Lurie An Introduction to the Mathematical Theory of Dynamic Materials

2. Auflage 2017
ISBN: 978-3-319-65346-4
Verlag: Springer Nature Switzerland
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 15, 287 Seiten

Reihe: Advances in Mechanics and Mathematics

This fascinating book is a treatise on real space-age materials. It is a mathematical treatment of a novel concept in material science that characterizes the properties of dynamic materials-that is, material substances whose properties are variable in space and time. Unlike conventional composites that are often found in nature, dynamic materials are mostly the products of modern technology developed to maintain the most effective control over dynamic processes.

Autoren/Hrsg.

Lurie, Konstantin A.

Inhaltsverzeichnis

1;Preface;6
2;Contents;8
3;List of Figures;12
4;1 A General Concept of Dynamic Materials;17
4.1;1.1 The Idea and Definition of Dynamic Materials;17
4.2;1.2 Two Types of Dynamic Materials;19
4.3;1.3 Implementation of Dynamic Materials in Mechanics and Electromagnetics;22
4.3.1;1.3.1 Realization of DM as Large Array of Coupled Micro/Nanoelectromechanical Structures;23
4.3.2;1.3.2 Electromagnetic Realization of DM Structures;33
4.3.3;1.3.3 Ferroelectric and Ferromagnetic Materials;33
4.3.4;1.3.4 Nonlinear Optics;37
4.4;1.4 Some Applications of Dynamic Materials;38
4.5;1.5 Dynamic Materials and Vibrational Mechanics;39
4.6;References;41
5;2 An Activated Elastic Bar:Effective Properties;48
5.1;2.1 Longitudinal Vibrations of Activated Elastic Bar;48
5.2;2.2 The Effective Parameters of Regular Activated Laminate;54
5.3;2.3 The Effective Parameters:Homogenization;59
5.4;2.4 The Effective Parameters: Floquet Theory;62
5.5;2.5 The Effective Parameters:Discussion;65
5.6;2.6 Balance of Energy in Longitudinal Wave Propagation Through an Activated Elastic Bar;71
5.7;2.7 Averaged and Effective Energy and Momentum;76
5.8;2.8 Homogenization of Regular Activated Laminates:Theoretical Motivation;81
5.9;References;84
6;3 Dynamic Materials in Electrodynamics of Moving Dielectrics;85
6.1;3.1 Preliminary Remarks;85
6.2;3.2 The Basics of Electrodynamics of Moving Dielectrics;85
6.3;3.3 Relativistic Form of Maxwell's System;87
6.4;3.4 Material Tensor s:Discussion—Two Types of Dynamic Materials;92
6.5;3.5 An Activated Dielectric Laminate: One-Dimensional Wave Propagation;94
6.6;3.6 A Spatio-Temporal Polycrystallic Laminate:One-Dimensional Wave Propagation;96
6.7;3.7 A Spatio-Temporal Polycrystallic Laminate: The Bounds;97
6.8;3.8 An Activated Dielectric Laminate: Negative Effective Material Properties;104
6.9;3.9 An Activated Dielectric Laminate:The Energy Considerations—Waves of Negative Energy;109
6.10;3.10 Numerical Examples and Discussion;115
6.11;3.11 Effective Properties of Activated Laminates Calculated via Lorentz Transform: Case of Spacelike Interface;121
6.12;References;123
7;4 G-Closures of a Set of Isotropic Dielectrics with Respect to One-Dimensional Wave Propagation;124
7.1;4.1 Preliminary Considerations: Terminology;124
7.2;4.2 Conservation of the Wave Impedance Through One-Dimensional Wave Propagation: A Stable G-Closure of a Single Isotropic Dielectric;126
7.3;4.3 A Stable G-Closure of a Set U of Two Isotropic Dielectrics with Respect to One-Dimensional Wave Propagation;129
7.4;4.4 The Second Invariant E/M as an Affine Function: A Stable G-Closure of an Arbitrary Set U of Isotropic Dielectrics;130
7.5;4.5 A Stable Gm-Closure of a Set U of Two IsotropicDielectrics;135
7.6;4.6 Comparison with an Elliptic Case;135
7.7;References;140
8;5 Rectangular Material Structures in Space-Time;141
8.1;5.1 Introductory Remarks;141
8.2;5.2 Statement of a Problem;142
8.3;5.3 Case of Separation of Variables;145
8.4;5.4 Checkerboard Assemblage of Materials with Equal Wave Impedance;148
8.5;5.5 Energy Transformation in the Presence of Limit Cycles;158
8.6;5.6 Numerical Analysis of Energy Accumulation;166
8.7;5.7 Energy Transformation in the Presence of Losses;170
8.8;5.8 Mathematical Analysis of the Energy Concentration in a Checkerboard: The Bounds Defining the ``Plateau Effect'';174
8.8.1;5.8.1 Analytic Characterization of the Limit Cycles and Plateau Zones;174
8.8.2;5.8.2 Conditions on Material Parameters Necessary and Sufficient for Energy Accumulation;182
8.8.3;5.8.3 Numerical Verification;186
8.8.4;5.8.4 Summary of Analytic Results;187
8.9;5.9 Propagation of Dilatation and Shear Waves Through a Dynamic Checkerboard Material Geometry in 1D Space + Time;190
8.9.1;5.9.1 Wave Propagation Through a Dynamic Elastic Checkerboard Assembly;191
8.9.2;5.9.2 Results;193
8.10;5.10 Coaxial Transmission Line as a Checkerboard;201
8.11;References;205
9;6 On Material Optimization in Continuum Dynamics;207
9.1;6.1 General Considerations;207
9.2;6.2 An Optimal Transportation of Masses;209
9.2.1;6.2.1 Statement of the Problem;209
9.2.2;6.2.2 Admissible Controls and the Propertiesof Solutions;210
9.2.3;6.2.3 Adjoint System;219
9.2.4;6.2.4 Application to Problem (6.5);223
9.2.5;6.2.5 Conclusions;228
9.3;6.3 Dynamic Material Optimization for Wave Equation;228
9.3.1;6.3.1 Preliminary Considerations;228
9.3.2;6.3.2 Statement and Solution of a Typical EllipticProblem;229
9.3.3;6.3.3 Some Properties of Polysaddlification;241
9.3.4;6.3.4 Additional Remarks;245
9.3.5;6.3.5 Application of Direct Approach to Material Optimization for the Wave Equation;245
9.4;6.4 A Plane Electromagnetic Wave Propagation Through an Activated Laminate in 3D;250
9.5;6.5 The Homogenized Equations: Elimination of the Cutoff Frequency in a Plane Waveguide;251
9.6;6.6 The Effective Material Tensor and Homogenized Electromagnetic Field;252
9.7;6.7 The Transport of Effective Energy;254
9.8;6.8 On the Necessary Conditions of Optimality in a Typical Hyperbolic Control Problem with Controls in theCoefficients;255
9.8.1;6.8.1 Introduction;255
9.8.2;6.8.2 Statement of the Problem;256
9.8.3;6.8.3 The Necessary Conditions of Optimality;258
9.9;6.9 Transformation of the Expression for ?I : The Strip Test;262
9.10;6.10 A Polycrystal in Space-Time;264
9.11;References;271
10;Appendix A;273
11;Appendix B;276
12;Appendix C;278
13;Appendix D;282
14;Index;285