Research in Mathematics of Materials Science

Preface
Acknowledgements
Contents
About the Editors
Part I Research Papers
Interaction Between Oscillations and Singular Perturbations in a One-Dimensional Phase-Field Model
1 Introduction
2 Setting of the Problem and Statement of the Main Result
3 Preliminary Results
3.1 The Optimal-Profile Problem
4 Oscillations on a Larger Scale than the Singular Perturbation
5 Oscillations on the Same Scale as the Singular Perturbation
6 Oscillations on a Smaller Scale than the Singular Perturbation
7 Limit Analysis of m
References
Grain Growth and the Effect of Different Time Scales
1 Introduction
2 Review of the Models with Single Triple Junction
3 Extension to Grain Boundary Network
4 Experiments and Numerical Simulations
4.1 Experimental Results: Grain Boundary Character Distribution
4.2 Numerical Experiments
5 Conclusion
References
Regularity of Minimizers for a General Class of Constrained Energies in Two-Dimensional Domains with Applications to Liquid Crystals
1 Introduction.
2 Continuity and H2loc Estimates for Minimizers in Two-Dimensional Domains
3 Proof of Theorem 1
4 Applications to Liquid Crystals
References
On Some Models in Radiation Hydrodynamics
1 Introduction
2 Compressible Viscous Radiation Fluid
2.1 Hypotheses and Main Results
2.2 Constitutive Equations
2.3 Weak Formulation
2.4 Existence Result
2.5 Semi-Relativistic Models
3 Inviscid Case
3.1 Euler System with Damping Term
3.1.1 Hypotheses
3.2 Non-isentropic Euler–Maxwell's System Coupled with Transport of Radiation
References
Poro-Visco-Elasticity in Biomechanics: Optimal Control
1 Introduction
2 Poro-Visco-Elasticity: Well-posedness Analysis
3 Optimal Control Problems: Well-Posedness
4 Necessary Optimality Condition
4.1 Adjoint System
4.2 First Order Necessary Optimality Conditions
References
Global Gradient Estimate for a Divergence Problem and Its Application to the Homogenization of a Magnetic Suspension
1 Introduction
2 Formulation
2.1 Notation
2.2 Setup of the Problem
3 Statement and Discussion of the Main Result
4 Interior Estimates
5 Boundary Estimates, Green Functions, Dirichlet Correctors, and Proof of Main Theorem
6 Application to Magnetic Suspensions
7 Conclusions
Appendix
References
On Static and Evolutionary Homogenization in Crystal Plasticity for Stratified Composites
1 Introduction
1.1 Notation
2 Minimizers of the Static Homogenized Limit Problem
3 Homogenization via Evolutionary -Convergence
3.1 The Case s=e2
3.2 The Case s=e1
References
On the Prescription of Boundary Conditions for Nonlocal Poisson's and Peridynamics Models
1 Introduction and Motivation
2 Preliminaries
2.1 The Nonlocal Poisson's Problem
2.2 The Linear Peridynamic Solid Model
3 Proposed Strategies
3.1 Dirichlet-to-Dirichlet Strategy
3.2 Dirichlet-to-Neumann Strategy
4 Convergence to the Local Limit
5 Numerical Tests
5.1 Consistency Tests for the Nonlocal Poisson's Equation
5.2 Convergence Tests for the Nonlocal Poisson's Equation
5.3 Numerical Tests for the LPS Model
6 Conclusion
References
Existence of Global Solutions for 2D Fluid–Elastic Interaction with Small Data
List of Definitions
1 Introduction
2 Local Existence of Solutions
3 Existence of Global Solutions for Small Data
Appendix
Definition of Spaces and Auxiliary Estimates
Estimates on (u·) u
Approximation of Data
References
Doubly Nonlocal Cahn–Hilliard Equations
1 Introduction
2 Nonlocal Vector Calculus
3 Asymptotic Behavior of Solutions to Doubly Nonlocal Cahn–Hilliard Systems
3.1 Decay Estimates for the Linearized System with Time-Dependent Coefficients
4 Steady-State Solutions
4.1 Well-posedness of Solutions
4.2 Regularity of Steady-State Solutions in the Nonlinear Settings
4.3 Higher Integrability of Steady-State Solutions
5 Conclusions and Future Directions
References
3D Image-Based Stochastic Micro-structure Modelling of Foams for Simulating Elasticity
1 Introduction
2 3D Image Analysis for Foams
2.1 Random Closed Sets and Their Characteristics
2.2 Image Analysis
3 Random Laguerre Tessellations and Fitting Them
3.1 Laguerre Tessellations Generated by Random Sphere Packings
3.2 Fitting a Tessellation Model
4 Numerical Simulation of Elastic Properties
4.1 Effective Properties of Micro-Structured Materials
4.2 Lippmann–Schwinger Fast Fourier Transform-Based Solver
5 Application Example
5.1 Material
5.2 Image Analysis and Model Fit
5.3 Prediction of Mechanical Properties
6 Conclusion
References
Machine Learning for Failure Analysis: A Mathematical Modelling Perspective
1 Introduction
2 Survival Analysis
3 Machine Learning
3.1 Discriminative Machine Learning
3.1.1 The Algorithms of Machine Learning
3.1.2 Evaluating a Machine Learning Model
3.1.3 Under-fitting and Over-fitting
3.2 Generative Machine Learning
4 Use Cases
4.1 Regression Models
4.1.1 Random Forest Regression
4.1.2 Survival Analysis
4.1.3 Random Survival Forests
4.1.4 Neural Networks
4.2 Classification Models
4.2.1 Support Vector Machines
4.2.2 Neural Networks
4.3 Anomaly Detection
4.4 Generative Models
4.4.1 Naïve Bayes
4.4.2 Bayesian Networks
5 Conclusions
References
Invertibility of Orlicz–Sobolev Maps
1 Introduction
2 Notation
3 Orlicz–Sobolev Spaces
3.1 Traces
4 Some Definitions and Preliminary Results
4.1 Degree for Orlicz–Sobolev Maps, Topological Image of a Set, and Geometric Image of a Set
5 The Class of Admissible Functions
5.1 Extension Properties
5.2 Regular Functions in A(Ω)
5.3 Some Properties of Orientation-Preserving Functions in A(Ω): Boundedness and Global Invertibility
6 Existence of Minimizers
References
Global Existence of Solutions for the One-Dimensional Response of Viscoelastic Solids Within the Context of Strain-Limiting Theory
1 Introduction
2 Preliminaries
2.1 Local Existence for the Displacement
3 Some Conventions
4 Global Existence
4.1 Energy Decay
5 Revisiting the Smallness Assumptions
References
GENERIC for Dissipative Solids with Bulk–Interface Interaction
1 Introduction
2 The GENERIC Formalism for Closed Systems
2.1 Hamiltonian Systems (Q,E,J)
2.2 Onsager Systems (Q,S,K) (Gradient Systems)
2.3 GENERIC Systems (Q,E,S,J,K)
3 GENERIC Formalism for Bulk–Interface Systems
3.1 Functional Calculus for Bulk–Interface Systems: Notation, Differentials, and *-Multiplication in the Setup of Definition1
3.2 Direct Implications for Geometric Structures
3.3 Weak Form of GENERIC as a Formalism for Bulk–Interface Systems
3.4 Tools for Dissipative Solids with Bulk–Interface Interaction
4 Delamination Processes in Thermo-viscoelastic Materials
4.1 Typical Choices for Interfacial Mechanical Energies for Delamination
4.2 Typical Choices of Dissipation Potentials for Delamination
References
Part II Review Papers
Phase Separation in Heterogeneous Media
1 Introduction
2 Phase Field Model
2.1 Sharp Interface Limit
2.2 Bounds on the Anisotropic Surface Tension σ
2.2.1 A Geometric Framework
2.2.2 Structure of Minimizers of the Cell Formula
2.2.3 The Planar Metric Problem
2.2.4 Bounds on the Anisotropic Surface Tension
2.3 Open Problems
2.3.1 Different Scales
2.3.2 Sharpness of Bounds and Inverse Homogenization
References
Some Recent Results on 2D Crystallization for Sticky Disc Models and Generalizations for Systems of Oriented Particles
1 Introduction
2 Preliminaries on Planar Graphs
3 The Sticky Disc Model: Minimizers and Quasi-minimizers
3.1 Minimizers of the Heitmann–Radin Sticky Disc Model: Single Crystals
3.2 Quasi-minimizers of the Heitmann–Radin Model: Polycrystalline Structures
4 Vectorial Crystallization and Collective Behavior
References
Pattern Formation for Nematic Liquid Crystals—Modelling, Analysis, and Applications
1 Introduction
2 The Landau–de Gennes Theory
3 Benchmark Example
4 Nematic Equilibria on 2D Polygons
5 Effects of Geometrical Anisotropy
6 Effects of Elastic Anisotropy
7 NLC Solution Landscapes on a Hexagon
8 Conclusions and Discussions
9 Supplement: Numerical Methods
References
On Applications of Herglotz-Nevanlinna Functions in Material Sciences, I: Classical Theory and Applications of Sum Rules
1 Introduction
2 Mathematical Background
2.1 Definition and First Examples
2.2 Integral Representation
2.3 Boundary Behavior
2.4 Subclasses
2.5 Other Representations
2.5.1 Operator Representations
2.5.2 Exponential Representation
2.6 Passive Systems
2.7 Asymptotic Behavior
2.8 Matrix- and Operator-Valued Herglotz-Nevanlinna Functions
3 Applications
3.1 Sum Rules and Physical Bounds in Electromagnetics
3.2 Physical Bounds via Convex Optimization
References
On Applications of Herglotz–Nevanlinna Functions in Material Sciences, II: Extended Applications and Generalized Theory
1 Introduction
2 Applications
2.1 Effective Properties of Two-Phase Composite Materials
2.1.1 Effective Properties of Composite Materials and Bounds by Using Theory of the Stieltjes Function
2.1.2 IRF for Permeability Tensors with Positive Matrix-Valued Measures
2.2 Numerical Treatment of Memory Terms in the Modeling of Materials
2.3 Broadband Passive Quasi-Static Cloaking
2.4 Hamiltonian Structure of Time Dispersive and Dissipative Systems
3 More General Classes of Functions
3.1 Quasi-Herglotz Functions
3.2 Generalized Nevanlinna Functions
3.3 Pseudo-Nevanlinna Functions
3.4 Functions in Several Variables
3.4.1 Loewner Functions
3.4.2 Herglotz–Nevanlinna Functions
4 Summary
References
Rigidity and Flexibility in the Modelling of Shape-Memory Alloys
1 Introduction
1.1 Shape-Memory Alloys: The Phenomenological Theory, Differential Inclusions, Rigidity, and Flexibility
1.2 Main Questions
2 Flexibility
3 Rigidity
4 Simulations
References