1st Edition

Mathematical Modelling of Waves in Multi-Scale Structured Media

    258 Pages 40 B/W Illustrations
    by Chapman & Hall

    262 Pages 40 B/W Illustrations
    by Chapman & Hall

    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.



    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


    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




    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