The homogenisation of Maxwell's equations with applications to photonic crystals and localised waveforms on metafilms

An asymptotic theory is developed to generate equations that model the global behaviour of electromagnetic waves in periodic photonic structures when the wavelength is not necessarily long relative to the periodic cell dimensions; potentially highly-oscillatory short-scale detail is encapsulated through integrated quantities. The theory we develop is then applied to two topical examples, the first being the case of aligned dielectric cylinders, which has great importance in the modelling of photonic crystal fibres. We then consider the propagation of waves in a structured metafilm, here chosen to be a planar array of dielectric spheres. At certain frequencies strongly directional dynamic anisotropy is observed, and the asymptotic theory is shown to capture the effect, giving highly accurate qualitative and quantitative results as well as providing interpretation for the underlying change from elliptic to hyperbolic behaviour

Mathematical modelling of the dynamic response of metamaterial structures

This thesis constitutes an exposition of the work carried out by the author whilst examining several physical problems under the broad theme of the dynamic response of metamaterial structures. An outline of the thesis is provided in chapter one. Chapter two introduces some notation and preliminary results on general lattice equations. Chapter three examines the dispersive behaviour of non-classical discrete elastic lattice systems. In particular, the effect of distributing the inertial properties of the lattice over the elastic rods, in addition to at the junctions, is considered. It is demonstrated that the effective material properties in the long wavelength limit are not what one would expect from the static response of the lattice. The effect of various interactions on the dispersive properties of the triangular cell lattice is considered, including so-called truss, frame, and micro-polar interactions. Compact analytical estimates for the band widths are presented, allowing the design of metamaterial structures possessing pass and/or stop bands at specific frequencies and in specified directions. The finite frequency response of several lattice structures is considered in chapter four. In particular, the dynamic anisotropy of both scalar and elastic lattices is examined. The resulting strongly anisotropic material response is linked, explicitly, to the dispersive properties of the lattice. A novel application of dynamic anisotropy to the focusing, shielding, and negative refraction of elastic waves using a flat discrete "metamaterial lens'' is presented. Chapter five is devoted to the analysis, using the dynamic Green's function, of a finite rectilinear inclusion in an infinite square lattice. Several representations of the Green's function are presented, including expression in terms of hypergeometric functions, which are employed in deriving band edge expansions. It is shown that localised defect modes, characterised by displacements which decay rapidly away from the defect, can be initiated by reducing the mass of one or more lattice nodes, whilst ensuring that the mass of the nodes remains positive. For one- and three-dimensional multi-atomic lattices, there exists a bound on the contrast in mass between the defect and ambient lattice such that localised defect modes exist. However, it is shown that for the two-dimensional lattice, no such bound exists, provided that the masses remain positive. The analysis of a finite-sized defect region is accompanied by the waveguide modes that may exist in a lattice containing an infinite chain of point defects. A numerical simulation illustrates that the solution of the problem for an infinite chain can be used to predict the range of eigenfrequencies of localised modes for a finite but, sufficiently long, array of masses representing a rectilinear defect in a square lattice. Continuing with the theme of defects, chapter six examines response of a triangular thermoelastic lattice, with an edge crack under mode I loading. The response of the triangular lattice is compared with that of the corresponding continuum. The model is related to the phenomenon of thermal striping, which occurs when a structure is exposed to periodic variations in temperature. In the thermal striping regime, crack propagation is a fatiguing processes with the rate of crack growth being proportional to some power of the peak-to-peak amplitude of the stress intensity factor. An "effective stress intensity factor'' for the lattice is introduced and it is demonstrated that, in the homogenised limit, the "effective stress intensity factor'' is lower than the stress intensity factor of the continuum for sufficiently long cracks and low frequencies. Finally, chapter seven presents a detailed analysis of a non-singular square cloak for acoustic, out-of-plane shear elastic, and electromagnetic waves. The propagation of waves through the cloak is examined analytically and is complemented with a range of numerical illustrations. The efficacy of the regularised cloak is demonstrated and an objective numerical measure of the quality of the cloaking effect is introduced. The results presented show that the cloaking effect persists over a sufficiently wide range of frequencies. To illustrate further the effectiveness of the regularised cloak, a Young's double slit experiment is presented. The stability of the interference pattern is examined when a cloaked and uncloaked obstacle are successively placed in front of one of the apertures. A significant advantage of this particular regularised square cloak is the straightforward connection with a discrete lattice. It is shown that an approximate cloak can be constructed using a discrete lattice structure. The efficiency of such a lattice cloak is analysed and several illustrative simulations are presented. It is demonstrated that effective cloaking can be achieved by using a relatively simple lattice, particularly in the low frequency regime. This discrete lattice structure provides a possible avenue toward the physical realisation of invisibility cloaks

Tilted resonators in a triangular elastic lattice: chirality, Bloch waves and negative refraction

We consider a vibrating triangular mass-truss lattice whose unit cell contains a resonator of a triangular shape. The resonators are connected to the triangular lattice by trusses. Each resonator is tilted, i.e. it is rotated with respect to the triangular lattice's unit cell through an angle $\vartheta_0$. This geometrical parameter is responsible for the emergence of a resonant mode in the Bloch spectrum for elastic waves and strongly affects the dispersive properties of the lattice. Additionally, the tilting angle $\vartheta_0$ triggers the opening of a band gap at a Dirac-like point. We provide a physical interpretation of these phenomena and discuss the dynamical implications on elastic Bloch waves. The dispersion properties are used to design a structured interface containing tilted resonators which exhibit negative refraction and focussing, as in a "flat elastic lens"

Transformation elastodynamics and cloaking for flexural waves

The paper addresses an important issue of cloaking transformations for fourth-order partial differential equations representing exural waves in thin elastic plates. It is shown that, in contrast with the Helmholtz equation, the general form of the partial differential equation is not invariant with respect to the cloaking transformation. The significant result of this paper is the analysis of the transformed equation and its interpretation in the framework of the linear theory of pre-stressed plates. The paper provides a formal framework for transformation elastodynamics as applied to elastic plates. Furthermore, an algorithm is proposed for designing a square cloak for exural waves, which employs a regularised push-out transformation. Illustrative numerical examples show high accuracy and efficiency of the proposed cloaking algorithm. In particular, a physical configuration involving a perturbation of an interference pattern generated by two coherent sources is presented. It is demonstrated that the perturbation produced by a cloaked defect is negligibly small even for such a delicate interference pattern. Keywords: cloaking, transformation elastodynamics, plates, invisibilit

An asymptotic hyperbolic-elliptic model for flexural-seismic metasurfaces

We consider a periodic array of resonators, formed from Euler-Bernoulli beams, attached to the surface of an elastic half-space. Earlier studies of such systems have concentrated on compressional resonators. In this paper we consider the effect of the flexural motion of the resonators, adapting a recently established asymptotic methodology that leads to an explicit scalar hyperbolic equation governing the propagation of Rayleigh-like waves. Compared with classical approaches, the asymptotic model yields a significantly simpler dispersion relation, with closed form solutions, shown to be accurate for surface wave-speeds close to that of the Rayleigh wave. Special attention is devoted to the effect of various junction conditions joining the beams to the elastic half-space which arise from considering flexural motion and are not present for the case of purely compressional resonators. Such effects are shown to provide significant and interesting features and, in particular, the choice of junction conditions dramatically changes the distribution and sizes of stop bands. Given that flexural vibrations in thin beams are excited more readily than compressional modes and the ability to model elastic surface waves using the scalar wave equation (i.e. waves on a membrane), the paper provides new pathways toward novel experimental set-ups for elastic metasurfaces

Tilted resonators in a triangular elastic lattice: chirality, Bloch waves and negative refraction

We consider a vibrating triangular mass-truss lattice whose unit cell contains a resonator of a triangular shape. The resonators are connected to the triangular lattice by trusses. Each resonator is tilted, i.e. it is rotated with respect to the triangular lattice's unit cell through an angle $\vartheta_0$. This geometrical parameter is responsible for the emergence of a resonant mode in the Bloch spectrum for elastic waves and strongly affects the dispersive properties of the lattice. Additionally, the tilting angle $\vartheta_0$ triggers the opening of a band gap at a Dirac-like point. We provide a physical interpretation of these phenomena and discuss the dynamical implications on elastic Bloch waves. The dispersion properties are used to design a structured interface containing tilted resonators which exhibit negative refraction and focussing, as in a "flat elastic lens"

Cymatics for the cloaking of flexural vibrations in a structured plate

Based on rigorous theoretical findings, we present a proof-of-concept design for a structured square cloak enclosing a void in an elastic lattice. We implement high-precision fabrication and experimental testing of an elastic invisibility cloak for flexural waves in a mechanical lattice. This is accompanied by verifications and numerical modelling performed through finite element simulations. The primary advantage of our square lattice cloak, over other designs, is the straightforward implementation and the ease of construction. The elastic lattice cloak, implemented experimentally, shows high efficiency

Chiral flexural waves in structured plates: Directional localisation and control

A new class of elastic waveforms, referred to as “chiral flexural waves”, is introduced for a multi-structure, which encompasses an elastic plate connected to a system of elastic flexural rods with gyroscopic spinners. The junction conditions describing the connection between the plate and the thin flexural rod require logarithmic asymptotics. The directional preference of the system is governed by the motion of gyroscopic spinners. For doubly-periodic chiral multi-structures studied here, parabolic modes associated with strong dynamic anisotropy of Bloch–Floquet waves are identified. Closed form analytical findings are accompanied by numerical simulations, which identify one-way flexural waves propagating along a straight interface in a flexural chiral system, without requiring the presence of Dirac cones on the dispersion surfaces. © 2020 Elsevier Lt