Energetic Relaxation to Structured Deformations: A Multiscale Geometrical Basis for Variational Problems in Continuum Mechanics

Description

This book is the first organized collection of some results that have been obtained by the authors, their collaborators, and other researchers in the variational approach to structured deformations. It sets the basis and makes more accessible the theoretical apparatus for assigning an energy to a structured deformation, thereby providing motivation to researchers in applied mathematics, continuum mechanics, engineering, and materials science to study the deformation of a solid body without committing at the outset to a specific mechanical theory. Researchers will benefit from an approach in which elastic, plastic, and fracture phenomena can be treated in a unified way. ​The book is intended for an audience acquainted with measure theory, the theory of functions of bounded variation, and continuum mechanics. Any students in their last years of undergraduate studies, graduate students, and researchers with a background in applied mathematics, the calculus of variations, and continuum mechanics will have the prerequisite to read this book. Preface Acknowledgements Contents 1 Introduction 1.1 First-Order Structured Deformations: L∞-Theory 1.1.1 Examples of First-Order Structured Deformations 1.1.2 Energetics: The L∞-Theory and Beyond 1.2 Second-Order Structured Deformations: L∞-Theory 1.2.1 Examples of Second-Order Structured Deformations 1.2.2 Energetics: The L∞-Theory and Beyond 1.3 Scope of the Present Contribution References 2 Mathematical Preliminaries 2.1 Measure Theory 2.2 Function Spaces 2.3 Γ-Convergence and Relaxation 2.4 The Global Method for Relaxation References 3 Energetic Relaxation to First-Order Structured Deformations 3.1 Spaces of First-Order Structured Deformations, Approximation Theorems, and Representation of Relaxed Energies 3.1.1 Relaxation in SBV 3.1.1.1 Some Properties of the Relaxed Energy 3.1.2 Relaxation in BV 3.1.3 Other Approximation Results 3.2 Applications 3.2.1 Relaxation of Purely Interfacial Energies 3.2.2 Other Special Settings for Relaxation 3.2.2.1 The Case of a Convex W 3.2.2.2 Motivation for Non-local Models 3.2.3 Dimension Reduction in the Context of Structured Deformations 3.2.4 Optimal Design of Fractured Media 3.2.5 Relaxation of Non-local Energies 3.2.6 Periodic Homogenization of Structured Deformations 3.2.7 Hierarchical First-Order Structured Deformations References 4 Energetic Relaxation to Second-Order Structured Deformations 4.1 Spaces of Second-Order Structured Deformations, Approximation Theorems, and Representation of Relaxed Energies 4.1.1 Relaxation in SBV2 4.1.2 Relaxation in SBH 4.2 Outlook for Applications References 5 Outlook for Future Research 5.1 Microdegree and Micromixing 5.2 Multiscale Differential Geometry 5.2.1 Toward a Multiscale Enrichment of Classical Hypersurfaces in RN+1 5.2.2 Structured Hypersurfaces in RN+1 in an L∞-Setting 5.2.3 L∞-Approximation of Structured Insertions 5.2.4 L∞-Approximation of Other Structured Hypersurfaces 5.2.5 SBV2- and SBH-Approximation of Structured Hypersurfaces in RN+1 References