As an equivalent counterpart of topological research on photonics and condensed matter physics, acoustic metamaterials create an opportunity to explore the topological behaviors in phononics and physics of programmable acoustics. This book introduces the topological behavior of acoustics through the novel design of metamaterials. It provides valuable insight into acoustic metamaterials, from multidisciplinary fundamentals to cutting-edge research. Serves as a single resource on acoustic metamaterials. Covers the fundamentals of classical mechanics, quantum mechanics, and state-of-the-art condensed matter physics principles so that topological acoustics can be easily understood by engineers. Introduces topological behaviors with acoustics and elastic wave through quantum anomalous Hall effects, quantum spin Hall effects and quantum valley Hall effects and their applications. Explains the pros and cons of different design methods and gives guidelines for selecting specific designs of acoustic metamaterials with specific topological behaviors. Includes MATLAB code for numerical analysis of band structures. This book is designed for graduate students, researchers, scientists, and professionals across materials, mechanical, civil, and aerospace engineering, and those who want to enhance their understanding and commence research in metamaterials.
Author(s): Sourav Banerjee
Publisher: CRC Press
Year: 2023
Language: English
Pages: 430
Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
About the Author
Chapter 1: Acoustic Metamaterials
1.1. Introduction
1.2. Metamaterials
1.3. Periodic Media and Phononic Crystals
1.4. Connecting Metamaterial and Phononic Crystals
1.5. Topological Phenomena
1.6. Future Direction
1.7. Summary
References
Chapter 2: Classical Mechanics and the Physics of Continua
2.1. Introduction: History of Classical Mechanics
2.2. Fundamental Concept of Classical Mechanics
2.3. Governing Equation from Classical Mechanics
2.4. Fundamentals of Continuum Mechanics
2.4.1. Lagrangian Coordinate or Material Coordinate System
2.4.2. Eulerian Coordinate or Spatial Coordinate System
2.5. Motion of a Deformable Body
2.5.1. Material Derivatives
2.5.2. Path Lines and Streamlines
2.6. Deformation and Strain in a Deformable Body
2.6.1. Cauchy’s and Green’s Deformation Tensors
2.6.2. Description of Strain in a Deformable Body
2.6.3. Strain in Terms of Displacement
2.7. Mass, Momentum, and Energy
2.7.1. Mass of a Body
2.7.2. Momentum of a Deformable Body
2.7.3. Angular Momentum of a Deformable Body
2.7.4. Kinetic Energy Stored in a Deformable Body
2.8. Fundamental Axiom of Continuum Mechanics
2.8.1. Axiom 1: Principle of Conservation of Mass
2.8.2. Axiom 2: Principle of Balance of Momentum
2.8.3. Axiom 3: Principle of Balance of Angular Momentum
2.8.4. Axiom 4: Principle of Conservation of Energy
2.9. Internal Stress State in a Deformable Body
2.10. External and Internal Load on a Deformable Body
2.11. Elastodynamic Equation or the Fundamental Wave Equation
2.12. Energy Concept of Continua and its Relation to the Classical Mechanics
2.12.1. Conservation of Local Energy from Global Energy Rule
2.12.2. Conservation of Mechanical Energy (Kinetic, Internal, and Potential Energy)
2.12.3. Internal Energy and Strain Energy
2.12.4. Energy Flux Density and Poynting Vector
2.13. Lagrangian Formulation of Wave Equation
2.14. Legendre Transform and Hamiltonian Formulation of Wave Equation
2.14.1. Graphical Representation of Legendre Transform
2.14.2. Hamiltonian Form with Legendre Transform
2.14.3. Hamilton’s Equation of Motion
2.15. Constitutive Law of Continua
2.15.1. Materials with One Plane of Symmetry: Monoclinic Materials
2.15.2. Materials with Two Planes of Symmetry: Orthotropic Materials
2.15.3. Materials with Three Planes of Symmetry and One Plane of Isotropy: Transversely Isotropic Materials
2.15.4. Materials with Three Planes and Three Axes of Symmetry: Isotropic Materials
2.16. Appendix
2.16.1. Important Equations in a Cartesian Coordinate System
2.16.2. Important Equations in a Cylindrical Coordinate System
2.16.2.1. Transformation to a Cylindrical Coordinate System
2.16.2.2. Gradient Operator in a Cylindrical Coordinate System
2.16.2.3. Strain-Displacement Relation in a Cylindrical Coordinate System
2.16.2.4. Governing Differential Equations of Motion in a Cylindrical Coordinate System
2.16.3. Important Equations in a Spherical Coordinate System
2.16.3.1. Gradient Operator in a Spherical Coordinate System
2.16.3.2. Strain-Displacement Relation in a Spherical Coordinate System
2.16.3.3. Governing Differential Equations of Motion in a Spherical Coordinate System
2.17. Summary
References
Chapter 3: Acoustics and Elastic Wave Propagation in Fluids and Anisotropic Solids
3.1. Wave Propagation in Fluid Media
3.1.1. Governing Differential Equation with Pressure in Fluid
3.1.2. Pressure Velocity Relation
3.1.3. Pressure and Velocity Potential in Fluid media
3.1.4. Wave Potential in Fluid
3.2. Wave Propagation in Solid Media
3.2.1. Navier’s Equation of Motion
3.2.2. Homogeneous Isotropic and Anisotropic Materials
3.2.2.1. Solving Navier’s Equation of Motion in Homogeneous Isotropic Materials
3.2.2.2. Solving Navier’s Equation of Motion in Homogeneous Anisotropic Materials
3.2.2.3. Understanding Wave Modes with Normal and Anomalous Polarity
3.2.2.4. Exploring Abnormal Polarities
3.2.3. Nonhomogeneous Isotropic and Anisotropic Materials
3.2.3.1. Wave Equations for Nonhomogeneous Isotropic Media
3.2.3.2. Wave Equations for Nonhomogeneous Anisotropic Media
3.2.3.3. Solution of Wave Equations for Nonhomogeneous Media
3.3. Appendix: Understanding Nabla Hamiltonian Operations
3.3.1. Nabla Hamiltonian
3.3.2. Strain Matrix Using Nabla Hamiltonian Form
3.3.3. Laplacian Using Nabla Hamiltonian
3.3.4. Nabla Operation on Stress Matrix and Spatially Varying Material Constants
3.4. Summary
References
Chapter 4: Electromagnetic Wave Propagation
4.1. Field and Field Theories
4.2. Electric and Magnetic Fields
4.2.1. Conservative and Nonconservative Fields
4.2.2. Coulomb’s Law and Gauss’s Law
4.2.3. Ampere’s Law and Ampere-Maxwell Equation
4.2.4. Faraday’s Induction Law
4.3. Maxwell’s Electromagnetic Wave Equation
4.3.1. Solution of Electromagnetic Wave Equations
4.4. Comparison of Electromagnetic and Elastic Acoustic Wave Equations
4.5. Appendix
4.5.1. Divergence Theorem
4.5.2. Stokes’ Theorem
4.6. Summary
References
Chapter 5: Quantum Mechanics for Engineers
5.1. Particle Waves and the Schrodinger Equation
5.1.1. Quantized Energy
5.1.2. Relativistic Particles
5.1.3. Wave Particle Duality
5.1.3.1. Momentum of a Relativistic Particle
5.1.3.2. Relativistic Energy in Terms of Momentum
5.1.4. Wave Function
5.1.5. The Schrodinger Equation
5.1.6. Time-Independent Schrodinger Equation
5.1.7. Time-Dependent Schrodinger Equation
5.1.8. Schrodinger Equation for a Particle in an Electromagnetic Field
5.2. Quantum Operators
5.2.1. Hamiltonian Operator
5.2.2. Ladder Operators and Properties
5.2.3. Angular Momentum Operators
5.2.3.1. Orbital Angular Momentum (OAM)
5.2.3.2. Spin Angular Momentum (SAM)
5.2.3.3. Total Angular Momentum (TAM)
5.2.3.4. Square Orbital Angular Momentum
5.2.3.5. Square Spin Angular Momentum
5.2.4. Observable Operators
5.2.4.1. Heisenberg Uncertainty Principle
5.2.4.2. Uncertainty Principle for Angular Momentum
5.2.4.3. Convention to Express the Angular Momentum
5.2.5. Pauli’s Matrix
5.3. Solution of Schrodinger Equation in Periodic Potential
5.3.1. Bloch Solution
5.3.2. Reduced-Order Solution and Brillouin Zone
5.3.3. Finding Energy Bands Using the Perturbation Method
5.3.3.1. Modified Hamiltonian in Periodic Media
5.3.3.2. k.p Perturbation Method
5.4. Relativistic Particles with Spin Zero and Spin Half
5.4.1. Klein-Gordon Equation for Spin 0 Relativistic Particles
5.4.2. Energy Square Root Parity
5.4.3. Dirac Equation for Spin Half Particles
5.4.4. Hamiltonian for Spin ½ Fermions and Spinors
5.5. Dirac Cones and Dirac-Like Cones
5.5.1. Dirac Cones
5.5.2. Dirac-Like Cones
5.6. Introduction to Topology and the Geometric Phase
5.6.1. What is Topology?
5.6.2. What Is the Geometric Phase?
5.6.3. How Are They Connected?
5.6.4. The Berry Phase
5.6.4.1. The Berry Phase in Bloch Media
5.6.4.2. The Chern Number at Degenerated Band Structure
5.6.5. The Zak Phase
5.7. Understanding Symmetry and Invariance
5.7.1. Geometric Symmetries
5.7.2. Time-Reversal Symmetries or T-symmetry
5.8. Connecting Symmetry Breaking and Geometric Phase
5.9. Quantum Hall Effects
5.10. Summary
References
Chapter 6: Waves in Periodic Media: Quantum Analogous Application of Acoustics and Elastic Waves
6.1. Periodic Media for Acoustic and Elastic Waves
6.1.1. Introduction
6.1.2. Periodicity and Symmetry
6.1.3. Brillouin Zones in Periodic Media
6.2. Acoustic Waves in Periodic Media
6.2.1. Governing Differential Equations
6.2.2. Bloch Solution for Acoustic Waves
6.2.2.1. One-Dimension Waves in a Continuous Periodic Chain
6.2.2.2. Plane Waves with Out-of-Plane Polarity
6.2.2.3. Plane Waves with In-Plane Polarity
6.2.2.4. Computer Code in MATLAB to Find Wave Dispersion
6.2.2.5. Dispersion Curves
6.3. Bloch Wave Vectors for Other Lattice Structures
6.3.1. Generalized Bloch Wave Solution
6.3.1.1. Fast-Plane Wave Expansion Method
6.3.1.2. Finite Element Simulation Method
6.4. Features of Wave Dispersion (ɷ − k)
6.4.1. Phonons
6.4.2. Equifrequency Contours
6.4.3. Band Degeneracies
6.4.4. Deaf Bands
6.4.5. Dirac Cones at K Point
6.4.6. Dirac-Like Cones
6.4.7. Weyl Point
6.4.8. Spawning Rings at Exceptional Points
6.4.9. Double Dirac Cones and Spinors
6.4.10. Topological Charge and Invariant
6.5. Examples: Counterintuitive Non-Topological Wave Phenomena
6.5.1. Acoustic Transparency, Beam Splitting, Negative Refraction, and Super Lensing
6.5.1.1. Butterfly Crystal Dispersion
6.5.1.2. Wave Bifurcation
6.5.1.3. Negative Refraction and Wave Focusing
6.5.1.4. Superlensing: Beyond the Diffraction Limit
6.5.2. Orthogonal Wave Transport at Dirac-like Cone
6.5.3. Acoustic Computing at Dirac-like Cone
6.6. Active Breaking of Time-Reversal Symmetry
6.6.1. Topological Band Gaps
6.6.1.1. Spatio-Temporal Modulation of Material Coefficients
6.6.1.2. Governing Differential Equations
6.6.1.3. Dispersion Bands with Directional Bandgaps
6.6.1.4. How Directional Band Gaps Are Topological
6.6.1.5. Manifold with Parallel Transport of the Wave Function
6.7. Quantum Analogous Elastic Waves
6.7.1. Hamiltonian and Ladder Operation for Elastic Waves
6.7.1.1. Elastic Hamiltonian
6.7.1.2. Super Symmetry (SUSY) Ladder Operations
6.7.2. Klein-Gordon Equation and Dirac Equation
6.7.2.1. Elastic Klein-Gordon Equation
6.7.2.2. Elastic Dirac Equation
6.7.2.3. Pseudospin State of Elastic Wave Modes
6.7.2.4. Spring-mass System for Topological Elastic Waves
6.7.3. Intrinsic Spin States of Elastic Wave
6.7.3.1. Elastic Spin Angular Momentum
6.7.3.2. Elastic Spin Operators
6.7.3.3. Topological Behavior with Wave Vortex
6.7.3.4. Power Flow with Spin Angular Momentum
6.7.3.5. Elastic and Acoustic Spin Mediated Skyrmion
6.8. Appendix
6.8.1. Circular Phononic Crystal in a Host Matrix
6.8.2. Square Phononic Crystal in a Host Matrix
6.8.3. Parallel Transport of a Vector
6.8.3.1. Christoffel Symbol and Geodesic
6.8.3.2. Parallel Transport along a Curved Path
6.8.3.3. Example: Parallel Transport
6.8.3.4. Notes on Parallel Transport for Wave Vectors
6.8.4. Computer Code to Explore Elastic Spin
6.9. Summary
References
Chapter 7: Topological Acoustics in Metamaterials
7.1. Topology
7.1.1. Topology in Crystals
7.1.2. Topology in Phononics
7.1.2.1. Breaking T-symmetry: QHE
7.1.2.2. Without Breaking T-symmetry: QSHE
7.1.2.3. Without Breaking T-symmetry: QVHE
7.2. Topological Phononics and Quantum Trio: A Gateway of Quantum Transportation
7.2.1. Berry Phase and Band Topology in Phononics
7.2.2. Phononic Topological States
7.2.2.1. Topological Insulators in Acoustics
7.2.2.2. Acoustic Spinning and Topological Edge State in Acoustics
7.3. Emergence of Topological Black Holes
7.3.1. Modal Analysis at Dirac Frequency
7.3.2. Tuning of Geometric Parameters for TBHs
7.3.3. Real-time Tunable Metamaterial for TBHs
7.4. Polarization Anomaly: Demonstration of Acoustic Spin
7.4.1. Formation of Black Hole-Like Sink
7.4.2. Counterinteractive Spin and OAM in TBH
7.4.2.1. Mathematical Treatment to Find Polarity Anomaly
7.4.2.2. A Possible Link to the Acoustic Skyrmions
7.5. Summary
References
Index
