Connecting Atomistic and Continuum Models of Nonlinear Elasticity Theory
Julian Braun
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Julian Braun, Connecting Atomistic and Continuum Models of Nonlinear Elasticity Theory (2016), Logos Verlag, Berlin, ISBN: 9783832591496
Beschreibung / Abstract
The nonlinear elastic behavior of solid materials is often described in the context of continuum mechanics. Alternatively, one can try to determine the behavior of every single atom in the material. Classically, the connection between these two types of models is made with the Cauchy-Born rule.
The aim of this book is to provide good criteria for the Cauchy-Born rule to be true and to make the connection between continuum and atomistic models precise. In particular, this includes rigorous proofs for the existence of solutions to the atomistic boundary value problem and their convergence to the corresponding continuum solutions in the limit of small interatomic distances.
The aim of this book is to provide good criteria for the Cauchy-Born rule to be true and to make the connection between continuum and atomistic models precise. In particular, this includes rigorous proofs for the existence of solutions to the atomistic boundary value problem and their convergence to the corresponding continuum solutions in the limit of small interatomic distances.
Inhaltsverzeichnis
- BEGINN
- 1 A Gamma-Convergence Result
- 1.1 Introduction and Results
- 1.2 The Model and Preliminaries
- 1.3 A General Representation Result
- 1.4 The Boundary Value Problem
- 1.5 Proof of the Main Results
- 2 The Models
- 2.1 The Continuum Model
- 2.2 The Atomistic Model
- 2.3 The Cauchy-Born Rule
- 3 Stability
- 3.1 Stability Constants
- 3.2 Representation Formulae
- 3.3 Examples
- 4 The Static Case
- 4.1 The Continuum Problem
- 4.2 Solutions to the Atomistic Equations
- 4.3 Residual Estimates
- 4.4 Proof of the Main Result
- 5 The Dynamic Case
- 5.1 Continuum Elastodynamics
- 5.2 Atomistic Elastodynamics
- 6 A Complimentary Result
- A Hyperbolic Regularity
- B Sobolev Functions
- C Elliptic Regularity