Mathematical Modelling of Waves in Multi-Scale Structured Media

Content: Preface Introduction Bloch-Floquet wavesStructured 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 dispersionElementary considerations for linear water waves Dispersion equation Asymptotics: deep and shallow water waves Bloch-Floquet wavesStanding wavesStop bandsAsymptotic 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: localisationWave 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 decayComparison with the exact approachWaves in structured media with thin ligaments and disintegrating junctionsStructures 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 junctionsBending problem Boundary layer at the junction Weight function and the junction condition Shear problem Representation of the junction condition in terms of the weight functionEffective 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 MembranesStop-band dynamic Green's functions and exponential localizationLocalised Green's function for the square latticeDispersion and dynamic anisotropyAsymptotics along the principal axes of the lattice Asymptotic approximation along the diagonal m = nLocalisation 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 propertiesForced problem in elastic structured media Localisation near cracks/inclusions in a lattice Finite inclusion in an infinite square latticeLocalised 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 latticeEquations of motion From an infinite inclusion to a large finite defect: The case of large NWaveguide modes versus waveforms around finite defectsCloaking 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 RIllustrative simulations Cloaking path information Cloaking with a lattice Geometry and governing equations for an inclusion cloaked by a globally orthogonal latticeIllustrative lattice simulations Basic lattice cloakRefined lattice cloak Boundary conditions on the interior contour of a cloak Cloaking in elastic plates Governing equations in the presence of in-plane forcesInterface 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 operatorQuality 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 SolidsStructured 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