Mathematical Modelling of Waves in Multi-Scale Structured Media

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