Eric Canon

Frédéric Chardard

Laurence Grammont

Ilya Kostin

Laetitia Paoli

Keywords: interaction fluide-structure, élasticité linéaire, équations de Stokes, fluide incompressible, interface, méthodes asymptotiques, modélisation, homogénéisation, traitement par laser, contraintes thermiques résiduelles, propriétés thermo-élastiques, fusion, la stabilité thermomécanique fluid-structure interaction, linear elasticity, Stokes equations, incompressible fluid, interface, asymptotic methods, modeling, homogenization, laser treatment, residual thermal stresses, thermoelastic properties, melting, thermomechanical stability

Le but de cette thèse pluridisciplinaire est d'étudier le problème de l'interaction fluide-structure à partir du point de vue mathématique et physique. Des problèmes d'interaction d'un fluide visqueux avec une structure élastique décrivent, par exemple, des interactions entre le manteau terrestre et de la croûte terrestre, le sang et la paroi vasculaire dans un vaisseau sanguin, etc. En génie l'interaction fluide visqueux-structure apparaît lors de la formation de solution colloïdale quand un laser passe à travers le fluide influençant le substrat (ablation laser dans un liquide). Fusion sélective au laser (FSL) est utilisée pour étudier le comportement des contraintes résiduelles en dépendance des propriétés thérmoélastiques et mécaniques du matériau et des formes variées des cordons rechargés. A partir du point de vue mathématique le système couplé "flux fluide visqueux -plaque mince élastique" en 3D lorsque l'épaisseur de la plaque, ε, tend vers zéro, tandis que la densité et le module de Young du matériau élastique sont d'ordre 1 et ε -3 , respectivement, est considéré. Le solide est couchée par le fluide qui occupe un domaine épais. Le développement asymptotique complet est construit lorsque ε tend vers zéro. L'existence, la régularité et l'unicité de la solution pour le problème initial sont étudiées au moyen de techniques variationnelles.

L'erreur de la méthode est évaluée.

Je garde un très bon souvenir de ma vie scientifique à Saint-Etienne. Je ne pourrai malheureusement pas citer toutes les personnes que je voudrais bien remercier ici, mais pourtant je prie d'agréer mes remerciements profonds à ceux qui ont été avec moi constamment ou juste un instant et qui ont apporté une contribution importante à ma vie durant ces années.

Mes premiers remerciements vont naturellement à mes directeurs de thèse, Grigory Panasenko et Andrey Gusarov. Il m'est difficile de devoir résumer en quelques lignes ma plus profonde gratitude et les sentiments que j'éprouve envers eux. Je tiens à adresser mes plus chaleureux remerciements à Grigory Petrovich pour les sujets intéressants qu'il m'a donnée et pour son encadrement tout au long de ma thèse. J'ai beaucoup appris grce à lui, j'ai progressé avec ses conseils spacieux et données toujours au moment propice. Sa patience et ses connaissances m'ont constamment permise de grandir en sagesse, en science et en vie. Merci pour tout. Je tiens à remercier chaleureusement mon co-directeur de thèse Andrey Vladimirovich qui n'a eu de cesse de m'encourager et de me soutenir tout au long de la thèse. J'ai pu apprécier non seulement sa dimension physique et mathématique, mais aussi sa non moins importante dimension humaine. Sa manière de vivre et travailler m'a donnée un moyen exemplaire comment être en équilibre et faire mieux.

Tout d'abord, je souhaiterais remercier mes rapporteurs pour le temps qu'ils ont accordé à la lecture de cette thèse et à l'élaboration de leur rapport. Je remercie de tout mon coeur Ruxandra Stavre qui était pour moi une "co-directrice" parce qu'elle m'a découverte des fonds de l'analyse fonctionnel et variationnel et je sais gré à elle également pour sa disponibilité et sa gentillesse. Je lui suis reconnaissante du temps qu'elle m'a accordée. Je remercie Julia Orlik et Valeria I have very good memories of my scientific life in Saint-Etienne. Unfortunately, I will not be able to mention all people I would like to thank here, but even so I please accept my deepest thanks to those who have been with me constantly or just for a moment and who made a significant contribution to my life during these years.

My first thanks go to my course supervisors, Grigory Panasenko and Andrey

Gusarov. It is difficult to summarize in a few lines have my deepest gratitude and my feelings towards them. I wish to extend my warmest thanks to Grigory Petrovich for interesting topics that he has given me and for his guidance throughout my thesis. I learned a lot from him, I progressed with its spacious advices which were given always at the propitious moment of the time. His patience and knowledge have consistently allowed me to grow in wisdom, science and life. Thank you for everything. I would like to thank my co-supervisor Andrey Vladimirovich who never stopped encouraging me and supporting me throughout the thesis. I could appreciate not only its physical and mathematical dimension, but also its equally important human dimension. His way of life and work gave me a exemplary manner how to be balanced and do better.

Firstly, I would like to thank my reviewers for the time they have given to the reading of this thesis and the preparation of their report. I thank with all my heart Ruxandra Stavre which for me was a "co-director" because she discovered funds of the functional and variational analysis and I appreciate her also for her availability and her kindness to me. I am grateful for the time she has given me. I thank Julia Orlik and Valeria Chiadò Piat for giving me the honor to judge my work carefully and for pleasant and very interesting discussions on scientific and personal questions. I would like to thank Professor Rachid Touzani for accepting the presidentship of the examination committee of this thesis and his welcome and his interest in my work.

A big thank you also to Professor Andrey Amosov for give another mind-set when researching. I measure at fair value the time he gave me. I am grateful to him to be a mentor, a "co-director" for me.

I especially thank the officers of laboratory UMR 5208 CNRS ICJ who have allowed me to quickly integrate and realize my projects, especially Elisabeth Mironescu. I also thank the UJM that funded this thesis and my travel conferences, as well as administrative staff, Franois Hennecart, Driss Essouabri, AS-TRE employees, who, through their greeting, helped me with my initiation and integration in this world of friendliness. My thoughts go in particular to Pascale Villet.

Then I would like to take this opportunity to say thank you to Marie-Claude and Eric Canon, Frederick Chardard, Laurence Grammont, Ilya Kostin, Laetitia Paoli and other members of the lab, where I worked during these three years.

Thank you all for the many informal discussions, for their warm welcome. I owe gratitude to Alexander Elbert for interesting discussions in mathematics.

Then, thank Imane, Roula, Domenico, Dalila, two Hananes, Nohra and many other colleagues in my office FST for professional and caring atmosphere for moments of rest required for a full scientific expression. In addition, a big thank you to Varia, Irina, Eduard, Alexander, Natalia, small Yaroslav, Olga who shared pleasant moments in Saint -Etienne.

It is an honor to be acquainted with Angelique Almeida, a computer technician, very kind and always ready to help with many problems that can happen. I would perhaps never have had the opportunity to work on my thesis, if my parents had not tied my importance to the study. I especially thank my mother for her extraordinary moral support, optimism and love. I want to honor the memory of Academician of the Academy of Sciences of Russia, Arlen Mikhailovich Il'in, which I was lucky to be one of his students. I am completely grateful to him for making me discover Asymptotic Analysis and beauty, meaning, passion for mathematical research. I will never forget his wise advices, his huge support and availability in all professional as well as personal Aussi, en génie l'interaction fluide visqueux-structure apparaît-elle lors de la formation de solution colloïdale. Plus précisément, un laser passe à travers le fluide influençant le substrat (ablation laser dans un liquide).

Problème aux limites élastique linéaire fournit l'informations complète sur les contraintes, déformations et déplacements [START_REF] Landau | Theory of Elasticity[END_REF] dans le matériau élastique considéré. Équations de Navier-Stokes [START_REF] Temam | Navier-Stokes Equations: Theory & Numerical Analysis[END_REF] décrivent le mouvement des substances fluides newtoniens. Le système couplé d'équations pour les mouvements élastiques et fluides avec l'égalité des vitesses et des contraintes normales à l'interface représente un modèle pour le problème d'interaction entre un fluide visqueux et une structure.

En général, le système d'équations de la théorie de l'élasticité est donné dans le système de coordonnées de Lagrange et le système de Navier-Stokes est représenté dans le système de coordonnées d'Euler. Les auteurs [START_REF] Čanić | Effective Equations Modeling the Flow of a Viscous Incompressible Fluid through a Long Elastic Tube Arising in the Study of Blood Flow through Small Arteries[END_REF][START_REF] Čanić | Effective equations describing the flow of a viscous incompressible fluid through a long elastic tube[END_REF] ont obtenu des résultats profonds quand un petit paramètre ε est le rapport entre le rayon et la longueur du vaisseau, le flux est rgi par une chute de pression en fonction du temps donné entre les frontières d'entrée et de sortie, la paroi vasculaire est flexible. Si l'épaisseur de la plaque élastique est négligeable par rapport à l'épaisseur du fluide visqueux, alors nous ne différencions pas des coordonnées lagrangiennes et euleriennes. Dans ce cas, il est beaucoup plus facile de construire un développement asymptotique complet [START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF][START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF].

Le sujet de la thèse est d'étudier le problème de l'interaction fluide-structure à partir du point de vue mathématique et physique. On commence avec le calcul des contraintes résiduelles sous fusion sélective au laser des poudres dans le chapitre 1. On considère un système d'équations de l'élasticité en présence des contraintes résiduelles, engendrées par un gradient des températures dans un matériau composite. Le développement asymptotique de la solution est construit et justifié. Ce problème est lié à l'application "traitement de surface par laser", plus précisément, à la fusion sélective au laser (FSL) des poudres ayant des propriétés thermoélastiques et mécaniques. On étudie le comportement des contraintes résiduelles avec dépendance des paramètres du procédé : du coefficient de

Poisson, du point de fusion, de la dilatation thermique, du module de Young, des formes variées des cordons rechargés. Le calcul fournit les contraintes résiduelles qui se seraient formées après refroidissement de la zone de traitement à l'état initial moyennant l'absence de déformation élastique et de défaillance à l'étape de refroidissement. Les résultats des calculs peuvent être utilisés pour évaluer la stabilité thermomécanique des matériaux dans la FSL. La représentation graphique des champs deux-dimensionnels est obtenue numériquement avec une base des données des propriétés thermoélastiques des matériaux métalliques, céramiques et polymères. Ces résultats sont publiés dans [START_REF] Gusarov | Raschyot ostatochnih napryazhenii pri selektivnom lazernom plavlenii poroshkov (Calculation of Residual Stresses under Selective Laser Melting of Powders)[END_REF] et [START_REF] Gusarov | Thermoelastic Residual Stresses and Deformations at Laser Treatment[END_REF].

La première étape de la fusion sélective au laser tandis qu'un liquide en fusion contacte avec un substrat élastique est étudié plus minutieusement dans la partie mathématique de la thèse (i.e. chapitres 2 et 3). La partie inférieure (élastique) est modélisée de manière plus détaillée dans la partie mathématique. Chapitre 2 représente l'analyse variationnelle du système couplé décrivant l'interaction fluide visqueux-plaque mince élastique en 3D. Résultats d'existence, d'unicité et de régularité sont obtenus pour le problème principal et le problème de limite. La méthode de Galerkin est appliquée. On généralise l'investigation du problème d'interaction du fluide visqueux avec la plaque mince élastique en 2D [START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF][START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF]. Une nouvelle idée est de considérer la même fonction pour la vitesse du fluide v ε et la vitesse dans la zone élastique ∂u ε ∂t . Cela permet de réduire le volume de la preuve en utilisant un ensemble des approximations de Galerkin à la place de deux (pour le milieu liquide et le milieu élastique). Le problème de limite est le problème aux limites de Stokes avec la condition spécifique sur une partie de la frontière

Introduction

Viscous fluid-structure interaction problems describe, for example, interactions between the Earth mantle and the Earth crust, the blood and the vascular wall in a blood vessels, the thin crust of ice and the water. There is a large number of articles on the subject and others. Also, in engineering viscous fluidstructure interaction appears during colloidal solution formation. More precisely, a laser pierce through the fluid influencing the substrate (laser ablation in a liquid).

Linear elastic boundary value problem provides complete information about stresses, strains and displacements [START_REF] Landau | Theory of Elasticity[END_REF] in the considered elastic material. Navier-Stokes equations [START_REF] Temam | Navier-Stokes Equations: Theory & Numerical Analysis[END_REF] describe the motion of Newtonian fluid substances. The coupled system of equations for elastic and fluid motions with the equality of velocities and normal stresses at the interface represents a model for viscous fluidstructure interaction problem. Generally, the system of equations of elasticity theory is given in the Lagrangian coordinate system and the Navier-Stokes system is putted down in the Euler coordinate system. The authors [START_REF] Čanić | Effective Equations Modeling the Flow of a Viscous Incompressible Fluid through a Long Elastic Tube Arising in the Study of Blood Flow through Small Arteries[END_REF][START_REF] Čanić | Effective equations describing the flow of a viscous incompressible fluid through a long elastic tube[END_REF] obtained profound results performing coordinate system changing, when a small parameter ε is the ratio between the radius and the length of the vessel, the flow is governed by a given time-dependent pressure drop between the inlet and the outlet boundary, vessel wall is compliant. If the thickness of elastic plate is negligibly small as compared to the thickness of viscous fluid, then we do not differentiate the Lagrangian and the Euler coordinates.

In this case it is a lot easier to construct a complete asymptotic expansion [START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF][START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF].

The subject of the thesis is to study the fluid-structure interaction problem from mathematical and physical viewpoints. We start with calculation of residual stresses under selective laser melting of powders in Chapter 1. A system of equations of the elasticity with the presence of residual stresses caused by a temperature gradient in a composite material is considered. The asymptotic expansion of the solution is constructed and justified. This problem is related to the application "surface laser treatment", more specifically, to the selective laser melting (SLM) of powders having thermoelastic and mechanical properties. We study the behavior of residual stresses with dependency of the process parameters: the Poisson's ratio, the melting point, the thermal expansion, the Young's modulus, the various forms of reloaded beads. The calculation provides the residual stresses that may have formed after cooling down of the treatment area to the initial state upon the absence of elastic deformation and failure of the cooling down stage. Calculation results can be used to evaluate the thermomechanical stability of the materials in the SLM process. The graphical representation of the two-dimensional fields is obtained numerically with a database of thermoelastic properties of metallic, ceramic and polymer materials. These results were published in [START_REF] Gusarov | Raschyot ostatochnih napryazhenii pri selektivnom lazernom plavlenii poroshkov (Calculation of Residual Stresses under Selective Laser Melting of Powders)[END_REF] and [START_REF] Gusarov | Thermoelastic Residual Stresses and Deformations at Laser Treatment[END_REF].

The first stage of selective laser melting while a liquid melt contacts with an elastic substrate is studied more minutely in the mathematical part of the thesis (i.e. Chapters 2 and 3). The bottom part (elastic) is modelled in more detail in the mathematical part. Chapter 2 represents the variational analysis of the coupled system describing the viscous fluid-thin elastic plate interaction in 3D. Existence, uniqueness and regularity results are obtained for the main and the limit problems.

Galerkin's method is applied. The investigation of the viscous fluid-thin elastic plate interaction in 2D [START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF][START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF] is generalized here. A new idea is to consider one function for the fluid velocity v ε and the velocity in the elastic area ∂u ε ∂t . This allows to reduce a proof volume using one set of Galerkin approximations in place of two (for the fluid and elastic mediums). The limit problem is the Stokes boundary value problem with the specific upper boundary condition corresponding to an elastic wall. The second derivative in time is absent in this condition in 3Dcase which makes the proof be different from that given in [START_REF] Panasenko | Asymptotic analysis of a periodic flow in a thin channel with visco-elastic wall[END_REF]. This system means that the plate degenerated in a wall becquse of contrast coefficients. There was obtained the transmission condition for Stokes equations with an elastic wall.

The core of the thesis is the Chapter 3 on the asymptotic analysis of the contact problem of a thin elastic stratified plate with a fluid layer. The mass force applied to the plate is assumed to be 1-periodic in "horizontal" variables, and the small parameter ε is the ratio of the elastic plate thicknesses to the fluid layer thicknesses. It is assumed that the latter is of the same order as the period of the applied force. All physical constants (the viscosity, the density, the characteristic time) are also of order 1, whereas the Young's modulus of the plate is of order ε -3 . This problem models the viscous-elastic wall flow interaction in many applications, for example, the flow of blood, the oil transportation, etc.

So, there are two mathematical "zooms" on physical and numerical solution to the problem: one of them when all cooled down (Section 1.5) and another case, when one part is molten (Chapters 2 and 3). Introduction

The basic concepts of the theory of elasticity are described in [START_REF] Landau | Theory of Elasticity[END_REF]. We are interested in the isotropic case (homogeneous solid) when three-dimensional stress state along the edges of the elementary cube isolated from the material is characterized by normal and tangential stresses (see Fig. 1). The principal stresses are normal stresses. Principal residual stresses for different materials are studied in this chapter. The results were in general published in [START_REF] Gusarov | Raschyot ostatochnih napryazhenii pri selektivnom lazernom plavlenii poroshkov (Calculation of Residual Stresses under Selective Laser Melting of Powders)[END_REF] and [START_REF] Gusarov | Thermoelastic Residual Stresses and Deformations at Laser Treatment[END_REF].

Selective laser melting (SLM) is one of fast developing trends of additive tech-nology. It is applied for rapid prototyping and manufacturing of functional parts from powders patterned after computer models. A part is shaped geometrically layer by layer, wherein mechanical deposition of thin flat layer powder is repeated several times (layer about 50 -100 µm thick) and its selective scanning by a focused laser beam (spot diameter about 50 -100 µm). In calculated for each computer patterned layer scanning zone, the powder is heated locally, resulting in its particles melted together along with the previously melted layer, forming a monolithic shape, dipped into granular powder. It remains only to remove the finished part of the technology container with the powder, and get the rest of it out of inside hollows. The details of SLM are described in monographs [START_REF] Shishkovsky | Lazernii sintez funktsional'no-gradientnih mezostruktur i ob'yomnih izdelii[END_REF][START_REF] Yadroitsev | Selective laser melting: Direct manufacturing of 3D-objects by selective laser melting of metal powders[END_REF][START_REF] Gladush | Physics of Laser Materials Processing: theory and experiment[END_REF][START_REF] Yu | Eksperimental'noe additivnoe pryamoe proizvodstvo s pomosch'yu lazera (Experimental additive direct manufacturing by laser)[END_REF][START_REF] Yu | Additivnoe proizvodstvo s pomosch'yu lazera. Provedenie eksperimental'nih rabot (Additive manufacturing with a laser. Experimental work)[END_REF][START_REF] Nazarov | Tehnologiya selektivnogo lazernogo spekaniya (Selective laser sintering technology)[END_REF].

SLM method gives an opportunity to create parts of intricate shapes, which cannot be achieved through other technology, thus minimizing the demanding assembly. Shape correction requires interference only at the stage of computer modeling, which comes in handy while producing one-of-a-kind pieces and small batches. At this point high quality is achieved while using ductile metals and alloys, e.g. austenitic steel. Due to absence of pores and cracks and very fine grain structure of those materials, obtained through SLM, often have improved strength characteristics [START_REF] Yadroitsev | Selective laser melting: Direct manufacturing of 3D-objects by selective laser melting of metal powders[END_REF]. At the same time, SLM of more fragile materials may cause unacceptable cracking. For a long time the usage of high-strength titanium alloys Ti-6Al-4V type in this technology wasn't successful, although recently there has been some progress with this alloy [START_REF] Thijs | A study of the microstructural evolution during selective laser melting of Ti-6Al-4V[END_REF].

A lot of technological parameters and wide diversity of materials employed along with economic factors, forcing to employ SLM only for one-of-a-kind pieces and small batches, put the issues of choice of stable processing modes and optimization at the top, which often take up a lot of production hours [START_REF] Yu | Specific features of ion-initiated processes during pulsed laser deposition of MoSe2 coatings in pulsed electric fields[END_REF][START_REF] Grigoriev | Experimental and numerical study of the chemical composition of WSex thin films obtained by pulsed laser deposition in vacuum and in a buffer gas atmosphere[END_REF]. Mathematical modeling is successfully applied for this [START_REF] Gusarov | Fizicheskie modeli vozdeistviya lazernogo izlucheniya na kondensirovannie veschestva v lazernoi tehnologii polucheniya materialov (Physical models of laser impact on condensed substances in laser technology of production of materials[END_REF]. The contemporary understanding of physics of laser radiation transport in powder layers and thermal processes at laser impingement point while SLM is reflected in works [START_REF] Shishkovsky | Lazernii sintez funktsional'no-gradientnih mezostruktur i ob'yomnih izdelii[END_REF][START_REF] Yadroitsev | Selective laser melting: Direct manufacturing of 3D-objects by selective laser melting of metal powders[END_REF][START_REF] Gladush | Physics of Laser Materials Processing: theory and experiment[END_REF][START_REF] Gusarov | Fizicheskie modeli vozdeistviya lazernogo izlucheniya na kondensirovannie veschestva v lazernoi tehnologii polucheniya materialov (Physical models of laser impact on condensed substances in laser technology of production of materials[END_REF]. Stability assessment issues of process and quality of micro-and macrostructure obtained through modeling are discussed in [START_REF] Gusarov | Model of Radiation and Heat Transfer in Laser-Powder Interaction Zone at Selective Laser Melting[END_REF][START_REF] Verhaeghe | A pragmatic model for selective laser melting with evaporation[END_REF][START_REF] Yadroitsev | Single track formation in selective laser melting of metal powders[END_REF][START_REF] Gusarov | Modeling the interaction of laser radiation with powder bed at selective laser melting[END_REF][START_REF] Doubenskaia | Comprehensive Optical Monitoring of Selective Laser Melting[END_REF][START_REF] Smurov | Optical Monitoring in Laser Cladding of Ti6Al4V[END_REF]. At the same time, the physics of residual stresses appearance, responsible for possible cracking and part shape deviation, is insufficiently studied.

The most general well-known method to reduce the residual stresses at a thermal treatment is to reduce the thermal gradient and the cooling rate [START_REF] Yu | Eksperimental'noe additivnoe pryamoe proizvodstvo s pomosch'yu lazera (Experimental additive direct manufacturing by laser)[END_REF][START_REF] Yu | Additivnoe proizvodstvo s pomosch'yu lazera. Provedenie eksperimental'nih rabot (Additive manufacturing with a laser. Experimental work)[END_REF][START_REF] Nazarov | Tehnologiya selektivnogo lazernogo spekaniya (Selective laser sintering technology)[END_REF][START_REF] Grigoriev | Features of micro-and nanostructures of Au Ni alloys obtained on nickel due to different modes of pulse laser alloying[END_REF].

This was confirmed, for example, at growing the monocrystals [START_REF] Brice | The cracking of Czochralski-grown crystals[END_REF]. The same approach works at laser treatment [START_REF] Triantafyllidis | Crack-free densification of ceramics by laser surface treatment[END_REF][START_REF] Glasser | Laser melting of refractory Al2O3-ZrO2 ceramics. British ceramic[END_REF], while it is difficult to implement because heating/cooling rate reduction means the proportional reducing of the productivity, and the reducing of the thermal gradient contradicts the local nature of the laser treatment. The preheating has recently become a universal method widely applied at the laser treatment [START_REF] Gurauskis | Laser-assisted, crack-free surface melting of large eutectic ceramic bodies[END_REF][START_REF] Hagedorn | Net shaped high performance oxide ceramic parts by selective laser melting[END_REF]. In order to reduce residual stresses during SLM, the part formation is going on in a heated container [START_REF] Hagedorn | Net shaped high performance oxide ceramic parts by selective laser melting[END_REF]. Preheating efficiency is commonly explained by following two factors [START_REF] Maruo | CO 2 Laser Welding of Ceramics (Report 1)[END_REF]. First, as the overall temperature difference in the treated part is decreased, the temperature gradients and the heating/cooling rates are proportionally decreased. Second, materials often become more plastic at elevated temperatures, so that thermal stresses can partially relax.

The conventional approach to calculate the residual stresses is to separate the thermal and the thermomechanical problems. The deformation due to the thermal expansion is usually small, so that its thermal effect is negligible compared to the laser energy. Therefore, the thermal problem can be considered independently. The temperature field and the shape of the melt pool calculated by the thermal model are the initial data for the thermomechanical model. Such a twostep scheme of calculation is useful to predict residual stresses for a given set of technological parameters [START_REF] Ghosh | Modeling and Experimental Verification of Transient/Residual Stresses and Microstructure Formation in Multi-Layer Laser Aided DMD Process[END_REF][START_REF] Brückner | Modeling the Influence of Process Parameters and Additional Heat Sources on Residual Stresses in Laser Cladding[END_REF], but can become too complicated for a theoretical parametric analysis and optimization of the technological process. A reasonable simplification of the thermomechanical model [START_REF] Gusarov | Residual Stresses at Laser Surface Remelting and Additive Manufacturing[END_REF] was shown to be sufficient to construct a single-step calculation scheme independent of the temperature distribution and its evolution. Instead of the analysis in terms of the technological parameters, an analysis in terms of remelted profiles was proposed [START_REF] Gusarov | Residual Stresses at Laser Surface Remelting and Additive Manufacturing[END_REF]. This method can be fruitful because the desired remelted profile is often given or the variety of acceptable profiles is restricted. In the scientific works [START_REF] Brückner | Modeling the Influence of Process Parameters and Additional Heat Sources on Residual Stresses in Laser Cladding[END_REF][START_REF] Gusarov | Residual Stresses at Laser Surface Remelting and Additive Manufacturing[END_REF] calculations are made for specific beads, obtained on a flat surface of half-infinite substrate, and in [START_REF] Ghosh | Modeling and Experimental Verification of Transient/Residual Stresses and Microstructure Formation in Multi-Layer Laser Aided DMD Process[END_REF] for several beads paralleled upon one another and on the substrate. More complex geometries weren't considered. Calculations were made for quarts glass and alumina and the capability to explain experimental data was shown [START_REF] Gusarov | Residual Stresses at Laser Surface Remelting and Additive Manufacturing[END_REF]. Despite the demonstrated satisfying alignment of experiment results

with the modeling [START_REF] Ghosh | Modeling and Experimental Verification of Transient/Residual Stresses and Microstructure Formation in Multi-Layer Laser Aided DMD Process[END_REF][START_REF] Brückner | Modeling the Influence of Process Parameters and Additional Heat Sources on Residual Stresses in Laser Cladding[END_REF][START_REF] Gusarov | Residual Stresses at Laser Surface Remelting and Additive Manufacturing[END_REF] and the confirmation in [START_REF] Brückner | Modeling the Influence of Process Parameters and Additional Heat Sources on Residual Stresses in Laser Cladding[END_REF][START_REF] Gusarov | Residual Stresses at Laser Surface Remelting and Additive Manufacturing[END_REF] technique of preliminary heating for residual stress decrease, there is still an open issue of generalization of the results, obtained in the above mentioned works, for geometric shapes and materials, different from those mentioned in these works.

This work is based on the preceding works [START_REF] Ghosh | Modeling and Experimental Verification of Transient/Residual Stresses and Microstructure Formation in Multi-Layer Laser Aided DMD Process[END_REF][START_REF] Brückner | Modeling the Influence of Process Parameters and Additional Heat Sources on Residual Stresses in Laser Cladding[END_REF][START_REF] Gusarov | Residual Stresses at Laser Surface Remelting and Additive Manufacturing[END_REF] and dedicated to development of more general technique of residual stresses assessment during SLM, applied to different materials (with arbitrary thermoelastic properties) and geometries.

Physical model

Strongly localized laser heating of the material generally leads to its expansion and compressive stresses appearance. With the passing of the laser beam, material compresses again and the appeared thermal stresses disappear. The practically observed formation of residual stresses as a result of thermal action is connected to full or partial relaxation of thermal stresses. With the cooling of the relaxation zone down to the initial temperature T a , the stretching residual stresses are supposed to appear, and, in the surrounding material, compensating residual compressive stresses should be formed. Such distribution of residual stresses is in fact typical for laser processing. Therefore, classical thermoelastic medium model, useful for thermal stresses estimation, becomes completely inapplicable for residual stresses calculation.

The formation of residual stresses is often calculated after elastoplastic or viscoelastic medium models. The above mentioned works [START_REF] Ghosh | Modeling and Experimental Verification of Transient/Residual Stresses and Microstructure Formation in Multi-Layer Laser Aided DMD Process[END_REF][START_REF] Brückner | Modeling the Influence of Process Parameters and Additional Heat Sources on Residual Stresses in Laser Cladding[END_REF] are the examples of such approach. Aside from complexity of numerical calculations such models contain a lot of parameters, including parameters of the moving and nonstationary heating source, thermophysical properties of a medium, temperature dependence of the viscosity and the yield limit. That is why the results are hard to analyze and extrapolate to the unstudied materials. This decreases practical usefulness of such calculations. On the other hand, thermophysical and viscoelastoplastic properties of materials under high temperatures are often unknown or have a poor accuracy.

Also, the measurement error of residual stresses calculation, validating mathematical model, can significantly exceed the computational accuracy. Therefore, the detailed description with the help of thermoviscoelastoplastic models often becomes excessive in respect to laser processing, particularly SLM.

Alternative approach, suggested in [START_REF] Gusarov | Residual Stresses at Laser Surface Remelting and Additive Manufacturing[END_REF], assumes explicit selection of thermal stresses relaxation zone while setting a problem. Given that laser melting and appearance of molten pool with free surface implies almost full stress relief in it, relaxation zone boundary may be defined in the thermal model as a boundary of maximum melting -the envelope of the melting front surface. The effective relaxation zone may considerably be larger than the remelting zone, provided that under elevated temperatures still in solid state affected by thermal stresses there is viscous and/or plastic flow. Then envelope of the generally transient isotherm of effective thermal stresses relaxation can be taken as the relaxation zone boundary.

The advantage of such setting is the absence of a rigid connection between the heat and mechanical problems. For instance, in many types of laser processing the geometry of the remelted area or the area heated to a given temperature is set as the input data. Then the necessary parameter adjustment of the heat source is carried out by the thermal model, and residual stresses evaluation -by the mechanical model, which, in fact, becomes independent. Such separation of thermal and mechanical parameters may be useful for theoretical analysis.

More generally, the principal assumption made in the known model of residual stresses formation [START_REF] Gusarov | Residual Stresses at Laser Surface Remelting and Additive Manufacturing[END_REF] consists in definition of a sharp boundary of the zone of complete relaxation of thermal stresses. After passage of the heat source and cooling down to the initial temperature, the relation between the tensors of stresses with components σ βγ and strains with components ε βγ outside the relaxation zone is given by the conventional generalized Hooke's law [START_REF] Landau | Theory of Elasticity[END_REF]

σ βγ = λθδ βγ + 2µε βγ , (1.2.1)
where λ -is Lamé's first parameter, µ -the shear modulus, θ = ε xx + ε yy + ε zz -the volumetric deformation, δ βγ the Kronecker symbol, and indices β and γ take the values x, y and z, corresponding to the Cartesian axes.

Remark. This is equivalent to

σ k i = 3 j,l=1 a kl ij ε l j , σ k i = σ ik ,
where the deformation is characterized by the vector field of displacement u, with the strain tensor components given by derivatives

ε l j = ε jl = 1 2 ∂u l ∂x j + ∂u j ∂x l and elastic constants a kl ij = E 2(1 + ν) 2ν 1 -2ν δ ik δ jl +δ ij δ kl +δ il δ jk verify standard properties: (i) a kl ij (x) = a il kj (x) = a lk ji (x) = a kj il (x), ∀ i, j, k, l ∈ {1, 2, 3}, ∀x ∈ Ω (symmetry), (ii) ∃ κ > 0 independent of ε such that 3 i,j,k,l=1 a kl ij (x)η l j η k i ≥ κ 3 j,l=1 (η l j ) 2 , ∀x ∈ Ω, ∀η = (η l j ) 1≤j,l≤3
, with η l j = η j l (coercivity).

Lamé's parameters λ and µ can be expressed in terms of E and ν, Young's modulus and Poisson's coefficient, respectively

λ = Eν (1 + ν)(1 -2ν) , µ = E 2(1 + ν) .
Another form is

σl j = a kl ij ∂u l ∂x j .
The second assumption of the model is the elastic deformation inside thermal stresses relaxation zone while cooling. Thus, after cooling down to the initial temperature, the generalized Hooke's law inside this zone is written as [START_REF] Landau | Theory of Elasticity[END_REF]

σ βγ = λθδ βγ + 2µε βγ + 3sKδ βγ , (1.2.2)
where K is the bulk modulus and s is the linear shrinkage while cooling from the thermal stresses relaxation temperature T m down to the initial temperature T a , which can be calculated according to linear thermal expansion coefficient α as Should the criteria be met, residual stresses shall be corrected accordingly.

s = T m T a α dT. ( 1
In general, calculation according to the described model gives the residual stresses, which would be formed after cooling the treatment zone down to the initial state, if deformation at the cooling stage were strictly elastic. The possible influence of plastic flow and destruction at cooling should be taken into account separately. The application example presented further.

Results

Given that the SLM technology is designed for the manufacture of parts of intricate shapes, it is almost impossible to cover their potential variety, so in sections

ε xx = 0, (1.3.1)
and away from the substrate there is no force acting in the transverse or vertical direction, so

σ yy = σ zz = 0. (1.3.2) Substrate (a)
σ xx = sE, (1.3.3)
where E is the Young's modulus.

Thus, the uniaxial tension is present in the computational domain of the vertical wall. The computational domain is shown in Fig. 2 (a). The points of this domain must be spaced from the substrate by the distance much greater than the wall thickness, and from the vertical edges -by the distance much greater than the height of the wall. If for the plastic material the stress σ xx , calculated according to equation (1.3.3), exceeds the yield limit, and the linear shrinkage s does not exceed the elongation at break, the plastic flow will lead to restriction of the longitudinal stress by the yield limit value. In case of a brittle material, equation

(1.3.3) can be used up to the tensile strength rupture limit. The resulting stresses in the vertical wall can be used for evaluating the possibility of its failure while constructing by the SLM method, and also for calculating the bending of the substrate, as it is important in selecting its thickness.

Horizontal plate

Horizontal plate on the substrate, as shown in Fig. 2 (b), simulates a coating, or an extended low part. As in the previous case, the hard substrate prevents the transversal displacement, that is why far from the edges of the plate

ε xx = ε yy = 0, (1.3.4)
and the free upper surface eliminates forces in the vertical direction, so

σ zz = 0. (1.3.5)
Simultaneous solution of equations (1.2.2), (1.3.4) and (1.3.5) gives the stress

σ xx = σ yy = sE 1 -ν , (1.3.6)
where ν is the Poisson's ratio.

This way the horizontal plate grown on a hard substrate is exposed to the isotropic biaxial tension. Since the Poisson's ratio is in the range from 0 to 1/2, the value of tensile stress is greater than in the vertical plate. The result is applicable to the computational domain shown in Fig.

1.3.3

Bead on a semi-infinite substrate

Typical for the SLM configurations of the remelted beads are shown in Figure 3.

The first configuration (3a) presents the shallow remelted band of finite width and infinite length on the surface of the half-space substrate. Material parameters are listed in Table 1. It is important that the lower part of the bead comes into the substrate, providing the metallurgical contact of the remelted powder with the substrate. Geometry (3b) corresponds to the extremely low amount of powder, as compared to the remelted substrate material, and in geometries (3c) and (3d) the proportion of the powder is increased successively. In the direction of the bead the problem is uniform and there is no displacement of the medium. Strain tensor is expressed through the displacement vector in the plane (Y Z), u = (u y , u z ):

ε xx = ε xy = ε xz = 0, ε yy = ∂u y ∂y , ε zz = ∂u z ∂z , ε yz = 1 2 ∂u y ∂z + ∂u z ∂y . (1.3.7)
In the general case, the three components of the displacement vector are to be found from the system of the three force balance equations. In the considered below case of a uniform in x-direction remelted profile, u x = 0, and the following two force balance (in the directions y and z) equations are sufficient:

∂σ yy ∂y + ∂σ yz ∂z = 0, ∂σ zz ∂z + ∂σ yz ∂y = 0, (1.3.8)
substitution of the Hooke's law in which in the form of (1.2.1) or (1.2.2) results in a system of equations for the displacement u. Example of the numerical solution of this system for the configurations shown in Fig. 3 (a-d) is given in Fig. 4567, respectively. The equivalent Mises stress is calculated by the formula

σ M ises = 1 2 [(σ 1 -σ 2 ) 2 + (σ 1 -σ 3 ) 2 + (σ 2 -σ 3 ) 2 ], (1.3.9)
where σ 1 , σ 2 and σ 3 are the principal stresses. The non-dimensional values here are calculated using the following formulas:

û = u u 0 , σβγ = σ βγ σ 0 , (1.3.10)
where the normalizing values of displacement u 0 and stress σ 0 correspond to the maximum values of these quantities in the horizontal plate of the unit thickness:

u 0 = 1 + ν 1 -ν s, σ 0 = sE 1 -ν . (1.3.11)
The above example of calculation for quartz glass suggests that the principal dependencies on the thermoelastic parameters are given by (1.3.11). Indeed, the dimensional analysis of the model equations indicates that dimensionless distributions (1.3.10) are functions of Poissons radio ν for a given shape of the remelted domain. In addition, the thermoelastic problem does not contain a characteristic space size. Therefore, the similar remelted domains of different scale form the similar displacement and stress fields.

Thermoelasticity problem does not contain the characteristic size, so the coordinates in Fig. 4, 5, 6 and 7 are given in arbitrary units. In the same units both u and u 0 are measured.

Discussion

Comparison of the relevant non-dimensional epures for isolated beads in Fig. 5a and 5b and also Fig. 6a and 6b obtained at various Poisson's ratios does not show neither qualitative nor quantitative significant differences between them.

This suggests that the main dependence on Poisson's ratio in the geometry of the insulated bead is the same as for the horizontal plate, and is given by equations (1.3.11). It can be also seen that the values of stresses in the region below the surface of the substrate, which are critical for the destruction and plastic flow, are weakly dependent on the bead height above the surface. Thus the conclusion of the work [START_REF] Gusarov | Residual Stresses at Laser Surface Remelting and Additive Manufacturing[END_REF], obtained for the quartz glass, can be generalized.

Conventional methods of reducing the residual stresses in case of the SLM, such as preheating and the choice of material with less thermal expansion, elastic modulus, or melting point result from equations (1.3.3) and (1.3.6), and from the second equation (1.3.11). It can be also seen that the increase of the Poisson's ratio is advantageous for reducing residual stresses in the individual beads and the horizontal plate. At the same time, the results of this work indicate that the residual stresses are not dependent on the spatial resolution and, hence, on the temperature gradient. The cooling rate is also not among the parameters of the model. Therefore, it seems that these findings contradict the lessons learned.

Though that is not quite true. For example, preheating decreases the total temperature difference in the part and thus reducing the gradient and the cooling rate. Model equation (1.2.2) implies that within the elastic deformation residual stresses are proportional to the linear thermal shrinkage s, which is roughly proportional to the temperature difference of the processed part in accordance with equation (1.2.3). In conventional furnace technologies of heat treatment this temperature difference is not fixed, but it is known that it is proportional to the temperature gradient in heterogeneous processes and the rate of heating/cooling in nonstationary processes. Therefore it is necessary to analyze these two parameters. In the laser technologies with their highly localized heating, the temperature gradient is almost always known in advance and is equal to T m -T a , that is why it is easier to work with one parameter -s. Thus, the conclusion about the independence of residual stresses in the SLM on the temperature gradient and cooling rate made in [START_REF] Gusarov | Residual Stresses at Laser Surface Remelting and Additive Manufacturing[END_REF] for quartz glass and generalized here for arbitrary materials within their elastic deformation during cooling should be seen not as a contradiction to previous experience, but rather as a proposal to use during the analysis of laser technologies only one parameter instead of two -the complete linear thermal shrinkage s.

Conclusion [START_REF] Gusarov | Residual Stresses at Laser Surface Remelting and Additive Manufacturing[END_REF] that the maximum longitudinal tensile residual stresses in the individual beads are approximately twice as large as the maximum transverse and confirmed for phosphate glass ones [START_REF] Ryzhkov | Obrazovanie treschin pri selektivnom lazernom spekanii keramiki (Crack formation in ceramics under selective laser sintering)[END_REF], are generalized here to arbitrary materials having a Poisson's ratio in the investigated range from 0.17 to 0.34 within their elastic deformation during cooling after laser manufacturing.

Stress distributions for three different geometries of the individual bead on a massive substrate shown in Figure 3, are very close. In contrast, in the vertical and horizontal plates shown in Figure 2, residual stresses are very different. Thus the question arises of how the fundamentally different stress states of parts constructed from individual beads are formed from the locally identical stress states at the stage of these beads. Model does not give an answer to it, as it is not considering the consecutive imposition of beads on each other. It can only be stated that the mechanical interaction of the part with the substrate on which it is built is crucial. In Figure 8 the calculated residual stresses are compared to the tensile strength rupture limit of non-metallic materials and the yield limits of the metallic ones.

The assumed values of material properties are given in Table 1. Stresses in Fig. 8 were made dimensionless involving a complete linear shrinkage s, which is dependent on the preheating temperature T a . This eliminates the temperature dependence of the calculations that are shown as lines and artificially introduces the dependency of the dimensionless strength limits (shown by dots) on it. That is why for each point the value T a is mentioned. Tab. 1. Thermoelastic and mechanical properties of different materials adopted for calculations.

Material ν µ (GPa) α (10 -6 K -1 ) T m ( • C)
Calculations for vertical plate in which the uniaxial tension is formed, can be directly compared with the results of uniaxial mechanical tests. For individual beads on a massive substrate the Mises equivalent stress is calculated as shown in Fig. 8 by the dashed line. Usage of other failure criteria or plastic flow under complex stress state is also not excluded as the principal stresses are given. In case of biaxial isotropic tension in the horizontal plate the Mises stress is equal to the stress along any of the equivalent axes in the plane of stretching.

The calculations predict the alumina oxide fracturing, even with the preheating up to 1600 • C, and absence of polystyrene fracturing. This corresponds to the known SLM practice. The fracturing of the volumetric quartz glass parts is also expected, but without longitudinal cracks in the individual beads thereof. However, in this experiment [START_REF] Gusarov | Residual Stresses at Laser Surface Remelting and Additive Manufacturing[END_REF] neither longitudinal nor transverse fracturing of the beads from this material was observed. This suggests its possible strengthening by rapid cooling after laser treatment. As for the high-impact metal alloys Ti-6Al-4V and H13, it is likely that the residual stresses in them reach the yield limits.

This conclusion may be affected by the preheating and possible laser strengthen-ing that is associated with grain refinement to submicron size characteristic for SLM [START_REF] Gusarov | Fizicheskie modeli vozdeistviya lazernogo izlucheniya na kondensirovannie veschestva v lazernoi tehnologii polucheniya materialov (Physical models of laser impact on condensed substances in laser technology of production of materials[END_REF].

Titanium alloys fracturing in the SLM is known to be prevented by processing in a protective atmosphere, eliminating oxidation [START_REF] Gusarov | Fizicheskie modeli vozdeistviya lazernogo izlucheniya na kondensirovannie veschestva v lazernoi tehnologii polucheniya materialov (Physical models of laser impact on condensed substances in laser technology of production of materials[END_REF]. That is why the mechanisms of its fracturing are not under consideration. Only the question about the ratio of elastic and plastic residual deformations is actual. Calculations shown in Fig. 8 suggest that the elastic and plastic deformation components depending on the particular geometry and on the preheating temperature are generally comparable.

The same can be said about the ratio of elastic and plastic deformations in the H13 steel.

Thus, the above calculation results may be used to estimate the thermomechanical stability of materials in the SLM.

1.5

Asymptotic analysis of the problem with residual stresses in the right-hand side 1.5.1

Statement of the problem

Let G be limited (or confined) s-measured domain with infinitely smooth boundary ∂G. Consider the system of the conductivity equations

L ε u ≡ s k,j=1 ∂ ∂x k A kj x ε ∂u ∂x j = f 0 (x) + ∇ α x ε T x, x ε , (1.5.1)
with boundary condition

u is 1 -periodic in R s . (1.5.2)
Here ε > 0 is a small parameter, which is the characteristic scale of microstructure of the environment. We require that ε -1 be an integer. Also

x = (x 1 , x 2 , . . . , x s ) ∈ R s , u = u(x 1 , x 2 , . . . , x s ) = (u 1 , u 2 , . . . , u s ) -tempera- ture, f 0 = (f 1 0 , f 2 0 , . . . , f s 0 ), A ij (ξ)
are periodic with respect to all variables ξ i with period 1 (s× s) matrix-functions describing heat-conducting properties of the material and satisfying properties (i) and (ii) p. 14, where

ξ i = ε -1 x i , ∀i = 1, . . . , s, ξ = (ξ 1 , ξ 2 , . . . , ξ s ) = (ε -1 x 1 , ε -1 x 2 , . . . , ε -1 x s ).
The external force f 0 (x) has zero averaged value:

f 0 (x) = 0; (1.5.3)
where • = (0,1) s dξ.

The thermal expansion coefficient α( x ε ) is a measurable bounded function on [0, 1], the temperature T(x, x ε ) has the following structure

T x, x ε = T 0 (x) + εT 1 x, x ε ,
where the solution of the homogenized thermal conductivity equation T 0 (x) is known function and the first corrector of the thermal conductivity is represented in the form

T 1 x, x ε = s i=1 N i (ξ) ∂T 0 (x) ∂x i , (1.5.4)
as it was introduced by N.S. Bakhvalov and G.P. Panasenko in [START_REF] Bakhvalov | Homogenisation: Averaging Processes in Periodic Media36volume, series, Mathematics and its Applications[END_REF].

A ij (ξ), f 0 (x), T 0 (x), T 1 x, x ε are infinitely differentiable functions in R s .
We assume that elements of the matrices A ij (ξ) satisfy conditions:

a il kj = a kl ij = a ij kl = a li jk , (1.5.5)
a il kj (ξ)η ik η lj κ η ik η ik (1.5.6)
for any symmetrical matrix η ik , where κ > 0 is constant independent of ε.

It is necessary to construct the asymptotic of the solution of the problem at ε tending to zero.

Existence and uniqueness of a solution

Denote G = (0, 1) s and consider the spaces

H 1 # (G) = f ∈ H 1 loc (R s ) f -1-periodic , f H 1 # (G) = f H 1 (G) , L 2 # (G) = f ∈ L 2 loc (R s ) f -1-periodic , f L 2 # (G) = f L 2 (G) .
(1.5.

       Find u ∈ H 1 # (G) such that - G s k,j=1 A kj ∂u ∂x j ∂ϕ ∂x k = G f 0 ϕ - G αT∇ϕ, ∀ϕ ∈ H 1 # (G).
(1.5.8)

Theorem 1.2. Let f 0 ∈ L 2 # (G)
, α and T satisfy the properties indicated above (measurable bounded fucntions). In addition the averaged value u = 0 and f 0 = 0. Then there exists a unique solution to problem (1.5.8) that satisfies the following estimate

u H 1 C f 0 L 2 + αT L 2 ,
(1.5.9)

with C a positive constant.

Proof. This is a classical result [START_REF] Bakhvalov | Homogenisation: Averaging Processes in Periodic Media36volume, series, Mathematics and its Applications[END_REF] that there exists a unique solution u ∈ H 1 # (G) to problem (1.5.8).

A priory estimates can be easily obtained with help of Korn inequality [START_REF] Oleinik | Mathematical Problems in Elasticity and Homogenization2volume[END_REF] and Cauchy-Schwarz-Bunyakovsky inequality [START_REF] Kolmogorov | Elements of the Theory of Functions and Functional Analysis[END_REF]. Namely,

κ ∇u 2 L 2 κ G s k,j=1 ∂u ∂x j ∂u ∂x k G s k,j=1 A kj ∂u ∂x j ∂u ∂x k f 0 L 2 u L 2 + αT L 2 ∇u L 2 f 0 L 2 + αT L 2 u H 1 , κ ∇u 2 L 2 f 0 L 2 + αT L 2 u H 1 ,
and using the equivalence of ∇u L 2 and u H 1 :

u H 1 ≤ (1 + C 2 ) ∇u L 2 ≤ (1 + C 2 ) u H 1 ,
where C > 0 is the constant appearing in the Poincaré-Friedrichs' inequality [START_REF] Abdessamad | Memory Effect In Homogenization Of A Viscoelastic Kelvin-Voigt Model With Time-Dependent Coefficients[END_REF] finally we get (1.5.9).

Construction of a solution

We introduce the following notation for the left-hand side of (1.5.1)

P u ≡ s k,j=1 ∂ ∂x k A kj x ε ∂u ∂x j , x ∈ G. (1.5.10)
We need to take into consideration that

∇ = ∇ x + ε -1 ∇ ξ ξ= x ε (1.5.11)
and that formula (1.5.4) is carried out. Then (1.5.10) becomes

P u ≡ s k,j=1 ∂ ∂x k A kj (ξ) ∂u ∂x j + 1 ε ∂ ∂ξ k A kj (ξ) ∂u ∂x j + 1 ε ∂ ∂x k A kj (ξ) ∂u ∂ξ j + + 1 ε 2 ∂ ∂ξ k A kj (ξ) ∂u ∂ξ j (1.5.12)
and the right-hand side is transformed as follows

f 0 (x) + ∇ α x ε T x, x ε = f 0 (x) + 1 ε s k=1 T 0 (x) ∂α(ξ) ∂ξ k + s j=1 α(ξ) + s k=1 ∂(αN j ) ∂ξ k • ∂T 0 (x) ∂x j + ε s k,j=1 α(ξ)N j (ξ) ∂ 2 T 0 (x) ∂x k ∂x j = f 0 (x) + 2 l=0 ε l-1 |i|=l G i (ξ)D i T 0 , (1.5.13)
where it is denoted:

G ∅ = ∂α(ξ) ∂ξ k , G i 1 = α(ξ) + ∂(αN j ) ∂ξ k , G i 1 i 2 = α(ξ)N j (ξ). (1.5.14)
A formal asymptotic solution to problem (1.5.1), (1.5.2) is searched in the following form

u (K) = K+1 l=0 ε l |i|=l N i x ε D i v (K) + Y i-1 x ε D i-1 T 0 , v (K) = K j=0 ε j v j (x), (1.5.15)
where i = (i 1 , . . . , i l ) is a multi-index, i j ∈ {1, . . . , s}, v(x) is a infinitely differentiable 1-periodic with respect to x 1 , . . . , x s vector function, N i (ξ), Y i (ξ) are (s × s) matrix functions that are 1-periodic with respect to ξ 1 , . . . , ξ s .

By substitution of series (1.5.15) instead of u in (1.5.12) we obtain

P u (K) = K+1 l=0 ε l |i|=l A kj (ξ)N i (ξ) ∂ 2 D i v (K) ∂x k ∂x j + A kj (ξ)Y i-1 (ξ) ∂ 2 D i-1 T 0 ∂x k ∂x j + +ε -1 K+1 l=0 ε l |i|=l ∂ ∂ξ k A kj (ξ)N i (ξ) ∂D i v (K) ∂x j + ∂ ∂ξ k A kj (ξ)Y i-1 (ξ) ∂D i-1 T 0 ∂x j + +ε -1 K+1 l=0 ε l |i|=l A kj (ξ) ∂N i (ξ) ∂ξ j ∂D i v (K) ∂x k + A kj (ξ) ∂Y i-1 (ξ) ∂ξ j ∂D i-1 T 0 ∂x k + +ε -2 K+1 l=0 ε l |i|=l ∂ ∂ξ k A kj (ξ) ∂N i (ξ) ∂ξ j D i v (K) + ∂ ∂ξ k A kj (ξ) ∂Y i-1 (ξ) ∂ξ j D i-1 T 0 , After index changing r = l + 2, i 1 = k, i 1 = l, q = l + 1, i 1 = j, q = l + 1, i 1 = k (1.5.16)
in the first, the second and the third sum respectively we obtain:

P u (K) = K+3 r=0 ε r-2 |i|=r A i 1 i 2 (ξ)N i 3 ...i r (ξ)D i v (K) + A i 1 i 2 (ξ)Y i 3 ...i r-1 (ξ)D i-1 T 0 + + K+2 q=0 ε q-2 |i|=q ∂ ∂ξ k A ki 1 (ξ)N i 2 ...i q (ξ) D i v (K) + ∂ ∂ξ k A ki 1 (ξ)Y i 2 ...i q-1 (ξ) D i-1 T 0 + + K+2 q=0 ε q-2 |i|=q A i 1 j (ξ) ∂N i 2 ...i q (ξ) ∂ξ j D i v (K) + A i 1 j (ξ) ∂Y i 2 ...i q-1 (ξ) ∂ξ j D i-1 T 0 + + K+1 l=0 ε l-2 |i|=l ∂ ∂ξ k A kj (ξ) ∂N i (ξ) ∂ξ j D i v (K) + ∂ ∂ξ k A kj (ξ) ∂Y i-1 (ξ) ∂ξ j D i-1 T 0 ,
and after identification of the indexes r, q, l:

P u (K) = K+1 l=0 ε l-2 |i|=l H i (ξ)D i v (K) + S i-1 (ξ)D i-1 T 0 + R (K) 1ε , (1.5.17)
where

R (K) 1ε = ε K |i|=K+2 ∂ ∂ξ k A ki 1 (ξ)N i 2 ...i K+2 (ξ) + A i 1 j (ξ) ∂N i 2 ...i K+2 (ξ) ∂ξ j + +A i 1 i 2 (ξ)N i 3 ...i K+2 (ξ) D i v (K) + ∂ ∂ξ k A ki 1 (ξ)Y i 2 ...i K+1 (ξ) + +A i 1 j (ξ) ∂Y i 2 ...i K+1 (ξ) ∂ξ j + A i 1 i 2 (ξ)Y i 3 ...i K+1 (ξ) D i-1 T 0 + +ε K+1 |i|=K+3 A i 1 i 2 (ξ)N i 3 ...i K+3 (ξ)D i v (K) + A i 1 i 2 (ξ)Y i 3 ...i K+2 (ξ)D i-1 T 0 .
(1.5.18)

Remark. In (1.5.18)

ε K |i|=K+2 ∂ ∂ξ k A ki 1 (ξ)N i 2 ...i K+2 (ξ) D i v (K) + ε K+1 |i|=K+3 A i 1 i 2 (ξ)N i 3 ...i K+3 (ξ)D i v (K) = = ε K+1 |i|=K+2 1 ε ∂ ∂ξ k A ki 1 (ξ)N i 2 ...i K+2 (ξ) D i v (K) + ∂ ∂x k A ki 1 (ξ)N i 2 ...i K+2 (ξ)D i v (K) = = ε K+1 ∂ ∂x k   |i|=K+2 A ki 1 x ε N i 2 ...i K+2 x ε D i v (K)   and ε K |i|=K+2 ∂ ∂ξ k A ki 1 (ξ)Y i 2 ...i K+1 (ξ) D i-1 T 0 + ε K+1 |i|=K+3 A i 1 i 2 (ξ)Y i 3 ...i K+2 (ξ)D i-1 T 0 = = ε K+1 ∂ ∂x k   |i|=K+2 A ki 1 x ε Y i 2 ...i K+1 x ε D i-1 T 0   .
There are at |i| = 0:

H ∅ = L ξξ N ∅ ≡ 0, N ∅ = 1,
at |i| = 1:

H i 1 = ∂ ∂ξ k A kj (ξ) ∂N i 1 (ξ) ∂ξ j + A ki 1 (ξ) , S ∅ = L ξξ Y ∅ , L ξξ Y ∅ = G ∅
and at |i| 2:

H i (ξ) = L ξξ N i + T i (ξ), (1.5.19) T i (ξ) = s k=1 ∂ ∂ξ k A ki 1 (ξ)N i 2 ...i l (ξ) + s j=1 A i 1 j (ξ) ∂N i 2 ...i l (ξ) ∂ξ j + A i 1 i 2 (ξ)N i 3 ...i l (ξ),
(1.5.20)

S i-1 (ξ) = L ξξ Y i-1 + W i-1 (ξ), (1.5.21) W i-1 (ξ) = s k=1 ∂ ∂ξ k A ki 1 (ξ)Y i 2 ...i l-1 (ξ) + s j=1 A i 1 j (ξ) ∂Y i 2 ...i l-1 (ξ) ∂ξ j + A i 1 i 2 (ξ)Y i 3 ...i l-1 (ξ) (1.5.22)
in formula (1.5.17).

Remark. We formally assume that there are zero

N i (ξ), Y i (ξ), S i (ξ), G i (ξ) with
negative multi-index length |i| in formulas (1.5.17) -(1.5.22).

Since P u should be equal to

f 0 (x) + ∇ α x ε T x, x ε rewritten in (1.5.13), we
suppose that

H i (ξ) = h i , S i-1 (ξ) -G i-1 (ξ) = s i-1 ,
where h i and s i are constants.

We obtain the following recurrent chain of problems of the form

L ξξ N i = -T i (ξ) + h i , N ∅ = 1 (1.5.23)
to determine N i and of the form

L ξξ Y i-1 = -W i-1 (ξ) + G i-1 (ξ) + s i-1 , L ξξ Y ∅ = G ∅ (1.5.24)
to determine Y i . N i and Y i are 1-periodic functions with respect to ξ. The constant matrices h i and s i are chosen from the solvability conditions for problems (1.5.23),

(1.5.24):

h i = T i (ξ) = s j=1 A i 1 j (ξ) ∂N i 2 ...i l (ξ) ∂ξ j + A i 1 i 2 (ξ)N i 3 ...i l (ξ) , l 2, h ∅ = 0, h i 1 = 0, (1.5.25) s i-1 = W i-1 (ξ) -G i-1 (ξ) = s j=1 A i 1 j (ξ) ∂Y i 2 ...i l-1 (ξ) ∂ξ j + A i 1 i 2 (ξ)Y i 3 ...i l-1 (ξ) -G i-1 (ξ) , l 2, s ∅ = 0.
Thus, the algorithm for constructing the functions N i and Y i is recurrent: they are solutions of problems (1.5.23), (1.5.24) for l > 0. The right-hand side in (1.5.23) contains N j with multi-indices j whose length is smaller than |i| and the right-hand side in (1.5.24) contains Y j with multi-indices j whose length is smaller than |i| -1.

Substituting series (1.5.15) 2 we obtain

P u (K) = K+1 l=0 ε l-2 |i|=l H i (ξ)D i v (K) + S i-1 (ξ)D i-1 T 0 + R (K) 1ε .
Then by reason of (1.5.25) previous relation will go over

P u (K) = K+1 l=2 ε l-2 |i|=l h i D i v (K) + (s i-1 + G i-1 )(ξ)D i-1 T 0 + R (K) 1ε = = K+1 l=2 |i|=l K j=0 ε l+j-2 h i D i v j + K+1 l=2 ε l-2 |i|=l (s i-1 + G i-1 )(ξ)D i-1 T 0 + R (K) 1ε ,
where

G i = 0 at |i| > 2.
Assuming that r = l + j -2 gives to us:

K+1 l=2 |i|=l K j=0 ε l+j-2 h i D i v j = K r=0 ε r   s i 1 ,i 2 =1 h i 1 i 2 ∂ 2 v r ∂x i 1 ∂x i 2 + r-1 j=0 |i|=r-j+2 h i D i v j   + R (K) 2ε ,
where

R (K) 2ε = 2K-1 r=K+1 ε r K j=0 |i|=min{r-j+2,K+1} h i D i v j = 2K-1 r=K+1 ε r |i|+j-2=r |i| K+1 j K h i D i v j (1.5.26)
because differentiation is possible while |i| K + 1 and the functions v j exist prior to j = K.

R (K) ε = R (K) 1ε + R (K) 2ε . (1.5.27)
To determine v j we have

K+1 l=2 ε l-2 |i|=l h i D i v (K) + s i-1 D i-1 T 0 = f 0 (x). s i 1 ,i 2 =1 h i 1 i 2 ∂ 2 v (K) ∂x i 1 ∂x i 2 + s i 1 ∂T 0 ∂x i 1 + K+1 l=3 ε l-2 |i|=l h i D i v (K) + s i-1 D i-1 T 0 -f 0 (x) = 0.
(1.5.28)

Problem (1.5.28) can be regarded as the homogenized equation with respect to s-dimensional vector v. Substituting series (1.5.15) 2 into (1.5.28), we obtain a recurrent chain of equations for the components v k j of the vectors v j in the form

s i 1 ,i 2 =1 h i 1 i 2 ∂ 2 v j ∂x i 1 ∂x i 2 = g j (x), (1.5.29)
where

g 0 = f 0 (x) -s i 1 ∂T 0 ∂x i 1
, the functions g j depend on v j 1 , j 1 < j, and on the derivatives of these functions. Thus the f.a.s. of problem (1.5.1), (1.5.2), and (1.5.15) is constructed.

Justification of the asymptotic expansion

After substituting u (K) in the left-hand side of equation (1.5.1) and repeating transformations as in the construction section we obtain

L ε u (K) = f 0 (x) + R (K) ε , where | R (K) ε | c 1 ε K , c 1 is positive constant independent on ε. So, the difference u (K) -u is a solution to equation L ε (u (K) -u) = R (K)
ε . A priory estimate (1.5.9) for this solution gives

u (K) -u H 1 C R (K) ε L 2 = O(ε (K) ).

Conclusions

Concerning the technical physical aspect we can conclude that calculation by the described model gives residual stresses that could form after cooling of the treatment area to its original state in case of the absence of inelastic deformation and fracturing during the cooling stage.

Vertical plate grown with the help of the SLM method on a hard substrate, is exposed to uniaxial tension and the horizontal plate is exposed to the isotropic biaxial one.

Maximum longitudinal tensile residual stresses in the individual beads are approximately twice as large as the maximum transverse one within the elastic deformation of materials during cooling after SLM.

Residual stresses in the SLM do not depend on the temperature gradient and the cooling rate under the condition of elastic deformation of the materials during cooling.

The results of the presented calculations are used to estimate the thermomechanical stability of materials in the SLM.

Concerning the mathematical aspect we can conclude that there exists a unique solution of the system of linear equations with residual stresses in the right-hand side.

The solution is constructed as a asymptotic series. Its coefficients are determined owing to the homogenization process. The difference between the asymptotic approach and the solution is small.

Chapter 2

Variational analysis of a viscous fluid-thin plate interaction problem

Introduction. Formulation of the problem

The viscous fluid -thin plate interaction problem arises in numerous applications:

the blood flow near the vessel wall, the fluid motion in the pipelines, the hydrodynamic resistance to the boat, etc. In the present paper we consider a viscous fluid-3D thin rigid stratified plate interaction problem. The small parameter ε stands for the ratio of the thicknesses of the plate and of the fluid layer. In the same time the Young modulus of the plate is supposed to be great having the order of ε -3 , while the viscosity of the fluid, as well as the densities of the fluid and solid are supposed to be finite (i.e. of order of 1). The right hand side functions are supposed 1-periodic with respect to the tangential variables of the plate. At the plate fluid interface the velocity and the normal stress are continuous. This problem is a three dimensional generalization of the two-dimensional setting considered in [START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF][START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF]. However the density of the plate was supposed to be much greater than the density of the fluid which is less natural for the applications. Moreover, the conclusions made on the two-dimensional modeling may be very different of the three-dimensional one, which is more realistic. That is why in the present paper the three-dimensional fluid-structure interaction is considered. Previously the problems of the fluid-structures interaction were considered in papers [START_REF] Panasenko | Asymptotic analysis of a non-periodic flow in a thin channel with visco-elastic wall[END_REF][START_REF] Stavre | Asymptotic analysis of the Stokes flow with variable viscosity in a thin elastic channel[END_REF][START_REF] Panasenko | Asymptotic analysis of a periodic flow in a thin channel with visco-elastic wall[END_REF][START_REF] Čanić | Effective Equations Modeling the Flow of a Viscous Incompressible Fluid through a Long Elastic Tube Arising in the Study of Blood Flow through Small Arteries[END_REF][START_REF] Čanić | Effective equations describing the flow of a viscous incompressible fluid through a long elastic tube[END_REF][START_REF] Stavre | Asymptotic and numerical modelling of a flow in a thin channel with visco-elastic wall[END_REF]. In [START_REF] Panasenko | Asymptotic analysis of a non-periodic flow in a thin channel with visco-elastic wall[END_REF][START_REF] Stavre | Asymptotic analysis of the Stokes flow with variable viscosity in a thin elastic channel[END_REF][START_REF] Panasenko | Asymptotic analysis of a periodic flow in a thin channel with visco-elastic wall[END_REF][START_REF] Stavre | Asymptotic and numerical modelling of a flow in a thin channel with visco-elastic wall[END_REF]] the deformable structure description was simplified by neglecting the thickness of the wall. In [START_REF] Mikelić | Fluid-structure interaction in a prestressed tube with thick elastic walls. I. The stationary Stokes problem[END_REF], the authors study the steady-state fluid-structure interaction between a three-dimensional, axi-symmetric elastic tube filled with an incompressible viscous fluid, when the thickness of the tube wall is of the same order of magnitude as the tube radius.

The elastic moduli of the wall were supposed to be of the same order that the fluid viscosity.

In the present paper the fluid flow is modeled by the 3D Stokes equations while the plate is described by the linear 3D elasticity equations. At the interface the velocity continuity and the normal stress continuity are imposed. Although the fluid and the solid phases are described in different variables (Eulerian and Lagrangian, respectively), it is supposed that the displacements and strains are small enough so that the values of the velocity and of the pressure are close in both variables.

The fluid occupies the horizontal layer L -= R 2 × (-1, 0) while the plate corresponds to the thin layer

L + ε = R 2 × (0, ε)
, where ε is a small positive parameter. The applied mass forces are supposed to be 1-periodic with respect to the "horizontal" variables x 1 and x 2 . Put x = (x 1 , x 2 ), x = (x 1 , x 2 , x 3 ) and denote the

x 1 x 2 x 3 0 ε -1 F + ε F 0 F - L + ε L -
F + ε = {x ∈ R 2 , x 3 = ε}. Denote D the square (0, 1) 2 .
Then the fluid-plate interaction problem can be described by the following boundary value problem:

                                                                     Find triplet (u ε , v ε , p ε ) such that ρ + ( x 3 ε ) ∂ 2 u ε ∂t 2 -ε -3 3 i,j=1 ∂ ∂x i A ij ( x 3 ε ) ∂u ε ∂x j = ε -1 g(x, t) in L + ε × (0, T ), ρ - ∂v ε ∂t -2ν div (D(v ε )) + ∇p ε = f (x, t) in L -× (0, T ), div v ε = 0 in L -× (0, T ), 3 j=1 A 3j (1) ∂u ε ∂x j = 0 on F + ε × (0, T ), v ε = 0 on F -× (0, T ),          v ε = ∂u ε ∂t , -p ε e 3 + 2νD(v ε )e 3 = ε -3 3 j=1 A 3j (0) ∂u ε ∂x j on F 0 × (0, T ), u ε , v ε , p ε D-periodic, u ε (x, 0) = ∂u ε ∂t (x, 0) = 0 in L + ε , v ε (x, 0) = 0 in L -, (2.1.1)
with a positive given constant T .

The characteristics of the elastic medium are described by the variable density ρ + (ξ 3 ), by 3×3 matrix-valued functions ε -3 A ij (ξ 3 ), i, j ∈ {1, 2, 3} of elastic moduli of the plate material, by the Young's modulus ε -3 E(ξ 3 ) and by the Poisson's coefficient ν(ξ 3 ), with

ξ 3 = x 3 ε . (2.1.2)
The plate is supposed to be heterogeneous, stratified, and so these coefficients depend on the variable x 3 . We study the case corresponding to ρ + and E of order one (the elastic moduli are of order ε -3 ). The coefficients ρ + and A ij are supposed to be piecewise smooth functions on [0, 1] i.e. there exist p ∈ N, p ≥ 2

and p + 1 real numbers ζ 0 , ζ 1 , ..., ζ p , with 0 = ζ 0 < ζ 1 < ... < ζ p = 1 such that ρ + , a kl ij ∈ C 1 ([ζ a , ζ a+1 ]
), a = 0, 1, ..., p -1, i, j, k, l ∈ {1, 2, 3}; moreover, there exist two positive constants ρ + min , ρ + max independent of ε such that

ρ + min ≤ ρ + (ξ 3 ) ≤ ρ + max , ∀ ξ 3 ∈ [0, 1]. (2.1.3)
The matrices A ij = (a kl ij ) 1≤k,l≤3 with the elements a kl ij defined via two piecewise-smooth functions E and ν by the following formula

a kl ij = E 2(1 + ν) 2ν 1 -2ν δ ik δ jl +δ ij δ kl +δ il δ jk satisfying the bounds -1 < ν < 1 / 2 , E > 0.
Hence, they satisfy the properties:

(i) a kl ij (ξ 3 ) = a il kj (ξ 3 ) = a lk ji (ξ 3 ), ∀ i, j, k, l ∈ {1, 2, 3}, ∀ξ 3 ∈ [0, 1], (ii) ∃ κ > 0 independent of ε such that 3 i,j,k,l=1 a kl ij (ξ 3 )η l j η k i ≥ κ 3 j,l=1 (η l j ) 2 , ∀ξ 3 ∈ [0, 1], ∀η = (η l j ) 1≤j,l≤3
, with η l j = η j l . The matrices A ij have the following expressions:

                                                                                 A 11 (ξ 3 ) =     E(ξ 3 )(1-ν(ξ 3 )) (1+ν(ξ 3 ))(1-2ν(ξ 3 )) 0 0 0 E(ξ 3 ) 2(1+ν(ξ 3 )) 0 0 0 E(ξ 3 ) 2(1+ν(ξ 3 ))     , A 22 (ξ 3 ) =     E(ξ 3 ) 2(1+ν(ξ 3 )) 0 0 0 E(ξ 3 )(1-ν(ξ 3 )) (1+ν(ξ 3 ))(1-2ν(ξ 3 )) 0 0 0 E(ξ 3 ) 2(1+ν(ξ 3 ))     , A 33 (ξ 3 ) =     E(ξ 3 ) 2(1+ν(ξ 3 )) 0 0 0 E(ξ 3 ) 2(1+ν(ξ 3 )) 0 0 0 E(ξ 3 )(1-ν(ξ 3 )) (1+ν(ξ 3 ))(1-2ν(ξ 3 ))     , A 12 (ξ 3 ) =     0 E(ξ 3 )ν(ξ 3 ) (1+ν(ξ 3 ))(1-2ν(ξ 3 )) 0 E(ξ 3 ) 2(1+ν(ξ 3 )) 0 0 0 0 0     , A 13 (ξ 3 ) =     0 0 E(ξ 3 )ν(ξ 3 ) (1+ν(ξ 3 ))(1-2ν(ξ 3 )) 0 0 0 E(ξ 3 ) 2(1+ν(ξ 3 )) 0 0     , A 21 (ξ 3 ) =     0 E(ξ 3 ) 2(1+ν(ξ 3 )) 0 E(ξ 3 )ν(ξ 3 ) (1+ν(ξ 3 ))(1-2ν(ξ 3 )) 0 0 0 0 0     , A 23 (ξ 3 ) =     0 0 0 0 0 E(ξ 3 )ν(ξ 3 ) (1+ν(ξ 3 ))(1-2ν(ξ 3 )) 0 E(ξ 3 ) 2(1+ν(ξ 3 )) 0     , A 31 (ξ 3 ) =     0 0 E(ξ 3 ) 2(1+ν(ξ 3 )) 0 0 0 E(ξ 3 )ν(ξ 3 ) (1+ν(ξ 3 ))(1-2ν(ξ 3 )) 0 0     , A 32 (ξ 3 ) =     0 0 0 0 0 E(ξ 3 ) 2(1+ν(ξ 3 )) 0 E(ξ 3 )ν(ξ 3 ) (1+ν(ξ 3 ))(1-2ν(ξ 3 )) 0     . (2.1.4)
The characteristics of the viscous fluid, independent of ε, are the positive constants ρ -and ν representing its density and its viscosity, respectively. In addition to the data ρ + , A ij , E, ν (for the elastic medium) and ρ -, ν (for the viscous fluid), we also know right hand side functions g and f , the scaled mass forces, which act on the elastic medium and on the fluid, respectively; they are supposed to be 1-periodic in x 1 and x 2 .

The unknown functions u ε , v ε , p ε are the plate displacement, the fluid velocity and the pressure, respectively.

In the Stokes equations the standard notation is used for the symmetrized gradient:

D(v) = 1 2 (∇v + (∇v) T ), (2.1.5)
and represents the velocity strain tensor. We will use as well the following notation for the strain tensor in the plate:

E(u) = 1 2 ∇u + (∇u) T . (2.1.6)
As usual, at the layer interfaces within the elastic plate the continuity conditions are satisfied for the displacement u ε as well as for the normal stress 3 j=1 A 3j ∂u ε ∂x j

(see [START_REF] Bakhvalov | Homogenisation: Averaging Processes in Periodic Media36volume, series, Mathematics and its Applications[END_REF]).

We emphasize that the plate's material Young modulus is great; it is of the order ε -3 and depends on the "vertical" fast variable ξ 3 = x 3 ε : it is equal to ε -3 E(ξ 3 ), where E is a function of order one, while the Poisson's ratio ν(ξ 3 ) is of order of one. This value of the Young's modulus is critical with respect to the small parameter ε that is the ratio of thicknesses of the plate and of the fluid layer (the fluid layer thickness is of the same order as the period of the right hand sides). For this value of the Young's modulus there is a coupling in the limit problem between the Stokes equations and the limit plate equation. This coupling generates a non-standard boundary condition for the Stokes equations. In other cases the Stokes equations and the limit plate equation may be decoupled. This boundary value problem for the coupled elasticity equations and the Stokes equations describing the fluid covered with a stratified plate may be used for the modeling of the blood flow -the vessel wall interaction in hemodynamics, as well as for the flow-wall interaction in the pipelines. In order to provide mathematical analysis of this problem, let us state the variational formulation and define a weak solution. To this end let us reduce the number of the unknown functions.

Extend formally the velocity v ε to the layer

L + ε : u ε (x, t) = t 0 v ε (x, s)ds and
exclude the pressure p ε by introducing the divergence free functional space for the test functions in the fluid part of the domain. This step is quite standard for the Stokes and the Navier-Stokes equations [START_REF] Temam | Navier-Stokes Equations: Theory & Numerical Analysis[END_REF], [START_REF] Ladyzhenskaya | The Boundary Value Problems of Mathematical Physics[END_REF]: normally the variational formulation starts without the pressure in the solenoidal functional space and then, if needed, the pressure is defined according to the De Rham theorem, so that both formulations are equivalent. Let us introduce the necessary functional spaces.

In what follows we call function f defined in R 2 ×(a, b) (and so depending on x 1 , x 2 and eventually x 3 ∈ (a, b)) D-periodic iff for any integer i 1 , i 2 , and for any real

x 1 , x 2 (and eventually for any real

x 3 ∈ (a, b)) the relation f (x 1 + i 1 , x 2 + i 2 , x 3 ) = f (x 1 , x 2 , x 3 ) holds.
Define the periodicity domains by

D -= D × (-1, 0), D + ε = D × (0, ε), D ε = D × (-1, ε), (2.1.7)
with D -the fluid part and D + ε the elastic part.

x 1

x 2

x 3 0 ε -1 1 Γ + ε Γ 0 Γ - D + ε D -

Fig. 10. Periodicity domain

Introduce the "horizontal" boundaries and interface in the periodic cell:

       Γ -= {(x, -1)/x ∈ D}, Γ 0 = {(x, 0)/x ∈ D}, Γ + ε = {(x, ε)/x ∈ D}. (2.1.8) Define C ∞ # (D) = f ∈ C ∞ (R 2 ) f -D-periodic , L 2 # (D) = f ∈ L 2 loc (R 2 ) f -D-periodic , f L 2 # (D) = f L 2 (D) , L 2 # (D -) = f ∈ L 2 loc (R 2 × (-1, 0)) f -D-periodic , f L 2 # (D -) = f L 2 (D -) , L 2 # (D + ε ) = f ∈ L 2 loc (R 2 × (0, ε)) f -D-periodic , f L 2 # (D + ε ) = f L 2 (D + ε ) , L 2 # (D ε ) = f ∈ L 2 loc (R 2 × (-1, ε)) f -D-periodic , f L 2 # (D ε ) = f L 2 (D ε ) , H 1 # (D -) = f ∈ H 1 loc (R 2 × (-1, 0)) f -D-periodic , f H 1 # (D -) = f H 1 (D -) , H 1 # (D + ε ) = f ∈ H 1 loc (R 2 × (0, ε)) f -D-periodic , f H 1 # (D + ε ) = f H 1 (D + ε ) , H 1 # (D ε ) = f ∈ H 1 loc (R 2 × (-1, ε)) f -D-periodic , f H 1 # (D ε ) = f H 1 (D ε ) .
The corresponding inner products are introduced in these Hilbert spaces in a standard way.

Let us define

V = ϕ ∈ (H 1 # (D ε )) 3 div ϕ = 0 in D -, ϕ = 0 on Γ - (2.1.9)
and

V + = (H 1 # (D + ε )) 3 , (2.1.10)
Note that since the space V introduced in (2.1.9) is a separable Hilbert space, we can select a countable orthogonal in L 2 # (D ε )) basis {ϕ j } ∞ j=1 . Assume that the right hand side functions have the following regularity

g ∈ H 1 (0, T ; (L 2 # (D + ε )) 3 ), f ∈ H 1 (0, T ; (L 2 # (D -)) 3 ), (2.1.11)
while ρ + , a kl ij are piecewise-smooth as it was formulated above.

We also put

H v = v ∈ L 2 (0, T ; V ) v ′ ∈ L 2 (0, T ; V ′ ), t 0 v dt ∈ L 2 (0, T ; V + ) .
Define a weak solution to problem (2.1.1) as a function v ε satisfying the following integral identity:

                       Find v ε ∈ H v d dt D ε ρ + χ(D + ε ) + ρ -χ(D -) v ε • ϕ + ε -3 D + ε 3 i,j=1 A ij ∂ ∂x j t 0 v ε dt • ∂ϕ ∂x i +2ν D - D(v ε ) : D(ϕ) = D ε ε -1 χ(D + ε )g + χ(D -)f • ϕ ∀ ϕ ∈ V a.e. in (0, T ), v ε (0) = 0 in D ε , (2.1.12)
Here χ(A) is the characteristic function of the set A. Remind that so defined weak solution corresponds to the velocity function only, while the displacement in the solid phase and the pressure are absent. In the next section we study this variational formulation from the point of view of the existence and uniqueness of its solution. Then, using the mentioned regularity of the data, we prove the corresponding regularity of function v ε which gives the regularity of the displacement and that of the pressure, previously introduced as a distribution. An important difference of this problem with respect to the standard Stokes or the Navier-Stokes equations is that the pressure is defined here uniquely. This gives us a possibility to prove that the triplet velocity -displacement -pressure satisfies the relations (2.1.1) pointwisely.

Remark. The function u ε is given by u

ε (x, t) = t 0 v ε (x, s)ds for (x, t) ∈ D + ε × (0, T )
, where v ε represents a continuous extension of the fluid velocity in ( Dε \ D-) × (0, T ), due to (2.1.1) 6 , and it was denoted in (2.1.12) also by v ε . So, the displacement is formally eliminated for performing the variational analysis of problem (2.1.1), it can be rewritten in the form:

                                                   ρ + ∂v ε ∂t -ε -3 3 i,j=1 ∂ ∂x i A ij ∂ ∂x j t 0 v ε (s)ds = ε -1 g in D + ε × (0, T ), ρ - ∂v ε ∂t -2ν div (D(v ε )) + ∇p ε = f in D -× (0, T ), div v ε = 0 in D -× (0, T ), 3 j=1 A 3j ∂ ∂x j t 0 v ε (s)ds = 0 on Γ + ε × (0, T ), v ε = 0 on Γ -× (0, T ), -p ε e 3 + 2νD(v ε )e 3 = ε -3 3 j=1 A 3j ∂ ∂x j t 0 v ε (s)ds on Γ 0 × (0, T ), v ε , p ε D-periodic, v ε (0) = 0 in D ε . (2.1.13)
After multiplication of (2.1.13) 1,2 by ϕ / D + ε , ϕ / D -, respectively and integration by parts we obtain, by means of the other equations and conditions of (2.1.13), the problem (2.1.12) that we will study from a variational viewpoint.

2.2

Existence and uniqueness of a solution

Main problem

Consider the variational formulation (2.1.12). The following theorem holds.

Theorem 2.1. Let g and f have the regularity given by (2.1.11). Then there exists a unique solution v ε ∈ H v of problem (2.1.12).

Proof. The proof of the existence and uniqueness of solution to problem (2.1.12) is based on the Galerkin's method.

System of Galerkin approximations is

                             D ε ρ + χ(D + ε ) + ρ -χ(D -) ∂v n ∂t • ϕ +ε -3 D + ε 3 i,j=1 A ij ∂ ∂x j t 0 v n (s) ds ∂ϕ ∂x i + 2ν D - D(v n ) : D(ϕ) = D ε ε -1 χ(D + ε )g + χ(D -)f • ϕ ∀ ϕ ∈ V n a.e. in (0, T ), v n (0) = 0 in D ε , (2.2.1)
where

V n = span{ϕ 1 , ϕ 2 , . . . , ϕ n }.
The solution is sought in the form

v n = n k=1 a k (t)ϕ k (x), in D ε × (0, T ), (2.2.2)
with the functions a k determined from the system of integro-differential equations:

                               n p=1 D ε ρ + χ(D + ε ) + ρ -χ(D -) ϕ p • ϕ k a ′ k (t) + n p=1 ε -3    D + ε 3 i,j=1 A ij ∂ϕ p ∂x j • ∂ϕ k ∂x i    t 0 a k (s)ds+ n p=1 2ν   D - D(ϕ p ) : D(ϕ k )   a k (t) = D ε ε -1 χ(D + ε )g + χ(D -)f • ϕ k a.e. in (0, T ), a k (0) = 0 k = 1, ...n. (2.2.3)
We denote

c k (t) = t 0 a k (s) ds. (2.2.4) So, as a k = c ′ k , system (2.2.
3) becomes a system of linear differential equations of order 2 that can be used for determination of functions c k (t), k = 1, . . . , n:

                               n p=1 D ε ρ + χ(D + ε ) + ρ -χ(D -) ϕ p • ϕ k c ′′ k (t) + n p=1 ε -3    D + ε 3 i,j=1 A ij ∂ϕ p ∂x j • ∂ϕ k ∂x i    c k (t)+ n p=1 2ν   D - D(ϕ p ) : D(ϕ k )   c ′ k (t) = D ε ε -1 χ(D + ε )g + χ(D -)f • ϕ k a.e. in (0, T ), c k (0) = c ′ k (0) = 0 k = 1, ...n. (2.2.5)
System (2.2.5) has a unique solution, since the matrix

M =   D ε χ(D + ε )ρ + + χ(D -)ρ -ϕ p • ϕ k   1≤p,k≤n (2.2.6)
is non-degenerate. Denoting

ρ ± = ρ + χ(D + ε ) + ρ -χ(D -), (2.2.7)
we obtain that

Mξ • ξ =   n p=1 D ε ρ ± ξ p ϕ p ϕ k   1≤k≤n • ξ = n p,k=1 D ε ρ ± ξ p ϕ p ξ k ϕ k = D ε ρ ± n j=1 ξ j ϕ j 2 ≥ 0. Thus Mξ • ξ = 0 if and only if ξ 1 = ξ 2 = • • • = ξ n = 0 or ξ = 0 (because the system ϕ k is a linear independent system).
Next, for passing to the limit in (2.2.1) 1 as n → ∞ (and to get weak formulation of the problem), we will prove the following estimates:

v n L ∞ (0,T ;(L 2 (D -)) 3 ) ≤ C, ∂v n ∂t L ∞ (0,T ;(L 2 (D ε )) 3 ) ≤ C, D + ε 3 i,j=1 A ij ∂ ∂x j t 0 v ε dt ∂ ∂x i t 0 v ε dt L ∞ (0,T ) ≤ C, D(v n ) L 2 (0,T ;(L 2 (D -)) 3×3 ) ≤ C. (2.2.8)
To this effect we take in (2.2.1) 1 ϕ = ϕ k , multiply by a k (t), sum up from k = 1

to n and integrate from 0 to t:

D ε ρ ± v 2 n + ε -3 D + ε 3 i,j=1 A ij ∂ ∂x j t 0 v n (s) ds ∂ ∂x i t 0 v n (s) ds +4ν t 0 D - D(v n ) 2 = 2 t 0 D ε ε -1 χ(D + ε ) g + χ(D -) f v n .
(2.2.9)

The function ρ ± defined in (2.2.7) can be evaluated owing to (2.1.3) as follows

ρ ± ≥ ρ + min χ(D + ε ) + ρ -χ(D -) ≥ c 1 ,
with c 1 = min ρ + min , ρ -. Denoting F ± = ε -1 χ(D + ε ) g + χ(D -) f , using the previous inequality and Cauchy-Schwarz-Bunyakovsky inequality we obtain from (2.2.9)

c 1 D ε v 2 n + ε -3 D + ε 3 i,j=1 A ij ∂ ∂x j t 0 v n (s) ds ∂ ∂x i t 0 v n (s) ds +4ν t 0 D - D(v n ) 2 ≤ 2 T 0 F ± (t) (L 2 (D ε )) 3 v n (t) (L 2 (D ε )) 3 dt.
(2.2.10)

The right-hand side in (2.2.10) can be evaluated as outlined below

2 T 0 F ± (t) (L 2 (D ε )) 3 v n (t) (L 2 (D ε )) 3 dt ≤ 2 F ± L 1 (0,T ;(L 2 (D ε )) 3 ) v n L ∞ (0,T ;(L 2 (D ε )) 3 ) ≤ c 1 2 v n 2 L ∞ (0,T ;(L 2 (D ε )) 3 ) + 2 c 1 F ± 2 L 1 (0,T ;(L 2 (D ε )) 3 ) .
Finally we obtain the following estimates

c 1 2 v n 2 L ∞ (0,T ;(L 2 (D ε )) 3 ) + 4ν D(v n ) 2 L 2 (0,T ;(L 2 (D -)) 3×3 ) +ε -3 D ε 3 i,j=1 A ij ∂ ∂x j t 0 v n (s) ds ∂ ∂x i t 0 v n (s) ds L ∞ (0,T ) ≤ f , g 1 , (2.2.11)
where

f , g 1 = 2 c 1 F ± 2 L 1 (0,T ;(L 2 (D ε )) 3 ) .
(2.2.12)

We differentiate next (2.2.1) 1 with respect to t (it is possible thanks to regularity (2.1.11) for g and f ). In order to get an estimate for v ′ n (0) we consider (2.2.1) 1 at the moment t = 0

D ε ρ ± v ′ n (0) • ϕ k = D ε F ± (0) • ϕ k k = 1, . . . , n; (2.2.13)
we calculate then n k=1

(2.2.13) • a ′ k (0), which yields

D ε ρ ± (v ′ n (0)) 2 = D ε F ± (0) • v ′ n (0). ( 2

.2.14)

From relation (2.2.14) with the inequality ρ ± ≥ c 1 it follows that

c 1 v ′ n (0) 2 (L 2 (D ε )) 3 ≤ 1 c 1 F ± (0) 2 (L 2 (D ε )) 3 def = f , g 0 c 1 . ( 2

.2.15)

We repeat the same technique as before; i.e. we compute

n k=1 (2.2.1) ′ 1 • a ′ k (t) and
we integrate in time:

D ε ρ ± (v ′ n ) 2 + ε -3 D + ε 3 i,j=1 A ij ∂v n ∂x j ∂v n ∂x i + 4ν t 0 D - D(v ′ n ) 2 = 2 t 0 D ε F ′ ± v ′ n + D ε ρ ± (v ′ n (0)) 2 ,
which gives the estimates:

c 1 2 v ′ n 2 L ∞ (0,T ;(L 2 (D ε )) 3 ) + ε -3 D ε 3 i,j=1 A ij ∂v n ∂x j ∂v n ∂x i L ∞ (0,T ) +4ν D(v ′ n ) 2 L 2 (0,T ;(L 2 (D -)) 3×3 ) ≤ f ′ , g ′ 1 + c 2 c 2 1 f , g 0 , (2.2.16)
where c 2 = max{ρ + max , ρ -}. So, the maximal regularity obtained for v ′ n from (2.2.16) is:

v ′ n ∈ L ∞ (0, T ; (L 2 # (D ε )) 3 ), v ′ n ∈ L 2 (0, T ; (H 1 # (D -)) 3 ). ( 2
           v n k → v * , v ′ n k → v ′ * weakly-* in L ∞ (0, T ; (L 2 # (D ε )) 3 ), 3 j=1 A 3j ∂ t 0 v n k dt ∂x j → 3 j=1 A 3j ∂ t 0 v * dt ∂x j weakly-* in L ∞ (0, T ; (L 2 # (D + ε )) 3 ), D(v n k ) → D(v * ) weakly in L 2 (0, T ; (L 2 # (D -)) 9 ). ( 2

.2.18)

We need to show that the limit v * ∈ H v is solution for (2.1.12).

For passing to the limit in (2.2.1), written for the subsequences n k , we take ϕ = ϕ k in (2.2.1) 1 , we consider an arbitrary function η ∈ L 2 (0, T ) and we compute

T 0 n k=1 a k (2.2.1) 1 • η dt, which gives T 0 D ε ρ ± ∂v n ∂t • ϕη + ε -3 T 0 D + ε 3 i,j=1 A ij ∂ ∂x j t 0 v n (s) ds ∂ϕ ∂x i η +2ν T 0 D - D(v n ) : D(ϕ)η = T 0 D ε F ± • ϕη, (2.2
T 0 D ε ρ ± ∂v * ∂t • ϕη + ε -3 T 0 D + ε 3 i,j=1 A ij ∂ ∂x j t 0 v * (s) ds ∂ϕ ∂x i η +2ν T 0 D - D(v * ) : D(ϕ)η = T 0 D ε F ± • ϕη ∀ ϕ ∈ V. ( 2
- T 0 D ε ρ ± v n • ϕη ′ + ε -3 T 0 D + ε 3 i,j=1 A ij ∂ ∂x j t 0 v n (s) ds ∂ϕ ∂x i η +2ν T 0 D - D(v n ) : D(ϕ)η = T 0 D ε F ± • ϕη ∀ ϕ ∈ V and - D ε ρ ± v * (0) • ϕη(0) - T 0 D ε ρ ± v * • ϕη ′ + 2ν T 0 D - D(v * ) : D(ϕ)η +ε -3 T 0 D + ε 3 i,j=1 A ij ∂ ∂x j t 0 v * (s) ds ∂ϕ ∂x i η = T 0 D ε F ± • ϕη ∀ ϕ ∈ V.
As a consequence of the previous two relations and (2.2.18) we obtain v * (0) = 0.

In order to show the uniqueness we take g = 0, f = 0 in (2.1.12) and we need to prove the absence of nontrivial solutions for this problem; we take v ε as a test function

                   D ε ρ ± ∂v ε ∂t v ε + ε -3 D + ε 3 i,j=1 A ij ∂ ∂x j t 0 v ε (s) ds ∂v ε ∂x i +2ν D - D(v ε ) : D(v ε ) = 0 a.e. in (0, T ), v ε (0) = 0 in D ε .
(2.2.21)

A priori estimates for v ε are obtained in a similar manner to that described above (see (2.2.11), with v n replaced by v ε ):

c 1 2 v ε 2 L ∞ (0,T ;(L 2 (D ε )) 3 ) + 4ν D(v ε ) 2 L 2 (0,T ;(L 2 (D -)) 3×3 ) +ε -3 D ε 3 i,j=1 A ij ∂ ∂x j t 0 v ε (s) ds ∂ ∂x i t 0 v ε (s) ds L ∞ (0,T ) ≤ f , g 1 , (2.2.22)
where f , g 1 is given by (2.2.12).

For an a priori estimate the term

ε -3 D ε 3 i,j=1 A ij ∂ ∂x j t 0 v ε (s) ds ∂ ∂x i t 0 v ε (s) ds L ∞ (0,T )
should be estimated below by the second Korn inequality via lower part and the trace as authors did in [62, Th. 1] using Lax-Milgram lemma [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF]. In (2.2.22) it is easy to get the lower estimate carrying out a change of variables and unknowns in D + ε , reducing it to the field, independent of epsilon, and using the periodicity in x 1 , x 2 with the continuation through the interface Γ 0 .

From estimates (2.2.22) the uniqueness follows when f = g = 0.

The uniqueness result gives convergences (2.2.18) not only on subsequences, but on the whole sequences.

The pressure gradient can be constructed from the de Rham theorem [START_REF] Temam | Navier-Stokes Equations: Theory & Numerical Analysis[END_REF] (as a distribution).

Limit problem

Generally the main question is the constructing of the limit problem that will be constructed in the chapter 3. For the asymptotic approach of the problem we need further regularity for the data. We suppose that (H1) the function g is independent of

x 3 and g ∈ C ∞ ([0, T ], (C ∞ # ( D)) 3 ); (H2) the function f is C ∞ D-periodic and ∂ l f ∂x s 1 . . . ∂x s l ∈ C ∞ ([0, T ], (L 2 (D -)) 3 ),
for any l ∈ N, s = (s 1 , . . . , s l ), s j ∈ {1, 2}, |s| = l;

(H3) ∃τ 0 < T such that f = 0 in D -× [0, τ 0 ], g = 0 in D × [0, τ 0 ].
Consider the following problem (for k = 0)

                                       Ĵ∆ 2 x(w 0 ) 3 -p 0 x 3 =0 = g 3 in D × (0, T ), ρ - ∂ v0 ∂t -ν∆v 0 + ∇p 0 = f , div v0 = 0 in D -× (0, T ), v0 (x, -1, t) = 0 in D × (0, T ), v0 (x, 0, t) = ∂(w 0 ) 3 ∂t (x, t)e 3 in D × (0, T ), (w 0 ) 3 , v0 , p 0 D -periodic, v0 (x, 0) = 0 in D -; (w 0 ) 3 (x, 0) = 0 in D (2.2.23)
with regularity for the data satisfying the hypothesis (H1)-(H3).

Problem (2.2.23) can be rewritten

                               ρ - ∂ v0 ∂t -ν∆v 0 + ∇p 0 = f , div v0 = 0 in D -× (0, T ), v0 (x, -1, t) = 0 in D × (0, T ), (v 0 ) 1 = (v 0 ) 2 = 0, Ĵ t 0 ∆ 2 x(v 0 ) 3 (x, 0, s) ds -p 0 Γ 0 = g 3 in D × (0, T ), v0 , p 0 D -periodic, v0 (x, 0) = 0 in D -, (2.2
                             ρ - v′′ 0 -ν∆v ′ 0 + ∇p ′ 0 = f ′ , div v′ 0 = 0 in D -× (0, T ), v′ 0 (x, -1, t) = 0 in D × (0, T ), (v 0 ) ′ 1 = (v 0 ) ′ 2 = 0, Ĵ∆ 2 x(v 0 ) 3 (x, 0, s) -p ′ 0 Γ 0 = g ′ 3 in D × (0, T ), v0 , p 0 D -periodic, v0 (0) = v′ 0 (0) = 0 in D -.
(2.2.25)

We notice that the condition v′ 0 (0) = 0 appears as a consequence of (H3).

Define

H = ω ∈ (H 1 # (D -)) 3 div ω = 0, ω = 0 on Γ -, ω 1,2 = 0 on Γ 0 , γ 0 ω 3 ∈ H 2 (Γ 0 ) , (2.2.26)
where γ : (H

1 (D -)) 3 → (H 1/2 (∂D -)) 3 , γ 0 = γ Γ 0
is the trace operator. Such spaces [START_REF] D'yakonov | Estimation of the Kolmogorov N-widths for certain compacts in the strengthened Sobolev spaces[END_REF][START_REF] Timerbaev | Spaces with a graph norm and strengthened Sobolev spaces[END_REF][START_REF] Timerbaev | Spaces with a graph norm and stengthened Sobolev spaces[END_REF] are called strengthened Sobolev spaces. We put

H v = v ∈ L 2 (0, T ; H) v ′ ∈ L 2 (0, T ; (H 1 # (D -)) 3 ), v ′′ ∈ L 2 (0, T ; (L 2 # (D -)) 3 ) . (2.2.27)
The space H is provided with the norm

ω 2 H = ω 2 (H 1 (D -)) 3 + ∆ xγ 0 ω 3 2 L 2 (Γ 0 ) . We compute D -(2.2.25) 1 • ϕ + Γ 0 (2.2.25) 4
• γ 0 ϕ 3 and after integration by parts we obtain the variational formulation for our problem

                 Find v0 ∈ H v such that ρ - D - v′′ 0 (t)ϕ + ν D - ∇v ′ 0 (t) : ∇ϕ + Ĵ Γ 0 ∆ x(γ 0 (v 0 ) 3 )(t)∆ x(γ 0 ϕ 3 ) = D - f ′ (t)ϕ + Γ 0 g ′ 3 (t)γ 0 ϕ 3 a.e. in (0, T ), ∀ϕ ∈ H, v0 (0) = v′ 0 (0) = 0 in (L 2 # (D -)) 3 .
(

v′ 0 ∈ W 1,∞ (0, T ; H), v′′ 0 ∈ L 2 (0, T ; (H 1 # (D -)) 3 ). ( 2

.2.29)

Proof. The space H being a separable Hilbert space with respect to • H we shall apply the Galerkin's method. For this purpose we consider a orthogonal basis {ϕ j } j∈N of the space H and we define the approximate functions v0n (x, t) = n j=1 c j (t)ϕ j (x). with the coefficients c j (t) : [0, T ] → R uniquely determined from

             ρ - D - v′′ 0n ϕ j + ν D - ∇v ′ 0n ∇ϕ j + Ĵ Γ 0 ∆ x(γ 0 (v 0n ) 3 )∆ x(γ 0 ϕ j3 ) = D - f ′ ϕ j + Γ 0 g ′ 3 γ 0 ϕ j3 ∀ j = 1, . . . , n a.e. in (0, T ), v0n (0) = v′ 0n (0) = 0 in (L 2 # (D -)) 3 .
(2.2.30)

Next we multiply (2.2.30) 1 by c ′ j and sum from 1 to n. We get

             ρ - D - v′′ 0n v′ 0n + D - ν∇v ′ 0n ∇v ′ 0n + Γ 0 Ĵ∆ x(γ 0 (v 0n ) 3 )∆ x(γ 0 (v ′ 0n ) 3 ) = D - f ′ v′ 0n + Γ 0 g ′ 3 γ 0 (v ′ 0n ) 3 a.e. in (0, T ), ∀ϕ ∈ H, v0n (0) = v′ 0n (0) = 0 in (L 2 # (D -)) 3 (2.2.31) knowing that n j=1 c ′ j ϕ j = v′ 0n , n j=1 c ′ j (ϕ j ) 3 = γ 0 (v ′ 0n ) 3 .
We majorate the right-hand side as follows

ρ - d dt D - (v ′ 0n ) 2 + 2ν D - (∇v ′ 0n ) 2 + Ĵ d dt Γ 0 (∆ x(γ 0 (v 0n ) 3 )) 2 ≤ 2 f ′ (L 2 (D -)) 3 v′ 0n (L 2 (D -)) 3 + 2 g ′ 3 L 2 (Γ 0 ) γ 0 (v ′ 0n ) 3 L 2 (Γ 0 ) ≤ 2c 1 f ′ (L 2 (D -)) 3 ∇v ′ 0n L 2 (D -) + 2 g ′ 3 L 2 (Γ 0 ) γ 0 (v ′ 0n ) 3 L 2 (Γ 0 ) ,
with a constant c 1 independent of ε given by Poincaré's inequality.

Integrating in time the previous inequality and using initial conditions (2.2.31)

2
we get

ρ - D - (v ′ 0n ) 2 (t) + 2ν t 0 D - (∇v ′ 0n ) 2 + Ĵ Γ 0 (∆ x(γ 0 (v 0n ) 3 )) 2 (t) ≤ 2c 1 t 0 f ′ (L 2 (D -)) 3 ∇v ′ 0n (L 2 (D -)) 3 + 2 t 0 g ′ 3 L 2 (Γ 0 ) γ 0 (v ′ 0n ) 3 L 2 (Γ 0 ) ≤ 2c 1 t 0 f ′ (L 2 (D -)) 3 ∇v ′ 0n (L 2 (D -)) 3x3 + 2c 2 t 0 g ′ 3 L 2 (Γ 0 ) ∇v ′ 0n (L 2 (D -)) 3x3 ≤ c 3 f ′ 2 L 2 (0,T ;(L 2 (D -)) 3 ) + g ′ 3 2 L 2 (0,T ;L 2 (Γ 0 )) + ν t 0 D - (∇v ′ 0n ) 2 , using γ 0 (v ′ 0n ) 3 L 2 (Γ 0 ) ≤ c 2 ∇v ′ 0n (L 2 (D -)) 3x3 .
So we obtain the first estimates as below max{ρ

1/2 - v′ 0n L ∞ (0,T ;(L 2 (D -)) 3 ) , ν1/2 ∇v ′ 0n L 2 (0,T ;(L 2 (D -)) 3x3 ) , Ĵ1/2 ∆ x(γ 0 (v 0n ) 3 ) L ∞ (0,T ;L 2 (Γ 0 )) } ≤ f ′ , g ′ 3 2 , (2.2.32) with f ′ , g ′ 3 2 = (2c 3 ) 1/2 f ′ L 2 (0,T ;(L 2 (D -)) 3 ) + g ′ 3 L 2 (0,T ;L 2 (Γ 0 )) .
Taking into account (H3) we obtain from (2.2.32) the second estimates max{ρ

1/2 - v′′ 0n L ∞ (0,T ;(L 2 (D -)) 3 ) , ν1/2 ∇v ′′ 0n L 2 (0,T ;(L 2 (D -)) 3x3 ) , Ĵ1/2 ∆ x(γ 0 (v ′ 0n ) 3 ) L ∞ (0,T ;L 2 (Γ 0 )) } ≤ f ′′ , g ′′ 3 2 , (2.2.33)
With these estimates we have the following convergences on subsequences

       v0n k ⇀ v0 * , v′ 0n k ⇀ v′ 0 * , v′′ 0n k ⇀ v′′ 0 * weakly in L 2 (0, T ; (L 2 # (D -)) 3 ), ∇v ′ 0n k ⇀ ∇v ′ 0 * weakly in L 2 (0, T ; (L 2 # (D -)) 3 ), ∆ x(γ 0 (v 0n k ) 3 ) ⇀ ∆ x(γ 0 (v 0 * ) 3 ) weakly -* in L 2 (0, T ; L 2 # (Γ 0 )) (2.2.34)
for k → ∞.

We need to show that the limit v0 * ∈ H v is the weak solution of (2.2.28).

For passing to the limit in (2.2.30), written for the subsequences n k , we consider arbitrary functions η ∈ L 2 (0, T ) and ϕ = n j=1 c j ϕ j ∈ H and we compute

T 0 n j=1 c j (2.2.30) 1 • η dt, which gives ρ - T 0 D - v′′ 0n ϕη + ν T 0 D - ∇v ′ 0n ∇ϕη + Ĵ T 0 Γ 0 ∆ x(γ 0 (v 0n ) 3 )∆ x(γ 0 ϕ 3 )η = T 0 D - f ′ ϕη + T 0 Γ 0 g ′ 3 γ 0 ϕ 3 η. (2.2.35)
We use next (2.2.34), which yields

ρ - D - v′′ 0 * ϕ + ν D - ∇v ′ 0 * ∇ϕ + Ĵ Γ 0 ∆ x(γ 0 (v 0 * ) 3 )∆ x(γ 0 ϕ 3 ) = D - f ′ ϕ + Γ 0 g ′ 3 γ 0 ϕ 3 a.e. in (0, T ), ∀ϕ ∈ H. ( 2

.2.36)

To get initial conditions (2.2.28) 2 we introduce the space

Y = {v ∈ L 2 (0, T ; (H 1 # (D -)) 3 ), v ′ ∈ L 2 (0, T ; (L 2 # (D -)) 3 )},
we use the fact [START_REF] Temam | Navier-Stokes Equations: Theory & Numerical Analysis[END_REF] that Y ⊂ L 2 (0, T ; (L 2 # (D -)) 3 ) is compact owing to the compactness of (H 1 # (D -)) 3 ⊂ (L 2 # (D -)) 3 . Then applying the Poincaré's inequality, estimates (2.2.32), (2.2.33) and using the hypothesis (H3) we get the following relations v′

0n L 2 (0,T ;(H 1 # (D -)) 3 ) ≤ C 1 f ′ , g ′ 3 2 L 2 (0,T ) , v′′ 0n L 2 (0,T ;(L 2 # (D -)) 3 ) ≤ C 2 f ′′ , g ′′ 3 2 L 2 (0,T ) .
(2.2.37)

It signifies belonging of v0n , v′

0n to C([0, T ]; (L 2 # (D -)) 3 ). The compactness of the embedding H 1 (0, T ; (H 1 # (D -)) 3 ) ⊂ C([0, T ]; (L 2 # (D -)) 3 ), initial condi- tions (2.2.30) give the strong convergences v0n → v0 and v′ 0n → v′ 0 in C([0, T ]; (L 2 # (D -)) 3
) from the weak convergences v0n ⇀ v0 and v′

0n ⇀ v′ 0 in L 2 (0, T ; (H 1 # (D -)) 3
). The limits v0 , v′ 0 are also found in C([0, T ]; (L 2 # (D -)) 3 ). Then the convergences v0n → v0 and v′ 0n → v′

0 in C([0, T ]; (L 2 # (D -)) 3 ) lead to v0n (0) → v0 (0) in (L 2 # (D -)) 3 and v′ 0n (0) → v′ 0 (0) in (L 2 # (D -)) 3 whence we have initial conditions (2.2.28) 2 .
All the assertions of the theorem are obtained following the steps of the corresponding proof of [START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF][START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF].

Uniqueness of v0 * . Multiply the integral identity by v′ 0 * . We use the energetic estimates for the difference v1 0 * -v2 0 * with right-hand side = 0.

Regularity results

Main problem

Improving the regularity [START_REF] Fichera | Existence Theorems in Elasticity[END_REF], [START_REF] Ladyzhenskaya | The Boundary Value Problems of Mathematical Physics[END_REF] is a very important question for the existence of the pressure trace on the interaction boundary Γ 0 and the further asymptotic analysis. We have already got the existence and uniqueness results for initial problem (2.1.12) in the weak sense (with the minimal regularity) by means of Galerkin's method with a priori estimates (2.2.22) (without pressure). We improve the data regularity for carrying out of the regularity study as follows:

g ∈ H 2 ([0, T ], (H 1 # (D + ε )) 3 ), f ∈ H 2 ([0, T ], (H 1 # (D -)) 3
). We introduce new domains

D εh i , D + εh i , D - h i , that are obtained from D ε , D + ε , D -, respectively, with a shift in x i at h i , i = 1, 2: D h 1 = (h 1 , 1 + h 1 ) × (0, 1), D h 2 = (0, 1) × (h 2 , 1 + h 2 ), D εh i = D h i × (-1, ε), D + εh i = D h i × (0, ε), D - h i = D h i × (-1, 0),
for small h 1 and h 2 . We consider next the variables x 1 , x 2 and t consecutively. For example, we start with x 1 : to improve the regularity in x 1 we consider problem (2.1.12) with x 1 replaced by

x 1 + h 1                      d dt D εh 1 ρ ± v ε (x 1 + h 1 ) • ϕ + ε -3 D + εh 1 3 i,j=1 A ij ∂ ∂x j t 0 v ε (x 1 + h 1 ) ds • ∂ϕ ∂x i +2ν D - h 1 D(v ε (x 1 + h 1 )) : D(ϕ) = D εh 1 F ± (x 1 + h 1 ) • ϕ ∀ ϕ ∈ V a.e. in (0, T ), v ε (0) = 0 in D ε , (2.3.1)
and the problem corresponding to the right-hand side ∂F ± ∂x 1 :

                   d dt D ε ρ ± v * ε • ϕ + ε -3 D + ε 3 i,j=1 A ij ∂ ∂x j t 0 v * ε ds • ∂ϕ ∂x i +2ν D - D(v * ε ) : D(ϕ) = D ε ∂F ± ∂x 1 • ϕ ∀ ϕ ∈ V a.e. in (0, T ), v * ε (0) = 0 in D ε . (2.3.2)
We remark that the test function ϕ in (2.3.1) would have to be equal to ϕ(x 1 +h 1 ), but (2.3.1) 1 remains true for all ϕ ∈ V . So, it can be taken ϕ(x 1 ). In this case problem (2.3.1) is equivalent to variational problem (2.1.12) on the score of the Dperiodicity. Convergence of the finite difference

v ε (x 1 + h 1 , x 2 , x 3 , t) -v ε (x, t) h 1 to the solution v * ε (x, t) of (2.3.
2) can be proved as follows. Evidently, the difference

v ε (x 1 + h 1 , x 2 , x 3 , t) -v ε (x, t) h 1 -v * ε (x, t
) satisfies the problem with the right-hand side

F ± (x 1 + h 1 , x 2 , x 3 , t) -F ± (x, t) h 1 - ∂F ± ∂x 1 (x, t).
We apply a priori estimates (2.2.22):

c 1 2 v ε (x 1 + h 1 ) -v ε (x 1 ) h 1 -v * ε 2 L ∞ (0,T ;(L 2 (D ε )) 3 ) +4ν D( v ε (x 1 + h 1 ) -v ε (x 1 ) h 1 -v * ε ) 2 L 2 (0,T ;(L 2 (D -)) 3×3 ) +ε -3 D ε 3 i,j=1 A ij ∂ ∂x j t 0 v ε (x 1 + h 1 ) -v ε (x 1 ) h 1 -v * ε ds • ∂ ∂x i t 0 v ε (x 1 + h 1 ) -v ε (x 1 ) h 1 -v * ε ds L ∞ (0,T ) ≤ f (x 1 + h 1 ) -f (x 1 ) h 1 - ∂f ∂x 1 , g(x 1 + h 1 ) -g(x 1 ) h 1 - ∂g ∂x 1 1 . (2.3.3)
Due to the improved regularity of f and g, the right-hand side of (2.3.3) goes to zero when h 1 → 0, so

v ε (x 1 + h 1 , x 2 , x 3 , t) -v ε (x, t) h 1 tends to v * ε (x, t), when h 1 → 0. From the convergence of v ε (x 1 + h 1 , x 2 , x 3 , t) -v ε (x, t) h 1 to v * ε (x, t
) we deduce that v ε is differentiable in x 1 and, within the meaning of the generalized derivative defined by Sobolev, that the derivative ∂v ε ∂x 1 is exactly the function v * ε . With the same ideas, the previous result holds also for the variables x 2 and t.

We consider now the following problem

                                 ρ + ∂ 2 u ε ∂t 2 -ε -3 3 i,j=1 ∂ ∂x i A ij ∂u ε ∂x j = ε -1 g in D + ε × (0, T ), 3 j=1 A 3j ∂u ε ∂x j = 0 on Γ + ε × (0, T ), u ε = t 0 v ε (s) ds Γ 0 on Γ 0 × (0, T ), u ε D-periodic, u ε (0) = u ′ ε (0) = 0 in D + ε . (2.3.4)
Repeating the same arguments as above we obtain the regularity

∂u ε ∂x i (t) ∈ (H 1 (D + ε )) 3 , i = 1, 2,
and

∂ 2 u ε ∂t 2 (t) ∈ (L 2 (D + ε )) 3 .
As the coefficients of A ij are piecewise smooth functions on [0, 1] as stated on p. 39, then relation (2.3.4) 1 gives

∂ 2 u ε ∂x 2 3 (t) ∈ (L 2 (D + ε )) 3 . It means that u ε (t) ∈ H 2 in a neighborhood of Γ 0 and (u ε ) / Γ 0 (t) ∈ H 3/2 (Γ 0 ).
Next, we consider the problem

                           ρ - ∂v ε ∂t -2ν div (D(v ε )) + ∇p ε = f in D -× (0, T ), div v ε = 0 in D -× (0, T ), v ε = 0 on Γ -× (0, T ), v ε = ∂u ε ∂t Γ 0 on Γ 0 × (0, T ), v ε , p ε D-periodic, v ε (0) = 0 in D -.
(2.3.5)

From (2.3.5) 4 we get v ε ∈ (H 3/2 (∂D -)) 3 . Using the regularity previously obtained for the derivatives in x 1 , x 2 and t for v ε , we obtain from equation (2.3.5) 3 . The ADN-estimates [START_REF] Temam | Navier-Stokes Equations: Theory & Numerical Analysis[END_REF] give the following regularity v ε (t) ∈ (H 2 (D -)) 3 , p ε (t) ∈ H 1 (D -). Consequently, the trace p ε Γ 0 has the regularity H 1/2 .

1 the condition ∇p ε (t) ∈ (H 1 # (D -))

Limit problem

The regularity question is very important in the asymptotic analysis because there are the higher-order derivatives taken with respect to t and x 1 , x 2 from the unknowns (w 0 ) 3 , v0 , p 0 (obtained in solving the limit problem) in the asymptotics construction.

Let us consider variational problem (2.2.28) at the moment t + ∆t ∈ (0, T ) (∆t is small)

             ρ - D - v′′ 0 (t+ ∆t)ϕ j + ν D - ∇v ′ 0 (t+ ∆t)∇ϕ j + Ĵ Γ 0 ∆ xγ 0 (v 0 ) 3 (t+ ∆t)∆ xγ 0 ϕ j3 = D - f ′ (t + ∆t)ϕ j + Γ 0 g ′ 3 (t + ∆t)γ 0 ϕ j3 , j = 1, . . . , n, v0 (0) = v′ 0 (0) = 0 in (L 2 # (D -)) 3 , (2.3.6)
and problem (2.2.30) replacing f ′ and g ′ 3 in the right-hand side by its derivatives

             ρ - D - v′′ 0 * ϕ j + ν D - ∇v ′ 0 * ∇ϕ j + Ĵ Γ 0 ∆ x(γ 0 (v 0 * ) 3 )∆ x(γ 0 ϕ j3 ) = D - f ′′ ϕ j + Γ 0 g ′′ 3 γ 0 ϕ j3 a.e. in (0, T ), j = 1, . . . , n, v0 * (0) = v′ 0 * (0) = 0 in (L 2 # (D -)) 3 .
(2.3.7)

Remark. The initial condition v′ 0 * (0) = 0 is obtained automatically from the fourth hypothesis for f and g 3 (see asymptotic analysis).

We consider the difference between

(2.3.6) 1 -(2.2.30) 1 ∆t
and (2.3.7) 1 :

ρ - D - v′′ 0 (t + ∆t) -v′′ 0 (t) ∆t -v′′ 0 * (t) ϕ j +ν D - ∇v ′ 0 (t + ∆t) -∇v ′ 0 (t) ∆t -∇v ′ 0 * (t) ∇ϕ j + Ĵ Γ 0 ∆ xγ 0 (v 0 ) 3 (t + ∆t) -∆ xγ 0 (v 0 ) 3 (t) ∆t -∆ xγ 0 (v 0 * ) 3 (t) ∆ xγ 0 ϕ j3 = D - f ′ (t + ∆t) -f ′ (t) ∆t -f ′′ (t) ϕ j + Γ 0 g ′ 3 (t + ∆t) -g ′ 3 (t) ∆t -g ′′ 3 (t) γ 0 ϕ j3 , j = 1, . . . , n, (2.3.8)
We obtain the estimates

v′ 0 (t + ∆t) -v′ 0 (t) ∆t -v′ 0 * (t) C([0,T ];(L 2 (D -)) 3 ) + ∇v ′ 0 (t + ∆t) -∇v ′ 0 (t) ∆t -∇v ′ 0 * (t) L 2 (0,T ;(L 2 (D -)) 3 ) + ∆ xγ 0 (v 0 ) 3 (t + ∆t) -∆ xγ 0 (v 0 ) 3 (t) ∆t -∆ xγ 0 (v 0 * ) 3 (t) L ∞ (0,T ;L 2 (Γ 0 )) ≤ C c f ′ (t + ∆t) -f ′ (t) ∆t -f ′′ (t), g ′ 3 (t + ∆t) -g ′ 3 (t) ∆t -g ′′ 3 (t) L 2 (0,T )
.

(2.3.9)

Knowing, that

f ′′ (t) = lim ∆t→0 f ′ (t + ∆t) -f ′ (t) ∆t , g ′′ 3 (t) = lim ∆t→0 g ′ 3 (t + ∆t) -g ′ 3 (t) ∆t we obtain the following convergences v′ 0 (t + ∆t) -v′ 0 (t) ∆t → v′ 0 * (t), ∆t → 0, ∇v ′ 0 (t + ∆t) -∇v ′ 0 (t) ∆t → ∇v ′ 0 * (t), ∆t → 0, ∆ xγ 0 (v 0 ) 3 (t + ∆t) -∆ xγ 0 (v 0 ) 3 (t) ∆t → ∆ xγ 0 (v 0 * ) 3 (t), ∆t → 0.
The limit of

v′ 0 (t+∆t)-v ′ 0 (t) ∆t will be the function in H v because v′ 0 (t+∆t)-v ′ 0 (t) ∆t ∈ H v
which is the closed Hilbert space and contains its limit points. In plus, v′ 0 * will be the second derivative v′′ 0 (t). In fact, if our sequence converges weakly, we can choose a convergent subsequence in the classical sense in L 2 and its limit will be needed derivative.

Analogically it is possible to get all derivatives in time of v0 by induction.

Next, we do the same to the derivatives with respect to the space variables x 1 and x 2 . This gives the estimates (for i = 1, 2)

v′ 0 (x i + ∆x i ) -v′ 0 (x i ) ∆x i - ∂ v′ 0 ∂x i C([0,T ];(L 2 (D -)) 3 ) + ∇v ′ 0 (x i + ∆x i ) -∇v ′ 0 (x i ) ∆x i - ∂∇v ′ 0 ∂x i L 2 (0,T ;(L 2 (D -)) 3 ) + ∆ xγ 0 (v 0 ) 3 (x i + ∆x i ) -∆ xγ 0 (v 0 ) 3 (x i ) ∆x i - ∂∆ xγ 0 (v 0 ) 3 ∂x i L ∞ (0,T ;L 2 (Γ 0 )) ≤ C c f ′ (x i + ∆x i ) -f ′ (x i ) ∆x i - ∂f ′ ∂x i , g ′ 3 (x i + ∆x i ) -g ′ 3 (x i ) ∆x i - ∂g ′ 3 ∂x i L 2 (0,T ) .
(2.3.10)

It means the convergences

v′ 0 (x i + ∆x i ) -v′ 0 (x i ) ∆x i → ∂ v′ 0 ∂x i , ∆x i → 0, ∇v ′ 0 (x i + ∆x i ) -∇v ′ 0 (x i ) ∆x i → ∂∇v ′ 0 ∂x i , ∆x i → 0, ∆ xγ 0 (v 0 ) 3 (x i + ∆x i ) -∆ xγ 0 (v 0 ) 3 (x i ) ∆x i → ∂∆ xγ 0 (v 0 ) 3 ∂x i , ∆x i → 0.
The derivative ∂ v′ 0 ∂x i , i ∈ {1, 2} will be in the our space too (remark, that 1periodicity is saved) and we obtain other derivatives of bigger order by induction.

In order to get the derivatives

∂ k v′ 0 ∂x k 3 , k ≥ 1

we replace the initial problem by

th Dirichlet problem and we apply for it the ADN-estimates [START_REF] Temam | Navier-Stokes Equations: Theory & Numerical Analysis[END_REF].

As stated above, there exist the unique solution to problem (2.2.23), but there is no information for the reverse transition from the variational problem (without pressure) to a physical problem (with pressure).

Remark. The formal differentiations in (2.2.23), (2.2.24), (2.2.25) become informal with the results of applying the smoothness increasing method.

In order to improve the regularity of the unknowns as stated in Theorem 3.5, for any q, l ∈ N we consider the problem

                                         Ĵ∆ 2 x(w q,l ) 3 -p q,l x 3 =0 = ∂ q+l g 3 ∂t q ∂x s 1 . . . ∂x s l in D × (0, T ), ρ - ∂ vq,l ∂t -ν∆v q,l + ∇p q,l = ∂ q+l f ∂t q ∂x s 1 . . . ∂x s l div vq,l = 0 in D -× (0, T ), vq,l (x, -1, t) = 0 in D × (0, T ), vq,l (x, 0, t) = ∂(w q,l ) 3 ∂t (x, t)e 3 in D × (0, T ), (w q,l ) 3 , v0 , p 0 D-periodic, vq,l (x, 0) = 0 in D -; (w q,l ) 3 (x, 0) = ∂(w q,l ) 3 ∂t (x, 0) = 0 in D.
(2.3.11)

Taking into account the assumptions (H1) and (H2) we obtain for the unknowns of (2.3.11) the same regularity as that for (w 0 ) 3 , v0 , p 0 .

We extend next problem (2.2.23) from the bounded domain D -to the layer corresponding to x ∈ R 2 :

                                 Ĵ∆ 2 x(w 0 ) 3 -p 0 x 3 =0 = g 3 in R 2 × (0, T ), ρ - ∂ v0 ∂t -ν∆v 0 + ∇p 0 = f div v0 = 0 in R 2 × (-1, 0) × (0, T ), v0 (x, -1, t) = 0 in R 2 × (0, T ), v0 (x, 0, t) = ∂(w 0 ) 3 ∂t (x, t)e 3 in R 2 × (0, T ), v0 (x, 0) = 0 in R 2 × (-1, 0); (w 0 ) 3 (x, 0) = ∂(w 0 ) 3 ∂t (x, 0) = 0 in R 2 .
(2.3.12)

Conclusion

We have proved the existence and uniqueness of solution to the viscous fluid-thin elastic plate interaction problem, assuming the interface continuity of velocity and normal stresses. Variational analysis of the weak formulation is effectuated.

The higher regularity of the solution is obtained. The existence of pressure follows from De Rham's theorem and its regularity. The same arguments and conclusions are right for the limit problem.

for engineers the main criteria are the stability of the material to loads, the melting point of the composite as a homogeneous material and other (chapter 1). For example, the coefficients of thermal conductivity in the system of equations of elasticity theory (stationary), which describes the temperature distribution in the material are rapidly oscillating functions that depend on the "fast variable ξ = x ε . The variable x is then called "slow. And they are considered to be independent. This is the basis of the method twoscale (and multiscale) expansions [START_REF] Nazarov | Asymptotic expansion of the solution of the Dirichlet problem for an equation with rapidly oscillating coefficients in a rectangle[END_REF][START_REF] Malakhova | Boundary value problem for an elliptic equation with rapidly oscillating coefficients in a rectangle[END_REF] with homogenisation [START_REF] Orlik | Two-scale homogenization in transmission problems of elasticity with interface jumps[END_REF][START_REF] Braides | Remarks on the homogenization of connected media[END_REF]. Typically, the solution of the problem is sought in the form of an asymptotic series (Ansatz). The main issue is to construct the limit problem when ε → 0 with constant coefficients (as averaging), it characterizes the general properties of the composite material as a uniform.

The method of partial asymptotic decomposition of the domain (see [START_REF] Panasenko | Multi-scale Modelling for Structures and Composites[END_REF]) allows to reduce the dimension in some part of a thin domain and to glue the models of different dimension at the interface. The idea of coupling models of different dimension or different scales ( [START_REF] Blanco | Modeling dimensionallyheterogeneous problems: analysis, approximation and applications[END_REF][START_REF] Angelo | Asymptotic-numerical derivation of the Robin type coupling conditions for the macroscopic pressure at a reservoir-capillaries interface[END_REF][START_REF] Panasenko | Method of asymptotic partial decomposition of domain[END_REF][START_REF] Panasenko | Partial asymptotic decomposition of domain: Navier-Stokes equation in tube structure[END_REF][START_REF] Panasenko | The partial homogenization: continuous and semi-discretized versions[END_REF]) is an important trend in the domain decomposition approach. It is applied in the blood circulation modelling (see e.g. [START_REF] Angelo | On the coupling of 1D and 3D diffusionreaction equations: application to tissue perfusion problems[END_REF][START_REF] Passerini | A 3D/1D geometrical multiscale model of cerebral vasculature[END_REF]) and in engineering [START_REF] Panasenko | Multi-scale Modelling for Structures and Composites[END_REF].

Construction of an asymptotic expansion of the solution to the problem

As it was suggested in the chapter 2 for the asymptotic approach of the problem we need regularity for the data given by four hypothesis (H1)-(H3), p. 52.

We look for an asymptotic solution of order J for (2.1.1) in the form

u (J) ε (x, t) = J q+l=0 ε q+l s:|s|=l s j ∈{1,2} N q, s 1 ...s l (ξ 3 ) ∂ q+l w (J) ε (x, t) ∂t q ∂x s 1 . . . ∂x s l + J q+l=0 ε q+l+2 s:|s|=l s j ∈{1,2} M q, s 1 ...s l (ξ 3 ) ∂ q+l ψ (J) ε (x, t) ∂t q ∂x s 1 . . . ∂x s l , (x, t) ∈ D + ε × (0, T ),              v (J) ε (x, t) = J k=0 ε k v k (x, t), p (J) ε (x, t) = J k=0 ε k p k (x, t), (x, t) ∈ D -× (0, T ), ψ (J) ε (x, t) = 2νD(v (J) ε (x, 0, t))e 3 -p (J) ε (x, 0, t)e 3 , (x, t) ∈ D × (0, T ), w (J) ε (x, t) = J k=0 ε k w k (x, t), (x, t) ∈ D × (0, T ), (3.2.1)
with w k , v k , p k D-periodic, for any k ∈ {0, 1, ..., J} and ξ 3 is introduced in (2.1.2).

Remark. For any fixed value of J the required smoothness of the data f and g may be reduced to the class C α(J) with respect to x 1 , x 2 and t variables, where α(J) is some finite number depending on J, chosen in such a way that all the derivatives of w (J) ε and ψ (J) ε in (3.2.2) exist and belong to C 2 . However, we consider in what follows the regularity given by (H1) and (H2), in order to ensure the necessary smoothness for an asymptotic solution of any arbitrary order.

To determine the asymptotic solution means to determine the matrices N q, s 1 ...s l = N q, s 1 ...s l (ξ 3 ), M q, s 1 ...s l = M q, s 1 ...s l (ξ 3 ), N q, s 1 ...s l , M q, s 1 ...s l ∈ M 3,3 and the

functions v k = v k (x, t), p k = p k (x, t), w k = w k (x, t)
. All these functions depend on ε but we omit the subscript ε.

We introduce the following notation for the left hand side of (2.1.1) 1

P ε u ε = ρ + (ξ 3 ) ∂ 2 u ε ∂t 2 -ε -3 3 i,j=1 ∂ ∂x i A ij (ξ 3 ) ∂u ε ∂x j . ( 3.2.2)
Let us replace in (3.2.2) u ε with its asymptotic expansion (3.2.1) 1 .

Applying the chain rule, producing index changing, identifying some indexes we obtain for J > 4

P ε u (J) ε = J q+l=0 ε q+l-5 s:|s|=l H N q, s 1 ...s l (ξ 3 ) ∂ q+l w (J) ε (x, t) ∂t q ∂x s 1 . . . ∂x s l + J q+l=0 ε q+l-3 s:|s|=l H M q, s 1 ...s l (ξ 3 ) ∂ q+l ψ (J) ε (x, t) ∂t q ∂x s 1 . . . ∂x s l + r (J),1 ε , (3.2.3)
where a.e. in [0, 1]

                 H N q, s 1 ...s l (ξ 3 ) = ε 3 ρ + (ξ 3 )N q-2, s 1 ...s l (ξ 3 ) -(A 33 (ξ 3 )N ′ q, s 1 ...s l (ξ 3 )) ′ +(A 3s 1 (ξ 3 )N q, s 2 ...s l (ξ 3 )) ′ + A s 1 3 (ξ 3 )N ′ q, s 2 ...s l (ξ 3 ) + A s 1 s 2 (ξ 3 )N q, s 3 ...s l (ξ 3 ) , H M q, s 1 ...s l (ξ 3 ) = ε 3 ρ + (ξ 3 )M q-2, s 1 ...s l (ξ 3 ) -(A 33 (ξ 3 )M ′ q, s 1 ...s l (ξ 3 )) ′ +(A 3s 1 (ξ 3 )M q, s 2 ...s l (ξ 3 )) ′ + A s 1 3 (ξ 3 )M ′ q, s 2 ...s l (ξ 3 ) + A s 1 s 2 (ξ 3 )M q, s 3 ...s l (ξ 3 ) (3.2.4)
and

                                                       r (J),1 ε (x, ξ 3 , t) = ε J-4 J+1 q+l=J+1 ε 3 ρ + (ξ 3 )N q-2, s 1 ...s l (ξ 3 ) -(A 3s 1 (ξ 3 )N q, s 2 ...s l (ξ 3 )) ′ +A s 1 3 (ξ 3 )N ′ q, s 2 ...s l (ξ 3 ) + A s 1 s 2 (ξ 3 )N q, s 3 ...s l (ξ 3 ) ∂ q+l w (J) ε (x, t) ∂t q ∂x s 1 . . . ∂x s l +ε J-3 J+2 q+l=J+2 ε 3 ρ + (ξ 3 )N q-2, s 1 ...s l (ξ 3 ) -A s 1 s 2 (ξ 3 )N q, s 3 ...s l (ξ 3 ) ∂ q+l w (J) ε (x, t) ∂t q ∂x s 1 . . . ∂x s l +ε J-2 J+1 q+l=J+1 ε 3 ρ + (ξ 3 )M q-2, s 1 ...s l (ξ 3 ) -(A 3s 1 (ξ 3 )M q, s 2 ...s l (ξ 3 )) ′ +A s 1 3 (ξ 3 )M ′ q, s 2 ...s l (ξ 3 ) + A s 1 s 2 (ξ 3 )M q, s 3 ...s l (ξ 3 ) ∂ q+l ψ (J) ε (x, t) ∂t q ∂x s 1 . . . ∂x s l +ε J-1 J+2 q+l=J+2 ε 3 ρ + (ξ 3 )M q-2, s 1 ...s l (ξ 3 ) -A s 1 s 2 (ξ 3 )M q, s 3 ...s l (ξ 3 ) ∂ q+l ψ (J) ε (x, t) ∂t q ∂x s 1 . . . ∂x s l , (3.2.5)
with s j ∈ {1, 2}.

Since

P ε u (J) ε
should be equal to ε -1 g(x, t), we seek the matrices N q, s 1 ...s l and M q, s 1 ...s l such that H N q, s 1 ...s l and H M q, s 1 ...s l have constant matrix values. In this way we obtain the equations for determining the matrices N q, s 1 ...s l , M q, s 1 ...s l as follows

H N q, s 1 ...s l (ξ 3 ) = -h N q, s 1 ...s l , H M q, s 1 ...s l (ξ 3 ) = -h M q, s 1 ...s l , (3.2.6)
where h N q, s 1 ...s l and h M q, s 1 ...s l are constant matrices.

The left hand side of (2.1.1) 4 written for u

(J)
ε has the expression

3 j=1 A 3j (1) ∂u (J) ε ∂x j (x, ε, t) = 3 j=1 A 3j (1) ∂u (J) ε ∂x j (x, ε, t) + ε -1 3 j=1 A 3j (1) ∂u (J) ε ∂ξ j (x, ε, t) = J q+l=0 ε q+l 3 j=1 s:|s|=l A 3j (1)N q, s 1 ...s l (1) ∂ q+l+1 w (J) ε (x, t) ∂t q ∂x j ∂x s 1 . . . ∂x s l + J q+l=0 ε q+l+2 3 j=1 s:|s|=l A 3j (1)M q, s 1 ...s l (1) ∂ q+l+1 ψ (J) ε (x, t) ∂t q ∂x j ∂x s 1 . . . ∂x s l + J q+l=0 ε q+l-1 s:|s|=l A 33 (1) ∂N q, s 1 ...s l ∂ξ 3 (1) ∂ q+l w (J) ε (x, t) ∂t q ∂x s 1 . . . ∂x s l + J q+l=0 ε q+l+1 s:|s|=l A 33 (1) ∂M q, s 1 ...s l ∂ξ 3 (1)
∂ q+l ψ (J) ε (x, t) ∂t q ∂x s 1 . . . ∂x s l .

After index changing l 1 = l + 1, s 1 = j and grouping terms of the same order

3 j=1 A 3j (1) ∂u (J) ε ∂x j (x, ε, t) = J q+l=0 ε q+l-1 s:|s|=l A 3s 1 (1)N q, s 2 ...s l (1) + A 33 (1)N ′ q, s 1 ...s l (1) ∂ q+l w (J) ε (x, t) ∂t q ∂x s 1 . . . ∂x s l + J q+l=0 ε q+l+1 s:|s|=l A 3s 1 (1)M q, s 2 ...s l (1) + A 33 (1)M ′ q, s 1 ...s l (1)
∂ q+l ψ (J) ε (x, t) ∂t q ∂x s 1 . . . ∂x s l +r (J),4 ε (x, t), (3.2.7)
where

r (J),4 ε (x, t) = ε J J+1 q+l=J+1 s:|s|=l A 3s 1 (1)N q, s 2 ...s l (1) ∂ J+1 w (J) ε (x, t) ∂t q ∂x s 1 . . . ∂x s l +ε J+2 J+1 q+l=J+1 s:|s|=l A 3s 1 (1)M q, s 2 ...s l (1) ∂ J+1 ψ (J) ε (x, t) ∂t q ∂x s 1 . . . ∂x s l . (3.2.8)
Satisfying boundary condition (2.1.1) 4 with the residual of order ε J we get

A 33 (1)N ′ q, s 1 ...s l (1) + A 3s 1 (1)N q, s 2 ...s l (1) = O 3 , A 33 (1)M ′ q, s 1 ...s l (1) + A 3s 1 (1)M q, s 2 ...s l (1) = O 3 , s j ∈ {1, 2}. (3.2.9)
Introducing expansion (3.2.1) 1 in the right hand side of (2.1.1) 7 and using (3.2.1)

4
we obtain, as before

-p (J) ε (x, 0, t)e 3 + 2νD(v (J) ε (x, 0, t))e 3 -ε -3 3 j=1 A 3j (0) ∂u (J) ε ∂x j (x, 0, t) = ψ (J) ε (x, t) -ε -3 J q+l=0 ε q+l 3 j=1 s:|s|=l A 3j (0)N q, s 1 ...s l (0) ∂ q+l+1 w (J) ε (x, t) ∂t q ∂x j ∂x s 1 . . . ∂x s l + J q+l=0 ε q+l+2 3 j=1 s:|s|=l A 3j (0)M q, s 1 ...s l (0) ∂ q+l+1 ψ (J) ε (x, t) ∂t q ∂x j ∂x s 1 . . . ∂x s l + J q+l=0 ε q+l-1 s:|s|=l A 33 (0) ∂N q, s 1 ...s l ∂ξ 3 (0) ∂ q+l w (J) ε (x, t) ∂t q ∂x s 1 . . . ∂x s l + J q+l=0 ε q+l+1 s:|s|=l A 33 (0) ∂M q, s 1 ...s l ∂ξ 3 (0) ∂ q+l ψ (J) ε (x, t) ∂t q ∂x s 1 . . . ∂x s l , wherefrom it follows that -p (J) ε (x, 0, t)e 3 + 2νD(v (J) ε (x, 0, t))e 3 -ε -3 3 j=1 A 3j (0) ∂u (J) ε ∂x j (x, 0, t) = ψ (J) ε (x, t) - J q+l=0 ε q+l-4 s:|s|=l A 3s 1 (0)N q, s 2 ...s l (0) + A 33 (0)N ′ q, s 1 ...s l (0) ∂ q+l w (J) ε (x, t) ∂t q ∂x s 1 . . . ∂x s l - J q+l=0 ε q+l-2 s:|s|=l A 3s 1 (0)M q, s 2 ...s l (0) + A 33 (0)M ′ q, s 1 ...s l (0) ∂ q+l ψ (J) ε (x, t) ∂t q ∂x s 1 . . . ∂x s l -r (J),7 ε (x, t), with r (J),7 ε (x, t) = ε J-3 J+1 q+l=J+1 s:|s|=l A 3s 1 (0)N q, s 2 ...s l (0) ∂ J+1 w (J) ε (x, t) ∂t q ∂x s 1 . . . ∂x s l +ε J-1 J+1 q+l=J+1 s:|s|=l A 3s 1 (0)M q, s 2 ...s l (0) ∂ J+1 ψ (J) ε (x, t) ∂t q ∂x s 1 . . . ∂x s l , (3.2.10)
and relation (2.1.1) 7 is satisfied with a small residual if

A 33 (0)N ′ q, s 1 ...s l (0) + A 3s 1 (0)N q, s 2 ...s l (0) = O 3 , ε -2 A 33 (0)M ′ q, s 1 ...s l (0) + A 3s 1 (0)M q, s 2 ...s l (0) = I 3 δ q0 δ l0 , s j ∈ {1, 2}. (3.2.11)
In what follows we shall use the notations

                 F = 1 0 F (s)ds, F (x) = x F - x 0 F (s)ds, F (x) = θ 0 F (s)ds - x 0 F (s)ds, (3.2.12)
where F : [0, 1] → R is an integrable function. The notation • will be used as well for functions of several variables and in this case it concerns the variable x 1 :

F (x 1 , t) = 1 0 F (s, t)ds.
Using equations (3.2.6) and conditions (3.2.9), (3.2.11) we obtain next the problems for N q, s 1 ...s l and M q, s 1 ...s l .

From (3.2.4) 1 , (3.2.6) 1 , (3.2.9) 1 , (3.2.11) 1 with the additional conditions

N q, s 1 ...s l = O 3 ∀ q + l > 0, N 0, ∅ = I 3 (3.2.13)
we get for N q, s 1 ...s l , ∀ q + l > 0 the second order differential problem

                   A 33 N ′ q, s 1 ...s l + A 3s 1 N q, s 2 ...s l ′ = -A s 1 3 N ′ q, s 2 ...s l -A s 1 s 2 N q, s 3 ...s l +ε 3 ρ + N q-2, s 1 ...s l + h N q, s 1 ...s l , h N q, s 1 ...s l = A s 1 3 N ′ q, s 2 ...s l + A s 1 s 2 N q, s 3 ...s l -ε 3 ρ + N q-2, s 1 ...s l , A 33 (0)N ′ q, s 1 ...s l (0) = -A 3s 1 (0)N q, s 2 ...s l (0), N q, s 1 ...s l = O 3 . (3.2.14)
In the same way, from (3.2.4) 2 , (3.2.6) 2 , (3.2.9) 2 , (3.2.11) 2 , with the additional condition

M q, s 1 ...s l (0) = O 3 ∀ q + l ≥ 0 (3.2.15)
we obtain for M q, s 1 ...s l the problem in the classical sense and in each point ζ α , α = 1, ..., p -1 two junction con-

                   A 33 M ′ q, s 1 ...s l + A 3s 1 M q, s 2 ...s l ′ = -A s 1 3 M ′ q, s 2 ...s l -A s 1 s 2 M q, s 3 ...s l +ε 3 ρ + M q-2, s 1 ...s l + h M q, s 1 ...s l , h M q, s 1 ...s l = A s 1 3 M ′ q, s 2 ...s l + A s 1 s 2 M q, s 3 ...s l -ε 3 ρ + M q-2, s 1 ...s l -ε 2 I 3 δ q0 δ l0 , A 33 (0)M ′ q, s 1 ...s l (0) = ε 2 I 3 δ q0 δ l0 , M q, s 1 ...s l (0) = O 3 . ( 3
ditions are satisfied: [N q, s 1 ...s l ] = O 3 , [A 33 N ′ q, s 1 ...s l + A 3s 1 N q, s 2 ...s l ] = O 3 , where [R(ξ 3 )] = lim ξ 3 →ζ α+0 R(ξ 3 ) -lim ξ 3 →ζ α-0 R(ξ 3 ).
The same remark holds for (3.2.16) 1 . These junction conditions, together with the smoothness of w (J) ε and ψ (J) ε imply the corresponding junction conditions for the function u

(J)
ε and for the normal stress

3 j=1 A 3j ∂u (J) ε ∂x j .
Solving recursively problems (3.2.14) and (3.2.16) we compute the matrices N q, s 1 ...s l , h N q, s 1 ...s l and M q, s 1 ...s l , h M q, s 1 ...s l , respectively. Concerning h N q, s 1 ...s l , we are interested to know explicitely h N q, s 1 ...s l , q + l ≤ 4. Using expression (3.2.14) 2 and the formula for N q, s 1 ...s l , easily obtained by integrating of relation (3.2.14) 1

A 33 N ′ q, s 1 ...s l + A 3s 1 N q, s 2 ...s l ′ = -A s 1 3 N ′ q, s 2 ...s l -A s 1 s 2 N q, s 3 ...s l + ε 3 ρ + N q-2, s 1 ...s l + h N q, s 1 ...s l , A 33 N ′ q, s 1 ...s l + A 3s 1 N q, s 2 ...s l = - ξ 3 0 A s 1 3 N ′ q, s 2 ...s l + A s 1 s 2 N q, s 3 ...s l -ε 3 ρ + N q-2, s 1 ...s l ds +h N q, s 1 ...s l ξ 3 = A s 1 3 N ′ q, s 2 ...s l + A s 1 s 2 N q, s 3 ...s l -ε 3 ρ + N q-2, s 1 ...s l , N ′ q, s 1 ...s l = A -1 33 -A 3s 1 N q, s 2 ...s l + A s 1 3 N ′ q, s 2 ...s l + A s 1 s 2 N q, s 3 ...s l -ε 3 ρ + N q-2, s 1 ...s l , N q, s 1 ...s l = A -1 33 A 3s 1 N q, s 2 ...s l -A s 1 3 N ′ q, s 2 ...s l + A s 1 s 2 N q, s 3 ...s l -ε 3 ρ + N q-2, s 1 ...s l we can find consecutively 1) h N 0, ∅ = O 3 , N 0, ∅ = I 3 , 2) h N 0, s 1 = O 3 , N 0, 1 = A -1 33 A 3s 1 N 0, ∅ , 2.1) s 1 = 1: N ′ 0, 1 (ξ 3 ) = -A -1 33 (ξ 3 )A 31 (ξ 3 ) = -     2(1+ν) E 0 0 0 2(1+ν) E 0 0 0 (1+ν)(1-2ν) E(1-ν)     •     0 0 E 2(1+ν) 0 0 0 Eν (1+ν)(1-2ν) 0 0     =     0 0 -1 0 0 0 -ν 1-ν 0 0     , N 0, 1 (ξ 3 ) =     0 0 1 2 -ξ 3 0 0 0 ν 1-ν 0 0     , 2.2) s 1 = 2: N ′ 0, = -A -1 33 (ξ 3 )A 32 (ξ 3 ) = -     2(1+ν) E 0 0 0 2(1+ν) E 0 0 0 (1+ν)(1-2ν) E(1-ν)     •     0 0 0 0 0 E 2(1+ν) 0 Eν (1+ν)(1-2ν) 0     =     0 0 0 0 0 -1 0 -ν 1-ν 0     , N 0, =     0 0 0 0 0 1 2 -ξ 3 0 ν 1-ν 0     , 3) h N 0, s 1 s 2 = A s 1 N ′ 0, s 2 + A s 1 s 2 , N ′ 0, s 1 s 2 = A -1 33 -A 3s 1 N 0, s 2 + A s 1 3 N ′ 0, s 2 + A s 1 s 2 , N 0, s 1 s 2 = A -1 33 A 3s 1 N 0, s 2 -A s 1 3 N ′ 0, s 2 + A s 1 s 2 , 3.1) s 1 = 1, s 2 = 1: h N 0, 11 =     E 1-ν 2 0 0 0 E 2(1+ν) 0 0 0 0     , N 0, 11 =      ν 1-ν -2(1+ν) E E 1-ν 2 0 0 0 -1+ν E E 1+ν 0 0 0 ν( 1 2 -ξ 3 ) 1-ν      , 3.2) s 1 = 1, s 2 = 2: h N 0, 12 =     0 Eν 1-ν 2 0 E 2(1+ν) 0 0 0 0 0     , N 0, 12 =      0 ν 1-ν -2(1+ν) E Eν 1-ν 2 -1+ν E E 1+ν 0 0 0      , 3.3) s 1 = 2, s 2 = 1: h N 0, 21 =     0 E 2(1+ν) 0 Eν 1-ν 2 0 0 0 0 0     , N 0, 21 =      0 -1+ν E E 1+ν ν 1-ν -2(1+ν) E Eν 1-ν 2 0 0 0      , 3.4) s 1 = 2, s 2 = 2: h N 0, 22 =     E 2(1+ν) 0 0 0 E 1-ν 2 0 0 0 0     , N 0, 22 =      -1+ν E E 1+ν 0 0 0 ν 1-ν -2(1+ν) E E 1-ν 2 0 0 0 ν( 1 2 -ξ 3 ) 1-ν      , so h N 0, 1, 1 = h N 0, 12 + h N 0, 21 =      0 E 2(1-ν) 0 E 2(1-ν) 0 0 0 0 0      ,
where the coefficient at

∂ q+r+p w (J) ε (x, t) ∂t q ∂x r 1 ∂x p 2
in the first sum of (3.2.3) is de-noted by h N q, r, p (here and below),

4) h N 0, s 1 s 2 s 3 = A s 1 3 N ′ 0, s 2 s 3 + A s 1 s 2 N 0, s 3 , N ′ 0, s 1 s 2 s 3 = A -1 33 -A 3s 1 N 0, s 2 + A s 1 3 N ′ 0, s 2 s 3 + A s 1 s 2 N 0, s 3 , N 0, s 1 s 2 s 3 = A -1 33 A 3s 1 N 0, s 2 -A s 1 3 N ′ 0, s 2 s 3 + A s 1 s 2 N 0, s 3 , 4.1) s 1 = 1, s 2 = 1, s 3 = 1: h N 0, 111 =      0 0 E( 1 2 -ξ 3 ) 1-ν 2 0 0 0 E 1-ν 2 0 0      , N 0, 111 =       0 0 ν( 1 2 -ξ 3 ) 1-ν -2(1+ν) E E( 1 2 -ξ 3 ) 1-ν 2 0 0 0 ν 1-ν ν 1-ν -2(1+ν) E E 1-ν 2 -(1+ν)(1-2ν) E(1-ν) E 1-ν 2 0 0       , 4.2) s 1 = 1, s 2 = 1, s 3 = 2: h N 0, 112 =      0 0 0 0 0 E( 1 2 -ξ 3 ) 2(1+ν) 0 Eν 1-ν 2 0      , N 0, 112 =       0 0 0 0 0 -1+ν E E( 1 2 -ξ 3 ) 1+ν 0 ν 1-ν ν 1-ν -2(1+ν) E Eν 1-ν 2 -(1+ν)(1-2ν) E(1-ν) Eν 1-ν 2 0       , 4.3) s 1 = 1, s 2 = 2, s 3 = 1: h N 0, 121 =      0 0 0 0 0 E( 1 2 -ξ 3 ) 2(1+ν) 0 E 2(1+ν) 0      , N 0, 121 =      0 0 0 0 0 -1+ν E E( 1 2 -ξ 3 ) 1+ν 0 -ν 1-ν 1+ν E E 1+ν -(1+ν)(1-2ν) E(1-ν) E 2(1+ν) 0      , 4.4) s 1 = 1, s 2 = 2, s 3 = 2: h N 0, 122 =      0 0 Eν( 1 2 -ξ 3 ) 1-ν 2 0 0 0 E 2(1+ν) 0 0      , N 0, 122 =       0 0 ν( 1 2 -ξ 3 ) 1-ν -2(1+ν) E Eν( 1 2 -ξ 3 ) 1-ν 2 0 0 0 -ν 1-ν 1+ν E E 1+ν -(1+ν)(1-2ν) E(1-ν) E 2(1+ν) 0 0       , 4.5) s 1 = 2, s 2 = 1, s 3 = 1: h N 0, 211 =      0 0 0 0 0 Eν( 1 2 -ξ 3 ) 1-ν 2 0 E 2(1+ν) 0      , N 0, 211 =       0 0 0 0 0 ν( 1 2 -ξ 3 ) 1-ν -2(1+ν) E Eν( 1 2 -ξ 3 ) 1-ν 2 0 -ν 1-ν 1+ν E E 1+ν -(1+ν)(1-2ν) E(1-ν) E 2(1+ν) 0       , 4.6) s 1 = 2, s 2 = 1, s 3 = 2: h N 0, 212 =      0 0 E( 1 2 -ξ 3 ) 2(1+ν) 0 0 0 E 2(1+ν) 0 0      , N 0, 212 =      0 0 -1+ν E E( 1 2 -ξ 3 ) 1+ν 0 0 0 -ν 1-ν 1+ν E E 1+ν -(1+ν)(1-2ν) E(1-ν) E 2(1+ν) 0 0      , 4.7) s 1 = 2, s 2 = 2, s 3 = 1: h N 0, 221 =      0 0 E( 1 2 -ξ 3 ) 2(1+ν) 0 0 0 Eν 1-ν 2 0 0      , N 0, 221 =       0 0 -1+ν E E( 1 2 -ξ 3 ) 1+ν 0 0 0 ν 1-ν ν 1-ν -2(1+ν) E Eν 1-ν 2 -(1+ν)(1-2ν) E(1-ν) Eν 1-ν 2 0 0       , 4.8) s 1 = 2, s 2 = 2, s 3 = 2: h N 0, 222 =      0 0 0 0 0 E( 1 2 -ξ 3 ) 1-ν 2 0 E 1-ν 2 0      , N 0, 222 =       0 0 0 0 0 ν( 1 2 -ξ 3 ) 1-ν -2(1+ν) E E( 1 2 -ξ 3 ) 1-ν 2 0 ν 1-ν ν 1-ν -2(1+ν) E E 1-ν 2 -(1+ν)(1-2ν) E(1-ν) E 1-ν 2 0       , so h N 0, 2, 1 = h N 0, 112 + h N 0, 121 + h N 0, 211 =      0 0 0 0 0 E( 1 2 -ξ 3 ) 1-ν 2 0 E 1-ν 2 0      , N 0, 2, 1 =       0 0 0 0 0 ν( 1 2 -ξ 3 ) 1-ν -2(1+ν) E E( 1 2 -ξ 3 ) 1-ν 2 0 ν 1-ν ν 1-ν -2(1+ν) E E 1-ν 2 -(1+ν)(1-2ν) E(1-ν) E 1-ν 2 0       , h N 0, 1, 2 = h N 0, 122 + h N 0, 212 + h N 0, 221 =      0 0 E( 1 2 -ξ 3 ) 1-ν 2 0 0 0 E 1-ν 2 0 0      , N 0, 1, 2 =       0 0 ν( 1 2 -ξ 3 ) 1-ν -2(1+ν) E E( 1 2 -ξ 3 ) 1-ν 2 0 0 0 ν 1-ν ν 1-ν -2(1+ν) E E 1-ν 2 -(1+ν)(1-2ν) E(1-ν) E 1-ν 2 0 0       , 5) h N 0, 1111 =      * 0 0 0 * 0 0 0 E( 1 2 -ξ 3 ) 1-ν 2      , h N 0, 2222 =      * 0 0 0 * 0 0 0 E( 1 2 -ξ 3 ) 1-ν 2      , h N 0, 3, 1 =     0 * 0 * 0 0 0 0 0     , h N 0, 2, 2 =     * 0 0 0 * 0 0 0 E(1-2ξ 3 ) 1-ν 2     , h N 0, 1, 3 =     0 * 0 * 0 0 0 0 0     , 6) h N 1, ∅ = O 3 , N 1, ∅ = O 3 , 7) h N 1, s 1 = O 3 , N 1, s 1 = O 3 , 8) h N 1, s 1 s 2 = O 3 , N 1, s 1 s 2 = O 3 , 9) h N 1, s 1 s 2 s 3 = O 3 , N 1, s 1 s 2 s 3 = O 3 , 10) h N 2, ∅ = -ε 3 ρ + I 3 , N 2, ∅ = ε 3     2(1+ν) E ρ + 0 0 0 2(1+ν) E ρ + 0 0 0 (1+ν)(1-2ν) E(1-ν) ρ +     , 11) h N 2, s 1 = A s 1 3 N ′ 2, ∅ -ε 3 ρ + N 0, s 1 ,
11.1) s 1 = 1:

h N 2, 1 =     0 0 -ε 3 ρ + ν 1-ν + ρ + ( 1 2 -ξ 3 ) 0 0 0 -ε 3 ρ + + ρ + ν 1-ν 0 0     , 11.2) s 1 = 2: h N 2, 2 =     0 0 0 0 0 -ε 3 ρ + ν 1-ν + ρ + ( 1 2 -ξ 3 ) 0 -ε 3 ρ + + ρ + ν 1-ν 0     , 12) h N 3, ∅ = O 3 , N 3, ∅ = O 3 , 13) h N 3, s 1 = O 3 , N 3, s 1 = O 3 .
We didn't write above the matrices h N 2, s 1 s 2 and h N 4, ∅ since they are of the form

h N 2, s 1 s 2 = O(ε 3 ), h N 4, ∅ = O(ε 6
). Concerning problems (3.2.16) we prove next the following result

Proposition 3.1. M q, s 1 ...s l = ε 2 Mq, s 1 ...s l , h M q, s 1 ...s l = ε 2 hM q, s 1 ...s l ∀ q + l ≥ 0, where Mq, s 1 ...s l = O(ε m ), hM q, s 1 ...s l = O(ε k ), m, k ≥ 0.
Proof. We obtain this result by induction. Taking q = l = 0 in (3.2.16) we get

M 0, ∅ = ε 2 M0, ∅ , h M 0, ∅ = -ε 2 I 3 , with M0, ∅ = O(1)
; hence, the previous assertion holds for q + l = 0.

We suppose that ∀ q, l with q + l < p the assertion of the proposition is true and we prove it for q + l = p. This result is obtained directly from (3.2.16), by using the property for all the terms containing a matrix M k, j with k + j < p.

To determine the asymptotic solution it remains to obtain the functions w k , v k , p k . For this purpose we define the new function

w(J) ε (x, t) = ε -1 w (J) ε (x, t) • e 1 e 1 + ε -1 w (J) ε (x, t) • e 2 e 2 + w (J) ε (x, t) • e 3 e 3
(3.2.17)

and we introduce the notations

                 Ê1 = E 1 -ν 2 , Ê2 = E 2(1 + ν) , Ê3 = E 2(1 -ν) , Ê1 = E 1 -ν 2 1 2 -ξ 3 , Ê2 = E 1 -ν 2 , Ĵ = E 1 -ν 2 1 2 -ξ 3 .
(3.2.18)

Using the expressions previously obtained for h N q, s 1 ...s l , q + l ≤ 4, applying Proposition 3.1 for h M q, s 1 ...s l , replacing w (J) ε by the new function defined in (3.2.17) and using (3.2.18)

(P ε u (J) ε ) 1 = -ε -2 Ê1 ∂ 2 ( w(J) ε ) 1 ∂x 2 1 + Ê3 ∂ 2 ( w(J) ε ) 2 ∂x 1 ∂x 2 + Ê2 ∂ 2 ( w(J) ε ) 1 ∂x 2 2 + Ê1 ∂ ∂x 1 ∆(w (J) ε ) 3 + R (J) ε • e 1 , (P ε u (J) ε ) 2 = -ε -2 Ê2 ∂ 2 ( w(J) ε ) 2 ∂x 2 1 + Ê3 ∂ 2 ( w(J) ε ) 1 ∂x 1 ∂x 2 + Ê1 ∂ 2 ( w(J) ε ) 2 ∂x 2 2 + Ê1 ∂ ∂x 2 ∆(w (J) ε ) 3 + R (J) ε • e 2 , (P ε u (J) ε ) 3 = -ε -1 Ê2 ∂ ∂x 1 ∆( w(J) ε ) 1 + ∂ ∂x 2 ∆( w(J) ε ) 2 + Ĵ∆ ∆(w (J) ε ) 3 -ψ (J) ε 3 + R (J) ε • e 3 , (3.2.19)
where the three components of R (J)

ε are R (J) ε e 1 = ε -1 ψ (J) ε 1 + ε 0 -(h N 0, 1111 ) 11 ∂ 4 ( w(J) ε ) 1 ∂x 4 1 -(h N 0, 3, 1 ) 12 ∂ 4 ( w(J) ε ) 2 ∂x 3 1 ∂x 2 -(h N 0, 2, 2 ) 11 ∂ 4 ( w(J) ε ) 1 ∂x 2 1 ∂x 2 2 -(h N 0, 1, 3 ) 12 ∂ 4 ( w(J) ε ) 2 ∂x 1 ∂x 3 2 -(h N 0, 2222 ) 11 ∂ 4 ( w(J) ε ) 1 ∂x 4 2 + ε ρ + ∂ 2 ( w(J) ε ) 1 ∂t 2 + ρ+ ν 1 -ν + ρ + 1 2 -ξ 3 ∂ 3 (w (J) ε ) 3 ∂t 2 ∂x 1 + ε 3 2 i 1 =1 ( hN 2, s 1 s 1 ) 11 ∂ 4 ( w(J) ε ) 1 ∂t 2 ∂x 2 s 1 + ( hN 2, 1, 1 ) 12 ∂ 4 ( w(J) ε ) 2 ∂t 2 ∂x 1 ∂x 2 + ε 6 ( hN 4, ∅ ) 11 ∂ 4 ( w(J) ε ) 1 ∂t 4 - J q+l=5 ε q+l-5 s:|s|=l h N q, s 1 ...s l ∂ q+l ε( w(J) ε ) 1 e 1 + ε( w(J) ε ) 2 e 2 + (w (J) ε ) 3 e 3 ∂t q ∂x s 1 . . . ∂x s l 1 - J q+l=1 ε q+l-1 s:|s|=l hM q, s 1 ...s l ∂ q+l ψ (J) ε ∂t q ∂x s 1 . . . ∂x s l 1 + r (J),1 ε • e 1 , R (J) ε e 2 = ε -1 ψ (J) ε 2 + ε 0 -(h N 0, 1111 ) 22 ∂ 4 ( w(J) ε ) 2 ∂x 4 1 -(h N 0, 3, 1 ) 21 ∂ 4 ( w(J) ε ) 1 ∂x 3 1 ∂x 2 -(h N 0, 2, 2 ) 22 ∂ 4 ( w(J) ε ) 2 ∂x 2 1 ∂x 2 2 -(h N 0, 1, 3 ) 21 ∂ 4 ( w(J) ε ) 1 ∂x 1 ∂x 3 2 -(h N 0, 2222 ) 22 ∂ 4 ( w(J) ε ) 2 ∂x 4 2 + ε ρ + ∂ 2 ( w(J) ε ) 2 ∂t 2 + ρ+ ν 1 -ν + ρ + 1 2 -ξ 3 ∂ 3 (w (J) ε ) 3 ∂t 2 ∂x 2 + ε 3 2 i 1 =1 ( hN 2, s 1 s 1 ) 22 ∂ 4 ( w(J) ε ) 2 ∂t 2 ∂x 2 s 1 + ( hN 2, 1, 1 ) 21 ∂ 4 ( w(J) ε ) 1 ∂t 2 ∂x 1 ∂x 2 + ε 6 ( hN 4, ∅ ) 22 ∂ 4 ( w(J) ε ) 2 ∂t 4 - J q+l=5 ε q+l-5 s:|s|=l h N q, s 1 ...s l ∂ q+l ε( w(J) ε ) 1 e 1 + ε( w(J) ε ) 2 e 2 + (w (J) ε ) 3 e 3 ∂t q ∂x s 1 . . . ∂x s l 2 - J q+l=1 ε q+l-1 s:|s|=l hM q, s 1 ...s l ∂ q+l ψ (J) ε ∂t q ∂x s 1 . . . ∂x s l 2 + r (J),1 ε • e 2 ,
(3.2.20)

R (J) ε e 3 = ε 0 ρ + ∂ 2 (w (J) ε ) 3 ∂t 2 + ε 2 2 j=1 ρ+ + ρ + ν 1 -ν ∂ 3 ( w(J) ε ) j ∂t 2 ∂x j + 2 i 1 =1 ( hN 2, s 1 s 1 ) 33 ∂ 4 (w (J) ε ) 3 ∂t 2 ∂x 2 s 1 + ε 5 ( hN 4, ∅ ) 33 ∂ 4 (w (J) ε ) 3 ∂t 4 - J q+l=5 ε q+l-5 s:|s|=l h N q, s 1 ...s l ∂ q+l ε( w(J) ε ) 1 e 1 + ε( w(J) ε ) 2 e 2 + (w (J) ε ) 3 e 3 ∂t q ∂x s 1 . . . ∂x s l 3 - J q+l=1 ε q+l-1 s:|s|=l hM q, s 1 ...s l ∂ q+l ψ (J) ε ∂t q ∂x s 1 . . . ∂x s l 3 + r (J),1 ε • e 3 .
with h N, M q, s 1 ...s l = ε 2 hN, M q, s 1 ...s l . We consider

w(J) ε = J k=-1 ε k wk (x, t), (3.2.21)
so that, together with (3.2.17) and (3.2.1) 5 , it gives

       (w k ) 1 = ( wk-1 ) 1 , (w k ) 2 = ( wk-1 ) 2 , (w k ) 3 = ( wk ) 3 ∀ k ≥ 0. (3.2.22)
Analyzing the order of the terms of (3.2) we can write in what follows, we establish the following result: Lemma 3.2. a) For any q ∈ N * the matrices h N q, ∅ , N q, ∅ have the form

                       R (J) ε • e 1 -r (J),1 ε • e 1 = J+r 1 k=0 ε k-1 R k • e 1 , R (J) ε • e 2 -r (J),1 ε • e 2 = J+r 2 k=0 ε k-1 R k • e 2 , R (J) ε • e 3 -r (J),1 ε • e 3 = J+r 3 k=0 ε k R k • e 3 , (3.
h N q, ∅ =     α q 0 0 0 β q 0 0 0 γ q     , N q, ∅ (ξ 3 ) =     σ q (ξ 3 ) 0 0 0 δ q (ξ 3 ) 0 0 0 ϕ q (ξ 3 )     . (3.2.24)
The same result holds for h M q, ∅ , M q, ∅ , with the elements denoted by α M q , β M q , γ M q and σ M q , δ M q , ϕ M q , respectively. b) For any q, j 1,2 ∈ N * we have

h N q, r, p =     α q, r, p 0 0 0 β q, r, p 0 0 0 γ q, r, p     , if r = 2j 1 , p = 2j 2 , h N q, r, p =     0 0 0 0 0 β q, 2j 1 , 2j 2 +1 0 γ q, 2j 1 , 2j 2 +1 0     , if r = 2j 1 , p = 2j 2 + 1, h N q, r, p =     0 0 α q, 2j 1 +1, 2j 2 0 0 0 γ q, 2j 1 +1, 2j 2 0 0     , if r = 2j 1 + 1, p = 2j 2 , h N q, r, p =     0 α q, 2j 1 +1, 2j 2 +1 0 β q, 2j 1 +1, 2j 2 +1 0 0 0 0 0     , if r = 2j 1 + 1, p = 2j 2 + 1 (3.2.25)
and the matrices N q, r, p , h M q, r, p , M q, r, p have the same form as h N q, r, p , where h N,M q, r, p = s:|s|=r+p, |{s j =1}|=r, |{s j =2}|=p

h N,M q, s 1 ...s r+p .

Proof. The result is obtained by induction using the relations of (3.2.14) for h N q, s 1 ...s l , N q, s 1 ...s l and the relations of (3.2.16) for h M q, s 1 ...s l , M q, s 1 ...s l .

Denoting by

ψ k (x, t) = 2νD v k (x,
R k • e 1 = (ψ k ) 1 + ρ + ∂ 2 ( wk-2 ) 1 ∂t 2 + ( hN 4, ∅ ) 11 ∂ 4 ( wk-7 ) 1 ∂t 4 - J q=5 α q ∂ q ( wk+3-q ) 1 ∂t q - J q=1 α M q ∂ q (ψ k-q ) 1 ∂t q + Rk • e 1 , R k • e 2 = (ψ k ) 2 + ρ + ∂ 2 ( wk-2 ) 2 ∂t 2 + ( hN 4, ∅ ) 22 ∂ 4 ( wk-7 ) 2 ∂t 4 - J q=5 β q ∂ q ( wk+3-q ) 2 ∂t q - J q=1 β M q ∂ q (ψ k-q ) 2 ∂t q + Rk • e 2 , R k • e 3 = ρ + ∂ 2 (w k ) 3 ∂t 2 + ( hN 4, ∅ ) 33 ∂ 4 (w k-5 ) 3 ∂t 4 - J q=5 γ q ∂ q (w k+5-q ) 3 ∂t q - J q=1 γ M q ∂ q (ψ k-q+1 ) 3 ∂t q + Rk • e 3 , (3.2.27)
where Rk are functions defined by ( wk ′′ ) 1 , ( wk ′′ ) 2 , (w

k ′ ) 3 , v k ′ , p k ′ , with k ′′ ≤ k -2, k ′ ≤ k -1
                                         -Ê1 ∂ 2 ( wk ) 1 ∂x 2 1 -Ê3 ∂ 2 ( wk ) 2 ∂x 1 ∂x 2 -Ê2 ∂ 2 ( wk ) 1 ∂x 2 2 -Ê1 ∂ 3 (w k ) 3 ∂x 3 1 + ∂ 3 (w k ) 3 ∂x 1 ∂x 2 2 = g 1 δ k1 -R k-1 • e 1 , -Ê2 ∂ 2 ( wk ) 2 ∂x 2 1 -Ê3 ∂ 2 ( wk ) 1 ∂x 1 ∂x 2 -Ê1 ∂ 2 ( wk ) 2 ∂x 2 2 -Ê1 ∂ 3 (w k ) 3 ∂x 2 1 ∂x 2 + ∂ 3 (w k ) 3 ∂x 3 2 = g 2 δ k1 -R k-1 • e 2 , -Ê2 ∂ 3 ( wk ) 1 ∂x 3 1 + ∂ 3 ( wk ) 2 ∂x 2 1 ∂x 2 + ∂ 3 ( wk ) 1 ∂x 1 ∂x 2 2 + ∂ 3 ( wk ) 2 ∂x 3 2 -Ĵ ∂ 4 (w k ) 3 ∂x 4 1 + ∂ 4 (w k ) 3 ∂x 2 1 ∂x 2 2 + ∂ 4 (w k ) 3 ∂x 4 2 + 2ν ∂(v k ) 3 ∂x 3 -p k x 3 =0 = g 3 δ k0 -R k-1 • e 3 . ( 3
( Ê1 • Ê2 -Ê1 • Ĵ ) ∆ ∆(w k ) 3 + 2ν ∂(v k ) 3 ∂x 3 -p k x 3 =0 = g 3 δ k0 -R k-1 • e 3 -Ê-1 1 • Ê2 ∂g 1 ∂x 1 δ k1 - ∂R k-1 ∂x 1 • e 1 + ∂g 2 ∂x 2 δ k1 - ∂R k-1 ∂x 2 • e 2 .
(3.2.29)

We introduce the notation

Ĵ = Ê-1 1 ( Ê1 • Ê2 -Ê1 • Ĵ ) ( 3
         ρ - ∂v k ∂t -2νdiv(D(v k )) + ∇p k = f δ k0 , div v k = 0 in D -× (0, T ), v k (x, -1, t) = 0. ( 3
v (J) ε (x, 0, t) - ∂u (J) ε ∂t (x, 0, t) = J k=0 ε k v k (x, 0, t) - J q+l=0 s:|s|=l N q, s 1 ...s l (0) ∂ q+l+1 w k-(q+l) (x, t) ∂t q+1 ∂x s 1 . . . ∂x s l -r (J),6 ε (x, t), (3.2.32)
where where α k-1 contains only ( wk ′ -1 ) 1 , ( wk ′ -1 ) 2 and (w k ′ ) 3 , with k ′ ≤ k -1.

r (J),6 ε (x, t) = J+q+l k=J+1 ε k J q+l=1 s:|s|=l N q, s 1 ...s l (0) ∂ q+l+1 w k-(q+l) (x, t) ∂t q+1 ∂x s 1 . . . ∂x s l . ( 3
In addition to the previous relations, we consider the periodicity, the initial conditions and the property

(w k ) 3 = 0, ∀ k ≥ 0. (3.2.36)
In this way, for every k ≥ -1 we obtain the following two problems:

                                                                         Ĵ∆ 2 x(w k ) 3 + 2ν ∂(v k ) 3 ∂x 3 -p k x 3 =0 = g 3 δ k0 -R k-1 • e 3 -Ê-1 1 • Ê2 ∂g 1 ∂x 1 δ k1 - ∂R k-1 ∂x 1 • e 1 + ∂g 2 ∂x 2 δ k1 - ∂R k-1 ∂x 2 • e 2 in D × (0, T ), ρ - ∂v k ∂t -2ν div(D(v k )) + ∇p k = f δ k0 , div v k = 0 in D -× (0, T ), v k (x, -1, t) = 0 in D × (0, T ), (v k ) 1 (x, 0, t) = ∂ (( wk-1 ) 1 ) ∂t (x, t) + α k-1 • e 1 in D × (0, T ), (v k ) 2 (x, 0, t) = ∂ (( wk-1 ) 2 ) ∂t (x, t) + α k-1 • e 2 in D × (0, T ), (v k ) 3 (x, 0, t) = ∂(w k ) 3 ∂t (x, t) + α k-1 • e 3 in D × (0, T ), (w k ) 3 , v k , p k D-periodic, v k (x, 0) = 0 in D -, (w k ) 3 (x, 0) = 0 in D, k ≥ 0 (3.2.37) and                      Ê1 ∂ 2 ( wk ) 1 ∂x 2 1 + Ê3 ∂ 2 ( wk ) 2 ∂x 1 ∂x 2 + Ê2 ∂ 2 ( wk ) 1 ∂x 2 2 = -(y k ) 1 , Ê2 ∂ 2 ( wk ) 2 ∂x 2 1 + Ê3 ∂ 2 ( wk ) 1 ∂x 1 ∂x 2 + Ê1 ∂ 2 ( wk ) 2 ∂x 2 2 = -(y k ) 2 , y k = Ê1 ∇ x ∆ x(w k ) 3 + gδ k1 -R k-1 , wk D-periodic, k ≥ -1, (3.2.38)
where the terms R k-1 , α k-1 depend on the functions ( wk

′ -1 ) 1 , ( wk ′ -1 ) 2 , (w k ′ ) 3 , v k ′ , p k ′ and
R k-1 • e 1 = g 1 δ k,1 , R k-1 • e 2 = g 2 δ k,1 ∀ k ≥ 0. ( 3
               Ê1 ∂ 2 ( ŵk ) 1 ∂x 2 1 + Ê3 ∂ 2 ( ŵk ) 2 ∂x 1 ∂x 2 + Ê2 ∂ 2 ( ŵk ) 1 ∂x 2 2 = -(y k ) 1 , Ê2 ∂ 2 ( ŵk ) 2 ∂x 2 1 + Ê3 ∂ 2 ( ŵk ) 1 ∂x 1 ∂x 2 + Ê1 ∂ 2 ( ŵk ) 2 ∂x 2 2 = -(y k ) 2 , ŵk D-periodic, k ≥ -1, (3.2.42)
where y k is defined by (3.2.38) 3 .

The main result of this section consists in the construction of some smooth functions (w k ) 3 , v k , p k , c k-1 that satisfy (3.2.37) and Moreover, the regularity of these functions is given by (

R k • e 1 = g 1 δ k,0 , R k • e 2 = g 2 δ k,0 ∀ k ≥ 0. ( 3
w k ) 3 ∈ C ∞ ([0, T ]; C ∞ # ( D)), v k ∈ C ∞ ([0, T ]; (C ∞ # ( D) ∩ H 2 (D -)) 3 ), p k ∈ C ∞ ([0, T ]; C ∞ # ( D) ∩ H 1 (D -)) and c k-1 ∈ C ∞ ([0, T ]).
Proof. The proof of the announced result is technical and will be obtained recursively, in several steps.

Step 1. This step is devoted to the presentation of two auxiliary problems.

We study these problems and we determine their solutions which are used for the construction of the functions v k , c k-1 that solve, together with (w k ) 3 , p k problems (3.2.37), (3.2.39 ′ ). The first auxiliary problem is: Find the functions (w k ) 3 , vk , p k , k ≥ 0, which satisfy

                                                                         Ĵ∆ 2 x(w k ) 3 + 2ν ∂(v k ) 3 ∂x 3 -p k x 3 =0 = g 3 δ k0 -R k-1 • e 3 -Ê-1 1 • Ê2 ∂g 1 ∂x 1 δ k1 - ∂R k-1 ∂x 1 • e 1 + ∂g 2 ∂x 2 δ k1 - ∂R k-1 ∂x 2 • e 2 in D × (0, T ), ρ - ∂ vk ∂t -2ν div(D(v k )) + ∇p k = f δ k0 , div vk = 0 in D -× (0, T ), vk (x, -1, t) = 0 in D × (0, T ), (v k ) 1 (x, 0, t) = ∂ (( ŵk-1 ) 1 ) ∂t (x, t) + α k-1 • e 1 in D × (0, T ), (v k ) 2 (x, 0, t) = ∂ (( ŵk-1 ) 2 ) ∂t (x, t) + α k-1 • e 2 in D × (0, T ), (v k ) 3 (x, 0, t) = ∂(w k ) 3 ∂t (x, t) + α k-1 • e 3 in D × (0, T ), (w k ) 3 , vk , p k D-periodic, vk (x, 0) = 0 in D -, (w k ) 3 (x, 0) = 0 in D, k ≥ 0. ( 3

.2.43)

We give below the second auxiliary problem. For any k ≥ 0, find

V k : [-1, 0] × [0, T ] → R 2 solution for                  ρ - ∂V k ∂t - ν ∂ 2 V k ∂x 2 3 = 0 in (-1, 0) × (0, T ), ν ∂V k ∂x 3 = γ k-1 on {x 3 = 0} × (0, T ), V k = 0 on {x 3 = -1} × (0, T ), V k (0) = 0 in (-1, 0) (3.2.44) with (γ k-1 ) i (t) = ge i δ k+1,1 -ρ + (c ′′ k-2 ) i -ν ∂(v k ) i ∂x 3 x 3 =0 -(h 4,0 ) 11 d 4 (c k-7 ) i dt 4 + J q=5 α q d q (c k+q-3 ) i dt q + J q=1 α M q d q dt q (ψ k-q ) i , i = 1, 2. ( 3
) 3 ∈ C ∞ ([0, T ]; C ∞ # ( D)), vk ∈ C ∞ ([0, T ]; (H 2 (D -)) 3 ), ∂ j vk ∂x j i ∈ C ∞ ([0, T ]; (L 2 (D -)) 3 ), p k ∈ C ∞ ([0, T ]; (H 2 (D -)) 3 ), ∂ j p k ∂x j i ∈ C ∞ ([0, T ]; L 2 (D -)), i = 1, 2, j ∈ N.
Theorem 3.6. For any k ≥ 0 problem (3.2.44) has a unique solution V k with the

regularity V k ∈ C ∞ ([0, T ]; (H 3 (-1, 0)) 2 ).
Step 2. This step contains the proofs of the previous two theorems for k = 0.

We mention that, even if problems (3.2.43) and (3.2.44) are not coupled, we must solve them together since, for determining the functions of the k approximation, we need the induction assumption concerning all the functions from the previous approximations.

Proof of Theorem 3.5 for k=0 is given above in section 2.2.2.

Proof of Theorem 3.6 for k=0. For k = 0 (3.2.44) becomes

                 ρ - ∂V 0 ∂t - ν ∂ 2 V 0 ∂x 2 3 = 0 in (-1, 0) × (0, T ), ν ∂V 0 ∂x 3 = γ -1 on {x 3 = 0} × (0, T ), V 0 = 0 on {x 3 = -1} × (0, T ), V 0 (0) = 0 in (-1, 0) (3.2.46) with (γ -1 ) i (t) = ge i -ν ∂(v 0 ) i ∂x 3 x 3 =0 , i = 1, 2. (3.2.47)
This problem is known in geophysics as the heat equation with heat transfer boundary condition. We introduce the space H = {ϕ ∈ (H 1 (-1, 0)) 2 /ϕ(-1) = 0} and by means of the variational formulation we obtain, using the Galerkin's method, the existence of V 0 ∈ H 1 (0, T ; H), the unique solution for (3.2.46). From (3.2.46) 1 it follows that V 0 ∈ H 1 (0, T ; (H -3 (-1, 0)) 2 ); finally, the C ∞ -regularity in t is a consequence of (H2) and of the regularity of v0 , via (3.2.47), which achieves the proof.

We mention that, in addition to the properties previously obtained, all the functions determined at this step are zero for t ∈ [0, τ 0 ), due to the assumption (H3), p. 52.

Step 3. The induction assumption is that all the properties obtained in Step 2. for k = 0 hold for any k ′ , k ′ < k. This step is devoted to prove these properties for k.

Proof of Theorem 3.

             div ζ k (t) = 0 in R 2 × (-1, 0), ζ k (t) = 0 on R 2 × {x 3 = -1}, ζ k (t) = a(t) on R × {x 3 = 0}, ζ k (t) D-periodic, (3.2
.48) with a 1 (x, t) = ∂(( ŵk-1 ) 1 ) ∂t (x, t) + α k-1 (x, t) • e 1 , a 2 (x, t) = ∂(( ŵk-1 ) 2 ) ∂t (x, t) + α k-1 (x, t) • e 2 , a 3 (x, t) = α k-1 (x, t) • e 3 .
       a 0 (t) = a (t), a m (t) = 2 a cos 2πmx (t), b m (t) = 2 a cos 2πmx (t) |m| ≥ 1, where m = (m 1 , m 2 ) is a multi-index, x = (x 1 , x 2 ) and mx = m 1 x 1 + m 2 x 2 .
It can be proved by direct computation that the function

ζ k (x, t) = (x 3 + 1)(3x 3 + 1)(a 0 ) 1 (t)e 1 + (x 3 + 1)(3x 3 + 1)(a 0 ) 2 (t)e 2 -(x 3 + 1) ∞ m 1 ,m 2 =1 (-3x 3 -1)(a m ) 1 (t) + 3 2πm 1 x 3 (b m ) 3 (t) cos 2πmx + (-3x 3 -1)(b m ) 1 (t) - 3 2πm 1 x 3 (a m ) 3 (t) sin 2πmx e 1 -(x 3 + 1) ∞ m 1 ,m 2 =1 (-3x 3 -1)(a m ) 2 (t) + 3 2πm 2 x 3 (b m ) 3 (t) cos 2πmx + (-3x 3 -1)(b m ) 2 (t) - 3 2πm 2 x 3 (a m ) 3 (t) sin 2πmx e 2 -(x 3 + 1) ∞ m 1 ,m 2 =1 2πm 1 x 3 (x 3 + 1)(b m ) 1 (t) + 2πm 2 x 3 (x 3 + 1)(b m ) 2 (t) + (2x 2 3 + x 3 -1)(a m ) 3 (t) cos 2πmx + -2πm 1 x 3 (x 3 + 1)(a m ) 1 (t) -2πm 2 x 3 (x 3 + 1)(a m ) 2 (t) + (2x 2 3 + x 3 -1)(b m ) 3 (t) sin 2πmx e 3 (3.2.49)
is a solution for (3.2.48). Moreover, the function ζ k has the additional properties We define next the function ω

ζ k ∈ C ∞ ([0, T ]; (C ∞ # ( D-)) 3 ), ζ k = 0 in D-× [0, τ 0 ). ( 3
k : D-× [0, T ] → R 3 , given by ω k (x, t) = vk (x, t) -ζ k (x, t), ∀ k ≥ 0. ( 3
) 3 , ω k , p k ) the following problem                                                Ĵ∆ 2 x( ŵk ) 3 -pk x 3 =0 = G k (x, t) in D × (0, T ), ρ - ∂ω k ∂t -ν∆ω k + ∇p k = F k , div ω k = 0 in D -× (0, T ), ω k (x, -1, t) = 0 in D × (0, T ), ω k (x, 0, t) = ∂(w k ) 3 ∂t (x, t) • e 3 in D × (0, T ), (w k ) 3 , ω k , p k D-periodic, ω k (x, 0) = 0 in D -, (w k ) 3 (x, 0) = 0 in D, k ≥ 0. (3.2.52) with                G k (x, t) = g 3 δ k0 -R k-1 • e 3 -Ê-1 1 • Ê2 ∂g 1 ∂x 1 δ k1 - ∂R k-1 ∂x 1 • e 1 + ∂g 2 ∂x 2 δ k1 - ∂R k-1 ∂x 2 • e 2 + 2ν ∂a 1 ∂x 1 + ∂a 2 ∂x 2 , F k (x, t) = f (x, t)δ k0 -ρ - ∂ζ k ∂t (x, t) + ν∆ζ k (x, t).
Compare now problem (3.2.52) with (2.2.23). We notice that all the equations, initial and boundary conditions are the same; moreover, the known right hand sides of (3. Step 4. The last step is devoted to the construction of some smooth functions while ((w 0 ) 3 , v 0 , p 0 ) is the unique solution for

                                       Ĵ∆ 2
x(w 0 ) 3p 0

Justification of asymptotics and error estimation

In this section we will obtain the error estimates which generally demonstrate the small difference between the real solution and the constructed solution to the main problem. From the previous section we obtain the problem for the asymptotic solution of order J in the form

                                                                           ρ + x 3 ε ∂ 2 u (J) ε ∂t 2 -ε -3 3 i,j=1 ∂ ∂x i A ij x 3 ε ∂u (J) ε ∂x j = ε -1 g -r(J),1 ε in D + ε ×(0, T ),
ρ - ∂v (J) ε ∂t -2νdiv D(v (J) ε ) + ∇p (J) ε = f in D -× (0, T ), div v (J) ε = 0 in D -× (0, T ), given by (3.2.5).

In order to replace the previous system with another one with homogeneous boundary conditions instead of (3.3.1) 4,7 we prove the following result Proposition 3.7. The problem We define next

                         Find ϕ (J) ε : D+ ε × [0, T ] → R 3 such that ϕ (J) ε D-periodic,
U (J) ε = u (J) ε -ϕ (J) ε (3.3.4)
and Û(J) ε , v(J) ε , p(J)

ε = (u ε , v ε , p ε )-(U (J) ε , v (J)
ε , p (J) ε ) and, from (2.1.1), (3.3.1) and (3.3.2) we obtain

                                                                           ρ + x 3 ε ∂ 2 Û(J) ε ∂t 2 -ε -3 3 i,j=1 ∂ ∂x i A ij x 3 ε ∂ Û(J) ε ∂x j = -g (J) ε in D + ε × (0, T ), ρ - ∂ v(J)
ε ∂t -2νdiv D(v (J) ε ) + ∇p (J) ε = 0 in D -× (0, T ), div v(J) ε = 0 in D -× (0, T ), The first estimates between the exact solution and the asymptotic solution of order J are given by Theorem 3.8. Let (u ε , v ε , p ε ) be the exact solution of (2.1.1) and (u

(J) ε , v (J) ε , p (J)
ε ) the asymptotic solution of order J, defined by (3.2.1). Then the following esti-

mates hold                                                ∂ ∂t u ε -u (J) ε L ∞ (0,T ;(L 2 (D + ε )) 2 )
= O(ε J-7 / 2 ),

∂ 2 ∂t 2 u ε -u (J) ε L ∞ (0,T ;(L 2 (D + ε )) 2 )
= O(ε J-7 / 2 ), in the same way as in [START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF][START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF] estimates (3.3.7) 1-6 . We notice the difference that appears in (3.3.7) 1,2 with respect to the corresponding estimates from [START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF][START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF] due to the fact that here the density of the elastic material is of order 1 where in [START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF][START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF] is O(ε -1 ). In what follows we establish a more precise estimate for the pressure than in [START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF][START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF].

E x (u ε -u (J) ε ) L ∞ (0,T ;(L 2 (D + ε )) 3×3 ) = O(ε J-2 ), v ε -v (J) ε L ∞ (0,T ;(L 2 (D -)) 2 ) = O(ε J-7 / 2 ), ∂ ∂t v ε -v (J)
Let us consider a pair (ϕ, ω) ∈ (H 1 (D + ε )) 3 × (H 1 (D -)) 3 with the properties: ϕ = 0 on ∂D + ε \Γ 0 , ω = 0 on ∂D -\Γ 0 , ϕ = ω on Γ 0 . Computing • ϕ + ε -3

D + ε 3 i,j=1 A ij ∂ Û(J) ε (t) ∂x j • ∂ϕ ∂x i + D + ε g (J) ε (t) • ϕ + ρ - D - ∂ v(J) ε (t) ∂t • ω + 2ν D -
D(v (J) ε (t)) : D(ω) a.e. in (0, T ).

(3.3.9)

We introduce the notation

       A(u(t)) = c 1 ∂ 2 u ∂t 2 (t) (L 2 (D + ε )) 3 + c 2 ε -3 E(u(t)) (L 2 (D + ε )) 9
+ c 3 g (J) ε (t) As one can see, the error between the exact solution and the asymptotic solution of order J is big if J ≤ 6. The last result of this section is devoted to the improvement of the previous estimates.

(L 2 (D + ε )) 3 , B(v(t)) =
Theorem 3.9. Let (u ε , v ε , p ε ) be the exact solution of (2.1.1) and

(u

(K) ε , v (K) ε , p (K)
ε ) the asymptotic solution of order K, defined by (3.2.1). Then the error between these two solutions is given by

                                               ∂ ∂t u ε -u (K) ε L ∞ (0,T ;(L 2 (D + ε )) 2 )
= O(ε K+3/2 ),

∂ 2 ∂t 2 u ε -u (K) ε L ∞ (0,T ;(L 2 (D + ε )) 2 )
= O(ε K+3/2 ),

E x (u ε -u (K) ε ) L ∞ (0,T ;(L 2 (D + ε )) 3×3 ) = O(ε K+1/2 ), v ε -v (K) ε L ∞ (0,T ;(L 2 (D -)) 2 ) = O(ε K+1 ), ∂ ∂t v ε -v (K) ε L ∞ (0,T ;(L 2 (D -)) 2 ) = O(ε K+1 ), D x (v ε -v (K) ε ) L 2 (0,T ;(L 2 (D -)) 3×3 ) = O(ε K+1 ), p ε -p (K) ε L 2 (0,T ;L 2 (D -)) = O(ε K+1 ). ( 3

Conclusion

We have constructed the complete asymptotic expansion of the solution. Homogenization is used. The limit problem describes the principal term of the asymptotic expansion.

Conclusion

Conclusion

Viscous fluid-structure interaction problem is considered. The investigation of the viscous fluid-thin elastic plate interaction in 2D [START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF][START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF] is generalized here to 3D case. In variational analysis a new idea is to consider one function for the fluid velocity v ε and the velocity in the elastic area ∂u ε ∂t . This allows to reduce a proof volume using one set of Galerkin approximations in place of two (for the fluid and elastic mediums). The main distinguishing feature of the coupled system "viscous fluid flow-thin elastic plate" in 3D from the 2D-case is in the asymptotic analysis: when we construct the asymptotic expansion we have no more terms that we can determine explicitly, we have now the systems for them.

And as before, we succeeded in differentiation between two problems: for the solid and fluid parts. Physical application results are following : the formation of residual stresses during the laser treatment of materials is studied using a thermoelastic model. Calculation results can be used to evaluate the thermomechanical stability of the materials in the SLM process. The graphical representation of the two-dimensional fields is obtained numerically with a database of thermoelastic properties of metallic, ceramic and polymer materials.

Introduction 1 1 9 1. 1 9 1. 2

 19192 Calcul des contraintes résiduelles sous fusion sélective au laser des poudres Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modèle physique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Résultats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Plaque verticale . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Plaque horizontale . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Cordon sur un substrat semi-infini . . . . . . . . . . . . . . . 1.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Analyse asymptotique du problème avec contraintes résiduelles dans le second-membre . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Énoncé du problème . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Existence et unicité d'une solution . . . . . . . . . . . . . . . 1.5.3 Construction d'une solution . . . . . . . . . . . . . . . . . . .

3

  Analyse asymptotique du problème d'interaction d'un fluide visqueux avec une plaque mince 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Construction d'un développement asymptotique de la solution du problème . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Justification du développement asymptotique et l'estimation d'erreur 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 'interaction d'un fluide visqueux avec une structure élastique décrivent, par exemple, des interactions entre le manteau terrestre et de la croûte terrestre, le sang et la paroi vasculaire dans un vaisseau sanguin, la mince croûte de glace et l'eau. Il existe un grand nombre d'articles sur le sujet [1-22] et d'autres.

Fig. 1 .

 1 Fig. 1. Elementary cube with components of the stress state acting on the faces of the cube.

Fig. 2 .

 2 Fig. 2. Vertical (a) and horizontal (b) plates grown on the substrate with the help of the SLM method: 1computational domain; 2 -transition region; 3 -boundary zone of discharge; 4 -substrate influence zone; (XY Z) -Cartesian basis.

Fig. 3 .

 3 Fig. 3. Individual remelted beads on a semi-infinite substrate: remelted band (a); lens-shaped remelted profile (b); double-lens bead (c); circle-shaped bead (d)

Fig. 4 .

 4 Fig. 4. Dimensionless residual displacements û, principal residual stresses σ1 = σxx , σ2 , and σ3 and equivalent Mises stresses σMises in remelted band with the configuration shown in Fig. 3a, and in the substrate adjacent areas for the Poisson's ratios ν = 0.17 (a) and 0.34 (b). The displacement direction is indicated by the arrows. The principal axes directions in the plane (Y Z) are shown by the dashes.

Fig. 5 .

 5 Fig. 5. Dimensionless residual displacements û, principal residual stresses σ1 = σxx , σ2 , and σ3 and equivalent Mises stresses σMises in insulated remelted beads with the configuration shown in Fig. 3b, and in the substrate adjacent areas for the Poisson's ratios ν = 0.17 (a) and 0.34 (b). The displacement direction is indicated by the arrows. The principal axes directions in the plane (Y Z) are shown by the dashes.

Fig. 6 .

 6 Fig. 6. Dimensionless residual displacements û, principal residual stresses σ1 = σxx , σ2 , and σ3 and equivalent Mises stresses σMises in insulated remelted beads with the configuration shown in Fig. 3c, and in the substrate adjacent areas for the Poisson's ratios ν = 0.17 (a) and 0.34 (b). The displacement direction is indicated by the arrows. The principal axes directions in the plane (Y Z) are shown by the dashes.

Fig. 7 .

 7 Fig. 7. Dimensionless residual displacements û, principal residual stresses σ1 = σxx , σ2 , and σ3 and equivalent Mises stresses σMises in insulated remelted beads with the configuration shown in Fig. 3d, and in the substrate adjacent areas for the Poisson's ratios ν = 0.17 (a) and 0.34 (b). The displacement direction is indicated by the arrows. The principal axes directions in the plane (Y Z) are shown by the dashes.

Fig. 8 .

 8 Fig. 8. Maximum principal tensile residual stresses, σ 1 and σ 2 , and the equivalent Mises stress σ Mises in the individual beads (bold lines) and the maximum tensile residual stresses in the horizontal and vertical plates (thin lines) in comparison with the limits of the tensile strength of non-metallic materials and yield limits of the metallic ones (points). Preheating temperatures T a are indicated near the points.

7 ) 1 . 1 .

 711 Definition The weak solution to problem (1.5.1), (1.5.2) is called a function u ∈ H 1 # (G) which satisfies the following variational problem:

Fig. 9 .

 9 Fig. 9. 3D layer

2 . 23 )

 223 with r 1 , r 2 , r 3 > 0 and R k independent of ξ 3 and bounded by a constant independent of ε. In order to obtain the expressions of R k •e 1 , R k •e 2 and R k •e 3 necessary

≥ 1 .

 1 2.52), G k , F k , have the same regularity as g 3 , f , respectively, regularity given by (H1) and (H2), via the induction assumption. Hence, the remaining part of the proof of the proof for an arbitrary value of k, k ≥ 1, is the same as the one for k = 0. The regularity of ω k corresponds to the regularity for v0 in Step 2. and vk has the same regularity as ω k , due to (3.2.50) 1 , which achieves the proof for k Proof of Theorem 3.6 for k ≥ 1. The problem for V k , (3.2.44), has the same form for any value of k. Noticing that the regularity of γ k-1 is (C ∞ ([0, T ])) 2 due to the induction assumption, all the results obtained in Step 2. for V 0 are still true for V k .

x 3 =0 = g 3

 33 in D × (0, T ),ρ - ∂v 0 ∂t -ν∆v 0 + ∇p 0 = f , div v 0 = 0 in D -× (0, T ), v 0 (x, -1, t) = 0 in D × (0, T ), v 0 (x, 0, t) = (c ′ -1 ) 1 (t)e 1 + (c ′ -1 ) 2 (t)e 2 + ∂(w 0 ) 3 ∂t (x, t)e 3 in D × (0, T ), (w 0 ) 3 , v 0 , p 0 D-periodic, v 0 (x, 0) = 0 in D -; (w 0 ) 3 (x, 0) = 0 in D. (3.2.56) Taking into account (3.2.22), coupling condition (3.2.56) 5 can be written as v 0 = ∂w 0 ∂t on Γ 0 . (3.2.57)

3 j=1A

 3 ε × (0, T ), v (J) ε = 0 on Γ -× (0, T ), v (J) ε -∂u (J) ε ∂t = -r (J),6 ε on Γ 0 × (0, T ), -p (J) ε e 3 +2νD(v (J) ε )e 3 = ε -3 3j (0) ∂u (J) ε ∂x j -r (J),7 ε on Γ 0 × (0, T ), u (J) ε , v (J) ε , p (J) 1 (R k ) j e j + J+r 3 k=J+1 ε k (R k ) 3 e 3 , r(J),1 ε

3 j=1A 3 j=1A) 3 (x, t) e 3 ( 3 . 3 . 3 )

 3333333 3j[START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF] ∂ϕ (J) ε ∂x j (x, ε, t) = r (J),4 ε (x, t) in D × [0, T ], 3j (0) ∂ϕ (J) ε ∂x j (x, 0, t) = ε 3 r (J),7 ε (x, t) in D × [0, T ] (3.3.2)has at least a solution.Proof. It can be easily verified that the functionϕ (J) ε (x, t) = x 3 (x 3ε)has the properties stated in (3.3.2).

  = 0 in D + ε , v(J) ε (0) = 0 in D -,

  ε L ∞ (0,T ;(L 2 (D -)) 2 ) = O(ε J-7 / 2 ), D x (v εv (J) ε ) L 2 (0,T ;(L 2 (D -)) 3×3 ) = O(ε J-7 / 2 ), p εp (J) ε L 2 (0,T ;L 2 (D -)) = O(ε J-13 / 2 ).

  3.5) 2 • ω and using (3.3.5) 7 we get L(ϕ, ω)(t) = D - p(J) ε (t) div ω a.e. in (0, T ), (3.3.8) with L(ϕ, ω)(t) =

η 2 L 2 2 2L 2 2 L 2 2 L 2 ∂t L 2

 222222222 c 4 ∂v ∂t (t) (L 2 (D -)) 3 + 2ν D(v(t)) (L 2 (D -)) 3×3 ,(3.3.10)where c 1 , c 2 , c 3 , c 4 are positive known constants independent of ε.Applying Poincaré's inequality and an obvious estimate for the second term of the right-hand side of (5.9) it follows that , ω)(t) ≤ A( Û(J)ε (t)) ∇ϕ (L 2 (D + ε )) 3×3 + B(v (J) ε (t)) ∇ω (L 2 (D -)) 3×3 a.e. in (0, T ), (∀)(ϕ, ω) ∈ (H 1 (D + ε )) 3 × (H 1 (D -)) 3 : ϕ = 0 on ∂D + ε \Γ 0 , ω = 0 on ∂D -\Γ 0 , ϕ = ω on Γ 0 . (3.3.11)We choose next some particular functions ω and ϕ in(3.3.11). We begin with the construction of ω. For this purpose, we consider an arbitrary function η : D → R with the properties: = 0 on ∂D, η ∈ C 1,1 ( D) (3.3.12) From (3.3.8), (3.3.9), (3.3.13) 2 , (3.3.14) and (3.3.17) we finally obtain p(J) ε(t) (0,T ;L 2 (D -)) ≤ a 1 ε -3 ∂ 2 Û(J) ε ∂t (0,T ;(L 2 (D + ε )) 3 ) +a 2 ε -9 E( Û(J) ε ) (0,T ;(L 2 (D + ε )) 3×3 ) + a 3 ε -3 g (J) ε (0,T ;(L 2 (D + ε )) 3 ) (0,T ;(L 2 (D -)) 3 ) + a 5 D(v (J) ε ) L 2 (0,T ;(L 2 (D -)) 3×3 ) ,(3.3.18) with a 1 , . . . , a 5 positive constants independent of ε. Estimate (3.3.7) 7 follows from (3.3.18), (3.3.7) 2,3,5,6 , which achieves the proof.

  Le problème d'interaction d'un fluide visqueux avec une structure élastique est considéré. On généralise l'investigation du problème d'interaction du fluide visqueux avec la plaque mince élastique en 2D[START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF][START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF] ici en cas 3D. Dans l'analyse variationnelle une nouvelle idée est de considérer la même fonction pour la vitesse du fluide v ε et la vitesse dans la zone élastique ∂u ε ∂t . Cela permet de réduire le volume de la preuve en utilisant un ensemble des approximations de Galerkin à la place de deux (pour le milieu liquide et le milieu élastique). La caractéristique distinctive principale du système couplé "flux fluide visqueux -plaque mince élastique" en 3D du cas deux-dimensionnel est dans l'analyse asymptotique : lorsque nous construisons le développement asymptotique nous n'avons plus de termes que nous pouvons déterminer explicitement, nous avons maintenant des systèmes pour eux. Et comme avant, nous avons réussi à la différenciation entre les deux problèmes : pour les parties solides et liquides. Les résultats d'application physiques sont suivent : la formation des contraintes résiduelles pendant le traitement des matériaux par laser est étudié utilisant un modèle thermo-élastique. Les résultats des calculs numériques peuvent être utilisés pour évaluer la stabilité thermomécanique des matériaux dans la FSL. La représentation graphique des champs deux-dimensionnels est obtenue numériquement avec une base des données des propriétés thermoélastiques des matériaux métalliques, céramiques et polymères.

  1.5.4 Justification du développement asymptotique . . . . . . . . . 1.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction. Énoncé du problème . . . . . . . . . . . . . . . . . . 2.2 Existence et unicité d'une solution . . . . . . . . . . . . . . . . . . . 2.2.1 Problème principal . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Problème de limite . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Résultats de régularité . . . . . . . . . . . . . . . . . . . . . . . . .

	2 Analyse variationnelle du problème d'interaction d'un fluide
	visqueux avec une plaque mince
	2.1

2.3.1 Problème principal . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Problème de limite . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  0, t) e 3p k (x, 0, t)e 3 (3.2.26) and using (3.2), (3.2.23) and (3.2.24) we obtain for any k ≥ 0

  and they have the mean value with respect to x 1 , x 2 equal to zero.

	Introducing expansions (3.2.21), (3.2.23) and (3.2.1) 4,2,3 in (3.2.19), neglecting
	the residuals, using (2.1.1)

[START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF]

and collecting together the terms of the same order with respect to the small parameter ε we obtain

  Introducing next expansions (3.2.1) 2,3 in (2.1.1) 2,3,5 we obtain (without residual)

		.2.30)
	and prove that
	Proposition 3.3. The constant Ĵ is strictly positive.
	Proof. Let us denote a(s) =	E(s) 1 -ν 2 (s)

  1 e 1 + ( wk-(q+l)-1 ) 2 e 2 + (w k-(q+l) ) 3 e 3 ∂t q+1 ∂x s 1 . . . ∂x s l ,

				.2.33)
	So,
	v k (x, 0, t) =	∂ ∂t	( wk-1 ) J
				N q, s 1 ...s l (0)
	•	q+l=1 s:|s|=l ∂ q+l+1 ( wk-(q+l)-1 ) (3.2.35)

1 e 1 + ( wk-1 ) 2 e 2 + (w k ) 3 e 3 (x, t) + α k-1 (x, t), (3.2.34) with α k-1 (x, t) =

  their derivatives, with k ′ ≤ k-1. From the smoothness and periodicity properties of the functions ( wk ) 1 , ( wk ) 2 and (w k ) 3 , it follows that problem(3.2.38) has a solution if and only if the following solvability condition is satisfied:

  .2.45) Notice that the right hand side of (3.2.44) contains one of the unknowns of (3.2.43); after solving (3.2.43), the right hand side of (3.2.44) becomes a known function. ) 3 , vk , p k ) with the regularity (w k

	We announce next the results that we shall obtain concerning (3.2.43) and
	(3.2.44).
	Theorem 3.5. For any k ≥ 0 problem (3.2.43) has a unique so-
	lution ((w k

  Notice that, due to properties (3.2.24) and (3.2.36) we have α k-1 • e 3 = 0, which represents the solvability condition for(3.2.48). We define the Fourier coefficients associated to the function a as follows

  = O(ε J-7 / 2 ) + O(ε K+3/2). Since J > K + 4, we get (3.3.19) 1 . All the other estimates from (3.3.19) are obtained in a same way, from (3.3.7) and (3.2.1), and the proof is completed.

	∂ ∂t	u ε -u (K) ε	(t)	(L 2 (D + ε )) 2

.

3.19)

Proof. We consider K ≥ 0 a fixed integer and J > K + 7. Let us prove, for instance,

(3.3.19)

1 . Computing the order of

∂ ∂t u (J) εu (K) ε from (3.2.2) 1 we obtain ∂ ∂t u (J) εu (K) ε = O(ε K+1

) which gives, together with (3.3.7) 1 ,

Remerciements

Remerciements

Acknowledgements

Acknowledgements ix

'il m'accordée. Je lui remercie vivement d'etre un maître à penser, un "co-directeur" pour moi.

Chapter 3

Asymptotic analysis of a viscous fluid-thin plate interaction problem

Introduction

In order to construct an asymptotic expansion the existence, uniqueness and regularity of solution to the elasticity problem coupled with Stokes equations are required. We will construct the asymptotic expansion solution compared to the small parameter ε and prove estimates for the difference of the exact solution and the partial sums of the asymptotic expansion. The principal question is to find the zero approximation (satisfying the existence, uniqueness and regularity results from Chapter 2). This problem generalizes the result in two-dimensional case obtained in [START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF][START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF]. Admittedly, the 3D model is more adequate, because in the mentioned articles it was assumed that the density of the plate is of order ε -1 while in the thesis densities of the plate and the fluid are of the same order, which is more realistic for applications. In section 3.2 an asymptotic solution of order J is constructed and the limit problem is obtained. This limit problem differs from that of [START_REF] Panasenko | Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow[END_REF], due to the change of the density order. The asymptotic expansion is completely justified in section 3.3, where the error between the exact and the asymptotic solutions is evaluated.

In this chapter we deal with the equations with rapidly oscillating coefficients [START_REF] Bakhvalov | Homogenisation: Averaging Processes in Periodic Media36volume, series, Mathematics and its Applications[END_REF][START_REF] Ilin | Asimptoticheskie metody v analize [Asymptotic methods in analysis[END_REF][START_REF] Jikov | Homogenization of Differential Operators and Integral Functionals[END_REF][START_REF] Papanicolau | Asymptotic Analysis for Periodic Structures[END_REF][START_REF] Panasenko | Multi-scale Modelling for Structures and Composites[END_REF][START_REF] Kolpakov | Stressed Composite Structures: Homogenized Models for Thin-Walled Nonhomogeneous Structures with Initial Stresses[END_REF]. Heterogeneous composite materials consist of a plurality of other substantially smaller (microscale) materials with different thermomechanical properties. But usually it is very important to know how composite will behave on the whole (macroscale) rather than each of its components separately:

v k and c k-1 that satisfy, togther with (w k ) 3 , p k already obtained in Theorem 3.5, problems (3.2.37) and (3.2.39 ′ ). Let us define for any k ≥ 0

which represents exactly (3.2.44) 2 .

We use next expression (3.2.53) 2 corresponding to k -1 to obtain ( wk-1 ) i , i = 1, 2. Note that at the k-th approximation, k ≥ 0 we determine the functions

We present next the leading term of asymptotic solution (3.2.2). For k = -1

and the problem:

We consider next the problem for ϕ: with the constant C(D + ) independent of ε.