Titre :	Variational And Asymptotic Analysis Of Different Boundary Value Problems With Friction And Memory In Thin Domains
Type de document :	document électronique
Auteurs :	Oussama Merabet, Auteur ; Mustafa Derguine, Directeur de thèse
Editeur :	Sétif:UFS
Année de publication :	2025
Importance :	1 vol (40 f.)
Format :	29 cm
Langues :	Anglais (eng)
Mots-clés :	Variational Asymptotic Analysis Different Boundary Value Problems Friction
Résumé :	In this thesis, we study the variational formulations and asymptotic behavior of dynamic problems for viscoelastic materials defined in thin domains involving friction and memory effects. Using a rigid framework of functional analysis, we successfully transform physical models of partial differential equations into variational problems, incorporating Tresca-type boundary conditions and dynamic friction laws. The primary focus of this study was to derive accurate initial approximations and convergence results as the thickness of the thin domain approaches zero. This allowed us to define a boundary problem characterized by a generalized Reynolds-type equation, which captures the essential properties of the original problem in a simplified form. The analytical techniques used here-including weak convergence, properties of Sobolev spaces, and coercivity arguments-are not only fundamental to understanding the mathematical structure of viscoelastic systems but also pave the way for numerical applications in lubrication theory, biomechanics, and the design of layered materials in engineering. This work calls for further exploration of numerical methods adapted to these problems, and for extending the theory to include more complex geometries or more general types of friction and memory behavior.
Note de contenu :	Table of contents Table of contents 1 GENERAL INTRODUCTION 2 1 PRELIMINAIRES 4 1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Equations of continuous medium mechanics . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Viscoelastic constitutive law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Frictional contact boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Functional spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5.1 Some reminders on functional analysis . . . . . . . . . . . . . . . . . . . . . . 7 1.5.2 Reminder on Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 Spaces of vector-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.7 Gronwall’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.8 Properties of lower semi-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 Asymptotic Study of a Viscoelastic Dynamic Problem with Short Memory 19 2.1 Introduction and problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Variational formulation of the problem 2.1–2.6 . . . . . . . . . . . . . . . . . . . . . 22 2.3 Asymptotic analysis of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.1 Change of domain and variational problem . . . . . . . . . . . . . . . . . . . 26 2.3.2 A priori estimations and convergence results . . . . . . . . . . . . . . . . . . . 27 2.3.3 Limit problem and the Generalized Reynolds equation . . . . . . . . . . . . . 32 2.3.4 Uniqueness of the solution to the limit problem . . . . . . . . . . . . . . . . . 37 Bibliography 40
Côte titre :	MAM/0820

Variational And Asymptotic Analysis Of Different Boundary Value Problems With Friction And Memory In Thin Domains [document électronique] / Oussama Merabet, Auteur ; Mustafa Derguine, Directeur de thèse . - [S.l.] : Sétif:UFS, 2025 . - 1 vol (40 f.) ; 29 cm.
Langues : Anglais (eng)

Mots-clés :	Variational Asymptotic Analysis Different Boundary Value Problems Friction
Résumé :	In this thesis, we study the variational formulations and asymptotic behavior of dynamic problems for viscoelastic materials defined in thin domains involving friction and memory effects. Using a rigid framework of functional analysis, we successfully transform physical models of partial differential equations into variational problems, incorporating Tresca-type boundary conditions and dynamic friction laws. The primary focus of this study was to derive accurate initial approximations and convergence results as the thickness of the thin domain approaches zero. This allowed us to define a boundary problem characterized by a generalized Reynolds-type equation, which captures the essential properties of the original problem in a simplified form. The analytical techniques used here-including weak convergence, properties of Sobolev spaces, and coercivity arguments-are not only fundamental to understanding the mathematical structure of viscoelastic systems but also pave the way for numerical applications in lubrication theory, biomechanics, and the design of layered materials in engineering. This work calls for further exploration of numerical methods adapted to these problems, and for extending the theory to include more complex geometries or more general types of friction and memory behavior.
Note de contenu :	Table of contents Table of contents 1 GENERAL INTRODUCTION 2 1 PRELIMINAIRES 4 1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Equations of continuous medium mechanics . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Viscoelastic constitutive law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Frictional contact boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Functional spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5.1 Some reminders on functional analysis . . . . . . . . . . . . . . . . . . . . . . 7 1.5.2 Reminder on Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 Spaces of vector-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.7 Gronwall’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.8 Properties of lower semi-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 Asymptotic Study of a Viscoelastic Dynamic Problem with Short Memory 19 2.1 Introduction and problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Variational formulation of the problem 2.1–2.6 . . . . . . . . . . . . . . . . . . . . . 22 2.3 Asymptotic analysis of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.1 Change of domain and variational problem . . . . . . . . . . . . . . . . . . . 26 2.3.2 A priori estimations and convergence results . . . . . . . . . . . . . . . . . . . 27 2.3.3 Limit problem and the Generalized Reynolds equation . . . . . . . . . . . . . 32 2.3.4 Uniqueness of the solution to the limit problem . . . . . . . . . . . . . . . . . 37 Bibliography 40
Côte titre :	MAM/0820