An Introduction To The Mathematical Theory Of Dynamic Materials (advances In Mechanics And Mathematics)

Mathematical treatment to properties of dynamic materials, material substances whose properties are variable in space and time are examined in this book. This new edition emphasizes the differences between material optimization techniques in statics and dynamics. Systems with one spatial coordinate and time are used to illustrate essentials of temporal property change in this setting and prompt forthcoming extensions and technical improvements. Since the release of the first edition, a number of new results have created a more complete picture of unusual effects hidden in spatio-temporal material geometry. This renewed look has revealed a conceptually new mechanism of relaxation of material optimization problems in dynamics, which has led to additional resources for optimization previously concealed in the property layouts. Dynamic materials are studied in this book from the following perspectives: ability to appear in dissimilar implementations, universality as formations that are thermodynamically open, and unusual effects supported by dynamic materials in mechanical and electromagnetic implementations. Special effects accompanying the wave propagation through material geometries in space-time are analyzed by dynamic (spatio-temporal) laminates for screening the extended domains. An extended classification is provided for activated and kinetic dynamic materials, based on the nonstandard exposition of Maxwell-Minkowski electrodynamics of moving bodies. Unique applications as well as fundamental optimization problems are listed within the discussion. This book is intended for applied mathematicians interested in optimal problems of material design for systems governed by hyperbolic differential equations. It will also be useful for researchers in the field of smart metamaterials and their applications to optimal material design in dynamics.-- Provided by publisher 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 ΔI : 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 This book has emerged from the study of a new concept in material science that has been realized about a decade ago. Before that, I had been working for more than 20 years on conventional composites assembled in space and therefore adjusted to optimal material design in statics. The reason for that adjustmentisthatsuchcompositesappearedtobecomenecessaryparticipants in almost any optimal material design related to a state of equilibrium. A theoretical study of conventional composites has been very extensive over a long period of time. It received stimulation through many engineering applications, and some of the results have become a part of modern industrial technology. But again, the ordinary composites are all about statics, or, at the utmost, are related to control over the free vibration modes, a situation conceptually close to a static equilibrium. The world of dynamics appears to be quite di?erent in this aspect. When it comes to motion, the immovable material formations distributed in space alone become insu?cient as the elements of design because they are incapable of getting fully adjusted to the temporal variation in the environment. To be able to adequately handle dynamics, especially the wave motion, the material medium must itself be time dependent, i.e. its material properties should vary in space and time alike. Any substance demonstrating such variation has been termed a dynamic material [1]. Front Matter ....Pages i-xvii A General Concept of Dynamic Materials (Konstantin A. Lurie)....Pages 1-31 An Activated Elastic Bar: Effective Properties (Konstantin A. Lurie)....Pages 33-69 Dynamic Materials in Electrodynamics of Moving Dielectrics (Konstantin A. Lurie)....Pages 71-109 G-Closures of a Set of Isotropic Dielectrics with Respect to One-Dimensional Wave Propagation (Konstantin A. Lurie)....Pages 111-127 Rectangular Material Structures in Space-Time (Konstantin A. Lurie)....Pages 129-194 On Material Optimization in Continuum Dynamics (Konstantin A. Lurie)....Pages 195-260 Back Matter ....Pages 261-277 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.