ χ

ε ) = ε Å ∂ 2 v ε1 (φ φ φ) ∂x 2 3 y ε (x ) 1 ε + ∂ 2 v ε2 (φ φ φ) ∂x 2 3 y ε (x ) 2 ε ã Ñ ν 1 0 0 0 ν 1 0 0 0 -1 é + ε 2 e e e x (q q q) Å x, y ε (x ) ε ã , (111)
hence,

E T 1 + ν 1 Å ν 1 1 -2ν 1
tr(e e e(χ χ χ ε ))I I I + e e e(χ χ χ ε )

ã = -ε Å ∂ 2 v ε1 (φ φ φ) ∂x 2 3 y ε (x ) 1 ε + ∂ 2 v ε2 (φ φ φ) ∂x 2 3 y ε (x ) 2 ε ã E T (e e e 3
⊗ e e e 3 )

+ ε 2 E T 1 + ν 1 Å ν 1 1 -2ν 1 tr
Å e e e x (q q q) Å x, y ε (x ) ε ãã I I I + e e e x (q q q) Å x, y ε (x ) ε ãã .

Taking (20) into account, we infer

1 ε m a T (u u u ε , χ χ χ ε ) = 1 ε 3 a T (χ χ χ ε , u u u ε ) = 1 ε 3 Tε ß E T 1 + ν 1 Å ν 1 1 -2ν 1
tr(e e e(χ χ χ ε ))I I I + e e e(χ χ ε )

ã :e e e(u u u ε )

™ dx = 1 ε 3 ε|S| ß E T 1 + ν 1 Å ν 1 1 -2ν 1
tr(e e e(χ χ χ ε ))I I I + e e e(χ χ χ ε )

ã :e e e(u u u ε )

™ dm ε = -|S|E T Å ∂ 2 v ε1 (φ φ φ) ∂x 2 3 y ε (x ) 1 ε + ∂ 2 v ε2 (φ φ φ) ∂x 2 3 y ε (x ) 2 ε ã Å 1 ε ∂u ε3 ∂x 3 ã dm ε
+ |S|e e e(u u u ε ) :

E T 1 + ν 1 Å ν 1 1 -2ν 1 tr
Å e e e x (q q q) Å x, y ε (x ) ε ãã I I I + e e e x (q q q) Å x, y ε (x ) ε

ãã dm ε . (112)
Let us remark that e e e x (q q q) Ä x, yε(x ) ε ä is uniformly bounded on T ε and so we have, by the estimate (65) established in Proposition 4, e e e(u u u ε ) :

E T 1 + ν 1 Å ν 1 1 -2ν 1 tr
Å e e e x (q q q) Å x, y ε (x ) ε ãã I I I + e e e x (q q q) Å x,

y ε (x ) ε ãã dm ε ≤ C |e e e(u u u ε )|dm ε ≤ C |e e e(u u u ε )| 2 dm ε ≤ C √ ε m-1 ≤ Cε → 0. (113)
By [START_REF] Murat | Effets non locaux dans le passage 3d-1d en élasticité linéarisée anisotrope hétérogène[END_REF] and by the uniform continuity of φ φ φ on Ω, the next estimate takes place:

∂ 2 v εα (φ φ φ) ∂x 2 3 - ∂ 2 φ α ∂x 2 3 L ∞ (Tε) ≤ Cε ∀ α ∈ {1, 2}.
Since, on the other hand, by Proposition 4 there holds

1 ε ∂uε3 ∂x3 2 dm ε ≤ C, taking (24) into account, we deduce that Å Å ∂ 2 v ε1 (φ φ φ) ∂x 2 3 y ε1 (x ) ε + ∂ 2 v ε2 (φ φ φ) ∂x 2 3 y ε2 (x ) ε ã - Å ∂ 2 φ 1 ∂x 2 3 y ε1 (x ) ε + ∂ 2 φ 2 ∂x 2 3 y ε2 (x ) ε ããÅ 1 ε ∂u ε3 ∂x 3 ã dm ε ≤ Cε 1 ε ∂u ε3 ∂x 3 dm ε ≤ Cε Ç 1 ε ∂u ε3 ∂x 3 2 dm ε å 1 2 (m ε (Ω)) 1 2 ≤ Cε. (114)
By Proposition 4 (see the first line of (70) and the last line of (71)), the sequence

Ä 1 ε ∂uε3 ∂x3 ä two-scale converges with respect to (m ε ) to ∂w ∂x3 (x)-∂ 2 γ Σ (u1) ∂x 2 3 (x)y 1 -∂ 2 γ Σ (u2) ∂x 2 3
(x)y 2 . Therefore, by ( 1), [START_REF] Cioranescu | The periodic unfolding method in homogenization[END_REF], and ( 27), we have

lim ε→0 Å ∂ 2 φ 1 ∂x 2 3 y ε1 (x ) ε + ∂ 2 φ 2 ∂x 2 3 y ε2 (x ) ε ãÅ 1 ε ∂u ε3 ∂x 3 ã dm ε = 1 |S| Σ×S -|S| E T 2 Å ∂ 2 φ 1 ∂x 2 3 y 1 + ∂ 2 φ 2 ∂x 2 3 y 2 ã Å ∂w ∂x 3 (x)- ∂ 2 γ Σ (u 1 ) ∂x 2 3 (x)y 1 - ∂ 2 γ Σ (u 2 ) ∂x 2 3 (x)y 2 ã dH 2 (x)dy = |S|E T 2 α,β=1 Σ J αβ ∂ 2 φ α ∂x 2 3 ∂ 2 γ Σ (u β ) ∂x 2 3 dH 2 (x). (115)
We deduce from (112), ( 113), (114), and (115) that

lim ε→0 I 3ε = |S|E T 2 α,β=1 Σ J αβ ∂ 2 φ α ∂x 2 3 ∂ 2 γ Σ (u β ) ∂x 2 3 dH 2 (x). ( 116
)
By passing to the limit as ε → 0 in (97), taking (98), (100), (101), and (116) into account, we obtain the variational formulation

a 0 (u u u, φ φ φ) + |S||E T 2 α,β=1 Σ J αβ ∂ 2 φ α ∂x 2 3 ∂ 2 γ Σ (u β ) ∂x 2 3 dH 2 (x) = Ω f f f .φ φ φdx, (117)
equivalent to [START_REF] Cherednichenko | Two-scale Γ -convergence of integral functionals and its application to homogenisation of nonlinear high-contrast periodic composites[END_REF].

Other cases.

If 0 < m < 1, we simply set χ χ χ ε = φ φ φ (thus φ φ φ ε = φ φ φ). Noticing that by and (65), we have The variational formulation obtained by passing to the limit in (97) as ε → 0 is given by substituting 0 for u 3 and 0 for φ 3 in (109).

If m > 3, then by Proposition 4 we have u u u = 0 on Σ, and we set simply

χ χ χ ε = 0. ( 120
)
Remark 2 The field χ χ χ ε is constructed in such a way that the behavior of the couple (χ χ χ ε , e e e(χ χ χ ε )) should mimic that of (u u u ε , e e e(u u u ε )) in the fibres, studied in Proposition 4.

In the case m = 1, the convergences (66) indicate that e e e(u u u ε ) Ξ Ξ Ξ 0 Ä x, yε(x) ε ä in T ε . This, joined with the relations (67) and (68), and by virtue of a minimization principle, suggests that the following approximation is likely to hold in T ε e e e(u u u ε ) ∂u 3 ∂x 3 (x)e e e 3 ⊗ e e e 3 + M M M Å ∂u 3

∂x 3 (x) ã in T ε , (121)
where M M M (a) is the solution of min

M M M ∈S 3 , M33=0
g (M M M + ae e e 3 ⊗ e e e 3 ) , g(A A A)

:= E T 2(1 + ν 1 ) Å ν 1 1 -2ν 1 (tr A A A) 2 + A A A : A A A ã ,
given by

M M M (a) = Ñ -ν 1 a -ν 1 a 0 é .
Accordingly, the field χ χ χ ε we are looking for should satisfy χ χ χ ε φ φ φ and e e e(χ χ χ ε ) ∂φ 3 ∂x 3 (x)e e e 3 ⊗ e e e 3 + M M M Å ∂φ 3

∂x 3 (x) ã in T ε .
These estimates are verified by χ χ χ ε defined by ( 102), (103).

In the case m = 3, the convergences (70) indicate that 1 ε e e e(u u u

ε ) Υ Υ Υ 0 Ä x, yε(x) ε ä in T ε .
The identification relations (71) can be improved, and further investigations show that in the linear isotropic case considered in this paper, there holds

Υ Υ Υ 0 (x, y) Å - ∂ 2 γ Σ (u 1 ) ∂x 2 3 y 1 - ∂ 2 γ Σ (u 2 ) ∂x 2 3 y 2
ã e e e 3 ⊗ e e e 3 + e e e y Å q q q Å ∂ 2 γ Σ (u 1 )

∂x 2 3 , ∂ 2 γ Σ (u 2 ) ∂x 2 3 , y ãã , (122)
where q q q(a 1 , a 2 , .) is the solution of min q q q∈H 1 (S;R 3 ) -S g (e e e y q q q + (-a 1 y 1 -a 2 y 2 )e e e 3 ⊗ e e e 3 ) dy, g(A A A)

:= E T 2(1 + ν 1 ) Å ν 1 1 -2ν 1 (tr A A A) 2 + A A A : A A A ã ,
given by

q q q(a 1 , a 2 , y) = ν 1 Ö a 1 y 2 1 -y 2 2 2 + a 2 y 1 y 2 a 2 y 2 2 -y 2 1 2 + a 1 y 1 y 2 0 è .
Accordingly, the field χ χ χ ε we are looking for should satisfy χ ε3 0 and 1 ε e e e(χ χ χ ε ) These estimates are verified by χ χ χ ε defined by (110).

Å - ∂ 2 γ Σ (u 1 ) ∂x 2 3 y ε1 ε - ∂ 2 γ Σ (u 2 ) ∂x 2

Strong convergence

To fix the ideas, we assume m = 1 (the other cases are similar). The space D 1 defined by [START_REF] Bessoud | Multi-materials with strong interface: variational modelings[END_REF], equipped with the inner product a 1 (., .) defined by [START_REF] Lukkassen | Wall: Two-scale convergence[END_REF], is a Hilbert space in which D(Ω; R 3 ) is dense. Hence, fixing η > 0, we can choose φ φ φ ∈ D(Ω; R 3 ) such that a 1 (u u u -φ φ φ, u u u -φ φ φ) < η,

and consider φ φ φ ε defined by (96). There holds (see [START_REF] Grisvard | Singularities in Boundary Values Problems[END_REF])

|u u u ε -φ φ φ ε | 2 H 1 0 (Ω;R 3 ) ≤ Ca ε,m (u u u ε -φ φ φ ε , u u u ε -φ φ φ ε ) = C a ε,m (u u u ε , u u u ε ) J1ε -2 a ε,m (u u u ε , φ φ φ ε ) J2ε + a ε,m (φ φ φ ε , φ φ φ ε ) J3ε . ( 124
)
Since u u u ε is the solution of ( 7), we have J 1ε = Ω f f f .u u u ε dx. We then deduce from the weak convergence of u u u ε to u u u in H

By the arbitrary choice of η, the strong convergence of (u u u ε ) to u u u in H 1 0 (Ω; R 3 ) is proved. Due to the discontinuity of σ σ σ 0 e e e 1 across Σ (see [START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF], ( 12)), we can not expect the derivative with respect to x 1 of the solution u u u of the limit problem to be continuous on Ω. However, for a sufficiently regular datum f f f , we can expect that u u u satisfies (132) and ( 133). The comparison of the results of Theorem 1 with some of the results obtained in [START_REF] Bessoud | Multi-materials with strong interface: variational modelings[END_REF], [START_REF] Bessoud | Serpilli: Plate-like and shell-like inclusions with high rigidity[END_REF] and [START_REF] Bessoud | Asymptotic analysis of shell-like inclusions with high rigidity[END_REF] for the case of an homogeneous layer is interesting: indeed in theses cases Σ behaves as a "material surface" of plate-like type with membrane Kirchhoff-Love energy for m = 1 and with bending energy for m = 3.

In the present case Theorem 1 means that Σ always behaves as a "material surface". For m = 1 the "material surface" is without membrane energy in the direction of the plane Σ orthogonal to the direction of the fibres. For m = 3 the "material surface" Σ is without bending energy in the direction orthogonal to the fibres. In the papers [START_REF] Bessoud | Serpilli: Plate-like and shell-like inclusions with high rigidity[END_REF] and [START_REF] Bessoud | Asymptotic analysis of shell-like inclusions with high rigidity[END_REF] it is also considered the more general situation of a surface Σ and of a shell-like inclusion; the results obtained in this situation suggest to study the case of fibres that

1+ν Ä ν 1 -ν 1 -

 11 2ν tr(e e e(u u u ε ))I I I + e e e(u u u ε ) ä converges weakly to E 1+ν Ä 2ν tr(e e e(u u u))I I I + e e e(u u u) ä in L 2 (Ω, S 3 ). On the other hand, since |T ε | → 0, the sequence e e e(φ φ φ)1 Ω\T ε converges strongly in L 2 (Ω, S 3 ) to e e e(φ φ φ). We infer lim ε→0 I 1ε = a 0 (u u u, φ φ φ) = tr(e e e(u u u))I I I + e e e(u u u) ã : e e e(φ φ φ)dx.

  |e e e(φ φ φ -χ χ χ ε )| L ∞ (T ε ) |e e e(φ φ φ ε )| ≤ C, (see (102), (110), (120)) therefore by (95) and (96) we have |e e e(φ φ φ ε )| ≤ C in T ε \ T ε , hence by Cauchy-Schwartz inequality, tr(e e e(u u u ε ))I I I + e e e(u u u ε ) ã : e e e(φ φ φ ε )dx

1 -2ν 1 ≤

 1 tr(e e e(u u u ε ))I I I + e e e(u u u ε ) ã |dm ε Cε |e e e(u u u ε )|dm ε ≤ Cε |e e e(u u u )| 2 dm ε ≤ Cε.

  the limit as ε → 0 in (105), taking into account Proposition 4, (106) and the uniform convergence of deduce from (99), (100), (101), and (107) that lim ε→0 a ε,m (u u u ε , φ φ φ ε ) = a 1 (u u u, φ φ φ),

Tε|I 3ε | = lim sup ε→0 1 ε

 1 |e e e(u u u ε )| 2 dx ≤ Cε m , m |a T (u u u ε , φ φ φ)| ), (101), (119), we obtain the variational formulationa 0 (u u u, φ φ φ) = Ω f f f .φ φ φdx, equivalent to[START_REF] Bouchitté | Fragalà: Homogenization of thin structures by two-scale method with respect to measures[END_REF]. If 1 < m < 3, then by Proposition 4 we have u 3 = 0 on Σ. We define χ χ χ ε by (102), setting φ 3 = 0. Since v ε3 (φ φ φ) = 0, we deduce from (99) and (104) that lim ε→0 I 3ε = 0.

3 y ε2 ε ã e e e 3 ⊗

 33 e e e 3 + e e e y Å q q q Å ∂ 2 γ Σ (u 1 )

1 0J 2 H 1 0 1 0

 1211 (Ω; R 3 ) established in Proposition 4, that lim ε→0J 1ε = Ω f f f .u u udx = a 1 (u u u, u u u).(125)By (108), there holdslim ε→0 J 2ε = a 1 (u u u, φ φ φ). 3ε = a 1 (φ φ φ, φ φ φ).(127)Collecting (123), (124), (125), (126), and (127), we obtain lim supε→0 |u u u ε -φ φ φ ε | (Ω;R 3 ) ≤ Ca 1 (u u u -φ φ φ, u u u -φ φ φ) ≤ Cη. (128)It is easy to check that lim supε→0 |φ φ φ -φ φ φ ε | H 1 0 (Ω;R 3 ) = 0, |u u u -φ φ φ| 2 H (Ω;R 3 ) ≤ Ca 1 (u u u -φ φ φ, u u u -φ φ φ) ≤ Cη.(129)We infer from (131) and (129) that lim supε→0 |u u u -u u u ε | H 1 0 (Ω;R 3 ) ≤ lim sup ε→0 |u u u -φ φ φ| H 1 0 (Ω;R 3 ) + |φ φ φ -φ φ φ ε | H 1 0 (Ω;R 3 ) + |φ φ φ ε -u u u ε | H 1 0 (Ω;R 3 ) ≤ C √ η.

Remark 3 ∂ 2 φ α ∂x 2 3 ∈

 33 In view of (124-127), we established indeed that lim supε→0 a ε,m (u u u ε -φ φ φ ε , u u u ε -φ φ φ ε ) ≤ Ca 1 (u u u -φ φ φ, u u u -φ φ φ),(131)for all field φ φ φ that can be used in the construction of the sequence of oscillating test fields (φ φ φ ε ) (defined by (96)). Looking back at the proof of the weak convergence, we notice that the proof remains unchanged if we only assume, instead of φ φ φ ∈ D(Ω; R 3 ), that φ φ φ satisfiesφ φ φ ∈ C(Ω; R 3 ) ∩ C 1 (Ω + ; R 3 ) ∩ C 1 (Ω -; R 3 ); ∂φ φ φ ∂x 3 ∈ C(Ω; R 3 ); φ φ φ = 0 on ∂Ω,(132)and C(Ω; R 3 ) for α ∈ {1, 2} if m = 3. (133)

1 Å ν 1 1

 11 ), we can substitute u u u for φ φ φ in the definition of φ φ φ ε . Let us denote by u u u ε the field obtained in this way (to avoid the confusion with the solution u u u ε of the elasticity problem (7)). We then deduce from (131) that lim supε→0 a ε,m (u u u ε -u u u ε , u u u ε -u u u ε ) = 0.In particular, by the definition[START_REF] Grisvard | Singularities in Boundary Values Problems[END_REF] of a ε,m , we get, since u u u ε = χ χ χ ε on T ε , denoting by χ χ χ ε the field deduced by substituting u u u for φ φ φ in (102), (110): -2ν 1 tr(e e e(u u u ε -χ χ χ ε ))I I I + e e e(u u u ε -χ χ χ ε ) ã :e e e(u u u ε -χ χ χ ε )}dx = 0. This implies, since |T ε | ε, that lim sup ε→0 1 ε m-1 -Tε |e e e(u u u ε -χ χ χ ε )| 2 dx = 0. (135) If m = 1, it follows from (

We thank an anonymous referee for his/her suggestions who permitted to notably simplify the original proof.

Acknowledgement. The work of G.G. and F.K. has been partially supported by the French Agence Nationale de la Recherche (ANR) under Grant ARAMIS (Projet "Blanc", ANR 12 BS01-0021) (Analysis of Robust Asymptotic Methods In numerical Simulation in mechanics).

are tubular neighbourhoods of a family of lines of principal curvature of the surface Σ.

2.

Here we have considered the case of isotropic materials and of fibres with "constant" section. In this case the study of asymptotic theory of slender beams gives models where torsion and traction are uncoupled . The case of anisotropic materials and/or fibres whose section "varies with ε" deserves special attention. Indeed in this case asymptotic theory of slender beams can present coupling phenomena between torsion and traction ( see, e.g., [START_REF] Geymonat | Stress distribution in anisotropic elastic composite beams[END_REF], [START_REF] Murat | Comportement asymptotique des solutions du système de l'élasticité linéarisée anisotrope hétérogène dans des cylindres minces[END_REF], [START_REF] Murat | Effets non locaux dans le passage 3d-1d en élasticité linéarisée anisotrope hétérogène[END_REF]), or non-local effects (as in homogenization, see, e.g., [START_REF] Geymonat | Stress distribution in anisotropic elastic composite beams[END_REF], [START_REF] Murat | Effets non locaux dans le passage 3d-1d en élasticité linéarisée anisotrope hétérogène[END_REF], [START_REF] Bellieud | Torsion effects in elastic composites with high contrast[END_REF], [START_REF] Bellieud | Homogenization of an elastic material reinforced by very stiff or heavy fibers. Non local effects. Memory effects[END_REF]). A research in this direction is actually done by the first author (M. B.).

3. The transmission conditions on Σ imply that some "singular behaviour" can appear there and at ∂Σ ∩ ∂Ω; these "singular behaviours" could be analysed for instance with the methods of [START_REF] Grisvard | Singularities in Boundary Values Problems[END_REF].