1st Edition
Mathematical Modelling of Waves in Multi-Scale Structured Media
Mathematical Modelling of Waves in Multi-Scale Structured Media presents novel analytical and numerical models of waves in structured elastic media, with emphasis on the asymptotic analysis of phenomena such as dynamic anisotropy, localisation, filtering and polarisation as well as on the modelling of photonic, phononic, and platonic crystals.
Preface
Introduction
Bloch-Floquet waves
Structured interfaces and localisation
Multi-physics problems and phononic crystal structures
Designer multi-scale materials
Dynamic anisotropy and defects in lattice systems
Models and physical applications in materials science
Structure of the book
Foundations, methods of analysis of waves and analytical approaches to modelling of multi-scale solids
Wave dispersion
Elementary considerations for linear water waves
Dispersion equation
Asymptotics: deep and shallow water waves
Bloch-Floquet waves
Standing waves
Stop bands
Asymptotic lattice approximations
Transmission and reflection
Transmission matrix
Reflected and transmitted energy
Defect modes and enhanced transmission
Wave localisation and dynamic defect modes
Localisation of waves in a flexural beam on an elastic foundation
Flexural plate on an elastic foundation: localisation
Wave localisation in a non-local material
Waves in a chain of particles on an elastic foundation
Higher frequency band gap
Lower frequency band gap
Dynamic localisation in a bi-atomic discrete chain
Point forces applied to the central cell
Localised vibration modes within the finite band gap
Perturbation of mass
Asymptotic homogenisation
Returning to the bi-atomic chain
Leading order problem
Next-to-leading order problem
Second order problem
Propagation and decay
Comparison with the exact approach
Waves in structured media with thin ligaments and disintegrating junctions
Structures with undamaged multi-scale resonators
Geometry and governing equations
Thermal pre-stress and Euler’s buckling
Asymptotic approximations for two standing wave modes
Fundamental translational mode
Fundamental rotational mode
Dispersion diagrams and stop bands
Singular perturbation analysis of fields in solids with disintegrating junctions
Bending problem
Boundary layer at the junction
Weight function and the junction condition
Shear problem
Representation of the junction condition in terms of the weight function
Effective stiffness of the junction
Comparison with other models
Structures containing damaged multi-scale resonators
Out-of-plane vibration of a periodic structure with multi-scale resonators
Asymptotic approximations for the lowest eigenfrequency
Dispersion
Filtering versus dispersion properties of out-of-plane shear Bloch-Floquet waves
Undamaged interface
Damaged interface
Plain strain vector problem
Asymptotic approximations for the fundamental translational and rotational modes
Dispersion diagrams
Applications of multi-scale resonators in filtering and localisation of vibrations
Dynamic response of elastic lattices and discretised elastic Membranes
Stop-band dynamic Green’s functions and exponential localization
Localised Green’s function for the square lattice
Dispersion and dynamic anisotropy
Asymptotics along the principal axes of the lattice
Asymptotic approximation along the diagonal m = n
Localisation exponents
Dynamic anisotropy and localisation near defects
Primitive waveforms in scalar lattices
Square monatomic lattice
Stationary point of a different kind
Triangular cell lattice
Diffraction in elastic lattices
Dispersive properties
Forced problem in elastic structured media
Localisation near cracks/inclusions in a lattice
Finite inclusion in an infinite square lattice
Localised modes
Asymptotic expansions in the far field
Band edge expansions
Illustrative examples
Single defect
Pair of defects
Triplet of defects
Infinite inclusion in an infinite square lattice
Equations of motion
From an infinite inclusion to a large finite defect: The case of large N
Waveguide modes versus waveforms around finite defects
Cloaking and channelling of elastic waves in structured solids
A cloak is not a shield
Cloaking as a channelling method for incident waves
Regularised transformation
Interface conditions
Cloaking problem
Ray equations
Negative refraction
Scattering measure
Choice of R
Illustrative simulations
Cloaking path information
Cloaking with a lattice
Geometry and governing equations for an inclusion cloaked by a globally orthogonal lattice
Illustrative lattice simulations
Basic lattice cloak
Refined lattice cloak
Boundary conditions on the interior contour of a cloak
Cloaking in elastic plates
Governing equations in the presence of in-plane forces
Interface conditions
Square cloak
Material parameters and pre-stress for the cloak
Principal directions of orthotropy
Implementation of the cloak for the flexural plate
Green’s functions and comparison with cloaking for the Helmholtz operator
Quality of cloaking
Measuring cloaking quality for interference patterns
Singular perturbation analysis of an approximate cloak
Push-out transformation
Physical interpretation of transformation cloaking for a membrane
Singular perturbation problem in a membrane
Model problem: scattering of a plane wave by a circular obstacle in a membrane
Boundary conditions and the cloaking problem in a membrane
Singular perturbation and cloaking action for the biharmonic problem
A model problem of scattering of a flexural wave by a circular scatterer
Boundary conditions and the cloaking problem in a Kirchhoff-Love plate
Structured interfaces and chiral systems in dynamics of elastic Solids
Structured interface as a polarising filter
Stratified domain
Lower-dimensional approximations within the interface
Incident, reflected and transmitted waves
The energy of transmitted and reflected waves
Trapped waveforms
Enhanced transmission
Vortex-type resonators and chiral polarisers of elastic waves
Governing equations
Evaluation of the spinner constants
Elastic Bloch-Floquet waves in the active chiral lattice
Dispersion properties of the monatomic lattice
Lattice of the vortex-type
Low frequency range
Bi-atomic lattice of the vortex-type
Discrete structured interface: shielding, negative refraction, and focusing
Equations of motion
Constructing the structured interface
Bibliography
Index
Biography
Alexander Movchan is a Professor at the University of Liverpool, Natasha Movchan is a Professor at the University of Liverpool, Ian Jones is a Professor at Liverpool John Moores University and an Honorary Fellow at the University of Liverpool, and Daniel Colquitt is a Lecturer at the University of Liverpool. The authors have worked on wave propagation in multi-scale elastic media over many years and have developed novel modelling approaches, which have opened efficient ways to design and study the dynamic response of multi-scale structures known as elastic metamaterials introduced within the last decade.
"This book is aimed at specialists in applied mathematics, physics and engineering. The material is based upon the authors’ research into waves in structured media, dealing with the dynamic response of elastic structures, cracks and interfaces. The mathematical techniques mostly used are Green’s function, asymptotic approximations and numerical simulations. Chapter 1 contains a brief introduction to some ideas and notions and a description of the material in the book. In Chapter 2, dispersion is discussed using linear water waves; also, Bloch-Floquet waves, standing waves and asymptotic lattice approximations are introduced. The elastic problems involving flexural waves on an elastic foundation and waves in chains of particles are discussed. Chapter 3 deals with waves in structured media and ligaments. The asymptotic problems arising from thin interfaces and disintegrating are also dealt with. In Chapter 4, dispersion in periodic structures, dynamic localization and defects in lattices are discussed. Chapter 5 deals with cloaking of waves in which the scattered wave is suppressed by an encompassing structure. In Chapter 6, the models of structured interfaces and chiral media are introduced. Although prerequisite notions are briefly discussed in Chapter 2, some knowledge of asymptotic and singular perturbations and waves in continuous media would be desirable."
-Fiazud Din Zaman (Lahore) - Zentralblatt MATH 1397 — 1