An Introduction to the Mathematical Theory of Dynamic Materials (eBook)

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